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--- abstract: 'We present a general framework to describe the simultaneous para-to-ferromagnetic and semiconductor-to-metal transition in electron-doped EuO. The theory correctly describes detailed experimental features of the conductivity and of the magnetization, in particular the doping dependence of the Curie temperature. The existence of correlation-induced local moments on the impurity sites is essential for this description.' author: - Michael Arnold and Johann Kroha title: ' Simultaneous ferromagnetic metal-semiconductor transition in electron-doped EuO' --- At room temperature stoichiometric europiumoxide (EuO) is a paramagnetic semiconductor which undergoes a ferromagnetic (FM) transition at the Curie temperature of $T_C=69~{\rm K}$. Upon electron doping, either by O defects or by Gd impurities, this phase transition turns into a simultaneous ferromagnetic and semiconductor-metal (SM) transition with nearly 100 % of the itinerant charge carriers polarized and a sharp resistivity drop of 8 to 13 orders of magnitude, depending on sample quality [@oliver1; @oliver2; @penney; @steeneken]. Concomitant with this transition is a huge colossal magnetoresistance (CMR) effect [@shapira], much larger than in the intensely studied manganates [@tokura]. These extreme properties make electron-doped EuO interesting for spintronics applications. Known since the 1970s, these features have therefore recently stimulated more systematic experimental studies with modern techniques and improved sample quality [@steeneken; @ott; @schmehl] as well as theoretical calculations [@schiller; @sinjukow]. In pure EuO the FM ordering is driven by the Heisenberg exchange coupling between the localized Eu 4$f$ moments with spin $S_f=7/2$ [@lee]. Upon electron doping, above $T_C$, the extra electrons are bound in defect levels situated in the semiconducting gap, and the transition to a FM metal occurs when the majority states of the spin-split conduction band shift downward to overlap with the defect levels. Although this scenario is widely accepted, several questions of fundamental as well as applicational relevance have remained poorly understood. (1) Why does the magnetic ordering of the Eu 4$f$ system occur simultaneously [@steeneken] with the SM transition of the conduction electron system? (2) What is the order of the transition? While the magnetic ordering of the 4$f$ system should clearly be of 2nd order, the metallic transition requires a [*finite*]{} shift of the conduction band and, hence, seems to favor a 1st order transition. (3) How can the critical temperature $T_C$ be enhanced by doping for spintronics applications? While in the Eu-rich compound EuO$_{1-x}$ a systematic $T_C$ increase due to the O defects (i.e. missing O atoms) is not observed experimentally [@oliver1; @oliver2], a minute Gd doping concentration significantly enhances $T_C$ [@matsumoto; @ott]. An O defect in EuO$_{1-x}$ essentially binds the two excess electrons from the extra Eu 6s orbital and, therefore, should not carry a magnetic moment. As shown theoretically in Ref. [@sinjukow], the presence of O defects with two-fold electron occupancy does not enhance $T_C$, in agreement with experiments [@oliver1; @oliver2]. In the present work we focus on the Gd-doped system Eu$_{1-y}$Gd$_y$ and calculate the temperature and doping dependent magnetization and resistivity from a microscopic model. We find that the key feature for obtaining a $T_C$ enhancement is that the impurities not only donate electrons but also carry a local magnetic moment in the paramagnetic phase. [*The model.*]{} — A Gd atom substituted for Eu does not alter the $S_f=7/2$ local moment in the Eu Heisenberg lattice but donates one dopant electron, which in the insulating high-temperature phase is bound in the Gd 5d level located in the gap. Therefore, the Gd impurities are Anderson impurities with a local level $E_d$ below the chemical potential $\mu$ and a [*strong*]{} on-site Coulomb repulsion $U>\mu - E_d$ which restricts their electron occupation essentially to one. The hybridization $V$ with the conduction band is taken to be site-diagonal because of the localized Gd 5d orbitals. The Hamiltonian for the Eu$_{1-y}$Gd$_y$O system then reads, $$\begin{aligned} \label{hamiltonian} H&=&\sum_{{\bf k}\sigma}\varepsilon_{{\bf k}} c_{{\bf k}\sigma}^{\dagger}c_{{\bf k}\sigma}^{\phantom{\dagger}}+H_{cd}+H_{cf}\\ \label{Hcd} H_{cd}&=&E_{d} \sum_{i=1 \dots N_I,\sigma} d_{i\sigma}^{\dagger}d_{i\sigma}^{\phantom{\dagger}} + V \sum_{i=1 \dots N_I,\sigma} (c_{i\sigma}^{\dagger} d_{i\sigma}^{\phantom{\dagger}} + H.c.)\nonumber\\ &+& U \sum_{i=1 \dots N_I} d_{i\uparrow}^{\dagger} d_{i\uparrow}^{\phantom{\dagger}} d_{i\downarrow}^{\dagger} d_{i\downarrow}^{\phantom{\dagger}} \\ \label{Hcf} H_{cf}&=&- \sum_{i,j} J_{ij} \vec S_{i}\cdot\vec S_{j} - J_{cf}\sum_{i}\vec \sigma_{i}\cdot\vec S_{i} \ ,\end{aligned}$$ where the first term in Eq. (\[hamiltonian\]) denotes conduction electrons with spin $\sigma$. The Eu 4$f$ moments $\vec S_i$ on the lattice sites $i=1,\dots, N$ are described in terms of a Heisenberg model $H_{cf}$ with FM nearest and next-nearest neighbor couplings $J_{ij}$ and an exchange coupling $J_{cf}$ to the conduction electron spin operators at site $i$, $\vec\sigma_{i}=(1/2)\sum_{\sigma\sigma'} c_{i\sigma}^{\dagger}\vec\tau_{\sigma\sigma'}c_{i\sigma'}^{\phantom{\dagger}}$, with $c_{i\sigma}=\sum_{\bf k} \exp(i{\bf k x_i})\,c_{{\bf k}\sigma}$ and $\vec \tau_{\sigma\sigma'}$ the vector of Pauli matrices. The Gd impurities at the random positions $i=1, ..., N_I$ are described by $H_{cd}$. For the numerical evaluations we take $U\to\infty$ for simplicity. For the present purpose of understanding the general form of the magnetization $m(T)$ and the systematic doping dependence of $T_C$ it is sufficient to treat the 4$f$ Heisenberg lattice, $H_{cf}$, on mean field level, although recent studies have shown that Coulomb correlations in the conduction band can soften the spin wave spectrum in similar systems [@golosov; @perakis]. The effect of the latter on $m(T)$ can be absorbed in the effective mean field coupling of the 4$f$ system, $J_{4f} \equiv \sum_{j}J_{ij}$. We therefore choose $J_{4f}$ such that for pure EuO it yields the experimental value of $T_C=69~{\rm K}$ [@oliver1; @oliver2; @shapira; @steeneken]. For simplicity, we don’t consider a direct coupling $J_{df}$ between the 4$f$ and the impurity spins, since this would essentially renormalize $J_{cf}$ only. The indirect RKKY coupling will also be neglected, since for the small conduction band fillings relevant here it is FM, like $J_{ij}$, but much smaller than $J_{ij}$. In the evaluations we use a semi-elliptical bare conduction band density of states (DOS) with a half width $D_0=8\, {\rm eV}$, (consistent with experiment [@steeneken]), centered around $\Delta _0\approx 1.05\, D_0$ above the (bare) defect level $E_d$. The other parameters are taken as $J_{4f} \equiv \sum_{j}J_{ij} = 7\cdot 10^{-5} D_{0}$, $J_{cf}=0.05 D_{0}$, $E_{d}=-0.4 D_{0}$, and $\Gamma=\pi V^{2}=0.05 D_{0}^{2}$, where $J_{cf}\gg J_{4f}$ because $J_{4f}$ involves a non-local matrix element. [*Selfconsistent theory.*]{} — The averaging over the random defect positions is done within the single-site $T$-matrix approximation, sufficient for dilute impurities. This yields for the retarded conduction electron Green‘s function $G_{c\sigma}({\bf k},\omega)$ in terms of its selfenergy $\Sigma _{c\sigma}(\omega)$, $$\begin{aligned} &&G_{c\sigma}({\bf k},\omega)=\left[\omega+\mu-\varepsilon_{\bf k}-\Sigma_{c\sigma}(\omega)\right]^{-1} \label{gc}\\ &&\Sigma_{c\sigma}(\omega)=n_{I} |V|^{2}G_{d\sigma}(\omega) -J_{cf}\langle S \rangle \sigma \label{se}\end{aligned}$$ where $G_{d\sigma}(\omega)$ is the defect electron propagator and $\langle S \rangle$ the average 4$f$–moment per site. In mean field theory it is obtained, together with the conduction electron magnetization $m$, as $$\begin{aligned} &&\langle S \rangle = \frac{\sum_{S} S e^{-\beta(2J_{4f}\langle S \rangle + J_{cf}m)S}}{\sum_{S}e^{-\beta(2J_{4f}\langle S \rangle + J_{cf}m)S}}\\ &&m=\frac{1}{2}\int d \omega f(\omega) [A_{c\uparrow}(\omega) - A_{c\downarrow}(\omega)]\label{magn}\end{aligned}$$ where $f(\omega)$ is the Fermi distribution function and $A_{c\sigma}(\omega)=- \sum_{{\bf k}} {\rm{Im}} G_{c\sigma}(k,\omega)/\pi$ the conduction electron DOS of the interacting system. \ In order to treat the strongly correlated spin and charge dynamics of the Anderson impurities without double occupancy beyond the static approximation, we use a slave particle representation and employ the non-crossing approximation (NCA) [@grewe]. For EuO the DOS at the Fermi level is so low or even vanishing that the Kondo temperature is well below $T_C$ and Kondo physics plays no role. In this high-energy regime the NCA has been shown to give quantitatively reliable results [@costi]. This remains true even for a finite magnetization, where the NCA would develop spurious potential scattering singularities near $T_K$ only [@kirchner]. One obtains the following set of equations for $G_{d\sigma}(\omega)$ in terms of the auxiliary fermion and boson propagators $G_{f\sigma}$, $G_{b}$, their spectral functions $A_{f\sigma}$, $A_{b}$ and their selfenergies $\Sigma_{f\sigma}, \Sigma_{b}$, $$\begin{aligned} \Sigma_{f\sigma}(\omega)&=&\Gamma \int {d\varepsilon}\left[1-f(\varepsilon)\right] A_{c\sigma}(\varepsilon)G_{b}(\omega-\varepsilon )\label{sigmaf}\\ \Sigma_{b}(\omega)&=&\Gamma \sum_{\sigma}\int {d\varepsilon} f(\varepsilon) A_{c\sigma}(\varepsilon)G_{f\sigma}(\omega+\varepsilon )\label{sigmab}\\ \nonumber G_{d\sigma}(\omega)&=&\int \frac{d\varepsilon} {e^{\beta \varepsilon}} \left[ G_{f\sigma}(\omega+\varepsilon )A_{b}(\varepsilon)-A_{f\sigma}(\varepsilon)G^{*}_{b}(\varepsilon-\omega)\right] \\ \label{Gd}\end{aligned}$$ Note that in Eqs. (\[sigmaf\], \[sigmab\]) $A_{c\sigma}(\varepsilon)$ is the interacting DOS, renormalized by the dilute concentration of Anderson impurities and the 4$f$–spins according to Eq. (\[gc\]). For details of the NCA and its evaluation see [@costi]. The equations (\[gc\]-\[Gd\]) form a closed set of selfconsistent integral equations. They are solved iteratively, fixing the total electron number per lattice site in the system, $$\begin{aligned} n= \sum_{\sigma}\int \!d\omega f(\omega)\, \left[A_{c\sigma}(\omega)+n_I\,A_{d\sigma}(\omega)\right]=n_I \label{pnumber}\end{aligned}$$ by the chemical potential $\mu$ in each step. \ [*Electrical conductivity.*]{} — The current operator $\hat{\bf j}$ can be derived from the continuity equation, $\partial\hat\rho_i/\partial t + \nabla\ \cdot \hat{\bf j} =0$, and the Heisenberg equation of motion for the total local charge operator $\hat\rho_i$ at site $i$. Because the impurity Hamiltonians $H_{cf}$, $H_{df}$ conserve $\hat\rho_i$, only $c$–electrons contribute to the current, and one obtains [@schweitzer], $ \hat{\bf j}=({e}/{\hbar}) \sum_{{\bf k}\sigma}{\partial \varepsilon_{\bf k}}/ {\partial {\bf k}} \ c_{{\bf k}\sigma}^{\dagger} c_{{\bf k}\sigma}^{\phantom{\dagger}} $. The linear response conductivity then reads for a local selfenergy [@schweitzer], $$\sigma=\frac{\pi e^{2}}{3 \hbar V} \sum_{{\bf k}\sigma} \int d\omega \left( -\frac{\partial f}{\partial \omega} \right) A_{c\sigma}^{2}({\bf k},\omega) \left( \frac{\partial \varepsilon_{\bf k}}{\partial {\bf k}} \right)^{2} \ . \label{cond1}$$ [*Results and discussion.*]{} — The results of the selfconsistent theory, Eqs. (\[gc\]–\[pnumber\]), and for the conductivity, Eq. (\[cond1\]), are presented in Figs. \[fig1\]–\[fig3\]. They allow to draw a complete picture of the FM semiconductor-metal transition in Gd-doped EuO. The spectral densities per lattice site above and below the transition are shown in Fig. \[fig1\]. In the paramagnetic, insulating phase the hybridization between $d$– and $c$–electrons necessarily implies the appearance of a conduction electron sideband (Fig. \[fig1\], inset), situated below $\mu$ and at the same energies inside the semiconducting gap as the impurity $d$–band. The $d$-band (not shown) has a similar width and shape as the $c$-sideband. The combined weight of the $c$–sideband and the $d$-band adjusts itself selfconsistently such that it just accommodates the total electron number, $n=n_I$. Note that the weight of the $d$–band per impurity and spin is $\lesssim 1/2$, because the doubly occupied weight is shifted to $U\to \infty$ [@costi]. \ The $c$–4$f$ exchange coupling $J_{cf}$ induces an effective FM coupling between the electrons of the $c$–$d$ system. Hence, either the 4$f$– or the $c$–$d$–electron system can drive a FM transition, depending on which of the (coupled) subsystems has the higher $T_C$. We have chosen $J_{cf}$ (see above) large enough that the transition is driven by the $c$–$d$–electrons, because this will yield detailed agreement with the experiments [@steeneken; @ott; @matsumoto]. In this case, $T_C$ is naturally expected to increase with the impurity density $n_I$. The results for the $T$-dependent conduction electron magnetization $m(T)$, Eq. (\[magn\]), and for the doping dependence of $T_C$ are shown in Fig. \[fig2\], lower panel, and in Fig. \[fig3\], right panel, respectively. It is seen that not only $T_C$ increases with the impurity concentration, in agreement with recent measurements on Eu$_{1-y}$Gd$_{y}$O$_{1-x}$ [@matsumoto; @ott], but also that $m(T)$ has a dome-like tail near $T_C$, before it increases to large values deep inside the FM phase. From our theory this feature is traced back to the mean-field-like 2nd order FM transition of the electron system, while the large dome in the magnetization further below $T_C$ is induced by the FM ordering of the 4$f$ system, whose magnetization is controlled by $J_{4f}$ and sets in at lower $T$. This distinct feature is again in agreement with the experimental findings [@matsumoto; @ott] and lends significant support for the present model for Eu$_{1-y}$Gd$_{y}$O. We note that the Eu-rich EuO$_{1-x}$ samples of Ref. [@matsumoto] also show a magnetization tail and a $T_C$ enhancement, suggesting (small) magnetic moments on the O defects. However, the nature of the O defects requires further experimental and theoretical studies. The conduction electron polarization $P(T)=m(T)/n_c(T)$ does not show this double-dome structure and below $T_C$ increases steeply to $P=1$ (not shown in Fig. \[fig2\]). The FM phase is connected with a spin splitting of the $c$– as well as the $d$–densities of states, as shown in Fig. \[fig1\]. The narrow $d$-band induces a Fano dip structure in the $c$ majority band and a small sideband in the $c$ minority band. Note that for the present scenario the existence of preformed local moments on the impurities, induced by strong Coulomb repulsion $U$, is essential. Without these moments the transition of the electron system would be purely Stoner-like, and, because of the extremely low conduction electron DOS at the Fermi level, its $T_C$ would be far below the Curie temperature of the 4$f$ system, so that no doping dependence would be expected [@sinjukow]. \ We now discuss the conductivity and the simultaneity of the FM and the SM transitions. In the paramagnetic phase, the system is weakly semiconducting, because $\mu$ lies in the gap (Fig. \[fig1\], inset). When the FM transition occurs, the impurity d-band must acquire a spin splitting in such a way that at least part of the minority $d$–spectral weight lies above the chemical potential $\mu$, in order to provide a finite magnetization. Since near the transition the spin splitting is small, the majority $d$–band must, therefore, also be shifted to have overlap with $\mu$ (Fig. \[fig1\]), and so must the hybridization-induced $c$-electron sideband (which eventually merges with the main conduction band for $T$ sufficiently below $T_C$). This immediately implies a transition to a metallic state, simultaneous with the FM transition, as seen in Fig. \[fig2\]. Because of the small, but finite thermal occupation of the states around $\mu$, we find that this shifting of spectral weight occurs continuously, which implies the FM semiconductor-metal transition to be of 2nd order (see Fig. \[fig2\]). The doping $n_I$ dependence of the conductivity is shown in Fig, \[fig3\], left panel. It is seen that the metallic transition can be driven by increasing $n_I$, if $T>T_C(n_I=0)$. As an alternative to Gd-doping the charge carrier concentration $n$ can be controlled independently of the impurity concentration $n_I$ by varying the chemical potential $\mu$, e.g. by applying a gate voltage to an EuO thin film. The conductivity $\sigma$ and magnetization $m$ as a function of $\mu$ are shown in Fig. \[fig4\] for two temperatures. To both sides of the ungated system ($n=n_I$) $\sigma$ increases exponentially upon changing $\mu$, characteristic for semiconducting behavior. By increasing $\mu$, the FM-metallic transition is finally reached. I.e. the magnetization can be switched, in principle, by a gate voltage. The non-monotonic behavior of $\sigma$ towards more negative $\mu$ reflects the energy dependence of the $c$ sideband. A more detailed study will be presented elsewhere. To conclude, our theory indicates that in Gd-doped EuO the existence of preformed local moments on the impurity levels inside the semicondicting gap is essential for understanding the distinct shape of the magnetization $m(T)$ near the ferromagnetic semiconductor-metal transition. The FM ordering is driven by these impurity moments which are superexchange coupled via the 4$f$ moments of the underlying Eu lattice. This scenario immediately implies an increase of the Curie temperature with the impurity concentration, in agreement with experiments. The double-dome shape of $m(T)$ arises because of the successive ordering of the dilute impurity and of the dense Eu 4$f$ systems, as $T$ is lowered. The dynamical accumulation of conduction spectral weight at the chemical potential, induced by the hybridization $V$ and the constraint of an emerging magnetization at the FM transition, implies the FM and the SM transition to be simultaneous and of 2nd order. The magnetization can be switched by applying a gate voltage. This might be relevant for spintronics applications. We wish to thank T. Haupricht, H. Ott, and H. Tjeng for useful discussions. J.K. is grateful to the Aspen Center for Physics where this work was completed. This work is supported by DFG through SFB 608. [10]{} M. Oliver [*et al.*]{}, Phys. Rev. Lett. [**[24]{}**]{}, 1064 (1970). M. Oliver [*et al.*]{}, Phys. Rev. B [**[5]{}**]{}, 1078 (1972). T. Penney, M. W. Shafer, and J. B. Torrance, Phys. Rev. B [**[5]{}**]{}, 3669 (1972). P. B. Steeneken et al., Phys. Rev. Lett. [**[88]{}**]{},047201 (2002). Y. Shapira, T. Foner, and S. B. Reed, Phys. Rev. B [**[8]{}**]{}, 2299 (1973). For a review see, e.g., M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. [**70**]{}, 1039 (1998). H. Ott [*et al.*]{}, Phys. Rev. B [**[73]{}**]{}, 094407 (2006). A. Schmehl [*et al.*]{}, Nature Materials doi:10.1038/nmat2012 (2007). R. Schiller, W. Müller, and W. Nolting, Phys. Rev. B [**[64]{}**]{},134409 (2001). P. Sinjukow and W. Nolting, Phys. Rev. B [**[68]{}**]{}, 125107 (2003); Phys. Rev. B [**[69]{}**]{}, 214432 (2004). V.-C. Lee and L. Liu , Phys. Rev. B [**[30]{}**]{}, 2026 (1984). T.Matsumoto et al., [*[J.Phys.]{}*]{} [**[16]{}**]{},6017 (2004). D. I. Golosov, Phys. Rev. B [**71**]{}, 014428 (2005). M. D. Kapetanakis, A. Manousaki, and I. E. Perakis, Phys. Rev. B [**73**]{}, 174424 (2006); M. D. Kapetanakis and I. E. Perakis, Phys. Rev. B [**75**]{}, 140401(R) (2007). N. Grewe and H. Keiter, Phys. Rev. B [**[24]{}**]{},4420 (1981); Y. Kuramoto, Z. Phys. B [**[53]{}**]{}, 37 (1983). T. A. Costi, J. Kroha, and P. Wölfle, Phys. Rev. B [**53**]{}, 1850 (1996). S. Kirchner and J. Kroha, J. Low Temp. Phys. [**126**]{}, 1233 (2002); arXiv:cond-mat/0202351. H. Schweitzer and G. Czycholl, Phys. Rev. Lett. [**[67]{}**]{}, 3724 (1991); T. Pruschke, M. Jarrell and J. Freericks, Adv. Phys. [**[44]{}**]{}, 187 (1995).

ArXiv

--- abstract: 'To what extent should we expect the syzygies of Veronese embeddings of projective space to depend on the characteristic of the field? As computation of syzygies is impossible for large degree Veronese embeddings, we instead develop an heuristic approach based on random flag complexes. We prove that the corresponding Stanley–Reisner ideals have Betti numbers which almost always depend on the characteristic, and we use this to conjecture that the syzygies of the $d$-uple embedding of projective $r$-space with $r\geq 7$ should depend on the characteristic for almost all $d$.' author: - 'Caitlyn Booms, Daniel Erman, and Jay Yang' bibliography: - 'bib.bib' title: 'Heuristics for $\ell$-torsion in Veronese Syzygies' --- Introduction ============ Imagine ${\mathbb{P}}^{10}$ embedded into a larger projective space by the $d$-uple Veronese embedding, where $d$ is some large integer like $d=100$ or $d=100000$. What should we expect about the syzygies? Such questions were raised by Ein and Lazarsfeld in [@ein-lazarsfeld-asymptotic] and later in [@ein-erman-lazarsfeld-random]. While they focused on quantitative behaviors that are independent of the ground field, we ask: [*To what extent should we expect the syzygies to depend on the characteristic, if at all? Given the impossibility of computing data for large $d$, how can we make a reasonable conjecture?*]{} The central idea in this paper is the development of an heuristic—based on a random flag complex construction—for modelling the syzygies of Veronese embeddings of projective space. The resulting conjectures propose that, when it comes to dependence on the characteristic of the ground field, pathologies are the norm. Let us make this more precise. For any integers $r,d\geq 1$ and any field $k$, we may consider the $d$-uple embedding of ${\mathbb{P}}^r_k$ into ${\mathbb{P}}^{\binom{r+d}{d}-1}$; the image is given by an ideal $I\subset S$, where $S$ is a polynomial ring in $\binom{r+d}{d}$ variables over $k$. We denote the algebraic Betti numbers of the image by $\beta_{i,j}({\mathbb{P}}^r_k;d) := \dim_k \operatorname{Tor}_i^S(S/I,k)_j$. These encode the number of degree $j$ generators for the $i$’th syzygies, and a major open question is to describe the Betti table $\beta({\mathbb{P}}^r_k;d)$, which is the collection of all these Betti numbers [@green-koszul2; @castryck-et-al; @big-computation; @anderson; @bouc; @jozefiak-pragacz-weyman; @reiner-roberts; @ottaviani-paoletti; @vu; @greco-martino; @ein-lazarsfeld-asymptotic; @ein-erman-lazarsfeld-quick; @raicu]. Since each individual Betti number is invariant under flat extensions, the Betti table is determined by the integers $r,d$ and the characteristic of $k$. For a prime $\ell$, we say that $\beta({\mathbb{P}}^r;d)$ [**has $\ell$-torsion**]{} if $\beta({\mathbb{P}}^r_{\mathbb F_\ell};d) \ne \beta({\mathbb{P}}^r_{\mathbb Q};d)$, and we say that $\beta({\mathbb{P}}^r;d)$ [**depends on the characteristic**]{} if this occurs for some $\ell$.[^1] There are two known cases. - For $r=1$ and any $d$, the Betti numbers in $\beta({\mathbb{P}}^r; d)$ do not depend on the characteristic, as any rational normal curve is resolved by an Eagon-Northcott complex. - If $r\geq 7$, Andersen’s thesis [@anderson] shows that $\beta_{5,7}(\mathbb P^r;2)$ has $5$-torsion. Very little else seems to be known or even conjectured about the dependence of Veronese syzygies on the characteristic, including no known examples of $\ell$-torsion for $\ell\ne 5$. One key challenge in this area is the difficulty of generating good data. For instance, the syzygies of ${\mathbb{P}}^2$ under the $5$-uple embedding were only recently computed [@castryck-et-al; @big-computation]. For larger values of $d$ and $r$, computation is essentially impossible: in the case of ${\mathbb{P}}^{10}$ and $d=100$ mentioned above, the computation would involve $\approx 4.68\times 10^{13}$ variables. Heuristics can provide an alternate route for generating conjectures, especially when computation is infeasible. (Such an approach is quite common for predicting properties of how the prime numbers are distributed, for instance.) In this paper, we use an heuristic model to motivate conjectures about $\ell$-torsion in $\beta({\mathbb{P}}^r;d)$. For instance, we are led to conjecture that dependence on the characteristic should be commonplace as $d\to \infty$. \[conj:dependence\] Let $r\geq 7$. For any $d\gg 0$, the Betti table of $\mathbb P^r$ under the $d$-uple embedding depends on the characteristic. This conjecture is based upon corresponding properties of the following model for Veronese syzygies. We let $\Delta\sim \Delta(n,p)$ denote a random flag complex on $n$ vertices with attaching probability $p$. (See §\[sec:background\] for details.) For a given field $k$, we let $I_\Delta$ be the corresponding Stanley–Reisner ideal in $S=k[x_1,\dots,x_n]$. Ein and Lazarsfeld showed that if $d\gg 0$, then almost all of the Betti numbers in rows $1,\dots, r$ of $\beta({\mathbb{P}}_k^r;d)$ are nonzero (see for instance [@erman-yang Theorem 1.1]). Theorem 1.3 of [@erman-yang] gives that a similar result holds for $I_\Delta$ as long as $n^{-1/(r-1)}\ll p \ll n^{-1/r}$ and $n\gg 0$. Thus, if $p$ is in the specified range, then the Betti table $\beta(S/I_\Delta)$ as $n\to \infty$ satisfies similar nonvanishing properties[^2] as $\beta({\mathbb{P}}_k^r;d)$ as $d\to \infty$; in this sense, the Betti tables $\beta(S/I_\Delta)$ determined by $\Delta(n,p)$ can act as a random model for Veronese syzygies. To predict how $\beta({\mathbb{P}}^r;d)$ depends on the characteristic, we will therefore consider the corresponding questions for $\beta(S/I_\Delta)$ for various fields $k$. As with Veronese syzygies, we say that the Betti table of the Stanley–Reisner ideal of $\Delta$ **has $\ell$-torsion** if this Betti table is different when defined over a field of characteristic $\ell$ than it is over $\mathbb Q$, and we say that this Betti table **depends on the characteristic** if this occurs for some $\ell$. We prove: \[thm:Delta depend\] Let $r\geq 7$, and let $\Delta\sim \Delta(n,p)$ be a random flag complex with $n^{-1/(r-1)} \ll p \ll n^{-1/r}$. With high probability as $n\to \infty$, the Betti table of the Stanley–Reisner ideal of $\Delta$ depends on the characteristic. In other words, if $p$ is in the range where the Betti table of the Stanley–Reisner ideal of $\Delta$ behaves like $\mathbb P^{r}$—in the sense of [@erman-yang Theorem 1.3]—then this Betti table will almost always depend on the characteristic for $n\gg 0$. This theorem is the basis of Conjecture \[conj:dependence\]. Since our $r \geq 7$ hypothesis in Conjecture \[conj:dependence\] is based upon properties of the $\Delta(n,p)$ model, the fact that this hypothesis lines up with Andersen’s example appears to be a coincidence; see Remarks \[rmk:r7\] and \[rmk:r bound\] for more details. Note also that, based on [@anderson], we might even find $\ell$-torsion in $\beta({\mathbb{P}}^r; d)$ for small values of $d$ as well; however, Theorem \[thm:Delta depend\] is asymptotic in nature, which motivates the $d\gg 0$ hypothesis in Conjecture \[conj:dependence\]. In fact, we prove the sharper result: \[thm:m torsion\] Let $m\geq 2$, and let $\Delta\sim \Delta(n,p)$ be a random flag complex with $n^{-1/6} \ll p \leq 1-\epsilon$ for some $\epsilon>0$. With high probability as $n\to \infty$, the Betti table of the Stanley–Reisner ideal of $\Delta$ has $\ell$-torsion for every $\ell$ dividing $m$. In particular, this holds for $\Delta\sim \Delta(n,p)$ where $n^{-1/(r-1)} \ll p \ll n^{-1/r}$ for any $r\geq 7$. The proof of Theorem \[thm:m torsion\] (which implies Theorem \[thm:Delta depend\]) proceeds as follows. By Hochster’s formula [@bruns-herzog Theorem 5.5.1], it suffices to show that some induced subcomplex of $\Delta$ has $m$-torsion in its homology. So, for each $m$, we construct a flag complex $X_m$ with a small number of vertices and with $m$-torsion in $H_1(X_m)$. This complex is derived from Newman’s construction of a two-dimensional simplicial complex $X$ where $H_1(X)$ has $m$-torsion [@newman §3], though we modify his work to ensure that $X_m$ is a flag complex and to lower the maximal vertex degree. We then apply Bollobás’s theorem on subgraphs of a random graph [@Bollobas-subgraph Theorem 8]—or rather a minor variant of that result for induced subgraphs—to prove that $X_m$ appears as an induced subcomplex of $\Delta$ with high probability as $n\to \infty$, yielding Theorem \[thm:m torsion\]. Theorem \[thm:m torsion\] fits into an emerging literature on random monomial ideals. Our current work seems to be the first application of random monomial ideal methods to generate new conjectures outside of the world of monomial ideals. Random monomial ideals first appeared in the work of De Loera-Petrović-Silverstein-Stasi-Wilburne [@random-monomial], which outlined an array of frameworks for studying random monomial ideals, including the model used in this paper, as well as models related to other types of random simplicial complexes such as [@costa-farber; @kahle]; they also proved threshold results for dimension and other invariants of these ideals. In [@average-random], similar methods are applied to study the average behavior of Betti tables of random monomial ideals and to compare these with certain resolutions of generic monomial ideals. Recent work of Banerjee and Yogeshwaran analyzes homological properties of the edge ideals of Erds-Rényi random graphs [@banerjee]. The forthcoming [@stilverstein-wilburne-yang] looks more closely at threshold phenomena in the phase transitions of the random models from [@random-monomial]. There is also the previously referenced [@erman-yang], which uses random monomial methods to demonstrate some asymptotic syzygy phenomena observed/conjectured in [@ein-lazarsfeld-asymptotic; @ein-erman-lazarsfeld-random]. There is also a great deal of literature on the study of $\ell$-torsion arising in random constructions. The most relevant such study is perhaps the recent work by Kahle-Lutz-Newman on $\ell$-torsion in the homology of random simplicial complexes [@torsion-burst], which conjectures the existence of bursts of torsion homology at specific thresholds. For comparison, those authors are interested in $\ell$-torsion in the global homology of a complex like $\Delta(n,p)$, whereas, due to Hochster’s formula, we analyze the simpler question of finding $\ell$-torsion in the homology of [*any*]{} induced subgraph of $\Delta(n,p)$. \[rmk:r7\] We note that the bound $r\geq 7$ in Theorem \[thm:m torsion\] is not necessarily sharp. In fact, we undertake a detailed investigation of the $2$-torsion of the Betti table of the Stanley–Reisner ideal of $\Delta$ in §\[sec:2torsion\], which yields a bound of $r\geq 4$. See Remarks \[rmk:sharpness\] and \[rmk:r bound\] for further discussion on restrictions on $r$ in both Conjecture \[conj:dependence\] and Theorem \[thm:m torsion\]. Theorem \[thm:m torsion\] also leads us to a stronger conjecture on Veronese $\ell$-torsion: \[conj:bad primes\] Let $r\geq 7$. As $d \to \infty$, the number of primes $\ell$ such that $\beta({\mathbb{P}}^r;d)$ has $\ell$-torsion will be unbounded. Regarding Conjectures \[conj:dependence\] and  \[conj:bad primes\], it is worth emphasizing the total lack of direct evidence. As noted above, [@anderson] appears to provide the only known instance of $\ell$-torsion for any Veronese embedding. These conjectures are based primarily upon the heuristic model and, to a lesser extent, upon the nonvanishing results of [@ein-lazarsfeld-asymptotic; @ein-erman-lazarsfeld-quick], both of which rely on an inductive structure where pathologies in $\beta({\mathbb{P}}^r;d)$ tend to propagate as $d\to \infty$, and both of which show that the asymptotic behavior of syzygies exhibits a strong uniformity. However, we do not expect our random flag complex model to be a perfect predictor of all properties of Veronese syzygies. In fact, the results in [@erman-yang] imply that while the Betti tables associated to $\Delta$ have similar overarching nonvanishing properties as Veronese embeddings, these Betti tables do not demonstrate more nuanced properties such as Green’s $N_p$-property [@green-koszul2]: our model will not give correct predictions about these properties. This is why Conjectures \[conj:dependence\] and  \[conj:bad primes\] echo certain qualitative aspects of Theorem \[thm:m torsion\] as opposed to more specific and quantitative predictions about $\ell$-torsion in $\beta({\mathbb{P}}^r; d)$. In a rather different direction, an alternate heuristic model for Veronese syzygies is considered in  [@ein-erman-lazarsfeld-random]. That model is based on Boij-Söderberg theory and is used to generate quantitative conjectures about the entries of $\beta({\mathbb{P}}^r_k;d)$ for $d\gg 0$. However, since this model does not take into account the characteristic of the field, it cannot be used to generate conjectures such as those above. See also the results of [@cjw], which provided a combinatorial parallel of the asymptotic results of [@ein-lazarsfeld-asymptotic]. \[rmk:other varieties\] Ein and Lazarsfeld’s asymptotic nonvanishing results are more-or-less uniform for any smooth variety of dimension $r$ [@ein-lazarsfeld-asymptotic Theorem A], and these were even expanded to integral varieties by Zhou [@zhou-integral]. In this paper, we restrict attention to ${\mathbb{P}}^r$ for concreteness, but we would expect that Conjecture \[conj:dependence\] would likely apply to the $d$-uple embeddings of any $r$-dimensional integral variety which is flat over ${\mathbb{Z}}$, including products of projective spaces, toric varieties, hypersurfaces, Grassmanians, and more. This paper is organized as follows. In §\[sec:background\], we review notation and background, including on Betti numbers, Hochster’s formula, and random flag complexes. §\[sec:construction\] contains our main construction in which we construct an explicit flag complex $X_m$ with $m$-torsion in homology; see Theorem \[thm:Xm\]. In §\[sec:subgraphs\], we apply a minor variant of Bollobás’s Theorem on subgraphs of a random graph to show that, with high probability, $X_m$ appears as an induced subgraph of $\Delta(n,p)$ for any $n^{-1/6}\ll p\ll 1$ and $m\geq 2$. In §\[sec:Betti numbers\], we then combine this result with Hochster’s formula to prove Theorem \[thm:m torsion\]. §\[sec:2torsion\] is a bit separate from the main results as we analyze the case of 2-torsion more closely. Finally, §\[sec:veronese\] returns to the geometric setting where we use our results to produce heuristics and conjectures about $\ell$-torsion in Veronese syzygies. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank Kevin Kristensen, Rob Lazarsfeld, Andrew Newman, Melanie Matchett Wood for helpful conversations. We thank Claudiu Raicu for thoughtful comments on an early draft. Background and Notation {#sec:background} ======================= Betti tables for Veronese embeddings ------------------------------------ For a given $r,d\geq 1$ and field $k$, we have the $d$-uple Veronese embedding ${\mathbb{P}}^r_k\to {\mathbb{P}}_k^{\binom{r+d}{d}-1}$. The image is determined by a homogeneous ideal $I\subset S$ where $S$ is a polynomial ring with coefficients in $k$ and $\binom{r+d}{d}$ variables. The homogeneous coordinate ring $S/I$ of the image is a graded $S$-module. We can thus take a minimal free resolution $F_0\gets F_1 \gets \cdots$ of $S/I$, where each $F_i$ is a graded free $S$-module, $F_i=\displaystyle\bigoplus_{j\in {\mathbb{Z}}} S(-j)^{\beta_{i,j}(S/I)}$. This provides one way to define the algebraic Betti numbers; an alternate definition is $\beta_{i,j}(S/I) := \dim_k \operatorname{Tor}_i^S(S/I,k)_j$. To emphasize the dependence on $r$ and $d$ (and to avoid referencing the ambient ring $S$ and the homogeneous coordinate ring $S/I$, both of which change with $d$), we will denote these Betti numbers by $\beta_{i,j}({\mathbb{P}}^r_k;d)$ instead of the more standard $\beta_{i,j}(S/I)$. Further, we write $\beta({\mathbb{P}}^r_k;d)$ for the Betti table of this embedding, which is the collection of all $\beta_{i,j}({\mathbb{P}}^r_k;d)$. Torsion in Betti tables {#subsec:22} ----------------------- Throughout this paper we will analyze graded algebras, all of which have the following form: there is an ideal $I$ in a polynomial ring $T$ with coefficients in ${\mathbb{Z}}$, where $T/I$ is flat over ${\mathbb{Z}}$, and we are interested in specializations $(T/I)\otimes_{{\mathbb{Z}}} k$ to various different fields $k$. We consider such graded algebras that arise in two ways: as the coordinate rings of Veronese embeddings of projective space and as the Stanley–Reisner rings of simplicial complexes. The central questions of this paper are concerned with when the Betti numbers of such algebras depend on the choice of the characteristic of $k$. First, we consider the Veronese embeddings. For any positive integers $r$ and $d$, we can embed ${\mathbb{P}}^r_{{\mathbb{Z}}}\to {\mathbb{P}}^{\binom{r+d}{d}-1}_{{\mathbb{Z}}}$ via the $d$-uple Veronese embedding. If $T$ is the polynomial ring for the larger projective space, then there is an ideal $I\subset T$ defining the image of this map. Since $T/I$ is flat over ${\mathbb{Z}}$, the coordinate ring of the Veronese embedding over a field $k$ is given by $(T/I)\otimes_{{\mathbb{Z}}} k$. As noted in the previous subsection (with $S=T\otimes_{{\mathbb{Z}}} k$), the algebraic Betti numbers are defined as $$\beta_{i,j}({\mathbb{P}}^r_k;d) := \dim_k \operatorname{Tor}^{T\otimes_{{\mathbb{Z}}} k}_i((T/I)\otimes_{{\mathbb{Z}}} k,k)_j.$$ Since field extensions are flat, algebraic Betti numbers are invariant under field extensions, and thus, $\beta({\mathbb{P}}^r_k;d)$ only depends on $r,d$ and the characteristic of $k$. Moreover, by semicontinuity, we have an inequality $ \beta_{i,j}({\mathbb{P}}^r_{\mathbb Q};d) \leq \beta_{i,j}({\mathbb{P}}^r_{\mathbb F_{\ell}};d) $ for any prime $\ell$ (with equality for all but finitely many $\ell$). As noted in the introduction, we will say that $\beta({\mathbb{P}}^r;d)$ [**has $\ell$-torsion**]{} if this inequality is strict for some $i,j$, and we will say that $\beta({\mathbb{P}}^r;d)$ [**depends on the characteristic**]{} if this inequality is strict for some $i,j$ and some $\ell$. \[rmk:torsion\] Let $I$ be an ideal in $T={\mathbb{Z}}[x_1,\dots,x_n]$ which is flat over ${\mathbb{Z}}$. Let $S' = T\otimes_{{\mathbb{Z}}} \mathbb F_{\ell}=\mathbb F_\ell[x_1,\dots,x_n]$ and $I'=IS'$. By a standard argument, it follows that $$\dim_{\mathbb{F}_\ell} \operatorname{Tor}^{S'}_i(S'/I',\mathbb F_\ell)_j = \dim_{\mathbb{F}_\ell} (\operatorname{Tor}^T_i(T/I,{\mathbb{Z}})_j\otimes_{\mathbb{Z}}\mathbb F_\ell) + \dim_{\mathbb{F}_\ell}(\operatorname{Tor}^{{\mathbb{Z}}}_1(\operatorname{Tor}^T_{i+1}(T/I,{\mathbb{Z}})_j, \mathbb F_\ell)).$$ In particular, the Betti table of such an ideal has $\ell$-torsion in the sense of the introduction if and only if one of the $\operatorname{Tor}_{i+1}^T(T/I,{\mathbb{Z}})_j$ has $\ell$-torsion as an abelian group. We next consider notation for monomial ideals since Stanley–Reisner ideals of simplicial complexes are monomial ideals. Let $J$ be a monomial ideal in $T={\mathbb{Z}}[x_1,\dots,x_n]$. For a field $k$, the algebraic Betti numbers of $(T/J)\otimes_{{\mathbb{Z}}} k$ are given by $$\beta_{i,j}((T/J)\otimes_{{\mathbb{Z}}} k) := \dim_k \operatorname{Tor}^{T\otimes_{{\mathbb{Z}}} k}_i((T/J)\otimes_{{\mathbb{Z}}} k,k)_j.$$ As in the Veronese case, these only depend on the characteristic of the field, and we have the same inequality $\beta_{i,j}((T/J)\otimes_{{\mathbb{Z}}} \mathbb Q)\leq \beta_{i,j}((T/J)\otimes_{{\mathbb{Z}}} \mathbb F_{\ell}).$ As in the introduction, we say that $\beta(T/J)$ [**has $\ell$-torsion**]{} if this inequality is strict for some $i,j$, and we say that $\beta(T/J)$ [**depends on the characteristic**]{} if it has $\ell$-torsion for some $\ell$. Graphs and simplicial complexes ------------------------------- For a simplicial complex $X$, we write $V(X),$ $E(X),$ and $F(X)$ for the set of vertices, edges, and (2-dimensional) faces of $X$, respectively. We use $|*|$ to denote the number of elements in these sets. The degree of a vertex $v$ (denoted $\deg(v)$) is the number of edges in $X$ containing $v$. We write $\operatorname{maxdeg}(X)$ for the maximum degree of any vertex of $X$, and we write ${\operatorname{avg}}(X)$ for the average degree of a vertex in $X$. For a pair of graphs $H,G$, we write $H\subset G$ if $H$ is a subgraph of $G$. We write $H \overset{ind}{\subset} G$ if $H$ is an induced subgraph of $G$, that is, if the vertices of $H$ are a subset of the vertices of $G$ and the edges of $H$ are precisely the edges connecting those vertices within $G$ (see Figure \[fig:induced graph\]). We use similar definitions and notations for a simplicial complex $\Delta'$ to be a subcomplex (or an induced subcomplex) of another complex $\Delta$. If $\alpha \subset V(\Delta)$, then we let $\Delta|_{\alpha}$ denote the induced subcomplex of $\Delta$ on $\alpha$. (0,0) circle \[radius=2pt\]; (0,1) circle \[radius=2pt\]; (1,0) circle \[radius=2pt\]; (1,1) circle \[radius=2pt\]; (0,0)–(1,0)–(1,1)–(0,0)–(0,1)–(1,1); (.5,-.5) node [$G$]{}; (-.25,-.1) node [$1$]{}; (-.25,1.1) node [$2$]{}; (1.25,1.1) node [$3$]{}; (1.25,-.1) node [$4$]{}; (0,0) circle \[radius=2pt\]; (0,1) circle \[radius=2pt\]; (1,1) circle \[radius=2pt\]; (0,0)–(0,1)–(1,1); (.5,-.5) node [$H$]{}; (-.25,-.1) node [$1$]{}; (-.25,1.1) node [$2$]{}; (1.25,1.1) node [$3$]{}; The following definitions, adapted from [@Bollobas-subgraph] and [@Analytic-bollobas], will be used in sections \[sec:subgraphs\], \[sec:2torsion\], and \[sec:Betti numbers\]. \[mG\] The **essential density** of a graph $G$ is $$m(G):= \max\left\{\frac{|E(H)|}{|V(H)|}\; : \; H\subset G,\, |V(H)|>0\right\},$$ and $G$ is **strictly balanced** if $m(H)<m(G)$ for all subgraphs $H\subset G$. From a simplicial complex $\Delta$ on $n$ vertices, there is a corresponding Stanley–Reisner ideal $I_\Delta\subset S=k[x_1,\dots,x_n]$. Since these $I_\Delta$ are squarefree monomial ideals, Hochster’s Formula [@bruns-herzog Theorem 5.5.1] relates the Betti table of $S/I_\Delta$ to topological properties of $\Delta$, providing our key tool for studying $\beta(S/I_\Delta)$ for various fields $k$. An immediate consequence of Hochster’s formula is the following fact, which characterizes when these Betti tables are different over a field of characteristic $\ell$ than over ${\mathbb{Q}}$. \[fact:depend\] For a simplicial complex $\Delta$, the Betti table of the Stanley–Reisner ideal $I_\Delta$ has $\ell$-torsion if and only if there exists a subset $\alpha \subset V(\Delta)$ such that $\Delta|_\alpha$ has $\ell$-torsion in one of its homology groups. Monomial ideals from random flag complexes ------------------------------------------ Our monomial ideals are Stanley–Reisner ideals associated to random flag complexes. Recall that a flag complex is a simplicial complex obtained from a graph by adjoining a $k$-simplex to every $(k+1)$-clique in the graph. In particular, a flag complex is entirely determined by its underlying graph, and the process of obtaining a flag complex from its underlying graph is called taking the clique complex. We write $\Delta \sim \Delta(n,p)$ to denote the flag complex which is the clique complex of an Erdős-Rényi random graph $G(n,p)$ on $n$ vertices, where each edge is attached with probability $p$. If $\alpha\subset V(\Delta)$, then we note that $\Delta|_{\alpha}$ is also flag. The properties of random flag complexes have been analyzed extensively, with [@kahle] providing an overview. As discussed in the introduction, the syzygies of Stanley–Reisner ideals of random flag complexes were first studied in [@erman-yang]. Probability ----------- We use the notation ${\mathbf{P}}[*]$ for the probability of an event. For a random variable $X$, we use ${\mathbf{E}}[X]$ for the expected value of $X$ and $\operatorname{Var}(X)$ for the variance of $X$. For functions $f(x)$ and $g(x)$, we write $f\ll g$ if ${\displaystyle}\lim_{x\to \infty} f/g \to 0$. We use $f\in O(g)$ if there is a constant $N$ where $|f(x)|\leq N|g(x)|$ for all sufficiently large values of $x$, and we use $f\in \Omega(g)$ if there is a constant $N'$ where $|f(x)|\geq N'|g(x)|$ for all sufficiently large values of $x$. Constructing a flag complex with $m$-torsion homology {#sec:construction} ===================================================== The goal of this section is to prove the following result: \[thm:Xm\] For every $m \geq 2$, there exists a two-dimensional flag complex $X_m$ such that the torsion subgroup of $H_1(X_m)$ is isomorphic to ${\mathbb{Z}}/m{\mathbb{Z}}$ and $\operatorname{maxdeg}(X_m)\leq 12$. This result is the foundation of our proof of Theorem \[thm:m torsion\] as we will show that this specific complex $X_m$ appears as an induced subcomplex of $\Delta(n,p)$ with high probability under the hypotheses of that theorem. Here is an overview of our proof of Theorem \[thm:Xm\], which is largely based on ideas from [@newman]. Given an integer $m\geq 2$, we write its binary expansion as $m=2^{n_1}+\cdots+2^{n_k}$ with $0\leq n_1<\cdots<n_k$. Note that $k$ is the Hamming weight of $m$ and $n_k= \lfloor \log_2(m) \rfloor$. With this setup, the “repeated squares presentation” of ${\mathbb{Z}}/m{\mathbb{Z}}$ is given by $${\mathbb{Z}}/m{\mathbb{Z}}= {\langle}\gamma_0,\gamma_1,\dots,\gamma_{n_k}\; |\; 2\gamma_0=\gamma_1, 2\gamma_1=\gamma_2,\dots, 2\gamma_{n_k-1}=\gamma_{n_k}, \gamma_{n_1}+\cdots+\gamma_{n_k}=0 {\rangle}.$$ We will construct a two-dimensional flag complex $X_m$ such that the torsion subgroup of $H_1(X_m)$ has this presentation. To do so, we follow Newman’s “telescope and sphere” construction in [@newman], where $Y_1$ is the telescope satisfying $$H_1(Y_1) \cong {\langle}\gamma_0,\gamma_1,\dots,\gamma_{n_k}\; |\; 2\gamma_0=\gamma_1, 2\gamma_1=\gamma_2,\dots, 2\gamma_{n_k-1}=\gamma_{n_k} {\rangle},$$ $Y_2$ is the sphere satisfying $$H_1(Y_2) \cong {\langle}\tau_1,\dots,\tau_k \; |\; \tau_1+\cdots+\tau_k=0 {\rangle},$$ and $X_m$ is created by gluing $Y_1$ and $Y_2$ together to yield a complex with the desired $H_1$-group. Because we want our construction to be a flag complex with $\operatorname{maxdeg}(X_m)\leq 12$, we cannot simply quote Newman’s results. Instead, we must alter the triangulations to ensure that $Y_1$, $Y_2$, and $X_m$ are flag complexes. Then, we must further alter the construction to reduce $\operatorname{maxdeg}(X_m)$. However, each of our constructions is homeomorphic to each of Newman’s constructions. \[notation:hamming etc\] Throughout the remainder of this section we assume that $m\geq 2$ is given. We write $m=2^{n_1}+\cdots+2^{n_k}$ with $0\leq n_1<\cdots<n_k$. To simplify notation, we also denote $X_m$ by $X$ for the remainder of this section. The telescope construction -------------------------- The telescope $Y_1$ that we construct will be homeomorphic to the $Y_1$ that Newman constructs in [@newman Proof of Lemma 3.1] for the $d=2$ case. We start with building blocks which are punctured projective planes; in contrast with [@newman], our blocks are triangulated so that each is a flag complex. Explicitly, for each $i=0,\dots,(n_k-1)$, we produce a building block which is a triangulated projective plane with a square face removed, with vertices, edges, and faces as illustrated in Figure \[fig:Y1\]. Our building blocks differ from Newman’s in order to ensure that $Y_1$ and the final simplicial complex $X$ are flag complexes; for instance, we need to add extra vertices $v'_{8i},\dots,v'_{8i+7}$. (-1,3)–(-3,1)–(-1,2)–(-1,3); (-3,1)–(-2,1)–(-1,2)–(-3,1); (-2,1)–(-1,1)–(-1,2)–(-2,1); (-3,1)–(-3,-1)–(-2,1)–(-3,1); (-2,1)–(-3,-1)–(-2,-1)–(-2,1); (-2,1)–(-2,-1)–(-1,-1)–(-2,1); (-2,1)–(-1,-1)–(-1,1)–(-2,1); (-3,-1)–(-1,-3)–(-2,-1)–(-3,-1); (-2,-1)–(-1,-3)–(-1,-2)–(-2,-1); (-2,-1)–(-1,-2)–(-1,-1)–(-2,-1); (-1,-2)–(-1,-3)–(1,-3)–(-1,-2); (-1,-2)–(1,-3)–(1,-2)–(-1,-2); (-1,-2)–(1,-2)–(1,-1)–(-1,-2); (-1,-2)–(1,-1)–(-1,-1)–(-1,-2); (1,-2)–(1,-3)–(3,-1)–(1,-2); (1,-2)–(3,-1)–(2,-1)–(1,-2); (1,-2)–(2,-1)–(1,-1)–(1,-2); (2,-1)–(3,-1)–(3,1)–(2,-1); (2,-1)–(3,1)–(2,1)–(2,-1); (2,-1)–(2,1)–(1,1)–(2,-1); (2,-1)–(1,1)–(1,-1)–(2,-1); (2,1)–(3,1)–(1,3)–(2,1); (2,1)–(1,3)–(1,2)–(2,1); (2,1)–(1,2)–(1,1)–(2,1); (1,2)–(1,3)–(-1,3)–(1,2); (1,2)–(-1,3)–(-1,2)–(1,2); (1,2)–(-1,2)–(-1,1)–(1,2); (1,2)–(-1,1)–(1,1)–(1,2); \(0) at (-1,3) ; (0’) at (1,-3) ; (1) at (-3,1) ; (1’) at (3,-1) ; (2) at (-3,-1) ; (2’) at (3,1) ; (3’) at (1,3) ; (3) at (-1,-3) ; (4) at (-1,1) ; (5) at (-1,-1) ; (6) at (1,-1) ; (7) at (1,1) ; (8) at (-1,2) ; (9) at (-2,1) ; (10) at (-2,-1) ; (11) at (-1,-2) ; (12) at (1,-2) ; (13) at (2,-1) ; (14) at (2,1) ; (15) at (1,2) ; at (-1,3) [$\scriptstyle v_{4i}$]{}; at (1,-3) [$\scriptstyle v_{4i}$]{}; at (-3,1) [$\scriptstyle v_{4i+1}$]{}; at (3,-1) [$\scriptstyle v_{4i+1}$]{}; at (-3,-1) [$\scriptstyle v_{4i+2}$]{}; at (3,1) [$\scriptstyle v_{4i+2}$]{}; at (1,3) [$\scriptstyle v_{4i+3}$]{}; at (-1,-3) [$\scriptstyle v_{4i+3}$]{}; at (-1,1) [$\scriptstyle v_{4i+4}$]{}; at (-1,-1) [$\scriptstyle v_{4i+5}$]{}; at (1,-1) [$\scriptstyle v_{4i+6}$]{}; at (1,1) [$\scriptstyle v_{4i+7}$]{}; at (-1,2) [$\scriptstyle v'_{8i}$]{}; at (-1.8,1) [$\scriptstyle v'_{8i+1}$]{}; at (-2,-1) [$\scriptstyle v'_{8i+2}$]{}; at (-1.05,-2.05) [$\scriptstyle v'_{8i+3}$]{}; at (0.98,-1.98) [$\scriptstyle v'_{8i+4}$]{}; at (2,-1.07) [$\scriptstyle v'_{8i+5}$]{}; at (1.9, 0.93) [$\scriptstyle v'_{8i+6}$]{}; at (0.69,2.2) [$\scriptstyle v'_{8i+7}$]{}; (-1,3) circle \[radius=2pt\]; (1,-3) circle \[radius=2pt\]; (-3,1) circle \[radius=2pt\]; (3,-1) circle \[radius=2pt\]; (-3,-1) circle \[radius=2pt\]; (3,1) circle \[radius=2pt\]; (1,3) circle \[radius=2pt\]; (-1,-3) circle \[radius=2pt\]; (-1,1) circle \[radius=2pt\]; (-1,-1) circle \[radius=2pt\]; (1,-1) circle \[radius=2pt\]; (1,1) circle \[radius=2pt\]; (-1,2) circle \[radius=2pt\]; (-2,1) circle \[radius=2pt\]; (-2,-1) circle \[radius=2pt\]; (-1,-2) circle \[radius=2pt\]; (1,-2) circle \[radius=2pt\]; (2,-1) circle \[radius=2pt\]; (2,1) circle \[radius=2pt\]; (1,2) circle \[radius=2pt\]; We construct $Y_1$ by identifying edges and vertices of these $n_k$ building blocks as labeled. The underlying vertex set is $V(Y_1) = \{v_0,v_1,v_2,\dots,v_{4n_k+3},v'_0,v'_1,\dots,v'_{8n_k-1}\}$, so we have $|V(Y_1)|=(4n_k+4)+8n_k=12n_k+4$. Since each building block has $44$ edges, $4$ of which are glued to the next building block, and $28$ faces, a similar computation yields $|E(Y_1)|=40n_k+4$ and $|F(Y_1)|=28n_k$. In addition, observe that the vertices of highest degree are those in the squares in the “middle” of the telescope, such as vertex $v_4$ when $n_k\geq 2$. In this case, $v_4$ is adjacent to $v_5, v_7, v'_0, v'_1, v'_7, v'_8, v'_{15}, v'_{11},$ and $v'_{12}$, so $\deg(v_4)=9$. By the symmetry of $Y_1$, we have that $\operatorname{maxdeg}(Y_1)=9$ when $n_k\geq 2$, and $\operatorname{maxdeg}(Y_1)=6$ when $n_k=1$ (when $m=2,3$). To compute $H_1(Y_1)$, we simply apply the identical argument from [@newman]. We order the vertices in the natural way, where $v_j>v_k$ if $j>k$, similarly for the $v_\ell'$, and where $v'_{\ell} > v_j$ for all $\ell,j$. We let these vertex orderings induce orientations on the edges and faces of $Y_1$. For each $i=0,\dots,n_k$, denote by $\gamma_i$ the 1-cycle of $Y_1$ represented by $[v_{4i},v_{4i+1}]+[v_{4i+1},v_{4i+2}]+[v_{4i+2},v_{4i+3}]-[v_{4i},v_{4i+3}]$. Then $2\gamma_i - \gamma_{i+1}$ is a 1-boundary of $Y_1$ for each $i=0,\dots,(n_k-1)$, and, as in Newman’s construction, we have that $H_1(Y_1)$ can be presented as ${\langle}\gamma_0,\gamma_1,\dots,\gamma_{n_k}\; |\; 2\gamma_0=\gamma_1, 2\gamma_1=\gamma_2,\dots, 2\gamma_{n_k-1}=\gamma_{n_k} {\rangle}$. The sphere construction ----------------------- The sphere part $Y_2$ is a flag triangulation of the sphere $S^2$ that has $k$ square holes such that the squares are all vertex disjoint and nonadjacent. Our $Y_2$ will be homeomorphic to the $Y_2$ that Newman constructs in [@newman] for the $d=2$ case, but our construction involves a few different steps. First, we will show that for any integer $k\geq 1$, there exists a flag triangulation $T_i$ of $S^2$ (here $i=\lfloor \frac{k-1}{4}\rfloor$) with at least $k$ faces such that $\operatorname{maxdeg}(T_i)\leq 6$. Then, we will insert square holes on $k$ of the faces of $T_i$, while subdividing the edges, and call the resulting flag complex $\widetilde{T_i}$. Finally, we describe a process to replace each vertex of degree 14 in $\widetilde{T_i}$ with two degree 9 vertices so that the resulting complex, $Y_2$, has $\operatorname{maxdeg}(Y_2)\leq 12$. Throughout these constructions, we will have four cases corresponding to the value of $k \mod 4$, and we carefully keep track of the degrees of each vertex in $T_i$, $\widetilde{T_i}$, and $Y_2$ for each case. ### $T_i$ and flag bistellar 0-moves We begin by constructing an infinite sequence $T_0, T_1, T_2, \dots$ of flag triangulations of $S^2$ such that $\operatorname{maxdeg}(T_i)\leq 6$ for all $i$. To do so, we adapt the bistellar 0-moves used in [@newman Lemma 5.6]. Let $T_0$ be the $3$-simplex boundary on the vertex set $\{w_0, w_1, w_2, w_3\}$. Note that each vertex of $T_0$ has degree 3. We will construct the remaining $T_i$ inductively. To build $T_1$, first remove the face $[w_1, w_2, w_3]$ and edge $[w_1, w_3]$. Then, add two new vertices $w_4$ and $w_5$ as well as new edges $[w_0, w_4], [w_1, w_4], [w_3, w_4], [w_1, w_5],$ $[w_2, w_5], [w_3, w_5],$ and $[w_4, w_5]$. Taking the clique complex will then give $T_1$. See Figure \[fig:triangulations\]. Essentially, this process is the same as making the face $[w_1,w_2,w_3]$ into a square face $[w_1,w_2,w_3,w_4]$, removing that square face, taking the cone over it, and then ensuring that the resulting complex is a flag triangulation of $S^2$. We will call such a move a **flag bistellar 0-move**. Each $T_{i+1}$ for $i\geq 0$ will be obtained from $T_i$ by performing a flag bistellar 0-move on the face $[w_{2i+1}, w_{2i+2}, w_{2i+3}]$ of $T_i$. Explicitly, to construct $T_{i+1}$, remove the face $[w_{2i+1}, w_{2i+2}, w_{2i+3}]$ and the edge $[w_{2i+1}, w_{2i+3}]$. Then, add new vertices $w_{2i+4}$ and $w_{2i+5}$ and new edges $[w_{2i}, w_{2i+4}], [w_{2i+1}, w_{2i+4}], [w_{2i+3}, w_{2i+4}], [w_{2i+1},w_{2i+5}],$ $[w_{2i+2}, w_{2i+5}], [w_{2i+3}, w_{2i+5}],$ $[w_{2i+4}, w_{2i+5}],$ and take the clique complex to get $T_{i+1}$. Note that each flag bistellar 0-move adds 2 vertices, 6 edges, and 4 faces. Since $|V(T_0)|=4, |E(T_0)|=6$, and $|F(T_0)|=4$, this means that $|V(T_i)|=2i+4$, $|E(T_i)|=6i+6$, and $|F(T_i)|=4i+4$. Further, Table \[tab:Ti\_vertices\] summarizes the degrees of the vertices in each $T_i$. $T_i$ Degree Vertices ----------- -------- -------------------------------- $T_0$ 3 $w_0, w_1, w_2, w_3$ $T_1$ 4 $w_0, w_1, w_2, w_3, w_5, w_6$ $T_2$ 4 $w_0, w_1, w_6, w_7$ 5 $w_2, w_3, w_4, w_5$ $T_i$ 4 $w_0, w_1, w_{2i+2}, w_{2i+3}$ $i\geq 3$ 5 $w_2, w_3, w_{2i}, w_{2i+1}$ 6 $w_4, \dots, w_{2i-1}$ : Degrees of the vertices in $T_i$.[]{data-label="tab:Ti_vertices"} To compute the degrees of vertices in $T_i$ for $i\geq 3$, observe that when the new vertices $w_{2i+2}$ and $w_{2i+3}$ are added, they have degree $4$ in $T_i$. For each of the next two iterations of the flag bistellar-0 move, the degree of these vertices increases by one, resulting in degree 6 in $T_{i+2}$. In the remaining triangulations $T_j$ with $j\geq i+3$, these vertices are not affected. Therefore, $\operatorname{maxdeg}(T_i)\leq 6$ for each $i$. (-1,0)–(1,0)–(0,1.732)–(-1,0); (1,0)–(1.25,1)–(0,1.732)–(1,0); (-1,0)–(1.25,1); (-1,0) circle \[radius=1pt\]; (1,0) circle \[radius=1pt\]; (0,1.732) circle \[radius=1pt\]; (1.25,1) circle \[radius=1pt\]; at (0,1.732) [$w_1$]{}; at (-1,0) [$w_2$]{}; at (1,0) [$w_3$]{}; at (1.25,1) [$w_0$]{}; at (0, -0.25) [$T_0$]{}; (2,0)–(4,0)–(3.2,0.7)–(2,0); (2,0)–(3.2,0.7)–(3,1.732)–(2,0); (4,0)–(4.25,1)–(4,0.889)–(4,0); (4,0)–(4,0.889)–(3.2,0.7)–(4,0); (3.2,0.7)–(4,0.889)–(3,1.732)–(3.2,0.7); (4,0.889)–(4.25,1)–(3,1.732)–(4,0.889); (2,0)–(4.25,1); (2,0) circle \[radius=1pt\]; (4,0) circle \[radius=1pt\]; (3,1.732) circle \[radius=1pt\]; (4.25,1) circle \[radius=1pt\]; (3.2,0.7) circle \[radius=1pt\]; (4,0.889) circle \[radius=1pt\]; at (3,1.732) [$w_1$]{}; at (2,0) [$w_2$]{}; at (4,0) [$w_3$]{}; at (4.25,1) [$w_0$]{}; at (3.2,0.7) [$w_5$]{}; at (4,0.889) [$w_4$]{}; at (3, -0.25) [$T_1$]{}; (5,0)–(7,0)–(6.4,0.2)–(5,0); (5,0)–(6.4,0.2)–(6.2,0.7)–(5,0); (5,0)–(6.2,0.7)–(6,1.732)–(5,0); (7,0)–(7.25,1)–(7,0.889)–(7,0); (6.2,0.7)–(7,0.889)–(6,1.732)–(6.2,0.7); (7,0.889)–(7.25,1)–(6,1.732)–(7,0.889); (7,0)–(6.7,0.5)–(6.4,0.2)–(7,0); (6.4,0.2)–(6.7, 0.5)–(6.2, 0.7)–(6.4, 0.2); (7,0)–(7,0.889)–(6.7, 0.5)–(7,0); (6.7,0.5)–(7,0.889)–(6.2,0.7)–(6.7,0.5); (5,0)–(7.25,1); (5,0) circle \[radius=1pt\]; (7,0) circle \[radius=1pt\]; (6,1.732) circle \[radius=1pt\]; (7.25,1) circle \[radius=1pt\]; (6.2,0.7) circle \[radius=1pt\]; (7,0.889) circle \[radius=1pt\]; (6.4,0.2) circle \[radius=1pt\]; (6.7, 0.5) circle \[radius=1pt\]; at (6,1.732) [$w_1$]{}; at (5,0) [$w_2$]{}; at (7,0) [$w_3$]{}; at (7.25,1) [$w_0$]{}; at (6.2,0.7) [$w_5$]{}; at (7,0.889) [$w_4$]{}; at (6.4, 0.22) [$w_6$]{}; at (6.68,0.5) [$w_7$]{}; at (6, -0.25) [$T_2$]{}; From this infinite sequence of flag triangulations of $S^2$ with bounded degree, we are interested in the particular $T_i$ with $i=\lfloor \frac{k-1}{4} \rfloor$ to use in our construction of $Y_2$, where $k$ is the Hamming weight of $m$ as in Notation \[notation:hamming etc\]. Note that this $T_i$ has vertex set $\{w_0,\dots,w_{2i+3}\}$ and has $4\lfloor \frac{k-1}{4}\rfloor+4$ faces. Let $\delta$ be the integer $0\leq \delta \leq 3$ where $\delta \equiv -k \mod 4$. Then $T_i$ has exactly $k+\delta$ faces. ### Constructing $\widetilde{T_i}$ Next, we insert square holes in the first $k$ faces of $T_i$ and subdivide the remaining faces in such a way that the squares will be vertex disjoint and nonadjacent. First, we will insert square holes in $k$ of the faces of $T_i$, making sure to triangulate the resulting faces and take the clique complex so that our simplicial complex remains flag. Let $[w_r,w_s,w_t]$ with $r<s<t$ be the $j$th of these $k$ faces with respect to a fixed ordering of the faces (where $j$ ranges from 1 to $k$). We remove this face and subdivide the edges by adding new vertices $w'_{r,s}, w'_{r,t},$ and $w'_{s,t}$ and new edges $[w_r, w'_{r,s}], [w_s, w'_{r,s}], [w_r, w'_{r,t}], [w_t, w'_{r,t}],$ $[w_s, w'_{s,t}],$ and $[w_t, w'_{s,t}]$. Then, we add vertices $u_{4j-4}, u_{4j-3}, u_{4j-2},$ and $u_{4j-1}$ to form a square inside the original face with indices increasing counterclockwise. Moreover, we add edges $$\begin{aligned} & [w_r, u_{4j-4}], [w_r, u_{4j-1}], [u_{4j-4}, w'_{r,s}], [u_{4j-3}, w'_{r,s}], [w_s, u_{4j-3}] \\ & [u_{4j-3}, w'_{s,t}], [u_{4j-2}, w'_{s,t}], [w_t, u_{4j-2}], [u_{4j-2}, w'_{r,t}], [u_{4j-1}, w'_{r,t}].\end{aligned}$$ After applying this process, we take the clique complex. The result of this operation on face $[w_r, w_s, w_t]$ is depicted in Figure \[fig:Y2\_faces\] (left). The remaining $\delta$ faces of $T_i$ will simply be subdivided and triangulated before taking the clique complex. Explicitly, this means that after removing the face $[w_{2i+1},w_{2i+2},w_{2i+3}]$ and its edges, we add vertices $w'_{2i+1,2i+2}, w'_{2i+1,2i+3},$ and $w'_{2i+2,2i+3}$ and edges $$\begin{aligned} &[w_{2i+1}, w'_{2i+1,2i+2}], [w_{2i+2}, w'_{2i+1,2i+2}], [w_{2i+1}, w'_{2i+1,2i+3}],\\ &[w_{2i+3}, w'_{2i+1,2i+3}], [w'_{2i+1,2i+2}, w'_{2i+1,2i+3}], [w_{2i+2}, w'_{2i+2,2i+3}],\\ &[w_{2i+3}, w'_{2i+2,2i+3}], [w'_{2i+1,2i+2}, w'_{2i+2,2i+2}], [w'_{2i+1,2i+3}, w'_{2i+2,2i+3}].\end{aligned}$$ This subdivision of face $[w_{2i+1},w_{2i+2},w_{2i+3}]$ is shown in Figure \[fig:Y2\_faces\] (right). We do similarly for the faces $[w_{2i-1},w_{2i+2}, w_{2i+3}]$ and $[w_{2i}, w_{2i+1}, w_{2i+3}]$, if necessary. The clique complex of this construction is a flag complex which is homeomorphic to $S^2$ with $k$ distinct points removed. Call this complex $\widetilde{T_i}$. (-1,0)–(0,0)–(-0.2,0.667)–(-1,0); (-1,0)–(-0.2,0.667)–(-0.5,0.866)–(-1,0); (-0.5,0.866)–(-0.2,0.667)–(-0.2,1.066)–(-0.5,0.866); (-0.5,0.866)–(-0.2,1.066)–(0,1.732)–(-0.5,0.866); (0,1.732)–(-0.2,1.066)–(0.2,1.066)–(0,1.732); (0.5,0.866)–(0.2,1.066)–(0,1.732)–(0.5,0.866); (0.5,0.866)–(0.2,0.667)–(0.2,1.066)–(0.5,0.866); (1,0)–(0.2,0.667)–(0.5,0.866)–(1,0); (1,0)–(0,0)–(0.2,0.667)–(1,0); (0,0)–(-0.2,0.667)–(0.2,0.667)–(0,0); (0,1.723)–(-0.5,0.866); (0,1.723)–(-0.2,1.066); (-0.2,1.066)–(-0.5,0.866); (-0.2,0.667)–(-0.5,0.866); (-0.2,1.066)–(-0.2,0.667); (-1,0)–(-0.2,0.667); (-1,0)–(-0.5,0.866); (-1,0)–(0,0); (-0.2,0.667)–(0,0); (1,0)–(0,0); (0.2,0.667)–(0,0); (1,0)–(0.2,0.667); (1,0)–(0.5, 0.866); (0.2,0.667)–(0.5, 0.866); (0.2,1.066)–(0.5, 0.866); (0,1.723)–(0.2,1.066); (0,1.723)–(0.5, 0.866); (0.2,0.667)–(0.2,1.066); (-0.2,1.066)–(0.2,1.066); (-0.2,0.667)–(0.2,0.667); (2,0)–(3,0)–(2.5,0.866)–(2,0); (3,0)–(2.5,0.866)–(3.5,0.866)–(3,0); (4,0)–(3.5,0.866)–(3,0)–(4,0); (2.5,0.866)–(3.5,0.866)–(3,1.732)–(2.5,0.866); (3,1.723)–(2.5,0.866); (2,0)–(2.5,0.866); (2,0)–(3,0); (4,0)–(3,0); (3.5, 0.866)–(3,0); (3,1.723)–(3.5, 0.866); (2.5, 0.866)–(3,0); (4, 0)–(3.5,0.866); (2.5, 0.866)–(3.5,0.866); (-1,0) circle \[radius=1pt\]; (0,0) circle \[radius=1pt\]; (1,0) circle \[radius=1pt\]; (0,1.732) circle \[radius=1pt\]; (-0.5,0.866) circle \[radius=1pt\]; (0.5,0.866) circle \[radius=1pt\]; (-0.2,1.066) circle \[radius=1pt\]; (0.2,1.066) circle \[radius=1pt\]; (-0.2,0.667) circle \[radius=1pt\]; (0.2,0.667) circle \[radius=1pt\]; (2,0) circle \[radius=1pt\]; (3,0) circle \[radius=1pt\]; (4,0) circle \[radius=1pt\]; (3,1.732) circle \[radius=1pt\]; (2.5,0.866) circle \[radius=1pt\]; (3.5,0.866) circle \[radius=1pt\]; at (0,1.732) [$w_r$]{}; at (-1,0) [$w_s$]{}; at (1,0) [$w_t$]{}; at (-0.5,0.866) [$w'_{r,s}$]{}; at (0,0) [$w'_{s,t}$]{}; at (0.5,0.866) [$w'_{r,t}$]{}; at (-0.2,1.06) [$\scriptstyle u_{4j-4}$]{}; at (-0.2,0.68) [$\scriptstyle u_{4j-3}$]{}; at (0.22,0.63) [$\scriptstyle u_{4j-2}$]{}; at (0.22,1.08) [$\scriptstyle u_{4j-1}$]{}; at (3,1.732) [$w_{2i+1}$]{}; at (2,0) [$w_{2i+2}$]{}; at (4,0) [$w_{2i+3}$]{}; at (2.5,0.866) [$w'_{2i+1,2i+2}$]{}; at (3,0) [$w'_{2i+2,2i+3}$]{}; at (3.5,0.866) [$w'_{2i+1,2i+3}$]{}; Let’s consider the degrees of the vertices of $\widetilde{T_i}$. We have that $\deg(w'_{m,n})=6$ for all $m,n$ and $\deg(u_\ell)\in \{4,5\}$ for all $\ell$, where the “top” $u_\ell$ have degree 4 and the “bottom” $u_\ell$ have degree 5. To determine the degrees of the $w_j$ vertices, we need to consider their degrees in $T_i$ and how their degrees increase during the subdivision and square face removal processes. As we are interested in bounding the maximum degree of the vertices of $\widetilde{T_i}$, we need only consider the case when $\delta=0$ and all $k$ faces of $T_i$ have a square removed from them. $\widetilde{T_i}$ Degree Vertices --------------------- -------- ------------------------ 6 $w_2, w_3$ ${\widetilde}{T_0}$ 7 $w_1$ $(k=4)$ 9 $w_0$ 8 $w_4, w_5$ ${\widetilde}{T_1}$ 9 $w_2, w_3$ $(k=8)$ 10 $w_1$ 12 $w_0$ 8 $w_6, w_7$ ${\widetilde}{T_2}$ 10 $w_1$ $(k=12)$ 11 $w_4, w_5$ 12 $w_0, w_2, w_3$ 8 $w_{2i+2}, w_{2i+3}$ $\widetilde{T_i}$ 10 $w_1$ $i\geq 3$ 11 $w_{2i}, w_{2i+1}$ $(k=4i+4)$ 12 $w_0, w_2, w_3$ 14 $w_4, \dots, w_{2i-1}$ : Degrees of the vertices in $\widetilde{T_i}$ when $k\equiv 0 \mod 4$.[]{data-label="tab:Ti_tilde"} Table \[tab:Ti\_tilde\] gives the degrees of each of the $w_j$ vertices in $\widetilde{T_i}$ when $\delta=0$. To verify the degrees of the $w_j$ in $\widetilde{T_i}$ when $i\geq 3$, we consider how the degrees of the vertices change as $i$ increases. Between ${\widetilde}{T}_{i-1}$ and $\widetilde{T_i}$ (with $\delta=0$ for both), the only vertices that change degree are $w_{2i-2}, w_{2i-1}, w_{2i}, w_{2i+1}$, each of which increase degree by 3. This is because they each get one new edge from the $T_i$ flag bistellar 0-move and two new edges from the square removal triangulation process (since each vertex is the smallest indexed and hence the “top” vertex of one new triangular face). Further, the new vertices $w_{2i+2}, w_{2i+3}$ in $\widetilde{T_i}$ have degree 8, and they increase degree by 3 in the next two iterations, resulting in degree 14 in ${\widetilde}{T}_{i+2}$ and all future iterations. The above argument shows that regardless of $m$ and $k$, $\operatorname{maxdeg}(\widetilde{T_i})\leq 14$, where $i=\lfloor \frac{k-1}{4} \rfloor$. Furthermore, the only vertices that could have degree 14 are $w_4,\dots, w_{2i-1}$, each of which is separated from the others by a $w'_{m,n}$ vertex, which only has degree 6. We want to know exactly which vertices in $\widetilde{T_i}$ have degree 14, for all possible $k$ with $i\geq 3$, because we plan to alter these vertices to decrease $\operatorname{maxdeg}(\widetilde{T_i})$. Note that as $\delta$ increases from 0 to 3, the degree of each $w_j$ vertex is nonincreasing. When $k=4i+4$ and $\delta=0$, the above table gives that $w_4,\dots, w_{2i-1}$ have degree 14. When $k=4i+3$ and $\delta=1$, the face $[w_{2i+1},w_{2i+2},w_{2i+3}]$ is subdivided instead of having a square removed, but this does not change the degrees of $w_4,\dots, w_{2i-1}$, so these all still have degree 14. When $k=4i+2$ and $\delta=2$, the faces $[w_{2i+1},w_{2i+2},w_{2i+3}]$ and $[w_{2i-1},w_{2i+2},w_{2i+3}]$ are subdivided. Therefore, $w_{2i-1}$ has two fewer edges than in the previous case since $w_{2i-1}$ is the smallest indexed vertex in $[w_{2i-1}, w_{2i+2},w_{2i+3}]$ and so would have two “top” $u_\ell$ adjacent to it if this face had a square removed from it. So, in this case, $w_4,\dots, w_{2i-2}$ have degree 14 and $w_0,w_2, w_3, w_{2i-1}$ have degree 12 in $\widetilde{T_i}$. Finally, if $k=4i+1$ and $\delta=3$, then additionally the face $[w_{2i},w_{2i+1},w_{2i+3}]$ is subdivided, which means that the degree 12 and 14 vertices are the same as in the previous cases. ### Replacing degree 14 vertices to construct $Y_2$ Having identified the vertices of $\widetilde{T_i}$ of the highest degree, we now describe a process by which we will replace each vertex of degree 14 by two vertices of degree 9 in order to ensure that $\operatorname{maxdeg}(\widetilde{T_i})\leq 12$ for all $k$ and $i$. The resulting flag complex, given by taking the clique complex of this construction, will be the final $Y_2$, and it will be homeomorphic to $\widetilde{T_i}$. The process is summarized by Figure \[fig:Replacing\_vertex\] and described in detail in the following paragraphs. Suppose $w_j$ is a vertex of degree 14 in $\widetilde{T_i}$. Locally, on a small neighborhood of $w_j$, $\widetilde{T_i}$ is homeomorphic to a $2$-manifold. Since $\deg(w_j)=14$, $w_j$ is surrounded by six triangular faces coming from $T_i$, all of which have had a square removed. By our construction, two of these squares (which are in adjacent triangular faces) have both of their “top” $u_\ell$ vertices connected to $w_j$, but the other four squares just have a single edge connecting one of their “bottom” $u_\ell$ vertices to $w_j$. So, $w_j$ has six $w'_{m,n}$ neighbors and eight $u_\ell$ neighbors, which form a 14-sided polygon with $w_j$ as its “star” point. Choose two $w'_{m,n}$ vertices which are across from each other in this 14-sided polygon, say $w'_{a,b}$ and $w'_{c,d}$. Next, we will remove $w_j$ and all of the 14 faces that it is contained in. Then, we add vertices $w_{j_1}$ and $w_{j_2}$ in place of $w_j$ and add edges in such a way that $\deg(w_{j_1})=\deg(w_{j_2})=9$, there are edges $[w_{j_1},w_{j_2}], [w_{j_1},w'_{a,b}], [w_{j_1},w'_{c,d}], [w_{j_2},w'_{a,b}],$ and $[w_{j_2},w'_{c,d}]$, and the 14-sided polygon is triangulated with 16 triangles. This process only changes the degree of $w'_{a,b}$ and $w'_{c,d}$, each of which now have degree 7. Therefore, the maximum degree of $w_{j_1}, w_{j_2}$, and the 14 vertices in the polygon is 9 (since $\deg(u_\ell)\in \{4,5\}$ and $\deg(w'_{m,n})=6$). To illustrate this construction, we consider the case when $k=20$. Then $i=4$, $\delta=0$, and $\deg(w_7)=14$ in ${\widetilde}{T_4}$. Figure \[fig:Replacing\_vertex\] depicts this process when $w'_{a,b}=w'_{3,7}$ and $w'_{c,d}=w'_{7,11}$. (0,0)–(0.2,0.67)–(-0.2,0.67)–(0,0); (0,0)–(0.2,0.67)–(0.5,0.866)–(0,0); (0.2,0.67)–(0.5,0.866)–(0.2, 1.07)–(0.2,0.67); (0.5,0.866)–(0.2, 1.07)–(1,1.73)–(0.5,0.866); (0.2, 1.07)–(1,1.73)–(0,1.73)–(0.2,1.07); (0,1.73)–(0.2,1.07)–(-0.2,1.07)–(0,1.73); (-0.2,1.07)–(0,1.73)–(-1,1.73)–(-0.2,1.07); (-1,1.73)–(-0.2,1.07)–(-0.5,0.866)–(-1,1.73); (-0.2,1.07)–(-0.5,0.866)–(-0.2,0.67)–(-0.2,1.07); (-0.5,0.866)–(-0.2,0.67)–(0,0)–(-0.5, 0.866); (0,0)–(0.68, 0.16)–(0.48, 0.5)–(0,0); (0,0)–(0.68, 0.16)–(1,0)–(0,0); (0.68, 0.16)–(1,0)–(1.02, 0.36)–(0.68, 0.16); (1,0)–(1.02, 0.36)–(2,0)–(1,0); (1.02, 0.36)–(2,0)–(1.5,0.866)–(1.02,0.36); (1.5,0.866)–(1.02,0.36)–(0.82,0.7)–(1.5,0.866); (0.82,0.7)–(1.5,0.866)–(1,1.73)–(0.82,0.7); (1,1.73)–(0.82,0.7)–(0.5,0.866)–(1,1.73); (0.82,0.7)–(0.5,0.866)–(0.48, 0.5)–(0.82,0.7); (0.5,0.866)–(0.48, 0.5)–(0,0)–(0.5, 0.866); (-2,0)–(-1.32, 0.16)–(-1.52, 0.5)–(-2,0); (-2,0)–(-1.32, 0.16)–(-1,0)–(-2,0); (-1.32, 0.16)–(-1,0)–(-0.98, 0.36)–(-1.32, 0.16); (-1,0)–(-0.98, 0.36)–(0,0)–(-1,0); (-0.98, 0.36)–(0,0)–(-0.5,0.866)–(-0.98,0.36); (-0.5,0.866)–(-0.98,0.36)–(-1.18,0.7)–(-0.5,0.866); (-1.18,0.7)–(-0.5,0.866)–(-1,1.73)–(-1.18,0.7); (-1,1.73)–(-1.18,0.7)–(-1.5,0.866)–(-1,1.73); (-1.18,0.7)–(-1.5,0.866)–(-1.52, 0.5)–(-1.18,0.7); (-1.5,0.866)–(-1.52, 0.5)–(-2,0)–(-1.5, 0.866); (-1,-1.73)–(-0.8,-1.06)–(-1.2,-1.06)–(-1,-1.73); (-1,-1.73)–(-0.8,-1.06)–(-0.5,-0.866)–(-1,-1.73); (-0.8,-1.06)–(-0.5,-0.866)–(-0.8, -0.66)–(-0.8,-1.06); (-0.5,-0.866)–(-0.8, -0.66)–(0,0)–(-0.5,-0.866); (-0.8, -0.66)–(0,0)–(-1,0)–(-0.8,-0.66); (-1,0)–(-0.8,-0.66)–(-1.2,-0.66)–(-1,0); (-1.2,-0.66)–(-1,0)–(-2,0)–(-1.2,-0.66); (-2,0)–(-1.2,-0.66)–(-1.5,-0.866)–(-2,0); (-1.2,-0.66)–(-1.5,-0.866)–(-1.2,-1.06)–(-1.2,-0.66); (-1.5,-0.866)–(-1.2,-1.06)–(-1,-1.73)–(-1.5, -0.866); (-1,-1.73)–(-0.32, -1.57)–(-0.52, -1.23)–(-1,-1.73); (-1,-1.73)–(-0.32, -1.57)–(0,-1.73)–(-1,-1.73); (-0.32, -1.57)–(0,-1.73)–(0.02, -1.37)–(-0.32, -1.57); (0,-1.73)–(0.02, -1.37)–(1,-1.73)–(0,-1.73); (0.02, -1.37)–(1,-1.73)–(0.5,-0.866)–(0.02,-1.37); (0.5,-0.866)–(0.02,-1.37)–(-0.18,-1.03)–(0.5,-0.866); (-0.18,-1.03)–(0.5,-0.866)–(0,0)–(-0.18,-1.03); (0,0)–(-0.18,-1.03)–(-0.5,-0.866)–(0,0); (-0.18,-1.03)–(-0.5,-0.866)–(-0.52, -1.23)–(-0.18,-1.03); (-0.5,-0.866)–(-0.52, -1.23)–(-1,-1.73)–(-0.5, -0.866); (1,-1.73)–(1.2,-1.06)–(0.8,-1.06)–(1,-1.73); (1,-1.73)–(1.2,-1.06)–(1.5,-0.866)–(1,-1.73); (1.2,-1.06)–(1.5,-0.866)–(1.2, -0.66)–(1.2,-1.06); (1.5,-0.866)–(1.2, -0.66)–(2,0)–(1.5,-0.866); (1.2, -0.66)–(2,0)–(1,0)–(1.2,-0.66); (1,0)–(1.2,-0.66)–(0.8,-0.66)–(1,0); (0.8,-0.66)–(1,0)–(0,0)–(0.8,-0.66); (0,0)–(0.8,-0.66)–(0.5,-0.866)–(0,0); (0.8,-0.66)–(0.5,-0.866)–(0.8,-1.06)–(0.8,-0.66); (0.5,-0.866)–(0.8,-1.06)–(1,-1.73)–(0.5, -0.866); (6,0.2)–(6.2,0.67)–(5.8,0.67)–(6,0.2); (6,0.2)–(6.2,0.67)–(6.5,0.866)–(6,0.2); (6.2,0.67)–(6.5,0.866)–(6.2, 1.07)–(6.2,0.67); (6.5,0.866)–(6.2, 1.07)–(7,1.73)–(6.5,0.866); (6.2, 1.07)–(7,1.73)–(6,1.73)–(6.2,1.07); (6,1.73)–(6.2,1.07)–(5.8,1.07)–(6,1.73); (5.8,1.07)–(6,1.73)–(5,1.73)–(5.8,1.07); (5,1.73)–(5.8,1.07)–(5.5,0.866)–(5,1.73); (5.8,1.07)–(5.5,0.866)–(5.8,0.67)–(5.8,1.07); (5.5,0.866)–(5.8,0.67)–(6,0.2)–(5.5, 0.866); (6,0.2)–(6.5,0.866)–(6,-0.2)–(6,0.2); (6,-0.2)–(6.5, 0.866)–(6.48, 0.5)–(6,-0.2); (6,-0.2)–(6.48,0.5)–(6.68,0.16)–(6,-0.2); (6,-0.2)–(6.68, 0.16)–(7,0)–(6,-0.2); (6.68, 0.16)–(7,0)–(7.02, 0.36)–(6.68, 0.16); (7,0)–(7.02, 0.36)–(8,0)–(7,0); (7.02, 0.36)–(8,0)–(7.5,0.866)–(7.02,0.36); (7.5,0.866)–(7.02,0.36)–(6.82,0.7)–(7.5,0.866); (6.82,0.7)–(7.5,0.866)–(7,1.73)–(6.82,0.7); (7,1.73)–(6.82,0.7)–(6.5,0.866)–(7,1.73); (6.82,0.7)–(6.5,0.866)–(6.48, 0.5)–(6.82,0.7); (4,0)–(4.68, 0.16)–(4.48, 0.5)–(4,0); (4,0)–(4.68, 0.16)–(5,0)–(4,0); (4.68, 0.16)–(5,0)–(5.02, 0.36)–(4.68, 0.16); (5,0)–(5.02, 0.36)–(6,0.2)–(5,0); (5.02, 0.36)–(6,0.2)–(5.5,0.866)–(5.02,0.36); (5.5,0.866)–(5.02,0.36)–(4.82,0.7)–(5.5,0.866); (4.82,0.7)–(5.5,0.866)–(5,1.73)–(4.82,0.7); (5,1.73)–(4.82,0.7)–(4.5,0.866)–(5,1.73); (4.82,0.7)–(4.5,0.866)–(4.48, 0.5)–(4.82,0.7); (4.5,0.866)–(4.48, 0.5)–(4,0)–(4.5, 0.866); (5,-1.73)–(5.2,-1.06)–(4.8,-1.06)–(5,-1.73); (5,-1.73)–(5.2,-1.06)–(5.5,-0.866)–(5,-1.73); (5.2,-1.06)–(5.5,-0.866)–(5.2, -0.66)–(5.2,-1.06); (5.5,-0.866)–(5.2, -0.66)–(6,0.2)–(5.5,-0.866); (5.2, -0.66)–(6,0.2)–(5,0)–(5.2,-0.66); (5,0)–(5.2,-0.66)–(4.8,-0.66)–(5,0); (4.8,-0.66)–(5,0)–(4,0)–(4.8,-0.66); (4,0)–(4.8,-0.66)–(4.5,-0.866)–(4,0); (4.8,-0.66)–(4.5,-0.866)–(4.8,-1.06)–(4.8,-0.66); (4.5,-0.866)–(4.8,-1.06)–(5,-1.73)–(4.5, -0.866); (5,-1.73)–(5.68, -1.57)–(5.48, -1.23)–(5,-1.73); (5,-1.73)–(5.68, -1.57)–(6,-1.73)–(5,-1.73); (5.68, -1.57)–(6,-1.73)–(6.02, -1.37)–(5.68, -1.57); (6,-1.73)–(6.02, -1.37)–(7,-1.73)–(6,-1.73); (6.02, -1.37)–(7,-1.73)–(6.5,-0.866)–(6.02,-1.37); (6.5,-0.866)–(6.02,-1.37)–(5.82,-1.03)–(6.5,-0.866); (5.82,-1.03)–(6.5,-0.866)–(6,-0.2)–(5.82,-1.03); (6,-0.2)–(5.82,-1.03)–(5.5,-0.866)–(6,-0.2); (5.82,-1.03)–(5.5,-0.866)–(5.48, -1.23)–(5.82,-1.03); (5.5,-0.866)–(5.48, -1.23)–(5,-1.73)–(5.5, -0.866); (5.5,-0.866)–(6,-0.2)–(6,0.2)–(5.5,-0.866); (7,-1.73)–(7.2,-1.06)–(6.8,-1.06)–(7,-1.73); (7,-1.73)–(7.2,-1.06)–(7.5,-0.866)–(7,-1.73); (7.2,-1.06)–(7.5,-0.866)–(7.2, -0.66)–(7.2,-1.06); (7.5,-0.866)–(7.2, -0.66)–(8,0)–(7.5,-0.866); (7.2, -0.66)–(8,0)–(7,0)–(7.2,-0.66); (7,0)–(7.2,-0.66)–(6.8,-0.66)–(7,0); (6.8,-0.66)–(7,0)–(6,-0.2)–(6.8,-0.66); (6,-0.2)–(6.8,-0.66)–(6.5,-0.866)–(6,-0.2); (6.8,-0.66)–(6.5,-0.866)–(6.8,-1.06)–(6.8,-0.66); (6.5,-0.866)–(6.8,-1.06)–(7,-1.73)–(6.5, -0.866); (0,0) circle (1pt) node\[below\] [$\scriptstyle w_7$]{}; (1,0) circle (1pt); (2,0) circle (1pt) node\[right\] [$\scriptstyle w_8$]{}; (1,1.73) circle (1pt) node\[above right\] [$\scriptstyle w_{11}$]{}; (0.5, 0.866) circle (1pt); (-1,0) circle (1pt); (-0.5, 0.866) circle (1pt); (-2,0) circle (1pt) node\[left\] [$\scriptstyle w_6$]{}; (-1,1.73) circle (1pt) node\[above left\] [$\scriptstyle w_{10}$]{}; (1,-1.73) circle (1pt) node\[below right\] [$\scriptstyle w_{4}$]{}; (0.5, -0.866) circle (1pt); (-0.5,-0.866) circle (1pt); (-1,-1.73) circle (1pt) node\[below left\] [$\scriptstyle w_{3}$]{}; (0, 1.73) circle (1pt) node\[above\] [$\scriptstyle w'_{10,11}$]{}; (1.5, 0.866) circle (1pt) node\[right\] [$\scriptstyle w'_{8,11}$]{}; (-1.5, 0.866) circle (1pt) node\[left\] [$\scriptstyle w'_{6,10}$]{}; (0, -1.73) circle (1pt) node\[below\] [$\scriptstyle w'_{3,4}$]{}; (1.5, -0.866) circle (1pt) node\[right\] [$\scriptstyle w'_{4,8}$]{}; (-1.5, -0.866) circle (1pt) node\[left\] [$\scriptstyle w'_{3,6}$]{}; (-0.2,1.07) circle (0.7pt); (-0.2,0.67) circle (0.7pt); (0.2,1.07) circle (0.7pt); (0.2,0.67) circle (0.7pt); (0.48,0.5) circle (0.7pt); (0.82,0.7) circle (0.7pt); (1.02,0.36) circle (0.7pt); (0.68,0.16) circle (0.7pt); (-1.52,0.5) circle (0.7pt); (-1.18,0.7) circle (0.7pt); (-0.98,0.36) circle (0.7pt); (-1.32,0.16) circle (0.7pt); (-1.2,-0.66) circle (0.7pt); (-1.2,-1.06) circle (0.7pt); (-0.8,-0.66) circle (0.7pt); (-0.8,-1.06) circle (0.7pt); (-0.52,-1.23) circle (0.7pt); (-0.18,-1.03) circle (0.7pt); (0.02,-1.37) circle (0.7pt); (-0.32,-1.57) circle (0.7pt); (0.8,-0.66) circle (0.7pt); (0.8,-1.06) circle (0.7pt); (1.2,-0.66) circle (0.7pt); (1.2,-1.06) circle (0.7pt); (6,0.2) circle (1pt) node\[above left\] [$\scriptstyle w_{7_1}$]{}; (6,-0.2) circle (1pt) node\[right\] [$\scriptstyle w_{7_2}$]{}; (7,0) circle (1pt); (8,0) circle (1pt) node\[right\] [$\scriptstyle w_8$]{}; (7,1.73) circle (1pt) node\[above right\] [$\scriptstyle w_{11}$]{}; (6.5, 0.866) circle (1pt); (5,0) circle (1pt); (5.5, 0.866) circle (1pt); (4,0) circle (1pt) node\[left\] [$\scriptstyle w_6$]{}; (5,1.73) circle (1pt) node\[above left\] [$\scriptstyle w_{10}$]{}; (7,-1.73) circle (1pt) node\[below right\] [$\scriptstyle w_{4}$]{}; (6.5, -0.866) circle (1pt); (5.5,-0.866) circle (1pt); (5,-1.73) circle (1pt) node\[below left\] [$\scriptstyle w_{3}$]{}; (6, 1.73) circle (1pt) node\[above\] [$\scriptstyle w'_{10,11}$]{}; (7.5, 0.866) circle (1pt) node\[right\] [$\scriptstyle w'_{8,11}$]{}; (4.5, 0.866) circle (1pt) node\[left\] [$\scriptstyle w'_{6,10}$]{}; (6, -1.73) circle (1pt) node\[below\] [$\scriptstyle w'_{3,4}$]{}; (7.5, -0.866) circle (1pt) node\[right\] [$\scriptstyle w'_{4,8}$]{}; (4.5, -0.866) circle (1pt) node\[left\] [$\scriptstyle w'_{3,6}$]{}; (5.8,1.07) circle (0.7pt); (5.8,0.67) circle (0.7pt); (6.2,1.07) circle (0.7pt); (6.2,0.67) circle (0.7pt); (6.48,0.5) circle (0.7pt); (6.82,0.7) circle (0.7pt); (7.02,0.36) circle (0.7pt); (6.68,0.16) circle (0.7pt); (4.48,0.5) circle (0.7pt); (4.82,0.7) circle (0.7pt); (5.02,0.36) circle (0.7pt); (4.68,0.16) circle (0.7pt); (4.8,-0.66) circle (0.7pt); (4.8,-1.06) circle (0.7pt); (5.2,-0.66) circle (0.7pt); (5.2,-1.06) circle (0.7pt); (5.48,-1.23) circle (0.7pt); (5.82,-1.03) circle (0.7pt); (6.02,-1.37) circle (0.7pt); (5.68,-1.57) circle (0.7pt); (6.8,-0.66) circle (0.7pt); (6.8,-1.06) circle (0.7pt); (7.2,-0.66) circle (0.7pt); (7.2,-1.06) circle (0.7pt); (2.5,0)–(3.5,0); After repeating the above process for each degree 14 vertex in $\widetilde{T_i}$, we take the clique complex and call the resulting flag complex $Y_2$. Observe that this process increases the number of vertices by 1, the number of edges by 3, and the number of faces by 2 each time a degree 14 vertex in $\widetilde{T_i}$ is replaced. Also, note that $\operatorname{maxdeg}(Y_2)\leq 12$ for all $m$. Now, we give the $w_j$, $w'_{m,n},$ and $u_\ell$ vertices their natural orderings and say that $w'_{m,n} > w_j$ and $w'_{m,n}>u_\ell$ for all $\ell, m,n,$ and $j$, and then let these vertex orderings induce orientations on the edges and faces of $Y_2$ (as shown in Figure \[fig:triangulations\]). Counting the vertices, edges, and faces of $Y_2$ we have that if $0\leq k\leq 12$, then there were no degree 14 vertices to remove, so $|V(Y_2)|=6k+2\delta +2$, $|E(Y_2)|=17k+6\delta$, and $|F(Y_2)|=10k+4\delta$. If $k\geq 13$, then $i\geq 3$ and at least one degree 14 vertex was removed to construct $Y_2$ from $\widetilde{T_i}$. Table \[tab:Y2counts\] gives the number of vertices, edges, and faces of $Y_2$ for all values of $k\geq 13$. $k$ $\delta$ $|V(Y_2)|$ $|E(Y_2)|$ $|F(Y_2)|$ -------- ---------- ----------------------------- ------------------------------ ------------ $4i+4$ 0 $\frac{13}{2}k-4$ $\frac{37}{2}k-18$ $11k-12$ $4i+3$ 1 $\frac{13}{2}k-\frac{3}{2}$ $\frac{37}{2}k-\frac{21}{2}$ $11k-7$ $4i+2$ 2 $\frac{13}{2}k$ $\frac{37}{2}k-6$ $11k-4$ $4i+1$ 3 $\frac{13}{2}k+\frac{5}{2}$ $\frac{37}{2}k+\frac{3}{2}$ $11k+1$ : Number of vertices, edges, and faces in $Y_2$ when $k\geq 13$.[]{data-label="tab:Y2counts"} ### Homology of $Y_2$ Since $Y_2$ is an oriented flag triangulation of $S^2$ with $k$ square holes, each of which are vertex disjoint and nonadjacent, our $Y_2$ is homeomorphic to Newman’s $Y_2$ in the $d=2$ case of [@newman Lemma 5.7], and we can apply the same argument to compute the homology of $Y_2$. We denote the 1-cycles that are the boundaries of the $k$ square holes by $\tau_1,\dots, \tau_k$. Explicitly, for $j=1, \dots, k$, we define $$\tau_{j}: = [u_{4j-4}, u_{4j-3}] + [u_{4j-3}, u_{4j-2}] + [u_{4j-2}, u_{4j-1}] - [u_{4j-4}, u_{4j-1}].$$ Then, by our construction, each $\tau_j$ is a positively-oriented 1-cycle in $H_1(Y_2)$, and exactly as in [@newman Proof of Lemma 5.7], we have that $ $$$H_1(Y_2) = \langle \tau_1, \ldots, \tau_k \vert \tau_1 + \cdots + \tau_k = 0 \rangle. $ $$ Construction of $X$ and proof of Theorem \[thm:Xm\] --------------------------------------------------- Now we attach $Y_1$ and $Y_2$ together to form the two-dimensional flag complex $X$ such that the torsion subgroup of $H_1(X)$ is isomorphic to $\mathbb{Z}/m\mathbb{Z}$. This part essentially follows [@newman §3], though we must confirm that the resulting complex is flag and satisfies the desired bound of vertex degree. For a given $m$, let $Y_1$ and $Y_2$ be the complexes constructed in the previous subsections. Let $S$ denote the subcomplex of $Y_2$ induced by the $4k$ vertices $u_0,\dots, u_{4k-1}$. Since the square holes in $Y_2$ are vertex-disjoint and have no edges between any two of them, $S$ is a disjoint union of $k$ square boundaries. Let $f: S \rightarrow Y_1$ be the simplicial map defined, for $j=1, \ldots, k$, by $$\begin{aligned} & u_{4j-4} \mapsto v_{4n_j}, & & u_{4j-3} \mapsto v_{4n_j + 1}, & u_{4j-2} \mapsto v_{4n_j + 2}, && u_{4j-1} \mapsto v_{4n_j + 3}.\end{aligned}$$ Following [@newman §3], let $X = Y_1 \sqcup_f Y_2$ and observe that this is a simplicial complex by the same argument as Newman gives. In addition, $X$ is a flag complex because $Y_1$ and $Y_2$ are flag, and we subdivided the edges of $Y_1$ and $Y_2$ to avoid the possibility that $X$ might contain a 3-cycle which doesn’t have a face. Furthermore, in $X$ the squares $\tau_j$ and $\gamma_{n_j}$ are identified by $f$ for $j=1, \ldots, k$, and, as in [@newman], $$H_1(X) \cong \mathbb{Z}^{k-1} \oplus \mathbb{Z}/m \mathbb{Z},$$ where ${\mathbb{Z}}/m{\mathbb{Z}}$ has the repeated squares representation given by $${\langle}\gamma_0,\gamma_1,\dots,\gamma_{n_k}\; |\; 2\gamma_0=\gamma_1, 2\gamma_1=\gamma_2,\dots, 2\gamma_{n_k-1}=\gamma_{n_k}, \gamma_{n_1}+\cdots+\gamma_{n_k}=0 {\rangle}.$$ Finally, using our counts for the number of vertices, edges, and faces of $Y_1$ and $Y_2$ and with $\delta$ defined as above, we have $$|V(X)|=2k+12n_k+6+2\delta, \; |E(X)|= 13k+40n_k+4+6\delta, \text{ and }|F(X)|=10k+28n_k+4\delta.$$ If $k\geq 13$, then Table \[tab:Xcounts\] gives the number of vertices, edges, and faces in $X$ (where $i=\lfloor \frac{k-1}{4} \rfloor$). $k$ $\delta$ $|V(X)|$ $|E(X)|$ $|F(X)|$ -------- ---------- ----------------------------------- ------------------------------------ ---------------- $4i+4$ 0 $\frac{5}{2}k+12n_k$ $\frac{29}{2}k+40n_k-14$ $11k+28n_k-12$ $4i+3$ 1 $\frac{5}{2}k+12n_k+\frac{5}{2}$ $\frac{29}{2}k+40n_k-\frac{13}{2}$ $11k+28n_k-7$ $4i+2$ 2 $\frac{5}{2}k+12n_k+4$ $\frac{29}{2}k+40n_k-2$ $11k+28n_k-4$ $4i+1$ 3 $\frac{5}{2}k+12n_k+\frac{13}{2}$ $\frac{29}{2}k+40n_k+\frac{11}{2}$ $11k+28n_k+1$ : Number of vertices, edges, and faces in $X$ when $k\geq 13$.[]{data-label="tab:Xcounts"} Additionally, recall that $\operatorname{maxdeg}(Y_1)\leq9$ and $\operatorname{maxdeg}(Y_2)\leq 12$. Since in $X$ we are only identifying the squares of $Y_2$ with $k$ of the squares of $Y_1$, to find the maximum degree of any vertex of $X$, we need only check the degrees of the identified vertices. In $Y_1$, we know that $\deg(v_j)\leq 9$ for each $j$, and in $Y_2$, we know that $\deg(u_\ell) \in \{4,5\}$ for each $\ell$. Let $v_j$ and $u_\ell$ be vertices that are identified in $X$. Since two of their adjacent edges in the squares are identified as well, in $X$ we see that $\deg(v_j)=\deg(u_\ell) \leq 12$. Thus, $\operatorname{maxdeg}(X)\leq 12$. We also note the following corollary: \[cor:flag-newman\] For every finite abelian group $G$ there is a two-dimensional flag complex $X$ such that the torsion subgroup of $H_1(X)$ is isomorphic to $G$ and $\operatorname{maxdeg}(X)\leq 12$. Let $G = {\mathbb{Z}}/m_1{\mathbb{Z}}\oplus {\mathbb{Z}}/m_2{\mathbb{Z}}\oplus \cdots \oplus {\mathbb{Z}}/m_r {\mathbb{Z}}$ with $m_1|m_2|\cdots|m_r$ be an arbitrary finite abelian group. By Theorem \[thm:Xm\], there exist two-dimensional flag complexes $X_{m_i}$ such that the torsion subgroup of $H_1(X_{m_i})$ is isomorphic to ${\mathbb{Z}}/m_i{\mathbb{Z}}$ and $\operatorname{maxdeg}(X_{m_i})\leq 12$. If $X$ is the disjoint union of all the $X_{m_i}$, then $X$ satisfies the hypotheses of the corollary. Appearance of subcomplexes in $\Delta(n,p)$ {#sec:subgraphs} =========================================== The goal of this section is to show, for attaching probabilities $p$ in an appropriate range, the flag complex $X_m$ from Theorem \[thm:Xm\] will appear with high probability as an induced subcomplex of $\Delta(n,p)$. See §\[sec:background\] for the relevant definitions and notation used throughout this section. Here is our main result: \[prop:high-probability\] Let $m\geq 2$, and let $X_m$ be as in Theorem \[thm:Xm\]. If $\Delta\sim \Delta(n,p)$ is a random flag complex with $n^{-1/6}\ll p\leq 1-\epsilon$ for some $\epsilon>0$, then ${\mathbf{P}}\left[X_m \overset{ind}{\subset} \Delta(n,p)\right]\rightarrow 1$ as $n \to \infty$. Our proof of this result will rely on Bollobás’s theorem on the appearance of subgraphs of a random graph, which we state here for reference. \[thm:Bollobás\] Let $G'$ be a fixed graph, let $m(G')$ be the essential density of $G'$ defined in Definition \[mG\], and let $G(n,p)$ be the Erdős-Rényi random graph on $n$ vertices with attaching probability $p$. As $n \to \infty$, we have $${\mathbf{P}}\left[G'\subset G(n,p)\right]\rightarrow \begin{cases} 0 & \text{if } p\ll n^{-1/m(G')}\\ 1 & \text{if } p\gg n^{-1/m(G')} \end{cases}.$$ Since any flag complex is determined by its underlying graph, we can almost apply this to prove Proposition \[prop:high-probability\]. However, Proposition \[prop:high-probability\] (and our eventual application of it via Hochster’s formula to Theorem \[thm:m torsion\]) requires $X_m$ to appear as an induced subcomplex, whereas Bollobás’s result is for not necessarily induced subgraphs. The following proposition, which is likely known to experts, shows that so long as $p$ is bounded away from $1$, this distinction is immaterial in the limit. \[prop:induced-bollobas\] Let $G'$ be a fixed graph, let $m(G')$ be the essential density of $G'$ defined in Definition \[mG\], and let $G(n,p)$ be the Erdős-Rényi random graph on $n$ vertices with attaching probability $p$. Suppose $p = p(n)\leq 1-\epsilon$ for some constant $\epsilon>0$. Then as $n \to \infty$, we have $${\mathbf{P}}\left[G'\overset{ind}{\subset} G(n,p)\right]\rightarrow \begin{cases} 0 & \text{if } p\ll n^{-1/m(G')}\\ 1 & \text{if } p\gg n^{-1/m(G')} \end{cases}.$$ Since an induced subgraph is a subgraph, if ${\mathbf{P}}[G'\subset G(n,p)]\rightarrow 0$, then\ ${\mathbf{P}}\left[G'\overset{ind}{\subset} G(n,p)\right]\rightarrow 0$. Thus, the first half of the threshold is a direct consequence of Theorem \[thm:Bollobás\], and all that needs to be shown is the second half of the threshold. So, suppose that $p\gg n^{-1/m(G')}$. We will mirror the proof of Bollobàs’s theorem from [@frieze-book Theorem 5.3] (originally due to [@ruc-vince]), which relies on the second moment method. Let $\Lambda(G',n)$ be the set containing all of the possible ways that $G'$ can appear as a induced subgraph of $G(n,p)$. Thus, an element $H\in \Lambda(G',n)$ corresponds to a subset of the $n$ vertices and specified edges among those vertices such that the resulting graph is a copy of $G'$. We want to count the number of times $G'$ appears as an induced subgraph of $G(n,p)$. For each $H\in \Lambda(G',n)$, we let $\mathbf{1}_{H}$ be the corresponding indicator random variable, where $\mathbf{1}_H = 1$ occurs in the event that restricting $G(n,p)$ to the vertices of $H$ is precisely the copy of $G'$ indicated by $H$. Note that the random variables $\mathbf{1}_{H}$ are not independent, as two distinct elements from $\Lambda(G',n)$ might have overlapping vertex sets. If we let $N_{G'}$ be the random variable for the number of copies of $G'$ appearing as induced subgraphs in $G(n,p)$, then we have $N_{G'} = {\displaystyle}\sum_{H \in \Lambda(G',n)} \mathbf{1}_{H}.$ Our goal is to show that ${\mathbf{P}}[N_{G'}\geq 1]\to 1$, or equivalently that ${\mathbf{P}}[N_{G'}= 0]\to 0$. Since $N_{G'}$ is non-negative, the second moment method as seen in [@alon-spencer-book Theorem 4.3.1] states that ${\mathbf{P}}[N_{G'}= 0]\leq \frac{\operatorname{Var}(N_{G'})}{{\mathbf{E}}[N_{G'}]^2}$, so it suffices to show that $\frac{\operatorname{Var}(N_{G'})}{{\mathbf{E}}[N_{G'}]^2}\rightarrow 0$. To start, we will bound the expected value. To simplify notation throughout the following computation, we let $v=|V(G')|$ and $e=|E(G')|$ denote the number of vertices and edges of $G'$. $$\begin{aligned} {\mathbf{E}}[N_{G'}]&= \sum_{H\in \Lambda(G',n)} {\mathbf{E}}[\mathbf{1}_H]\\ &= \sum_{H\in \Lambda(G',n)} p^{e}(1-p)^{\binom{v}{2}-e}\\ &= \Omega(n^{v})\cdot p^{e}(1-p)^{\binom{v}{2}-e}.\end{aligned}$$ Now let us repeat this with the variance instead. $$\begin{aligned} \operatorname{Var}(N_{G'}) &= \sum_{H,H'\in \Lambda(G',n)} {\mathbf{E}}[\mathbf{1}_{H}\mathbf{1}_{H'}] - {\mathbf{E}}[\mathbf{1}_{H}]{\mathbf{E}}[\mathbf{1}_{H'}]\\ &= \sum_{H,H'\in \Lambda(G',n)} {\mathbf{P}}[\mathbf{1}_{H}=1\text{ and }\mathbf{1}_{H'}=1] - {\mathbf{P}}[\mathbf{1}_{H}=1]{\mathbf{P}}[\mathbf{1}_{H'}=1]\\ &= \sum_{H,H'\in \Lambda(G',n)} {\mathbf{P}}[\mathbf{1}_{H}=1]\left({\mathbf{P}}[\mathbf{1}_{H'}=1 \mid \mathbf{1}_{H}=1]- {\mathbf{P}}[\mathbf{1}_{H'}=1]\right)\\ &= p^{e}(1-p)^{\binom{v}{2}-e} \sum_{H,H'\in \Lambda(G',n)}{\mathbf{P}}[\mathbf{1}_{H'}=1 \mid \mathbf{1}_{H}=1]- {\mathbf{P}}[\mathbf{1}_{H'}=1]\\ \intertext{If $H$ and $H'$ don't share at least two vertices, $\mathbf{1}_H$ and $\mathbf{1}_{H'}$ are independent of each other, so we can restrict to the case where they share at least two vertices, which gives} &= p^{e}(1-p)^{\binom{v}{2}-e} \sum_{i=2}^{v}\sum_{\substack{H,H'\in \Lambda(G',n) \\ |V(H)\cap V(H')|=i}}{\mathbf{P}}[\mathbf{1}_{H'}=1 \mid \mathbf{1}_{H}=1]- {\mathbf{P}}[\mathbf{1}_{H'}=1].\end{aligned}$$ We now come to the key observation, which is also at the heart of the proof in [@frieze-book Theorem 5.3]: ${\mathbf{P}}[\mathbf{1}_{H'}=1 \mid \mathbf{1}_{H}=1]$ is maximized if those edges and non-edges in $H$ are exactly those that are required by $H'$. Thus, by applying the fact that any subgraph of $G'$ with $i$ vertices, has at most $i\cdot m(G')$ edges and at most $\binom{i}{2}$ non-edges we get the following bound for $H,H'\in \Lambda(G',n)$ sharing $i$ vertices: $${\mathbf{P}}[\mathbf{1}_{H'}=1 \mid \mathbf{1}_{H}=1]\leq {\mathbf{P}}[\mathbf{1}_{H'}=1]\cdot p^{-i\cdot m(G')}(1-p)^{-\binom{i}{2}}$$ From here, it is a standard computation. Substituting this back into the previous equation and simplifying, we get $$\begin{aligned} \operatorname{Var}(N_{G'}) &\leq p^{e}(1-p)^{\binom{v}{2}-e} \sum_{i=2}^{v}\sum_{\substack{H,H'\in \Lambda(G',n) \\ |V(H)\cap V(H')|=i}}{\mathbf{P}}[\mathbf{1}_{H'}=1]\left(p^{-i\cdot m(G')}(1-p)^{-\binom{i}{2}}-1\right)\\ &\leq \left(p^{e}(1-p)^{\binom{v}{2}-e}\right)^2 \sum_{i=2}^{v} O\left(n^{2v-i}\right)\left(p^{-i\cdot m(G')}(1-p)^{-\binom{i}{2}}-1\right).\\ \intertext{And since $p$ is bounded away from $1$ and $1-p$ is bounded away from $0$, we get} &\leq \left(p^{e}(1-p)^{\binom{v}{2}-e}\right)^2 \sum_{i=2}^{v} O\left(n^{2v-i}p^{-i\cdot m(G')}\right). \end{aligned}$$ Finally, applying the second moment method gives $${\mathbf{P}}[N_{G'}\leq 0]\leq \frac{\operatorname{Var}(N_{G'})}{{\mathbf{E}}[N_{G'}]^2}=\frac{{\displaystyle}\sum_{i=2}^{v} O\left(n^{2v-i}p^{-i\cdot m(G')}\right)}{\Omega(n^{2v})} =\sum_{i=2}^{v} O\left(n^{-i}p^{-i\cdot m(G')}\right). $$ Since $p\gg n^{-1/m(G')}$, we conclude that $np^{m(G')}\rightarrow \infty$, and therefore, ${\mathbf{P}}[N_{G'}=0 ]\rightarrow 0$. It follows that ${\mathbf{P}}\left[G'\overset{ind}{\subset} G(n,p)\right]\rightarrow 1$. We now turn to the proof of Proposition \[prop:high-probability\]. Recall that $X_m$ is the complex from Theorem \[thm:Xm\], and let $H_m$ be its underlying graph. Moreover, the underlying graph of $\Delta(n,p)$ is the Erdős-Rényi random graph $G(n,p)$. Since a flag complex is uniquely determined by its 1-skeleton, it suffices to show that ${\mathbf{P}}\left[H_m\overset{ind}{\subset}G(n,p)\right]\rightarrow 1$. Since $\operatorname{maxdeg}(H_m)\leq 12$, every subgraph has average degree at most $12$. Thus, the essential density $m(H_m)$ satisfies $m(H_m)\leq 6$. Since $p\gg n^{-1/6}$, we have $p\gg n^{-1/m(H_m)}$. Applying Proposition \[prop:induced-bollobas\] gives ${\mathbf{P}}\left[H_m \overset{ind}{\subset} G(n,p)\right]\rightarrow 1$; thus, ${\mathbf{P}}\left[X_m\overset{ind}{\subset}\Delta(n,p)\right]\rightarrow 1$. \[rmk:sharpness\] Explicitly computing the essential density $m(H_m)$ seems difficult in general, and our chosen bound $m(H_m)\leq 6$, which is determined by the fact that $6 = \frac{1}{2}\operatorname{maxdeg}(X_m)$, is likely too coarse. It would be interesting to see a sharper result on $m(H_m)$, as this could potentially provide an heuristic for decreasing the bound on $r$ in Conjecture \[conj:dependence\]. Might it even be the case that $m(H_m)$ is half the average degree, $\frac{1}{2}{\operatorname{avg}}(H_m)$? In any case, $\frac{1}{2}{\operatorname{avg}}(H_m)$ at least provides a lower bound on $m(H_m)$. Due to the detailed nature of the constructions in §\[sec:construction\], we can estimate this value. Let $k\geq 13$ and $m\gg 0$. By Table \[tab:Xcounts\], $n_k=\lfloor \log_2(m)\rfloor$ will be much larger than $\delta$, and so the number of vertices will be approximately $\frac{5}{2}k+12n_k$ and the number of edges will be approximately $\frac{29}{2}k+40n_k$. The smallest the ratio of edges to vertices can be is when $n_k\gg k$, in which case the ratio will be approximately $3\frac{1}{3}$. A similar computation holds for $k\leq 12$ and for $m\gg 0$. We can conclude that $m(H_m)\geq 3 \frac{1}{3}-\epsilon$, where $\epsilon$ is a positive constant that goes to $0$ as $m\to \infty$. A detailed analysis of 2-torsion {#sec:2torsion} ================================ The goal of this section is to provide a more detailed analysis of what happens in the case of 2-torsion. We use a known flag triangulation of ${\mathbb{R}}P^2$ that minimizes the number of vertices and where we can easily compute its essential density to produce induced subcomplexes of $\Delta(n,p)$ with $2$-torsion. In [@bibby2019minimal], the authors find two (nonisomorphic) minimal flag triangulations of ${\mathbb{R}}P^2$, each of which have 11 vertices and 30 edges and differ by a single bistellar 0-move. One of these flag triangulations is depicted in Figure \[fig:flagRP2\]. (0,2)–(0.85,1.3)–(0,1)–(0,2); (0,2)–(-0.85,1.3)–(0,1)–(0,2); (0,1)–(0.85,1.3)–(0.8,0.3)–(0,1); (0,1)–(-0.85,1.3)–(-0.8,0.3)–(0,1); (0,1)–(0.8,0.3)–(0,0)–(0,1); (0,1)–(-0.8,0.3)–(0,0)–(0,1); (0.85,1.3)–(1.7,0.6)–(0.8,0.3)–(0.85,1.3); (0.85,1.3)–(1.7,0.6)–(0.8,0.3)–(0.85,1.3); (-0.85,1.3)–(-1.7,0.6)–(-0.8,0.3)–(-0.85,1.3); (1.7,0.6)–(0.8,0.3)–(1.4,-0.5)–(1.7,0.6); (-1.7,0.6)–(-0.8,0.3)–(-1.4,-0.5)–(-1.7,0.6); (0.8,0.3)–(1.4,-0.5)–(0.5,-0.75)–(0.8,0.3); (0.8,0.3)–(0,0)–(0.5,-0.75)–(0.8,0.3); (-0.8,0.3)–(-1.4,-0.5)–(-0.5,-0.75)–(-0.8,0.3); (-0.8,0.3)–(0,0)–(-0.5,-0.75)–(-0.8,0.3); (0,0)–(-0.5,-0.75)–(0.5,-0.75)–(0,0); (1.4,-0.5)–(1,-1.6)–(0.5,-0.75)–(1.4,-0.5); (-1.4,-0.5)–(-1,-1.6)–(-0.5,-0.75)–(-1.4,-0.5); (1,-1.6)–(0.5,-0.75)–(0,-1.6)–(1,-1.6); (-1,-1.6)–(-0.5,-0.75)–(0,-1.6)–(-1,-1.6); (0,-1.6)–(0.5,-0.75)–(-0.5,-0.75)–(0,-1.6); (0,0) circle (2pt) node\[above right\] [$v_8$]{}; (0,1) circle (2pt) node\[above right\] [$v_9$]{}; (0,2) circle (2pt) node\[above\] [$v_3$]{}; (0.85,1.3) circle (2pt) node\[above right\] [$v_2$]{}; (1.7,0.6) circle (2pt) node\[right\] [$v_6$]{}; (-0.85,1.3) circle (2pt) node\[above left\] [$v_{10}$]{}; (-1.7,0.6) circle (2pt) node\[left\] [$v_5$]{}; (-1,-1.6) circle (2pt) node\[below left\] [$v_2$]{}; (-0.5,-0.75) circle (2pt) node\[below left\] [$v_7$]{}; (0.5,-0.75) circle (2pt) node\[below right\] [$v_4$]{}; (1,-1.6) circle (2pt) node\[below right\] [$v_{10}$]{}; (-1.4,-0.5) circle (2pt) node\[below left\] [$v_6$]{}; (1.4,-0.5) circle (2pt) node\[below right\] [$v_5$]{}; (0.8,0.3) circle (2pt) node\[above right\] [$v_1$]{}; (-0.8,0.3) circle (2pt) node\[left\] [$v_{11}$]{}; (0,-1.6) circle (2pt) node\[below\] [$v_3$]{}; For the remainder of this section, let $G$ denote the underlying graph of this flag triangulation of ${\mathbb{R}}P^2$, which we denote by $\Delta(G)$ as it is the clique complex of $G$. To understand the probability that this particular triangulation of ${\mathbb{R}}P^2$ appears as an induced subcomplex of $\Delta(n,p)$, we need to compute the essential density $m(G)$. \[lem:bollobas of G\] For the graph $G$ underlying the flag triangulation of ${\mathbb{R}}P^2$ exhibited in Figure \[fig:flagRP2\], the essential density $m(G)$ is $30/11$. This amounts to an exhaustive computation, which is summarized in Table \[tab:RP2 counts\]. In particular, Table \[tab:RP2 counts\] identifies the maximal number of edges that a subgraph $H\subset G$ on $|V(H)|$ vertices can have, for each $|V(H)|\leq 11$. One can see from the table that $m(G)$ is maximized by the entire graph, and thus $m(G) = |E(G)|/|V(G)| = 30/11$. Lemma \[lem:bollobas of G\] shows that the graph $G$ is strongly balanced in the sense of Definition \[mG\]. While we expect the essential density of our complexes $X_m$ to be lower than the coarse bound of $\frac{1}{2}\operatorname{maxdeg}(X_m)$ (see Remark \[rmk:sharpness\]), we note that in the case of the graph $G$, this difference is not very large. In fact, we have $\frac{1}{2}\operatorname{maxdeg}(G)=3$ and $m(G)=30/11\approx 2.72$. $|V(H)|$ $\max\{|E(H)|\}$ $V(H)$ $\max\left\{\frac{|E(H)|}{|V(H)|}\right\}$ ---------- ------------------ -------------------------------------------------- -------------------------------------------- 1 0 $\{v_1\}$ 0 2 1 $\{v_1, v_2\}$ $\frac12$ 3 3 $\{v_1, v_2, v_6\}$ 1 4 5 $\{v_1,v_2,v_5,v_6\}$ $\frac54$ 5 7 $\{v_1,v_2,v_4,v_5,v_6\}$ $\frac75$ 6 10 $\{v_1,v_4,v_7,v_8,v_9,v_{11}\}$ $\frac53$ 7 13 $\{v_1,v_2,v_4,v_7,v_8,v_9,v_{11}\}$ $\frac{13}{7}$ 8 17 $\{v_1,v_2,v_4,v_6,v_7,v_8,v_9,v_{11}\}$ $\frac{17}{8}$ 9 21 $\{v_1,v_2,v_3,v_4,v_6,v_7,v_8,v_9,v_{11}\}$ $\frac{7}{3}$ 10 25 $\{v_1,v_2,v_3,v_4,v_5,v_6,v_7,v_8,v_9,v_{11}\}$ $\frac{5}{2}$ 11 30 $\{v_1,\dots,v_{11}\}$ $\frac{30}{11}$ : With $G$ as the underlying graph of the complex in Figure \[fig:flagRP2\], this table computes the maximal number of edges of subgraphs $H\subset G$ with varying number of vertices.[]{data-label="tab:RP2 counts"} Combining Lemma \[lem:bollobas of G\] and Theorem \[thm:Bollobás\] we obtain an analogue of Proposition \[prop:high-probability\]. \[prop:high-prob 2tor\] if $\Delta\sim \Delta(n,p)$ is a random flag complex with $n^{-11/30}\ll p\leq 1-\epsilon$ for some $\epsilon>0$, then ${\mathbf{P}}\left[\Delta(G) \overset{ind}{\subset} \Delta(n,p)\right]\rightarrow 1$ as $n \to \infty$. The proof is nearly identical to that of Proposition \[prop:high-probability\], so we omit the details. \[q:2torsion threshold\] It would be interesting to know whether $p\ll n^{-11/30}$ is a sharp threshold for the appearance of 2-torsion in the homology of $\Delta(n,p)$. A closely related question is whether there exists a flag complex $X$ with 2-torsion homology and a smaller essential density. Torsion in the Betti tables associated to $\Delta$ {#sec:Betti numbers} ================================================== We now prove Theorem \[thm:m torsion\]. The hard work was done in the previous sections. Assume $n^{-1/6}\ll p \leq 1-\epsilon$ and let $\Delta\sim \Delta(n,p)$. Let $X_m$ be as in Theorem \[thm:Xm\]. By Proposition \[prop:high-probability\], $\Delta$ contains $X_m$ as an induced subcomplex, with high probability. Since $H_1(X_m)$ has $m$-torsion, Hochster’s Formula (see Fact \[fact:depend\]) gives that the Betti table of the Stanley–Reisner ideal of $\Delta$ has $\ell$-torsion for every $\ell$ dividing $m$. We can also apply the more detailed study of $2$-torsion from §\[sec:2torsion\] to obtain a result on the appearance of $2$-torsion in the Betti tables of random flag complexes. \[prop:2tors\] Let $r\geq 4$, and let $\Delta\sim \Delta(n,p)$ be a random flag complex with $n^{-1/(r-1)}\ll p \ll n^{-1/r}$. With high probability as $n\to \infty$, the Betti table of the Stanley–Reisner ideal of $\Delta$ has $2$-torsion. The proof is the same as the proof of Theorem \[thm:m torsion\], but utilizing Proposition \[prop:high-prob 2tor\] in place of Proposition \[prop:high-probability\] since $r\geq 4$ and $n^{-1/(r-1)}\ll p$ gives $n^{-11/30}\ll p$. Note that the bound on $r$ for the appearance of $2$-torsion in Proposition \[prop:2tors\] is lower than in Theorem \[thm:m torsion\]. This is due to our ability to sharply compute the essential density in this case; in contrast, for Theorem \[thm:m torsion\], we work with a bound on the essential density. See Question \[q:threshold\] and Remark \[rmk:r bound\] for more on the possibility of lowering the bound on $r$ in Theorem \[thm:m torsion\]. It would be interesting to understand a precise threshold on the attaching probability $p$ such that the Betti table of the Stanley–Reisner ideal of $\Delta$ does not depend on the characteristic. A related question is posed in Question \[q:threshold\]. We also note that our constructions are based entirely on torsion in the $H_1$-groups, and thus we obtain Betti tables where the entries in the second row of the Betti table (that is the row of entries of the form $\beta_{i,i+2}$) depend on the characteristic. Since Newman’s work also produces small simplicial complexes where the $H_i$-groups have torsion, for any $i\geq 1$ [@newman Theorem 1], one could likely apply the methods of §\[sec:construction\] to produce thresholds for where the other rows of the Betti table would depend on the characteristic, and it might be interesting to explore the resulting thresholds. $\ell$-torsion in Veronese syzygies {#sec:veronese} =================================== Finally, we return to the question of $\ell$-torsion in Veronese syzygies. Since there is very little computational evidence either in favor or in opposition to Conjecture \[conj:dependence\], we base the conjecture upon an heuristic model. As noted in the introduction, one of the central results of [@erman-yang] is that for $\Delta \sim \Delta(n,p)$ with $n^{-1/(r-1)}\ll p \ll n^{-1/r}$ and $S=k[x_1,\dots,x_n]$, the Betti table $\beta(S/I_\Delta)$ as $n\to \infty$ will exhibit the known nonvanishing properties of the Betti table of the Veronese embeddings $\beta({\mathbb{P}}_k^r;d)$ as $d\to \infty$. Based on this connection, we use Theorem \[thm:m torsion\] as an heuristic for understanding the behavior of $\beta({\mathbb{P}}^r;d)$, in particular, when these Betti tables depend on the characteristic. For Conjecture \[conj:dependence\], we set $r\geq 7$ and use the framework of Theorem \[thm:m torsion\]. With these hypotheses, as $n\to \infty$, the Betti table associated to $\Delta$ will depend on the characteristic with high probability. We thus conjecture a corresponding statement for $\beta({\mathbb{P}}^r;d)$ with $r\geq 7$ and $d\to \infty$. While we conjecture that this dependence on characteristic should be quite widespread, the only known examples of such behavior come from [@anderson]. It would thus be very interesting to produce any new examples (or non-examples!) of torsion in Veronese syzygies. For instance: Can one find any new examples of Veronese embeddings whose Betti tables depend on the characteristic? For a given $\ell$, can one produce a Betti table with $\ell$-torsion? Can one find some $\beta({\mathbb{P}}^r;d)$ which has $\ell$-torsion for two (or more) distinct primes? We find it especially surprising that there are no known examples of $2$-torsion. Conjecture \[conj:bad primes\] represents one way to sharpen Conjecture \[conj:dependence\]. In particular, since Theorem \[thm:m torsion\] shows that, with $r\geq 7$ and within the given framework, $m$-torsion appears with high probability as $n\to \infty$ in the Betti table of the Stanley–Reisner ideal of $\Delta$, we conjecture that $m$-torsion should appear frequently in the Betti tables of the $d$-uple Veronese embeddings for ${\mathbb{P}}^r$ as $d\to \infty$. There are many follow-up questions one might ask, and we assemble some of these below. What is the minimal value of $r$ such that $\beta({\mathbb{P}}^r;d)$ depends on the characteristic for some $d$? (It is known that $1<r\leq 6$.) To develop an heuristic for this question, along the lines of this paper, one would need to consider the following question, which seeks to sharpen Theorem \[thm:m torsion\]. \[q:threshold\] Let $m\geq 2$. For a random flag complex $\Delta\sim \Delta(n,p)$, what is the threshold on $p$ such that the Betti table of the Stanley–Reisner ideal of $\Delta$ has $m$-torsion with high probability as $n\to \infty$? \[rmk:r bound\] We know of two natural ways that one could improve the bound on $r$ in Theorem \[thm:m torsion\]. First, one could perform a more detailed study of the essential density $m(H_m)$, as that value is surely lower than our chosen bound $\frac{1}{2}\operatorname{maxdeg}(X_m)$. Second, one could aim to produce flag complexes $X_m'$ with torsion homology (not necessarily in $H_1$) which have a lower essential density than $X_m$. Of course, following the heuristic at the heart of this paper, any such improvement of the bound on $r$ in Theorem \[thm:m torsion\] would suggest a corresponding improvement of the bound on $r$ in Conjectures \[conj:dependence\] and \[conj:bad primes\]. In a different direction, one might ask about how large $n$ needs to be before we expect to see that the Betti table associated to $\Delta$ has $\ell$-torsion. Fix a prime $\ell$ and integer $r\geq 7$. Let $\Delta\sim\Delta(n,p)$ be a random flag complex with $n^{-1/(r-1)}\ll p \ll n^{-1/r}$. For a constant $0<\epsilon<1$, approximately how large does $n$ need to be to guarantee that $${\mathbf{P}}\left[\text{ Betti table associated to $\Delta$ has $\ell$-torsion }\right] \geq 1 - \epsilon?$$ It would be interesting to even answer this question for $2$-torsion, where the concrete constructions from §\[sec:2torsion\] make the question seemingly more tractable. The corresponding question for Veronese embeddings would be the following: Fix a prime $\ell$ and integer $r\geq 7$. Can one provide lower/upper bounds on the minimal value of $d$ such that $\beta({\mathbb{P}}^r; d)$ has $\ell$-torsion? We could turn to even more quantitative questions related to Conjecture \[conj:bad primes\] as well. Fix a prime $\ell$ and an integer $r\geq 7$. Can one describe the set of $d\in {\mathbb{Z}}$ such that $\beta({\mathbb{P}}^r; d)$ has $\ell$-torsion? Can one bound or estimate the density of that set? Can one estimate or bound the growth rate of the number of primes $\ell$ such that $\beta({\mathbb{P}}^r; d)$ has $\ell$-torsion as $d\to \infty$? Even a compelling heuristic for these last two questions could be quite interesting. [^1]: This is equivalent to a certain integral Tor group having $\ell$-torsion: see Remark \[rmk:torsion\]. [^2]: See [@ein-lazarsfeld-asymptotic; @ein-erman-lazarsfeld-quick] for more on these nonvanishing properties.

ArXiv

[**On more general forms of proportional fractional operators**]{} .20in Fahd Jarad$^{a}$, Manar A. Alqudah$^{b}$, Thabet Abdeljawad$^{c,d}$\ $^{a}$Department of Mathematics, Çankaya University, 06790 Ankara, Turkey\ email: [email protected]\ $^{b}$ Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University\ P.O. Box 84428, Riyadh 11671, Saudi Arabia.\ email:[email protected].\ $^{c}$Department of Mathematics and Physical Sciences, Prince Sultan University\ P. O. Box 66833, 11586 Riyadh, Saudi Arabia\ email:[email protected]\ $^d$ Department of Medical Research, China Medical University, 40402, Taichung, Taiwan .2in Introduction ============ The fractional calculus, which is engaged in integral and differential operators of arbitrary orders, is as old as the conceptional calculus that deals with integrals and derivatives of non-negative integer orders. Since not all of the real phenomena can be modeled using the operators in the traditional calculus, researchers searched for generalizations of these operators. It turned out that the fractional operators are excellent tools to use in modeling long-memory processes and many phenomena that appear in physics, chemistry, electricity, mechanics and many other disciplines. Here, we invite the readers to read [@podlubny; @Samko; @f1; @f222; @f2; @f3] and the reference cited in these books. However, for the sake of better understanding and modeling real world problems, researchers were in need of other types of fractional operators that were confined to Riemann-Liouville fractional operators. In the literature, one can find many works that propose new fractional operators. We mention [@had; @Kat1; @Kat2; @fahd3; @fahd1; @fahd11]. Nonetheless, the fractional integrals and derivatives which were proposed in these works were just particular cases of what so called fractional integrals/derivatives of a function with respect to another function [@Samko; @f2; @fahd10]. There are other types of fractional operators which were suggested in the literature. On the other hand, due to the singularities found in the traditional fractional operators which are thought to make some difficulties in the modeling process, some researches recently proposed new types of non-singular fractional operators. Some of these operators contain exponential kernels and some of them involve the Mittag-Leffler functions. For such types of fractional operators we refer to [@FCaputo; @Losada; @TD; @ROMP; @Abdon; @TD; @JNSA]. All the fractional operators considered in the references in the first and the second paragraphs are non-local. However, there are many local operators found in the literature that allow differentiation to a non-integer order and these are called local fractional operators. In [@kh], Khalil et al. introduced the so called conformable (fractional) derivative. The author in [@T11] presented other basic concepts of conformable derivatives. We would like to mention that the fractional operators proposed in [@Kat1; @Kat2] are the non-local fractional version of the local operators suggested in [@kh]. In addition, the non-local fractional version of the ones in [@T11] can be seen in [@fahd11]. It is customary that any derivative of order 0 when performed to a function should give the function itself. This essential property is dispossessed by the conformable derivatives. Notwithstanding, in [@Anderson1; @Anderson2], the authors introduced a newly defined local derivative that tend to the original function as the order tends to zero and hence improved the conformable derivatives. In addition to this, the non-local fractional operators that emerge from iterating the above-mentioned derivative were held forth in [@fahd12]. Motivated by the above mentioned background, we extend the work done in [@fahd12] introduce a new generalized fractional calculus based on the proportional derivatives of a function with respect to another function in paralel with the definition discussed in [@Anderson1]. The kernel obtained in the fractional operators which will be proposed contains an exponential function and is function dependent. The semi–group properties will be discussed. The article is organized as follows: Section 2 presents some essential definitions for fractional derivatives and integrals. In Section 3 we present the general forms of the fractional proportional integrals and derivatives. In section 4, we present the general form of Caputo fractional proportional derivatives. In the end, we conclude our results. Preliminaries ============= In this section, we present some essential definitions of some fractional derivatives and integrals. We first present the traditional fractional operators and then the fractional proportional operators. The conventional fractional operators and their general forms ------------------------------------------------------------- For $\alpha \in \mathbb{C},~Re(\alpha)>0$, the left Riemann–Liouville fractional integral of order $\alpha $ has the f form $$\label{001} (_{a}I^\alpha f)(x)=\frac{1}{\Gamma(\alpha)}\int_a^x (x-u)^{\alpha-1}f(u)du.$$ The right Riemann–Liouville fractional integral of order $\alpha >0$ is defined by $$\label{002} (I_b^\alpha f)(x)=\frac{1}{\Gamma(\alpha)}\int_x^b (u-x)^{\alpha-1}f(u)du.$$ The left Riemann–Liouville fractional derivative of order $\alpha, Re(\alpha)\geq 0 $ is given as $$\label{003} (_{a}D^\alpha f)(x)=\Big(\frac{d}{dx}\Big)^n(_{a}I^{n-\alpha} f)(x),~~n=[\alpha]+1.$$ The right Riemann–Liouville fractional derivative of order $\alpha, Re(\alpha)\geq 0 $ reads $$\label{004} (D_b^\alpha f)(t)=\Big(-\frac{d}{dt}\Big)^n(I_b^{n-\alpha} f)(t).$$ The left Caputo fractional derivative has the following form $$\label{005} (_{a}^{C}D^\alpha f)(x)=\big(_{a}I^{n-\alpha} f^{(n)}\big)(x),~~n=[\alpha]+1.$$ The right Caputo fractional derivative becomes $$\label{006} (^CD_b^\alpha f)(x)=\big(I_b^{n-\alpha}(-1)^nf^{(n)}\big)(x).$$ The generalized left and right fractional integrals in the sense of Katugampola [@Kat1] are given respectively as $$\label{015} (_{a}\textbf{I}^{\alpha,\rho} f)(x)=\frac{1}{\Gamma(\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{\alpha-1} f(u)\frac{du}{u^{1-\rho}}$$ and $$\label{016} (\textbf{I}_{b}^{\alpha,\rho}f)(x)=\frac{1}{\Gamma(\alpha)}\int_t^b (\frac{u^\rho- x^\rho}{\rho})^{\alpha-1} f(u)\frac{du}{u^{1-\rho}}.$$ The generalized left and right fractional derivatives in the sense of Katugampola [@Kat2] are defined respectively as $$\begin{aligned} \label{017}\nonumber (_{a}\textbf{D}^{\alpha,\rho} f)(x)&=&\gamma^n(_{a}\textbf{I}^{n-\alpha,\rho} f)(t)\\&=&\frac{\gamma^n}{\Gamma(n-\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{n-\alpha-1} f(u)\frac{du}{u^{1-\rho}}\end{aligned}$$ and $$\begin{aligned} \label{018}\nonumber (\textbf{D}_{b}^{\alpha,\rho} f)(x)&=& (-\gamma)^n(\textbf{I}_b^{n-\alpha,\rho} f)(x)\\ &=&\frac{(-\gamma)^n}{\Gamma(n-\alpha)}\int_x^b(\frac{u^\rho-x^\rho}{\rho})^{n-\alpha-1} f(u)\frac{du}{u^{1-\rho}}, \end{aligned}$$ where $\rho>0$ and $\gamma=x^{1-\rho}\frac{d}{dx}$. The Caputo modification of the left and right generalized fractional derivatives in the sense of Jarad et al. [@fahd3] are presented respectively as $$\begin{aligned} \label{019}\nonumber (_{a}^C\textbf{D}^{\alpha,\rho} f)(x)&=&(_{a}\textbf{I}^{n-\alpha,\rho}\gamma^n f)(x)\\&=&\frac{1}{\Gamma(n-\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{n-\alpha-1}\gamma^n f(u)\frac{du}{u^{1-\rho}},\end{aligned}$$ and $$\begin{aligned} \label{020}\nonumber (^C\textbf{D}_{b}^{\alpha,\rho} f)(x)&=& (_{a}\textbf{I}^{n-\alpha,\rho}(-\gamma)^n f)(x)\\ &=&\frac{1}{\Gamma(n-\alpha)}\int_x^b(\frac{u^\rho-x^\rho}{\rho})^{n-\alpha-1} (-\gamma)^nf(u)\frac{du}{u^{1-\rho}}.\end{aligned}$$ For $\alpha \in \mathbb{C},~Re(\alpha)>0$ the left Riemann-Liouville fractional integral of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ has the following form [@Samko; @f2] $$\label{3} ~_{a}I^{\alpha,g} f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{\alpha-1}f(u)g'(u)du.$$ For $\alpha \in \mathbb{C},~Re(\alpha)>0$ the right Riemann-Liouville fractional integral of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ has the following form [@Samko; @f2] $$\label{333} I_b^{\alpha,g} f(x)=\frac{1}{\Gamma(\alpha)}\int_{x}^b \Big(g(u)-g(x)\Big)^{\alpha-1}f(u)g'(u)du.$$ For $\alpha \in \mathbb{C},~Re(\alpha)\geq 0$, the generalized left and right Riemann-Liouville fractional derivative of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ have respectively the form [@Samko; @f3] $$\begin{aligned} \label{4}\nonumber ~_{a}D^{\alpha,g} f(x)&=&\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^n(~_{a}I^{n-\alpha,g} f)(x)\\&=&\frac{\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^n}{\Gamma(n-\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{n-\alpha-1}f(u)g'(u)du \end{aligned}$$ and $$\begin{aligned} \label{444}\nonumber D_b^{\alpha,g} f(x)&=&\Big(-\frac{1}{g'(x)}\frac{d}{dx}\Big)^n(I_b^{n-\alpha,g} f)(x)\\&=&\frac{\Big(-\frac{1}{g'(x)}\frac{d}{dx}\Big)^n}{\Gamma(n-\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{n-\alpha-1}f(u)g'(u)du, \end{aligned}$$ where $n=[\alpha]+1$. It is easy to observe that if we choose $g(x)=x$, the integrals in (\[3\]) and (\[333\]) becomes the left and right Riemann-Liouville fractional integrals respectively and (\[4\]) and (\[444\]) becomes the left and right Riemann-Liouville fractional derivatives. When $g(x)=\ln x$, the Hadamard fractional operators are obtained [@Samko; @f2]. While if one considers $g(x)=\frac{x^\rho}{\rho}$, the fractional operators in the settings of Katugampola [@Kat1; @Kat2] are derived. In left and right generalized Caputo derivatives of a function with respect to another function are presented respectively as [@fahd10] $$\label{555} ~_{a}^CD^{\alpha,g} f(x)=\Big(~_aI^{n-\alpha,g} f^{[n]}\Big)(x)$$ and $$\label{666} ^CD_b^{\alpha,g} f(x)=\Big(~_aI^{n-\alpha,g} (-1)^nf^{[n]}\Big)(x),$$ where $\displaystyle f^{[n]}(x)=\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^nf(x)$. The proportional derivatives and their fractional integrals and derivatives --------------------------------------------------------------------------- The conformable derivative was first introduced by Khalil et al. in [@kh] and then explored by the current author in [@T11]. In his distinctive paper [@Anderson1], Anderson et al. modified the conformable derivative by using the proportional derivative. Indeed, he gave the following definition. \[D1\] *(Modified conformable derivatives)* For $\rho \in [0,1]$, let the functions $\kappa_0, \kappa_1:[0,1]\times \mathbb{R}\rightarrow [0,\infty)$ be continuous such that for all $t \in \mathbb{R}$ we have $$\lim_{\rho\rightarrow 0^+}\kappa_1(\rho,t)=1,~\lim_{\rho\rightarrow 0^+}\kappa_0(\rho,t)=0, \lim_{\rho\rightarrow 1^-}\kappa_1(\rho,t)=0,~\lim_{\rho\rightarrow 1^-}\kappa_0(\rho,t)=1,$$ and $\kappa_1(\rho,t)\neq 0,~~\rho \in [0,1),~~\kappa_0(\rho,t)\neq 0,~~\rho \in (0,1]$. Then, the modified conformable differential operator of order $\rho$ is defined by $$\label{anndy} D^\rho f(t)=\kappa_1(\rho,t) f(t)+\kappa_0(\rho,t) f^\prime(t).$$ The derivative given in (\[anndy\]) is called a proportional derivative. For more details about the control theory of the proportional derivatives and its component functions $\kappa_0$ and $\kappa_1$, we refer the reader to [@Anderson1; @Anderson2]. Of special interest, we shall restrict ourselves to the case when $\kappa_1(\rho,t)=1-\rho$ and $\kappa_0(\rho,t)=\rho$. Therefore, (\[anndy\]) becomes $$\label{prop derivative} D^\rho f(t)=(1-\rho) f(t)+\rho f^\prime(t).$$ Notice that $\lim_{\rho \rightarrow 0^+}D^\rho f(t)= f(t)$ and $\lim_{\rho \rightarrow 1^-}D^\rho f(t)= f^\prime(t)$. It is clear that the derivative (\[prop derivative\]) is somehow more general than the conformable derivative which does not tend to the original function as $\rho$ tends to $0$. The associated fractional proportional integrals are defined as [@fahd12] \[left and right integrals\]For $\rho>0$ and $\alpha \in \mathbb{C},~~Re(\alpha)>0$, the left fractional proportional integral of $f$ reads $$\label{LPI} (_{a}I^{\alpha,\rho} f)(x)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)} (x-\tau)^{\alpha-1} f(\tau)d\tau$$ and the right one reads $$\label{RPI} (I_b^{\alpha,\rho} f)(x)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)} (\tau-x)^{\alpha-1} f(\tau)d\tau.$$ \[Prop fractional derivatives\][@fahd12] For $\rho>0$ and $\alpha \in \mathbb{C},~~Re(\alpha)\geq 0$, the left fractional proportional derivative is defined as $$\begin{aligned} \label{LPD}\nonumber (_{a}D^{\alpha,\rho}f)(x)&=& D^{n,\rho} ~_{a}I^{n-\alpha,\rho} f(x)\\&=&\frac{D_x^{n,\rho}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)}(x-\tau)^{n-\alpha-1} f(\tau)d \tau.\end{aligned}$$ The right proportional fractional derivative is defined by [@fahd12] $$\begin{aligned} \label{RPD}\nonumber (D_b^{\alpha,\rho}f)(x)&=& ~_{\ominus}D^{n,\rho} I_b^{n-\alpha,\rho} f(x)\\&=&\frac{~_{\ominus}D^{n,\rho}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)}(\tau-x)^{n-\alpha-1} f(\tau)d \tau, \end{aligned}$$ where $n=[Re(\alpha)]+1$ and $\displaystyle (_{\ominus}D^\rho f)(t)=(1-\rho)f(t)-\rho f^\prime(t)$. Lastly, the left and right fractional proportional derivatives in the Caputo settings respectively read [@fahd12] $$\begin{aligned} \label{LPDC}\nonumber (_{a}^CD^{\alpha,\rho}f)(x)&=& \Big(~_{a}I^{n-\alpha,\rho}D^{n,\rho}f\Big)(x)\\\nonumber &=&\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)}(x-\tau)^{n-\alpha-1} (D^{n,\rho}f)(\tau)d \tau \\\end{aligned}$$ and $$\begin{aligned} \label{RPDC}\nonumber (^CD_b^{\alpha,\rho}f)(x)&=& \Big( I_b^{n-\alpha,\rho}{~_\ominus}D^{n,\rho}f\Big)(x)\\\nonumber &=&\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)}(\tau-x)^{n-\alpha-1} (~_{\ominus}D^{n,\rho}f)(\tau)d\tau.\\ \end{aligned}$$ The fractional proportional derivative of a function with respect to another function ===================================================================================== \[D2\] *(The proportional derivative of a function with respect to anothor function)*\ For $\rho \in [0,1]$, let the functions $\kappa_0, \kappa_1:[0,1]\times \mathbb{R}\rightarrow [0,\infty)$ be continuous such that for all $t \in \mathbb{R}$ we have $$\lim_{\rho\rightarrow 0^+}\kappa_1(\rho,t)=1,~\lim_{\rho\rightarrow 0^+}\kappa_0(\rho,t)=0, \lim_{\rho\rightarrow 1^-}\kappa_1(\rho,t)=0,~\lim_{\rho\rightarrow 1^-}\kappa_0(\rho,t)=1,$$ and $\kappa_1(\rho,t)\neq 0,~~\rho \in [0,1),~~\kappa_0(\rho,t)\neq 0,~~\rho \in (0,1]$. Let also $g(t)$ be a strictly increasing continuous function. Then, the proportional differential operator of order $\rho$ of $f$ with respect to $g$ is defined by $$\label{eq1} D^{\rho,g} f(t)=\kappa_1(\rho,t) f(t)+\kappa_0(\rho,t)\frac{f^\prime(t)}{g'(t)}.$$ we shall restrict ourselves to the case when $\kappa_1(\rho,t)=1-\rho$ and $\kappa_0(\rho,t)=\rho$. Therefore, (\[eq1\]) becomes $$\label{eq2} D^{\rho,g} f(t)=(1-\rho) f(t)+\rho \frac{f^\prime(t)}{g'(t)}.$$ The corresponding integral of $$\label{eq3} _{a}I^{1,\rho,g}f(t)=\frac{1}{\rho}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(s))}f(s)g'(s)ds,$$ where we accept that $~_{a}I^{0,\rho}f(t)=f(t)$. To produce a generalized type fractional integral depending on the proportional derivative, we proceed by induction through changing the order of integrals to show that $$\begin{aligned} \label{eq4} \nonumber (_{a}I^{n,\rho,g} f)(t) &=& \frac{1}{\rho}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau_1))} g'(\tau_1)d \tau_1 \frac{1}{\rho} \int_a^{\tau_1} e^{\frac{\rho-1}{\rho}(g(\tau_1)-g(\tau_2))}g'(\tau_2)d \tau_2\cdot\cdot\cdot\\\nonumber&\cdot\cdot\cdot&\frac{1}{\rho} \int_a^{\tau_{n-1}} e^{\frac{\rho-1}{\rho}(g(\tau_{n-1})-g(\tau_n))}f(\tau_n)g'(\tau_n) d\tau_n\\ &=& \frac{1}{\rho^n \Gamma(n)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{n-1} f(\tau)g'(\tau)d\tau. \end{aligned}$$ Based on (\[eq4\]), we can present the following general proportional fractional integral. \[general left and right integrals\]For $\rho \in (0,1]$, $\alpha \in \mathbb{C},~~Re(\alpha)>0$, we define the left fractional integral of $f$ with respect to $g$ by $$\label{eq5} (_{a}I^{\alpha,\rho,g} f)(t)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{\alpha-1} f(\tau)g'(\tau)d\tau.$$ The right fractional proportional integral ending at $b$ can be defined by $$\label{eq6} (I_b^{\alpha,\rho,g} f)(t)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_t^b e^{\frac{\rho-1}{\rho}(g(\tau)-g(t))} (g(\tau)-g(t))^{\alpha-1} f(\tau)g'(\tau)d\tau.$$ To deal with the right proportional fractional case we shall use the notation $$(_{\ominus}D^{\rho,g} f)(t):=(1-\rho)f(t)-\rho \frac{f^\prime(t)}{g'(t)}.$$ We shall also write $$(_{\ominus}D^{n,\rho,g} f)(t)= (\underbrace{_{\ominus}D^{\rho,g}~ _{\ominus}D^{\rho,g}\ldots~_{\ominus}D^{\rho,g}}_{\texttt{n times}} f)(t).$$ \[general left and right derivatives\] For $\rho>0$, $\alpha \in \mathbb{C},~~Re(\alpha)\geq 0$ and $g\in C[a,b]$, where $g'(t)>0$, we define the general left fractional derivative of $f$ with respect to $g$ as $$\begin{aligned} \label{eq7}\nonumber (_{a}D^{\alpha,\rho,g}f)(t)&=& D^{n,\rho,g} ~_{a}I^{n-\alpha,\rho,g} f(t)\\\nonumber &=&\frac{D_t^{n,\rho,g}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{n-\alpha-1} f(\tau)g'(\tau)d \tau\\ \end{aligned}$$ and the general right fractional derivative of $f$ with respect to $g$ as $$\begin{aligned} \label{eq8}\nonumber (D_b^{\alpha,\rho,g}f)(t)&=& ~_{\ominus}D^{n,\rho,g} I_b^{n-\alpha,\rho,g} f(t)\\\nonumber &=&\frac{~_{\ominus}D_t^{n,\rho,g}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_t^b e^{\frac{\rho-1}{\rho}(g(\tau)-g(t))}(g(\tau)-g(t))^{n-\alpha-1} f(\tau)g'(\tau)d \tau,\\ \end{aligned}$$ where $n=[Re(\alpha)]+1$. \[reduction\] Clearly, if we let $\rho=1$ in Definition \[general left and right integrals\] and Definition \[general left and right derivatives\], we obtain the - the Riemann-Liouville fractional operators , , and if $g(t)=t$. - the fractional operators in the Katugampola setting, , and if $\displaystyle g(t)=\frac{t^{\mu}}{\mu}$. - The Hadamard fractional operators if $g(t)=\ln t$ [@Samko; @f2]. - The fractional operators mentioned in [@fahd11] if $\displaystyle g(t)=\frac{(t-a)^{\mu}}{\mu}$. \[2.4\] Let $\alpha, \beta \in \mathbb{C}$ be such that $Re(\alpha)\geq 0$ and $Re(\beta)>0$. Then, for any $\rho>0$ we have - \(a)  $\big(_{a}I^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)\rho^\alpha}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\alpha+\beta-1},$\ $~~~Re(\alpha)>0.$ - (b)  $\big(I_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)\rho^\alpha}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-g(t))^{\alpha+\beta-1},$\ $~~~Re(\alpha)>0.$ - (c)  $\big(_{a}D^{\alpha,\rho} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha},$\ $~~~Re(\alpha)\geq 0.$ - (d) $\big(D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-G g(t))^{\beta-1-\alpha},$\ $~~~Re(\alpha)\geq 0.$ The proofs of relations (a) and (b) are very easy to handle. We will prove (c) while the proof of (d) is analogous. By the definition of the left proportional fractional derivative and relation (a), we have $$\begin{aligned} &\Big(_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\Big)(t)\\ &=D^{n,\rho,g} \Big(_{a}I^{n-\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\Big)(t)\\ &=D^{n,\rho,g}\frac{\Gamma(\beta)}{\Gamma(\beta+n-\alpha)\rho^{n-\alpha}}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{n-\alpha+\beta-1} \\ &=\frac{\rho^n\Gamma(\beta)(n-\alpha+\beta-1)(n-\alpha+\beta-1)\cdot \cdot \cdot (\beta-\alpha)}{\rho^{n-\alpha}\Gamma(n-\alpha+\beta)} \times e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}\\ & =\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}.\end{aligned}$$ Here, we have used the fact that $\displaystyle D^{\rho,g} \Big(h(t) e^{\frac{\rho-1}{\rho}g(t)} \Big)=\rho \frac{h'(t)}{g^\prime (t)} e^{\frac{\rho-1}{\rho}g(t)} $. Below we present the semi–group property for the general fractional proportional integrals of a function with respect to another function. \[THM1\] Let $\rho\in (0,1],~Re(\alpha)>0$ and $Re(\beta)>0$. Then, if $f$ is continuous and defined for $t \geq a$ or $t\le b$, we have $$\label{Left Semi integrals} ~_aI^{\alpha,\rho,g} (_{a}I^{\beta,\rho,g} f)(t)= ~_aI^{\beta,\rho,g} (_{a}I^{\alpha,\rho} f)(t)=(~_{a}I^{\alpha+\beta,\rho,g} f)(t)$$ and $$\label{Right Semi integrals} I_b^{\alpha,\rho,g} (I_b^{\beta,\rho,g} f)(t)= ~I_b^{\beta,\rho,g} (I_b^{\alpha,\rho} f)(t)=(I_b^{\alpha+\beta,\rho,g} f)(t).$$ We will prove . is proved similarly. Using the definition, interchanging the order and making the change of variable $z=\frac{g(u)-g(\tau)}{g(t)-g(\tau)}$, we get $$\begin{aligned} &~_aI^{\alpha,\rho,g} (_{a}I^{\beta,\rho,g} f)(t)\\ &= \frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)} \int_a^t \int_a^ue^{\frac{\rho-1}{\rho}(g(t)-g(u))}e^{\frac{\rho-1}{\rho}(g(u)-g(\tau))}(g(t)-g(u))^{\alpha-1}\\ & \times(g(u)-g(\tau))^{\beta-1}f(\tau)g'(\tau)d\tau g'(u)du\\ &=\frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} f(\tau) \int_\tau^t (g(t)-g(u))^{\alpha-1} (g(u)-g(\tau))^{\beta-1}\\ &\times g'(u)dug'(\tau) d\tau \\ &= \frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)} \int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{\alpha+\beta-1} f(\tau)g'(u)d\tau \\ &\times \int_0^1 (1-z)^{\alpha-1} z^{\beta-1} dz\\ &=\frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha+\beta)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{\alpha+\beta-1} f(\tau)g'(\tau)d\tau\\ &=(_{a}I^{\alpha+\beta,\rho} f)(t).\end{aligned}$$ \[THM2\]Let $0\leq m< [Re(\alpha)]+1$. Then, we have $$\label{D on L} D^{m,\rho,g} (_{a}I^{\alpha,\rho,g}f)(t)=(_{a}I^{\alpha-m,\rho,g}f)(t)$$ and $$\label{LD on L} ~_{\ominus}D^{m,\rho,g} (I_b^{\alpha,\rho,g}f)(t)=(I_b^{\alpha-m,\rho,g}f)(t)$$ Here we prove , while one can prove likewise. Using the fact that $D_t^{\rho,g}e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} =0$), we have $$\begin{aligned} & D^{m,\rho,g} (_{a}I^{\alpha,\rho,g}f)(t) D^{m-1,\rho,g} (D^{\rho,g}~_{a}I^{\alpha,\rho,g}f)(t) \\ &= D^{m-1,\rho,g} \frac{1}{\rho^{\alpha-1}\Gamma(\alpha-1)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{\alpha-2}f(\tau)g'(\tau)d \tau.\end{aligned}$$ Proceeding $m-$times in the same manner we obtain (\[D on L\]). \[D on I\] Let $0<Re(\beta) < Re(\alpha)$ and $m-1<Re(\beta)\leq m$. Then, we have $$\label{LD on LI} _{a}D^{\beta, \rho,g} ~_{a}I^{\alpha,\rho,g} f(t)=~_{a}I^{\alpha-\beta,\rho,g} f(t)$$ and $$\label{RD on RI} D_b^{\beta, \rho,g} I_b^{\alpha,\rho,g} f(t)=I_b^{\alpha-\beta,\rho,g} f(t).$$ By the help of Theorem \[THM1\] and Theorem \[THM2\], we have $$\begin{aligned} ~_{a}D^{\beta, \rho,g} ~_{a}I^{\alpha,\rho,g} f(t)&=& D^{m,\rho,g} _{a}I^{m-\beta,\rho,g} _{a}I^{\alpha,\rho,g}f(t)\\ &=& D^{m,\rho,g}~_{a}I^{m-\beta+\alpha,\rho,g} f(t)=~_{a}I^{\alpha-\beta,\rho,g} f(t).\end{aligned}$$ This was the proof of . One can prove in a similar way. \[THM4\] Let $f$ be integrable on $t\geq a$ or $t\le b$ and $Re[\alpha]>0, ~\rho \in (0,1],~~n=[Re(\alpha)]+1$. Then, we have $$\label{LD on LI sameorder} ~_{a}D^{\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)=f(t)$$ and $$\label{RD on RI sameorder} D_b^{\alpha, \rho,g} I_b^{\alpha, \rho,g} f(t)=f(t).$$ By the definition and Theorem \[THM1\], we have $$~_{a}D^{\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)=D^{n,\rho,g}~_{a}I^{n-\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)= D^{n,\rho,g}~_{a}I^{n, \rho,g} f(t)=f(t).$$ The Caputo fractional proportional derivative of a function with respect to another function ============================================================================================= For $\rho \in (0,1]$ and $\alpha \in \mathbb{C}$ with $Re(\alpha)\geq 0$ we define the left derivative of Caputo type as $$\begin{aligned} \label{CFP} &(^{C}_{a}D^{\alpha,\rho,g} f)(t)=_{a}I^{n-\alpha,\rho,g} (D^{n,\rho,g}f)(t)\\\nonumber &=\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(s))}(g(t)-g(s))^{n-\alpha-1}(D^{n,\rho,g}f)(s)g'(s)ds.\end{aligned}$$ Similarly, the right derivative of Caputo type ending is defined by $$\begin{aligned} \label{rCFP} &(^{C}D_b^{\alpha,\rho} f)(t)= I_b^{n-\alpha,\rho,g} (_{\ominus}D^{n,\rho,g}f)(t)\\\nonumber &= \frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)}\int_t^b e^{\frac{\rho-1}{\rho}(g(s)-g(t))}(g(s)-g(t))^{n-\alpha-1}(~_{\ominus}D^{n,\rho,g}f)(s)g'(s)ds,\end{aligned}$$ where $n=[Re(\alpha)]+1$. \[4.2\] Let $\alpha, \beta \in \mathbb{C}$ be such that $Re(\alpha)> 0$ and $Re(\beta)>0$. Then, for any $\rho \in (0,1]$ and $n=[Re(\alpha)]+1$ we have 1. $\big(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha},$\ $~~~Re(\beta)> n.$ 2. $\big(^{C}D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-g(t))^{\beta-1-\alpha},$\ $~~~Re(\beta)> n.$ For $k=0,1,\ldots,n-1$, we have $$\big(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a)^{k}\big)(t)=0\quad \mbox{and}\quad \big(^{C}D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{k}\big)(t)=0.$$ In particular, $(~^{C}_{a}D^{\alpha,\rho} e^{\frac{\rho-1}{\rho}g(x})(t)=0$ and $(^{C}D_b^{\alpha,\rho} e^{-\frac{\rho-1}{\rho}g(x)})(t)=0$. We only prove the first relation. The proof of the second relation is similar. We have $$\begin{aligned} &(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1})(t)= ~_{a}I^{n-\alpha,\rho,g} D^{n,\rho,g} \left[e^{\frac{\rho-1}{\rho}g(t)} (g(t)-g(a))^{\beta-1} \right]\\ &=~_{a}I^{n-\alpha,\rho,g} \left[ \rho^n (\beta-1)(\beta-2)\ldots(\beta-1-n) (g(t)-g(a))^{\beta-n-1} e^{\frac{\rho-1}{\rho}g(t)}\right] \\ &= \frac{\rho^n (\beta-1)(\beta-2)\ldots(\beta-1-n)\Gamma(\beta-n)} {\Gamma(\beta-\alpha)\rho^{n-\alpha}} (g(t)-g(a))^{\beta-\alpha-1} e^{\frac{\rho-1}{\rho}g(t)}\\ &= \frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}.\end{aligned}$$ Conclusions =========== We have used the proportional derivatives of a function with respect to another to obtain left and right generalized type of fractional integrals and derivatives involving two parameters $\alpha$ and $\rho$ and depending on a kernel function . The Riemann–Liouville and Caputo fractional derivatives in classical fractional calculus can obtained as $\rho$ tends to $1$ and by choosing $g(x)=1$. The integrals have the semi–group property and together with their corresponding derivatives have exponential functions as part of their kernels. It should be noted that other properties of these new operators can be obtained by using the Laplace transform proposed in [@fahd10]. Moreover, for a specific choice of $g$, the proportional fractional operators in the settings of Hadamard and Katugamplola can be extracted. **Funding** This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.\ **Availability of data and materials** Not applicable.\ **Competing interests** The authors have no competing interest regarding this article.\ **Authors contributions** All authors have done equal contribution in this article. 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ArXiv

--- author: - 'Felix Kahlhoefer,' - 'Kai Schmidt-Hoberg,' - Thomas Schwetz - and Stefan Vogl title: Implications of unitarity and gauge invariance for simplified dark matter models --- Introduction ============ After the successful discovery of a Higgs Boson consistent with the predictions of the Standard Model (SM), the focus of the current and upcoming runs of the Large Hadron Collider (LHC) at 13 TeV will be to discover evidence for physics beyond the SM. Among the prime targets of this search is dark matter (DM), which has so far only been observed via its gravitational interactions at astrophysical and cosmological scales. Since no particle within the SM has the required properties to explain these observations, DM searches at the LHC are necessarily searches for new particles. In fact, LHC DM searches are also likely to be searches for new interactions. Given the severe experimental constraints on the interactions between DM and SM particles, it is a plausible and intriguing possibility that the DM particle is part of a (potentially rich) hidden sector, which does not couple directly to SM particles or participate in the known gauge interactions. In this setup, the visible sector interacts with the hidden sector only via one or several new mediators, which have couplings to both sectors. In the simplest case the mass of these mediators is large enough that they can be integrated out and interactions between DM particles and the SM can be described by higher-dimensional contact interactions [@Beltran:2008xg; @Beltran:2010ww]. This effective field theory (EFT) approach has been very popular for the analysis and interpretation of DM searches at the LHC [@Goodman:2010ku; @Fox:2011pm; @Rajaraman:2011wf]. Nevertheless, as any effective theory it suffers from the problem that unitarity breaks down if the relevant energy scales become comparable to the cut-off scale of the theory [@Shoemaker:2011vi; @Fox:2012ee; @Busoni:2013lha; @Busoni:2014sya; @Xiang:2015lfa] (for other examples of applying unitarity arguments in the DM context see refs. [@Griest:1989wd; @Walker:2013hka; @Endo:2014mja; @Hedri:2014mua]). The easiest way to avoid this problem appears to be to explicitly retain the (lightest) mediator in the theory. The resulting models are referred to as simplified DM models, in which couplings are only specified after electroweak symmetry breaking (EWSB) and no ultraviolet (UV) completion is provided [@Abdallah:2015ter]. Compared to the EFT approach, simplified models have a richer phenomenology [@Busoni:2013lha; @Buchmueller:2013dya; @Buchmueller:2014yoa; @Harris:2014hga; @Garny:2014waa; @Buckley:2014fba; @Jacques:2015zha; @Alves:2015dya; @Choudhury:2015lha], including explicit searches for the mediator itself [@Frandsen:2012rk; @Fairbairn:2014aqa; @Chala:2015ama]. Moreover, it is possible to achieve the DM relic abundance in large regions of parameter space [@Busoni:2014gta; @Chala:2015ama; @Blennow:2015gta]. Constraining the parameter space of simplified DM models is therefore a central objective of experimental collaborations [@Khachatryan:2014rra; @Aad:2015zva; @Abercrombie:2015wmb]. In the present work we focus on the case of a spin-1 $s$-channel mediator [@Dudas:2009uq; @Fox:2011qd; @Frandsen:2012rk; @Alves:2013tqa; @Arcadi:2013qia; @Jackson:2013pjq; @Jackson:2013rqp; @Duerr:2013lka; @Duerr:2014wra; @Lebedev:2014bba; @Hooper:2014fda; @Martin-Lozano:2015vva; @Alves:2015pea; @Alves:2015mua; @Blennow:2015gta; @Duerr:2015wfa; @Heisig:2015ira]. Our central observation is that the simplified model approach is not generally sufficient to avoid the problem of unitarity violation at high energies and that further amendments are required if the model is to be both simple and realistic. In particular, a spin-1 mediator with axial couplings violates perturbative unitarity at large energies, pointing towards the presence of additional new physics to restore unitarity. Indeed, the simplest way to restore unitarity is to assume that the spin-1 mediator is the gauge boson of an additional $U(1)'$ gauge symmetry [@Holdom:1985ag; @Babu:1997st] and that its mass as well as the DM mass are generated by a new Higgs field in the hidden sector. The famous Lee-Quigg-Thacker bound [@Lee:1977eg] implies that the additional Higgs boson cannot be arbitrarily heavy and may therefore play an important role for LHC and DM phenomenology. In particular, it can mix with the SM-like Higgs boson and mediate interactions between DM particles and quarks. Furthermore, we require for a consistent simplified DM model that the coupling structure respects gauge invariance of the full SM gauge group before EWSB (see [@Bell:2015sza] for a similar discussion in the EFT context). If the mediator has axial couplings to quarks, this requirement implies that the new mediator will also have couplings to leptons and mixing with the SM $Z$ boson, both of which are tightly constrained by experiments. Much weaker constraints are obtained for the simplified DM model containing a spin-1 mediator with vectorial couplings to quarks. Constraints from direct detection can be evaded if the mediator has only axial couplings to DM, which naturally arises in the case that the DM particle is a Majorana fermion. We discuss the importance of loop-induced mixing effects in this context, which can play a crucial role for both direct detection experiments and LHC phenomenology. The outline of the paper is as follows. Starting from a simplified model for a mediator, we explore in section \[sec:unitarity\] the implications of perturbative unitarity, deriving a number of constraints on the model parameters and in particular an upper bound on the scale of additional new physics. In section \[sec:higgs\] we then consider the case where this additional new physics is a Higgs field in the hidden sector and derive an upper bound on the mass of the extra Higgs boson. We then discuss additional constraints on the SM couplings implied by gauge invariance. Section \[sec:axial\] focuses on the case of non-zero axial couplings between SM fermions and the mediator, whereas in section \[sec:vector\] we assume that the SM couplings of the mediator are purely vectorial. Finally, we discuss the experimental implications of a possible mixing between the SM Higgs and the hidden sector Higgs in section \[sec:higgsmixing\]. A discussion of our results and our conclusions are presented in section \[sec:discussion\]. Unitarity constraints on simplified models {#sec:unitarity} ========================================== Brief review of $S$ matrix unitarity constraints ------------------------------------------------ Consider the scattering matrix element $\mathcal{M}_{if}(s, \cos \theta)$ between 2-particle initial and final states ($i,f$), with $\sqrt{s}$ and $\theta$ being the centre of mass energy and scattering angle, respectively. We define the helicity matrix element for the $J$th partial wave by $$\label{eq:Jexpansion} \mathcal{M}_{if}^J(s) = \frac{1}{32\pi} \beta_{if} \int_{-1}^1 \mathrm{d}\cos \theta \, d^J_{\mu \mu'}(\theta) \, \mathcal{M}_{if}(s, \cos \theta) \,,$$ where $d^J_{\mu \mu'}$ is the $J$th Wigner d-function, $\mu$ and $\mu'$ denote the total spin of the initial and the final state (see e.g. [@Chanowitz:1978mv]), and $\beta_{if}$ is a kinematical factor. In the high-energy limit $s \to \infty$, which we are going to consider below, $\beta_{if} \to 1$. The right-hand side of eq.  is to be multiplied with a factor of $1/\sqrt{2}$ each if the initial or final state particles are identical [@Schuessler:2007av]. Unitarity of the $S$ matrix implies $$\begin{aligned} {\rm Im}(\mathcal{M}_{ii}^J) & = \sum_f | \mathcal{M}_{if}^J|^2 \nonumber\\ &= | \mathcal{M}_{ii}^J|^2 + \sum_{f \neq i} | \mathcal{M}_{if}^J|^2 \ge | \mathcal{M}_{ii}^J|^2 \label{eq:unity}\end{aligned}$$ for all $J$ and all $s$. The sum over $f$ in the first line runs over all possible final states. Restricting these to be all possible 2-particle states leads to a conservative bound. If the relation is strongly violated for matrix elements calculated at leading order in perturbation theory one can conclude that either higher-order terms in perturbation theory restore unitarity (i.e. break-down of perturbativity) or that the theory is not complete and additional contributions to the matrix element are needed. From eq.  one obtains the necessary conditions $$\label{eq:unitary-bound} 0 \le {\rm Im}({\mathcal{M}}_{ii}^J) \le 1\,, \quad \left| \text{Re}({\mathcal{M}}_{ii}^J) \right| \le \frac{1}{2} \,.$$ In the following we will apply these inequalities to leading-order matrix elements in order to identify regions in parameter space where perturbative unitarity is violated. Since these matrix elements are always real in the present context, only the second constraint will be relevant. If the matrix $\mathcal{M}_{if}^J$ is diagonalized the inequality in eq.  becomes an equality. Hence, stronger constraints can be obtained by considering the full transition matrix connecting all possible 2-particle states with each other (or some submatrix thereof) and calculating the eigenvalues of that matrix. Then the bounds from eq.  have to hold for each of the eigenvalues [@Schuessler:2007av]. We note that $d^J_{00}(\theta) = P_J(\cos \theta)$, where $P_J$ are the Legendre polynomials. If initial and final state both have zero total spin, eq.  therefore becomes identical to the familiar partial wave expansion of the matrix element. In the following we will focus on the $J=0$ partial wave, which typically provides the strongest constraint. Since $d^0_{\mu\mu'}$ is non-zero only for $\mu = \mu' = 0$, we then obtain from eq.  $$\mathcal{M}_{if}^0(s) = \frac{1}{64\pi} \beta_{if} \, \delta_{\mu0} \delta_{\mu'0} \int_{-1}^1 \mathrm{d}\cos \theta \, \mathcal{M}_{if}(s, \cos \theta) \,.$$ Application to a simplified model with a $Z'$ mediator ------------------------------------------------------ Let us consider a simplified model for a spin-1 mediator $Z'^\mu$ with mass $m_{Z'}$ and a Dirac DM particle $\psi$ with mass $m_\text{DM}$.[^1] The most general coupling structure is captured by the following Lagrangian: $$\label{eq:L_VA} \mathcal{L} = - \sum_{f = q,l,\nu} Z'^\mu \, \bar{f} \left[ g_{f}^V \gamma_\mu + g_f^A \gamma_\mu \gamma^5 \right] f - Z'^\mu \, \bar{\psi} \left[ g_\text{DM}^V \gamma_\mu + g_\text{DM}^A \gamma_\mu \gamma^5 \right] \psi \; .\\$$ Although these interactions appear renormalisable, the presence of a massive vector boson implies that perturbative unitarity may be violated at large energies. In the following, we will study this issue in detail and derive constraints on the parameter space of the model. Let us first consider diagrams between 2-fermion states with the $Z'$ as mediator. The appropriate propagator for the mediator is $$\langle Z'^\mu(k)Z'^\nu(-k)\rangle = \frac{1}{k^2 - m_{Z'}^2} \left(g^{\mu\nu} - \frac{k^\mu k^\nu}{m_{Z'}^2}\right) \;,$$ where $k^\mu$ is the momentum of the mediator. For the case of a gauge boson this corresponds to unitary gauge in which the Goldstone boson has been absorbed. Since we are interested in the high-energy behaviour of the theory we concentrate on the second term, which does not vanish in the limit $k \rightarrow \infty$. This corresponds to restricting to the longitudinal component of the mediator, $Z'_L$, which dominates at high energy [@Chanowitz:1978mv].[^2] For instance, considering DM annihilations, we can contract the longitudinal part of the propagator with the DM current. Making use of $k = p_1 + p_2$, where $p_1$ and $p_2$ are the momenta of the two DM particles in the initial state, leads to a factor $$\begin{aligned} k^\mu \bar{v}(p_2) \left( g^V_\text{DM} \gamma_\mu + g^A_\text{DM} \gamma_\mu \gamma^5 \right) u(p_1) & = \bar{v}(p_2) \left[ g^V_\text{DM} (\slashed{p}_2 + \slashed{p}_1) + g^A_\text{DM} (\slashed{p}_2 \gamma^5 - \gamma^5 \slashed{p}_1) \right] u(p_1) \nonumber \\ & = - 2 \, g^A_\text{DM} \, m_\text{DM} \, \bar{v}(p_2) \gamma^5 u(p_1) \;.\end{aligned}$$ Hence, the second term in the propagator behaves exactly like a pseudoscalar with mass $m_{Z'}$ and couplings to DM equal to $2 \, g^A_\text{DM} \, m_\text{DM} / m_{Z'}$, just like the Goldstone boson present in Feynman gauge. Note that the term is independent of the vector couplings. The same argument holds for the quark couplings, which are found to be given by $2 \, g^A_f \, m_f / m_{Z'}$. This consideration suggests that perturbative unitarity will not only lead to bounds on $g^{V,A}$, but also on the combination $g^A_f \, m_f / m_{Z'}$. We can make this statement more precise by applying the methods outlined in the previous subsection to the self-scattering of two DM particles or two SM fermions. We obtain for any fermion $f$ with axial couplings $g^A_f \neq 0$ that the fermion mass must satisfy the bound $$m_f \lesssim \sqrt{\frac{\pi}{2}} \frac{m_{Z'}}{g^A_f} \; . \label{eq:DMmass}$$ Here $f$ can be any fermion, including SM fermions and the DM particle. As suggested by the above discussion we do not obtain any bound on the masses of fermions with purely vectorial couplings, nor on the scale of new physics. Let us now turn to the discussion of processes involving $Z'$ in the external state, in particular $Z'$ with longitudinal polarisation. For concreteness, we study the process $\psi \bar{\psi} \rightarrow Z'_L Z'_L$.[^3] At large momenta, $k^2 \gg m_{Z'}^2$, the polarisation vectors of the gauge bosons can be replaced by $\epsilon_L^\mu(k) = k^\mu / m_{Z'}$. One might therefore expect the matrix element for this process to grow proportional to $s / m_{Z'}^2$. However, such a term is absent due to a cancellation between the $t$- and $u$-channel diagram. To obtain a non-zero contribution, one needs to include a mass insertion along the fermion line [@Shu:2007wg]. It turns out that the contribution proportional to $g^V_\text{DM}$ still cancels in this case and that the leading contribution at high energies becomes proportional to $(g^A_\text{DM})^2 \sqrt{s} \, m_\text{DM} / m_{Z'}^2$. As a result, perturbative unitarity is violated unless [@Hosch:1996wu; @Shu:2007wg; @Babu:2011sd][^4] $$\label{eq:s} \sqrt{s} < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}} \; .$$ For larger energies new physics must appear to restore unitarity. This can be accomplished by including an additional diagram with an $s$-channel Higgs boson, since both contributions have the same high-energy behaviour. The consideration above implies an upper bound on the mass of the Higgs that breaks the $U(1)'$ and gives mass to the $Z'$: $$m_s < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}} \; . \label{eq:higgsmass}$$ We will discus the consequences of such an extension of the minimal model in section \[sec:higgs\]. ![Parameter space forbidden by the requirement of perturbative unitarity in the $\sqrt{s}-m_{Z'}$ plane for $m_\text{DM}=500\:\text{GeV}$. The constraint resulting from DM scattering is shown in grey (solid), the constraint resulting from DM annihilation into $Z'$s is shown in blue (dashed). Thick (thin) lines correspond to $g^A_\text{DM}=1$ ($g^A_\text{DM}=0.1$). In these cases, the $Z'$ can never be lighter than about $400\:\text{GeV}$ ($40\:\text{GeV}$) irrespective of the UV completion.[]{data-label="fig:ubound"}](uni.pdf){height="0.3\textheight"} In summary we have found that there are two different types of constraints on the parameters of this simplified model, even for perturbative couplings. For non-vanishing axial couplings there is an energy scale for which the theory violates perturbative unitarity and needs to be UV completed, see eq. . In addition, imposing that the coupling between the longitudinal component of the vector mediator and the DM particle remain perturbative, we find that the vector mediator cannot be much lighter than the DM, see eq. . This constraint is not related to missing degrees of freedom and is therefore completely independent of the UV completion. We illustrate both constraints in figure \[fig:ubound\] for different axial couplings and a DM mass $m_\text{DM} = 500\:\text{GeV}$. To conclude this section, we emphasise that for pure vector couplings of the $Z'$ ($g^A_\text{DM} = g^A_f = 0$) the simplified model considered in this section is well-behaved in the UV in the sense that there is no problem with perturbative unitarity.[^5] Indeed in this specific case a bare mass term for the dark matter is allowed such that it is sufficient to generate the vector boson mass via a Stueckelberg mechanism without the need for additional degrees of freedom [@Stueckelberg:1900zz; @Kors:2005uz]. However, this specific coupling configuration is highly constrained, since it is very difficult to evade bounds from direct detection experiments and still reproduce the observed DM relic abundance. This is illustrated in figure \[fig:vector\] where we show the parameter region excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX [@Akerib:2013tjd] and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe. One can clearly see that only a finely-tuned region of parameter space close to the resonance $m_\text{DM} = m_{Z'}/2$ is still allowed. For the rest of the paper, we will therefore not consider this case further and always assume that at least one of the vector couplings vanishes such that direct detection constraints can be weakened. ![Vector(SM)–Vector(DM): Parameter space excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX (green, dashed) and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe (red, solid).[]{data-label="fig:vector"}](vectorvector.pdf){height="0.3\textheight"} Including an additional Higgs field {#sec:higgs} =================================== As we have seen in the previous section, for non-zero axial couplings the simplified model violates perturbative unitarity at high energies, implying that additional new physics must appear below these scales. This observation motivates a detailed discussion of how to generate the vector boson mass from an additional Higgs mechanism. To restore unitarity let us therefore now consider the case that the $Z'$ is the gauge boson of a new $U(1)'$ gauge group. To break this gauge group and give a mass to the $Z'$, we introduce a dark Higgs singlet $S$, which needs to be complex in order to allow for a $U(1)'$ charge. We then obtain the following Lagrangian $$\mathcal{L} = \mathcal{L}_\text{SM} + \mathcal{L}_\text{DM} + \mathcal{L}'_\text{SM} + \mathcal{L}_\text{S} \; ,$$ where the first term is the usual SM Lagrangian and the second term describes the interactions of DM. The third term contains the interactions between SM states and the new $Z'$ gauge boson while the fourth term contains the extended Higgs sector. Implications for the dark sector -------------------------------- As mentioned above, it is well-motivated from a phenomenological perspective to consider the case that vector couplings to the $Z'$ mediator vanish in at least one of the two sectors, so that direct detection is suppressed. On the DM side this is naturally achieved for a Majorana fermion, which we will focus on from now. We therefore write $$\psi = \left( \begin{array}{c} \chi \\ \epsilon \chi^\ast \end{array} \right) \; ,$$ where $\chi$ is a Weyl spinor. We assume that $\chi$ carries a charge $q_\text{DM}$ under the new $U(1)'$ gauge group, such that under a gauge transformation $$\psi \rightarrow \exp\left[i \, g' q_\text{DM} \, \alpha(x) \, \gamma^5\right] \psi \; ,$$ where $g'$ is the gauge coupling of the new $U(1)'$. The kinetic term for $\psi$ can hence be written as $$\mathcal{L}_\text{kin} = \frac{1}{2} \bar{\psi} (i \slashed{\partial} - g' \, q_\text{DM} \, \gamma^5 \slashed{Z}') \psi = \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi \; ,$$ with $g_\text{DM}^A \equiv g' q_\text{DM}$. The $U(1)'$ charge forbids a Majorana mass term. Nevertheless, if the Higgs field $S$ carries charge $q_S = - 2 q_\text{DM}$, we can write down the gauge-invariant combination $$\mathcal{L}_\text{mass} = -\frac{1}{2} y_\text{DM} \bar{\psi} (P_L S + P_R S^\ast) \psi \; .$$ Including the kinetic and potential terms for the Higgs singlet, the full dark Lagrangian therefore reads $$\begin{aligned} \mathcal{L}_\text{DM} = & \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi - \frac{1}{2} y_\text{DM} \bar{\psi} (P_L S + P_R S^\ast) \psi \,, \nonumber \\ \mathcal{L}_S = & \left[ (\partial^\mu + i \, g_S \, Z'^\mu) S \right]^\dagger \left[ (\partial_\mu + i \, g_S \, Z'_\mu) S \right] + \mu_s^2 \, S^\dagger S - \lambda_s \left(S^\dagger S \right)^2 \,.\end{aligned}$$ Once the Higgs singlet aquires a vacuum expectation value (vev), it will spontaneously break the $U(1)'$ symmetry, thus giving mass to the $Z'$ gauge boson and the DM particle. After symmetry breaking, we obtain the following Lagrangian (defining $S = 1/\sqrt{2} (s + w)$ and using $g_S \equiv g'q_S = -2g_\text{DM}^A$) $$\begin{aligned} \mathcal{L} = & \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{4} F'^{\mu\nu}F'_{\mu\nu} - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi - \frac{m_\text{DM}}{2} \bar{\psi} \psi - \frac{y_\text{DM}}{2\sqrt{2}} s \bar{\psi} \psi \nonumber \\ & + \frac{1}{2} m_{Z'}^2 \, Z'^\mu Z'_\mu + \frac{1}{2} \partial^\mu s \partial_\mu s + 2 (g_\text{DM}^A)^2 \, Z'^\mu Z'_\mu (s^2 + 2\,s\,w) + \frac{\mu_s^2}{2} (s+w)^2 - \frac{\lambda_s}{4} (s+w)^4 \; ,\end{aligned}$$ with $F'^{\mu\nu} = \partial^\mu Z'^\nu - \partial^\nu Z'^\mu$ and $$\label{eq:masses} m_\text{DM} = \frac{1}{\sqrt{2}} \, y_\text{DM} \, w\,,\quad m_{Z'} \approx 2 g_\text{DM}^A \, w \,.$$ If the SM Higgs is charged under the $U(1)'$ the $Z'$ mass will receive an additional contribution from the SM Higgs vev, see eq.  below. Electroweak precisison data requires that this contribution is small, and therefore we neglect this term in eq.  and for the rest of this subsection. Note that without loss of generality we can choose $w$ and $y_\text{DM}$ to be real (ensuring real masses) by absorbing complex phases in the field definitions for $S$ and $\psi$.[^6] As discussed above, the mass of the additional Higgs particle must satisfy $$m_s < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}}$$ in order for perturbative unitarity to be satisfied, which when substituting the masses of the $Z'$ and DM becomes $$m_s < \frac{4\sqrt{2} \pi w}{y_\text{DM}} \;.$$ Once we include such a new particle coupling to the $Z'$, however, there are additional scattering processes such as $s s \rightarrow s s$ that need to be taken into account when checking perturbative unitarity [@Basso:2011na]. Here we consider the scattering of the states $ss/\sqrt{2}$ and $Z'_L Z'_L/\sqrt{2}$. In the limit $\sqrt{s} \gg m_s \gg m_{Z'}$, the $J=0$ partial wave of the scattering matrix takes the form [@Lee:1977eg] $$\lim_{\sqrt{s} \rightarrow \infty} \mathcal{M}^0_{if} = - \frac{(g^A_\text{DM})^2 m_s^2}{8 \pi m_{Z'}^2} \begin{pmatrix} 3 & 1\\ 1 & 3 \end{pmatrix} \; .$$ Partial wave unitarity requires the real part of the largest eigenvalue, which corresponds to the eigenvector $(ss + Z'_L Z'_L)/2$, to be smaller than $1/2$. We hence obtain the inequality $$m_s \leq \frac{\sqrt{\pi} \, m_{Z'}}{g_\text{DM}^A} = \sqrt{4 \pi} w \; . \label{eq:perturb}$$ This inequality together with eq. (\[eq:DMmass\]) gives a stronger bound on the Higgs mass than the one obtained in eq. (\[eq:higgsmass\]). In other words, the bound in (\[eq:higgsmass\]) can never actually be saturated in this UV completion. We note that eqs.  and can be unified to $$\label{eq:bound_w} \sqrt{\pi} \, \frac{m_{Z'}}{g_\text{DM}^A} \ge \text{max}\left[ m_s , \sqrt{2} m_\text{DM}\right] \,.$$ Implications for the visible sector ----------------------------------- For the discussion above we only needed to consider the DM part of the Lagrangian. Let us now also look at the coupling to the SM, see e.g. [@Carena:2004xs]. The interactions between SM states and the new $Z'$ gauge boson can be written as $$\begin{aligned} \mathcal{L}'_\text{SM} = & \left[ (D^\mu H)^\dagger (-i \, g' \, q_H \, Z'_\mu \, H) + \text{h.c.} \right] + g'^2 \, q_H^2 \, Z'^\mu Z'_\mu \, H^\dagger H \nonumber \\ & - \sum_{f = q,\ell,\nu} g' \, Z'^\mu \, \left[ q_{f_L} \, \bar{f}_L \gamma_\mu f_L + q_{f_R} \, \bar{f}_R \gamma_\mu f_R \right] \; ,\end{aligned}$$ where $D^\mu$ denotes the SM covariant derivative. We can now immediately write down a list of relations between the different charges $q$ required by gauge invariance of the SM Yukawa terms:[^7] $$\begin{aligned} \label{eq:charges} q_H = q_{q_L} - q_{u_R} = q_{d_R} - q_{q_L} = q_{e_R} - q_{\ell_L} \; .\end{aligned}$$ After electroweak symmetry breaking, we obtain $$\begin{aligned} \mathcal{L}'_\text{SM} & = \frac{1}{2} \frac{e \, g' \, q_{H}}{ s_\mathrm{W} \, c_\mathrm{W}} (h+v)^2 \, Z^\mu Z'_\mu + \frac{1}{2} g'^2 \, q_{H}^2 \, (h+v)^2 \, Z'^\mu Z'_\mu \nonumber \\ & \quad - \sum_{f = q,l,\nu} \frac{1}{2} g' Z'^\mu \, \bar{f} \left[ (q_{f_R} + q_{f_L}) \gamma_\mu + (q_{f_R} - q_{f_L}) \gamma_\mu \gamma^5 \right] f \; . \label{eq:LprSM}\end{aligned}$$ Comparing the second line of eq.  with eq.  we can read off the vector and axial vector couplings of the fermions: $$\label{eq:g_f_VA} g_f^V = \frac{1}{2}g'(q_{f_R} + q_{f_L}) \,,\quad g_f^A = \frac{1}{2}g'(q_{f_R} - q_{f_L}) \,.$$ It is well known that a $U(1)'$ under which only SM fields are charged is in general anomalous, unless the SM fields have very specific charges (e.g. $U(1)_{B-L}$ is anomaly free). The relevant anomaly coefficients can e.g. be found in [@Carena:2004xs]. The presence of these anomalies implies that the theory has to include new fermions to cancel the anomalies. While these fermions can be vectorlike with respect to the SM, they will then need to be chiral with respect to the $U(1)'$. The mass of the additional fermions is therefore constrained by the breaking scale of the $U(1)'$. In particular, the bound from eq.  applies to these fermions as well and therefore they cannot be decoupled from the low-energy theory. It is however interesting to note that the anomaly involving two gluons and a $Z'$ is proportional to $$A_{ggZ'} = 3 \left( 2q_{q_L} - q_{u_R} - q_{d_R} \right) \; ,$$ which always vanishes if we restrict the charges based on gauge invariance of the Yukawa couplings (see eq. ). This implies that no new coloured states are needed to cancel the anomalies, greatly reducing the sensitivity of colliders to these new states.[^8] In any case, there are many different possibilities for cancelling the anomalies via new fermions. While the existence of additional fermions will lead to new signatures, a detailed investigation of these is beyond the scope of this work. If the SM Higgs is charged under $U(1)'$ ($q_H \neq 0$) the mass of the $Z'$ receives a contribution from both Higgses: $$\label{eq:mZpr} m_{Z'}^2 = (g'q_H v)^2 + 4(g_\text{DM}^A w)^2 \,,$$ and we obtain a mass mixing term of the form $\delta m^2 \, Z^\mu Z'_\mu$ with $$\begin{aligned} \delta m^2 = \frac{1}{2} \frac{e \, g' \, q_{H}}{ s_\mathrm{W} \, c_\mathrm{W}} v^2 \; ,\end{aligned}$$ where $s_\mathrm{W} \,(c_\mathrm{W})$ is the sine (cosine) of the Weinberg angle. As we are going to discuss below, electroweak precision data requires $|\delta m^2| \ll |m_Z-m_{Z'}|$ (see also App. \[app:Z\]). Using $m_Z = e v / (2 s_\mathrm{W} c_\mathrm{W})$, $g'q_s = -2g_\text{DM}^A$, and neglecting order one factors this requirement implies either $g'q_H \ll e$ or $q_s w \gg v$. In the parameter regions of interest it follows from those conditions that the first term in eq.  is small and hence the mass of the $Z'$ is dominated by the vev of the dark Higgs. Taking into account eqs.  and , the condition $|\delta m^2| \ll |m_Z - m_{Z'}|$ then implies either small axial couplings ($g_f^A \ll 1$) or $m_{Z'} \gg m_Z$. We are going to present more quantitative results in the next section and discuss a number of interesting experimental signatures resulting from the new interactions due to eq. . To conclude this section, it should be noted, that the Lagrangian introduced above is UV-complete (up to anomalies) and gauge invariant but does not correspond to the most general realization of this model. In particular the term $$\mathcal{L} \supset - \lambda_{hs} (S^*S)(H^\dagger H) \,,$$ which will lead to mixing between the SM Higgs $h$ and the dark Higgs $s$ can be expected to be present at tree level. Furthermore, the term $$\label{eq:kin-mix} \mathcal{L} \supset -\frac{1}{2} \sin \epsilon F'^{\mu\nu} B_{\mu\nu} \; ,$$ which generates kinetic mixing between the $Z'$ and the $Z$-boson, respects all symmetries of the Lagrangian. It can be argued that $\epsilon$ might vanish at high scales in certain UV-completions, but even in this case kinetic mixing is necessarily generated at the one-loop level and can have a substantial impact on EWPT. We will return to these issues and the resulting phenomenology of the model in sections \[sec:vector\] and \[sec:higgsmixing\]. For the moment, however, we are going to neglect these additional effects and focus on the impact of $\delta m \neq 0$, which necessarily leads to mass mixing between the neutral gauge bosons in the case of non-vanishing axial couplings. Non-zero axial couplings to SM fermions {#sec:axial} ======================================= Let us start with the case that axial couplings on the SM side are non-vanishing. An immediate consequence is that the SM Higgs is charged under the $U(1)'$, which follows from eqs.  and for $g_f^A \neq 0$. Note that these equations also imply that it is inconsistent to set the vectorial couplings for all quarks equal to zero. For example, if we impose that the vectorial couplings of up quarks vanish, i.e. $g_u^V = 0$, eq.  implies $q_{u_R} = - q_{q_L}$, which using eq.  leads to $g_d^V = 2 g' q_{q_L}$. In the following, whenever $g^A_f \neq 0$, we always fix $g^V_f = g^A_f$, which corresponds to setting $q_{q_L} = q_{\ell_L} = 0$. Furthermore, eq.  requires that $Z'$ couplings are flavour universal and leptons couple with the same strength to the $Z'$ as quarks. This conclusion could potentially be modified by considering an extended Higgs sector, e.g. a two-Higgs-doublet model. Here we focus on the simplest case where a single Higgs doublet generates all SM fermion masses. This implies that the leading search channel at the LHC will be dilepton resonances, which give severe constraints. In principle also electron-positron colliders can constrain this scenario efficiently. Limits on a $Z'$ lighter than 209 GeV derived from LEP data imply $g \lesssim 10^{-2}$ [@Agashe:2014kda] (see also [@Appelquist:2002mw; @LEP:2003aa]). We do not include LEP constraints here since other constraints will turn out to be at least equally strong. For general couplings, the partial decay width of the mediator into SM fermions is given by $$\Gamma(Z'\rightarrow f\bar{f}) = \frac{m_{Z'} N_c}{12\pi} \sqrt{1-\frac{4 m_f^2}{m_{Z'}^2}} \, \left[(g^{V}_{f})^2+(g^{A}_{f})^2 + \frac{m_f^2}{m_{Z'}^2}\left(2 (g^{V}_{f})^2 - 4 (g^{A}_{f})^2\right) \right] \; ,$$ where $N_c = 3 \;(1)$ for quarks (leptons). The decay width into DM pairs is $$\Gamma(Z'\rightarrow \psi\psi) = \frac{m_{Z'}}{24\pi} (g^{A}_\text{DM})^2 \left(1-\frac{4 m_\text{DM}^2}{m_{Z'}^2}\right)^{3/2} \; .$$ Consequently, for $m_\text{DM} \ll m_{Z'}$ and $g^A_{\ell} = g^A_{q} \ll g^A_\text{DM}$ the branching ratio into $\ell = e,\,\mu$ is given by $\text{BR}(R\rightarrow \ell \ell) \approx 8 (g^A_{\ell})^2 / (g^A_\text{DM})^2$. For $m_\text{DM} > m_{Z'} / 2$, on the other hand, the branching ratio is given by $\text{BR}(R\rightarrow \ell \ell) \approx 0.08\text{--}0.10$ depending on the ratio $m_{Z'} / m_t$. We implement the latest ATLAS dilepton search [@Aad:2014cka], complemented by a Tevatron dilepton search [@Jaffre:2009dg] for the low mass region, and show the resulting bounds in figure \[fig:axialaxial\]. One can see that the bounds strongly depend on the assumed branching ratio of the $Z'$. As a conservative limiting case we show $g_\text{DM}^A=1$ and $m_\text{DM}=100$ GeV, which leads to a rather large branching fraction into DM and hence suppressed bounds. The second benchmark, $m_\text{DM}=500$ GeV, allows for $Z'$ decays to DM only for rather heavy $Z'$s, leading to correspondingly more restrictive dilepton constraints. Overall the bounds turn out to be very stringent and the $Z'$ coupling to leptons and quarks needs to be significantly smaller than unity for $100 \; \text{GeV} \lesssim m_{Z'} \lesssim 4 \; \text{TeV}$, so that dijet constraints are basically irrelevant in this case given that $g_q=g_l$. ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), electroweak precision observables (blue, dotted) and DM overproduction (red, solid) in the $m_{Z'}-g_{q,l}^A$ parameter plane for two exemplary DM masses 100 GeV (left) and 500 GeV (right). In the shaded region to the left of the vertical grey line the $Z'$-mass violates the bound from perturbative unitarity from eq. .[]{data-label="fig:axialaxial"}](axialaxial_100_label.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), electroweak precision observables (blue, dotted) and DM overproduction (red, solid) in the $m_{Z'}-g_{q,l}^A$ parameter plane for two exemplary DM masses 100 GeV (left) and 500 GeV (right). In the shaded region to the left of the vertical grey line the $Z'$-mass violates the bound from perturbative unitarity from eq. .[]{data-label="fig:axialaxial"}](axialaxial_500.pdf "fig:"){height="0.3\textheight"} The fact that the SM Higgs is charged also implies potentially large corrections to electroweak precision observables. In particular we obtain the non-diagonal mass term $\delta m^2 \, Z^\mu Z'_\mu$ leading to mass mixing between the SM $Z$ and the new $Z'$. The diagonalisation required to obtain mass eigenstates is discussed in the appendix. In the absence of kinetic mixing between the $U(1)'$ and the SM $U(1)$ gauge bosons ($\epsilon = 0$), the resulting effects can be expressed in terms of the mixing parameter $\xi = \delta m^2 /(m_Z^2-m_{Z'}^2)$ (see eq.  in the appendix with $\epsilon=0$). In particular, we can calculate the constraints from electroweak precision measurements, which are encoded in the $S$ and $T$ parameters. To quadratic order in $\xi$ we find [@Frandsen:2011cg] $$\begin{aligned} \alpha S = & - 4 c_\mathrm{W}^2 s_\mathrm{W}^2 \xi^2 \; , \nonumber\\ \alpha T = & \xi^2\left(\frac{m_{Z'}^2}{m_{Z}^2}-2\right) \; , \label{eq:ST}\end{aligned}$$ where $\alpha=e^2/4\pi$. The resulting bounds are shown in figure \[fig:axialaxial\]. To infer our bounds we use the $90\%$ CL limit on the $S$ and $T$ parameters as given in [@Agashe:2014kda]. Note that the bound from electroweak precision data is completely independent of the $Z'$ couplings to the DM as well as the DM mass. Hence, the same bound would apply also in the case of Dirac DM with vector couplings to the $Z'$. Note however that since vectorial couplings to quarks are necessarily non-zero if $g_q^A \neq 0$ (see eqs. , ), there will be very stringent bounds from direct detection experiments on any model with $g^V_\text{DM} \neq 0$ due to unsuppressed spin-independent scattering. For Majorana DM, the vectorial coupling always vanishes and the constraints from direct detection are much weaker. ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm1_monojet.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm025_label_monojet.pdf "fig:"){height="0.3\textheight"} ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm01_monojet.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm0025_monojet.pdf "fig:"){height="0.3\textheight"} In figure \[fig:axialaxial2\] we show the constraints from electroweak precision data as well as LHC dilepton searches in the $m_\text{DM}-m_{Z'}$ plane for different values of the axial vector coupling to fermions. In the lower right corner of the plots (grey area) the perturbative unitarity condition from eq.  is violated. We also show the region excluded by direct detection searches (dark region in the lower left corners). For the axial-axial couplings DM-nucleus scattering proceeds through spin-dependent interactions, with a scattering cross section given by $$\sigma^\text{SD}_N = \frac{3 \, a_N^2 \, (g^A_\text{DM})^2 \, (g^A_q)^2}{\pi} \frac{\mu^2}{m_{Z'}^4} \; ,$$ where $\mu$ is the DM-nucleon reduced mass, $N = p,\,n$ and $a_p = -a_n = 1.18$ is the effective nucleon coupling [@Agashe:2014kda]. This is the dominant contribution in this case as the vector-axial coupling combination is even further suppressed. In the plots we show the bound on the spin-dependent scattering cross section that can be calculated from the published LUX results [@Akerib:2013tjd], following the method described in [@Feldstein:2014ufa]. We observe that in this case direct detection is never competitive with other constraints. The red solid curves in Figs. \[fig:axialaxial\] and \[fig:axialaxial2\] show the parameter values that lead to the correct relic abundance. In order to calculate the relic abundance we have implemented the model in micrOMEGAs\_v4 [@Belanger:2014vza], assuming that the mass of the Higgs singlet saturates the unitarity bound and setting the mixing with the SM Higgs to zero.[^9] In the regions shaded in red (to the right/above the solid curve) there is overproduction of DM. In this region additional annihilation channels are required to avoid overclosure of the Universe, since the interactions provided by the $Z'$ are insufficient to keep DM in thermal equilibrium long enough. Such additional interactions could be obtained for instance from the scalar mixing discussed in section \[sec:higgsmixing\]. Conversely, to the left/below the red solid curve the model does not provide all of the DM matter in the Universe, since the annihilation rate is too high. Let us briefly discuss the various features that can be observed in the relic abundance curve. First there is a significant decrease of the predicted abundance as the DM mass crosses the top-quark threshold, $m_\chi > m_t$, resulting from the fact that the $s$-wave contribution to the annihilation cross section is helicity suppressed and hence annihilation into top-quarks becomes the dominant annihilation channel as soon as it is kinematically allowed. The second feature occurs at $m_\chi \sim m_{Z'}$ and reflects the resonant enhancement of the annihilation process $\chi \chi \rightarrow q \bar{q}$ as the mediator can be produced on-shell. A third visible feature is a very narrow resonance at $2 \, m_\chi \sim m_s = \sqrt{\pi} \, m_{Z'} / g^A_\text{DM}$ due to a resonant enhancement of the process $\chi \chi \rightarrow s \rightarrow Z' Z$. The position and magnitude of this effect depends on the mass of the dark Higgs, which has been (arbitrarily) fixed to saturate the unitarity bound. However, even for this extreme choice, it turns out to give a non-negligible contribution to the relic abundance. For $m_\chi > m_{Z'}$ direct annihilation into two mediators becomes possible, leading to a significant decrease of the predicted relic abundance. Finally, the fact that the relic abundance curve in figure \[fig:axialaxial2\] touches the unitarity bound for high DM masses reflects the well-known unitarity bound on the mass of a thermally produced DM particle [@Griest:1989wd]. All in all we find the case with non-vanishing axial couplings on the SM side to be strongly constrained by dilepton searches as well as electroweak precision observables, implying that in a UV complete model this is where a signal should first be seen. For comparison, we show recent bounds from LUX as well as from the CMS monojet search [@Khachatryan:2014rra].[^10] We find that these searches, as well as searches for dijet resonances, are not competitive. Note that in figure \[fig:axialaxial2\] we assume $g_\text{DM}^A = 1$. We comment on smaller couplings on the DM side later in the context of figure \[fig:smallgA\]. Let us now look at the case where axial couplings to quarks are taken to be zero, which will turn out to be somewhat less constrained. Purely vectorial couplings to SM fermions {#sec:vector} ========================================= Let us now consider the case with purely vectorial couplings on the SM side, i.e. $g^A_q = g^A_\ell = g^A_\nu = 0$. In this case the SM Higgs does not carry a $U(1)'$ charge and therefore the charges of quarks and leptons are independent. In particular, it is conceivable that $g^V_q \gg g^V_\ell$, so that constraints from dilepton resonance searches can be evaded. Also there can in principle be a flavour dependence of the $Z'$ couplings to quarks. Nevertheless, to avoid large flavour-changing neutral currents, we will always assume the same coupling for all quark families in what follows [@Abdallah:2015ter]. Finally, in contrast to the case discussed above, tree-level $Z-Z'$ mass mixing is absent. It therefore seems plausible that the $Z'$ is the only state coupling to both the visible and the dark sector. Nevertheless, as mentioned above, potentially important effects in this scenario can be kinetic mixing of the $U(1)$ gauge bosons as well as effects induced by the dark Higgs, which we are going to discuss below. Let us just mention that all these effects will also be present in the scenario discussed in the previous section. They are, however, typically less important than the effects of tree-level $Z$-$Z'$ mixing. We first consider the effects of kinetic mixing between the $Z'$ and the SM hypercharge gauge boson $B$: $$\mathcal{L} \supset -\frac{1}{2} \sin \epsilon \, F'^{\mu\nu} B_{\mu\nu} \; ,$$ where $F'^{\mu\nu} = \partial^\mu Z'^\nu - \partial^\nu Z'^\mu$ and $B^{\mu\nu} = \partial^\mu B^\nu - \partial^\nu B^\mu$. A non-zero value of $\epsilon$ leads to mixing between the $Z'$ and the neutral gauge bosons of the SM (see App. \[app:Z\]). As in the case of mass mixing discussed above, there are strong constraints on kinetic mixing from searches for dilepton resonances and electroweak precision observables. ![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed), electroweak precision observables (blue, dotted) and relic DM overproduction (red, solid) in the $m_{Z'}$-$\epsilon$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right). In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:va"}](vectoraxial_100.pdf "fig:"){height="0.3\textheight"}![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed), electroweak precision observables (blue, dotted) and relic DM overproduction (red, solid) in the $m_{Z'}$-$\epsilon$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right). In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:va"}](vectoraxialmm.pdf "fig:"){height="0.3\textheight"} The dilepton couplings induced via the kinetic mixing parameter $\epsilon$ can be inferred from the mixing matrices and are given in the appendix, cf. eq. . The $S$ and $T$ parameters are given by $$\begin{aligned} \alpha S = & 4 c_\mathrm{W}^2 s_\mathrm{W} \xi (\epsilon - s_\mathrm{W} \xi) \; , \nonumber\\ \alpha T = & \xi^2\left(\frac{m_{Z'}^2}{m_{Z}^2}-2\right)+2 s_\mathrm{W} \xi \epsilon \; , \label{eq:ST2}\end{aligned}$$ where for $\delta m^2 = 0$ the mixing parameter $\xi$ is given by $\xi = m_{Z}^2 s_\mathrm{W} \epsilon / (m_{Z}^2 - m_{Z'}^2)$ at leading order. If $\epsilon$ is sizeable, i.e. if mixing is present at tree level, the resulting bounds can be quite strong. This expectation is confirmed in figure \[fig:va\]. Note that the relic density curves shown in figure \[fig:va\] are basically independent of $\epsilon$, because freeze-out is dominated by direct $Z'$ exchange for the adopted choice of couplings. While tree-level mixing is tightly constrained, it is reasonable to expect that $\epsilon$ vanishes at high scales, for example if both $U(1)$s originate from the same underlying non-Abelian gauge group, as in Grand Unified Theories. Since quarks carry charge under both $U(1)'$ and $U(1)_Y$, quark loops will still induce kinetic mixing at lower scales [@Holdom:1985ag], but the magnitude of $\epsilon$ can be much smaller than what we considered above. The precise magnitude of the kinetic mixing depends on the underlying theory, but if we assume that $\epsilon(\Lambda) = 0$ at some scale $\Lambda \gg 1\:\text{TeV}$, the kinetic mixing at a lower scale $\mu > m_t$ will be given by [@Carone:1995pu] $$\epsilon(\mu) = \frac{e \, g^V_q}{2 \pi^2 \, \cos \theta_\text{W}} \log \frac{\Lambda}{\mu} \simeq 0.02 \, g^V_q \, \log \frac{\Lambda}{\mu} \; .$$ We can use this equation (setting $\mu = m_{Z'}$) to translate the bounds from figure \[fig:va\] into constraints on $g^V_q$. The results of such an analysis are shown in figure \[fig:va\_loop\] assuming $\Lambda = 10$ TeV. As can be seen in figure \[fig:va\_loop\] (left), searches for dilepton resonances give again stringent constraints, implying $g_q^V < 0.1$ for $m_{Z'} = 200\:\text{GeV}$ and $g_q^V < 1$ for $m_{Z'} = 1\:\text{TeV}$. ![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed) and electroweak precision observables (blue, dotted) in the $m_{Z'}$-$g^V_q$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right), assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ so that kinetic mixing is only induced at the one-loop level. In the right panel we show also the region excluded by LHC monojet (orange, dashed) and dijet (violet, dot-dashed) searches due to tree-level $Z'$ exchange for the adopted coupling choice. In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound. []{data-label="fig:va_loop"}](vectoraxial_loop_100.pdf "fig:"){height="0.3\textheight"}![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed) and electroweak precision observables (blue, dotted) in the $m_{Z'}$-$g^V_q$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right), assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ so that kinetic mixing is only induced at the one-loop level. In the right panel we show also the region excluded by LHC monojet (orange, dashed) and dijet (violet, dot-dashed) searches due to tree-level $Z'$ exchange for the adopted coupling choice. In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound. []{data-label="fig:va_loop"}](vectoraxialmm_loop.pdf "fig:"){height="0.3\textheight"} In the right panel of figure \[fig:va\_loop\] we also show the constraints coming from LHC searches for monojets (i.e. jets in association with large amounts of missing transverse energy) and for dijet resonances, adopted from ref. [@Chala:2015ama].[^11] These limits are independent of the kinetic mixing $\epsilon$ since they originate from the tree-level $Z'$ exchange and probe larger values of $m_{Z'}$ and smaller values of $m_\text{DM}$. Nevertheless, dilepton resonance searches and EWPT give relevant constraints for small $m_{Z'}$ and large $m_\text{DM}$, which are difficult to probe with monojet and dijet searches. We conclude from figure \[fig:va\_loop\] that the combination of constraints due to loop-induced kinetic mixing and bounds from LHC DM searches leave only a small region in parameter space (a small strip close to the $Z'$ resonance), where DM overabundance is avoided. While this result depends somewhat on our choice $\Lambda = 10$ TeV and $\mu = m_{Z'}$, it is only logarithmically sensitive to these choices. It is worth emphasising that the unitarity constraints shown in figures \[fig:axialaxial2\]– \[fig:va\_loop\] depend sensitively on the choice of $g^A_\text{DM}$ (cf. figure \[fig:ubound\]). We therefore show in figure \[fig:smallgA\] how these constraints change if we take $g^A_\text{DM} = 0.1$ rather than $g^A_\text{DM} = 1$. In this case both $m_\text{DM}$ and $m_s$ can be much larger than $m_{Z'}$. At the same time, however, relic density constraints become significantly more severe, excluding almost the entire parameter space with $m_\text{DM} < m_{Z'}$ (apart from the resonance region $m_\text{DM} \sim m_{Z'}$). Even for $m_\text{DM} > m_{Z'}$ is it difficult to reproduce the observed relic abundance, because the annihilation channel $\chi \chi \rightarrow Z' Z'$ is significantly suppressed due to the smallness of $g^A_\text{DM}$. It only becomes relevant close to the unitarity bound, where also the process $\chi \chi \rightarrow Z Z'$ mediated by the dark Higgs gives a sizeable contribution. While in the case with axial couplings on the SM side the region compatible with thermal freeze-out becomes fully excluded by dilepton resonance searches, the case with vector couplings on the SM side is very difficult to probe at colliders and direct detection, leading to a small allowed parameter region close to the bound from perturbative unitarity. ![Constraints for small DM couplings ($g^A_\text{DM} = 0.1$). The left panel considers the case Axial+Vector(SM)–Axial(DM) and should be compared to figure \[fig:axialaxial2\]. The right panel considers the case Vector(SM)–Axial(DM) with loop-induced kinetic mixing, assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ (cf. figure \[fig:va\_loop\]).[]{data-label="fig:smallgA"}](aamm_gAx01_gAq01_label.pdf "fig:"){height="0.3\textheight"}![Constraints for small DM couplings ($g^A_\text{DM} = 0.1$). The left panel considers the case Axial+Vector(SM)–Axial(DM) and should be compared to figure \[fig:axialaxial2\]. The right panel considers the case Vector(SM)–Axial(DM) with loop-induced kinetic mixing, assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ (cf. figure \[fig:va\_loop\]).[]{data-label="fig:smallgA"}](vectoraxialmm_loop_gax01_gvq01.pdf "fig:"){height="0.3\textheight"} In addition to the effects of kinetic mixing, we have shown above that for $g^A_\text{DM} \neq 0$ the dark sector necessarily contains a new Higgs particle. The presence of this additional Higgs can change the phenomenology of the model in two important ways. First, loop-induced couplings of the dark Higgs to SM states may give an important contribution to direct detection signals. And second, there may be mixing between the SM Higgs and the dark Higgs, leading to pertinent modifications of the properties of the SM Higgs as well as opening another portal for DM-SM interactions. We will discuss loop-induced couplings in this section and then return to a detailed study of the Higgs potential in the next section. For $g^A_q = g^V_\text{DM} = 0$, scattering in direct detection experiments is momentum-suppressed in the non-relativistic limit and the corresponding event rates are very small. This conclusion may change if loop corrections induce unsuppressed scattering [@Haisch:2013uaa]. Indeed, at the one-loop level the dark Higgs can couple to quarks and can therefore mediate unsuppressed spin-independent interactions. The resulting interaction can be written as $\mathcal{L} \propto \sum_q \, m_q \, s \, \bar{q} q$. After integrating out heavy-quark loops as well as the dark Higgs this interaction leads to an effective coupling between DM and nucleons of the form $\mathcal{L} \propto f_N \, m_N \, m_\text{DM} \, \bar{N}N \, \bar{\psi}\psi$, where $m_N$ is the nucleon mass, $N = p, n$ and $f_N \approx 0.3$ is the effective nucleon coupling. In the non-relativistic limit, the diagram in the left of figure \[fig:SI\] induces the effective interaction $$\mathcal{L}_\text{eff} \supset \frac{(g^A_\text{DM})^2 \, (g^V_q)^2}{\pi^2} \frac{1}{m_s^2 \, m_{Z'}^2} \times m_\text{DM} \, f_N \, m_N \, \bar{N} N \, \bar{\psi} \psi \; .$$ The corresponding spin-independent scattering cross section is given by $$\sigma_N^\text{SI} = \frac{m_\text{DM}^2 \, f_N^2 \, m_N^2 \, \mu^2}{\pi} \frac{(g^A_\text{DM})^4 \, (g^V_q)^4}{\pi^4} \frac{1}{m_s^4 \, m_{Z'}^4} \; ,$$ where $\mu$ is the DM-nucleon reduced mass. For masses of order $300\:\text{GeV}$ and couplings of order unity this expression yields $\sigma_N^\text{SI} \sim 10^{-46} \:\text{cm}^2$, which is below the current bounds from LUX but well within the potential sensitivity of XENON1T. ![Contributions to spin-independent scattering.[]{data-label="fig:SI"}](diagrams.pdf){width="75.00000%"} We note that there are two additional diagrams (shown in the second and third panel of figure \[fig:SI\]) that also lead to unsuppressed spin-independent scattering of DM particles [@Haisch:2013uaa]. For $m_\text{DM} \gg m_N$, the resulting contribution is given by $$\mathcal{L}_\text{eff} \supset \frac{(g^A_\text{DM})^2 \, (g^V_q)^2}{4 \pi^2} \frac{m_\text{DM}^2 - m_{Z'}^2 + m_{Z'}^2 \log (m_{Z'}^2 / m_\text{DM}^2)}{m_{Z'}^2 (m_{Z'}^2 - m_\text{DM}^2)^2} \times m_\text{DM} \, f_N \, m_N \, \bar{N} N \, \bar{\psi} \psi \; .$$ If there is no large hierarchy between $m_\text{DM}$, $m_{Z'}$ and $m_s$, this contribution is of comparable magnitude to the one from dark Higgs exchange and interference effects can be important. Moreover, there may be a relevant contribution from loop-induced spin-dependent scattering. We leave a detailed study of these effects to future work. Mixing between the two Higgs bosons {#sec:higgsmixing} =================================== In addition to the loop-induced couplings of the dark Higgs to SM fermions discussed in the previous section, such couplings can also arise at tree-level from mixing. In fact, an important implication of the presence of a second Higgs field is that the two Higgs fields will in general mix, thus modifying the properties of the mostly SM-like Higgs. Furthermore, the mixing opens up the so-called Higgs portal between the DM and SM particles, leading to a much richer DM phenomenology than in the case of DM-SM interactions only via the vector mediator. The mixing between the scalars is due to an additional term in the scalar potential: $$V(S, H) \supset \lambda_{hs} (S^*S)(H^\dagger H) \,.$$ The coupling $\lambda_{hs}$ is a free parameter, independent of the vector mediator. For non-zero $\lambda_{hs}$, the scalar mass eigenstates $H_{1,2}$ are given by $$\begin{aligned} H_1 = s \sin \theta + h \cos \theta \nonumber \\ H_2 = s \cos \theta - h \sin \theta \end{aligned}$$ where, as shown in App. \[app:scalar\], $$\theta \approx - \frac{\lambda_{hs} \, v \, w}{m_s^2 - m_h^2} + \mathcal{O}(\lambda_{hs}^3) \; .$$ We emphasise that perturbative unitarity implies that $m_s$ cannot be arbitrarily large (for given $m_{Z'}$ and $g^A_\text{DM}$) and hence it is impossible to completely decouple the dark Higgs. The resulting Higgs mixing leads to three important consequences. First, the (mostly) dark Higgs obtains couplings to SM particles, enabling us to produce it at hadron colliders and to search for its decay products (or monojet signals). Second, the properties of the (mostly) SM-like Higgs, in particular its total production cross section and potentially also its branching ratios, are modified. And finally, both Higgs particles can mediate interactions between DM and nuclei, leading to potentially observable signals at direct detection experiments. Higgs portal DM has been extensively studied, see for instance [@Djouadi:2011aa; @Lebedev:2012zw; @LopezHonorez:2012kv; @Baek:2012uj; @Walker:2013hka; @Esch:2013rta; @Freitas:2015hsa] for an incomplete selection of references. A full analysis of Higgs mixing effects is beyond the scope of the present paper. Nevertheless, to illustrate the magnitude of potential effects, let us consider the induced coupling of the SM-like Higgs $H_1 \approx h$ to DM particles $$\mathcal{L} \supset - \frac{m_\text{DM} \, \sin \theta}{2 \, w} h \, \bar{\psi} \psi \simeq \frac{m_\text{DM} \, \lambda_{hs} \, v}{2 (m_s^2 - m_h^2)} h \, \bar{\psi} \psi \; .$$ For small $\lambda_{hs}$, the resulting direct detection cross section is given by [@Djouadi:2011aa] $$\sigma_N^\text{SI} \simeq \frac{\mu^2}{\pi \, m_h^4} \frac{f_N^2 \, m_N^2 \, m_\text{DM}^2\,\lambda_{hs}^2}{(m_s^2 - m_h^2)^2} \; ,$$ where we can neglect an additional contribution from the exchange of a dark Higgs provided $m_s^4 \gg m_h^4$. The parameter regions excluded by the LUX results [@Akerib:2013tjd] are shown in figure \[fig:higgsmixing\] (green regions). ![Constraints on $m_s$ and $\lambda_{hs}$ from bounds on the Higgs invisible branching ratio (blue, dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green, dashed). In the grey parameter region unitarity constraints are in conflict with the stability of the potential.[]{data-label="fig:higgsmixing"}](HiggsPlot1.pdf "fig:"){height="0.3\textheight"}![Constraints on $m_s$ and $\lambda_{hs}$ from bounds on the Higgs invisible branching ratio (blue, dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green, dashed). In the grey parameter region unitarity constraints are in conflict with the stability of the potential.[]{data-label="fig:higgsmixing"}](HiggsPlot2.pdf "fig:"){height="0.3\textheight"} We note that (in the linear approximation) the direct detection cross section is independent of $w$ and does therefore not depend on $m_{Z'}$ or $g^A_\text{DM}$. Nevertheless $w$ is not arbitrary, because unitarity gives a lower bound $\sqrt{4\pi} w > \text{max}\left[\sqrt{2} m_\text{DM}, m_s \right]$. At the same time, stability of the Higgs potential requires $4 \lambda_s \, \lambda_h > \lambda_{hs}$. These two inequalities can only be satisfied at the same time if $$m_\text{DM} < \frac{\sqrt{2 \pi} \, m_s \, m_h}{v} \; .$$ In figure \[fig:higgsmixing\], we show the parameter region where unitarity and stability are in conflict in grey. Finally, if the DM mass is sufficiently small, the SM-like Higgs can decay into pairs of DM particles, with a partial width given by [@Djouadi:2011aa] $$\Gamma^\text{inv} = \frac{1}{8 \pi} \frac{m_\text{DM}^2 \, \lambda_{hs}^2}{(m_s^2 - m_h^2)^2} v^2 m_h \left(1-\frac{4 \, m_\text{DM}^2}{m_h^2}\right)^{3/2} \; .$$ The invisible branching fraction is tightly constrained by LHC measurements: $\text{BR}(h\rightarrow \text{inv}) < 0.27$ [@Khachatryan:2014jba]. Furthermore, a combined fit from ATLAS and CMS yields $\mu = 1.09^{+0.11}_{-0.10}$ for the total Higgs signal strength [@ATLAS-CONF-2015-044], which can be used to deduce $\text{BR}(h\rightarrow \text{inv}) < 0.11$ at 95% CL. The resulting constraints, compared to the ones on $\sigma_N^\text{SI}$ from LUX, are shown in figure \[fig:higgsmixing\] (blue regions). The crucial observation is that the necessary presence of a dark Higgs will in general induce additional signatures and therefore lead to new ways to constrain models with a $Z'$ mediator using both direct detection experiments and Higgs measurements. However, since $\lambda_{hs}$ and $m_s$ are effectively free parameters, it is difficult to directly compare the constraints shown in figure \[fig:higgsmixing\] to the ones obtained from monojet and dijet searches at the LHC. Nevertheless, we can conservatively estimate the relevance of these effects by fixing the dark Higgs mass $m_s$ to the largest value consistent with perturbative unitarity. ![Constraints in the $m_{Z'}$-$m_\text{DM}$ plane for different values of $\lambda_{hs}$, taking the mass of the hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded by bounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict with bounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) region shows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). In the grey parameter region unitarity constraints are in conflict with the stability of the potential, the red region corresponds to DM overproduction. Note the change of scale in these figures.[]{data-label="fig:higgsmixing2"}](HiggsPlot_01_label.pdf "fig:"){height="0.3\textheight"}![Constraints in the $m_{Z'}$-$m_\text{DM}$ plane for different values of $\lambda_{hs}$, taking the mass of the hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded by bounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict with bounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) region shows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). In the grey parameter region unitarity constraints are in conflict with the stability of the potential, the red region corresponds to DM overproduction. Note the change of scale in these figures.[]{data-label="fig:higgsmixing2"}](HiggsPlot_02.pdf "fig:"){height="0.3\textheight"} The resulting constraints in the conventional $m_{Z'}$-$m_\text{DM}$ parameter plane with fixed couplings are shown in figure \[fig:higgsmixing2\]. For comparison we show the constraints from the CMS monojet search [@Khachatryan:2014rra] and a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). We find that the additional constraints due to Higgs mixing provide valuable complementary information in the parameter region with small $m_{Z'}$ and large $m_\text{DM}$, which is difficult to probe with monojet or dijet searches. Note that for $m_{Z'} > 2\:\text{TeV}$ (not shown in figure \[fig:higgsmixing2\]) there is still an allowed parameter region if either $m_\text{DM} \approx m_{Z'} / 2$ or $m_\text{DM} > m_{Z'}$ (cf. figure 6). Furthermore, it is worth emphasising that for smaller values of $m_s$ significantly stronger constraints are expected from Higgs mixing. Moreover, these constraints are independent of the SM couplings of the $Z'$ and will therefore become increasingly important in the case of small $g_q$. Discussion and Outlook {#sec:discussion} ====================== In this paper we have studied the so-called simplified model approach to DM used to parametrise the interactions of a DM particle with the SM via one or several new mediators. It should be clear that simplified models are considered merely as an effective description, used as a tool to combine different DM search strategies. Nevertheless, it is important that such models fulfil basic requirements, such as gauge invariance and that perturbative unitarity is guaranteed in the regions of the parameter space where the model is used to describe data. To ensure gauge invariance, one needs to impose certain relations between the different couplings, whereas it is necessary to introduce additional states in order to restore perturbative unitarity. We have illustrated these issues by considering a simplified model consisting of a fermionic DM particle and a vector mediator, which may for example be the $Z'$ gauge boson of a new $U(1)'$ gauge symmetry in the hidden sector. The phenomenology of this model depends decisively on whether the couplings of the mediator are purely vectorial or whether there are non-zero axial couplings (implying that left- and right-handed fields are charged differently under the new $U(1)'$). Since the coupling structure on the SM side may be different from the one of the DM side, there are four different cases of interest: purely vectorial couplings on both sides, non-zero axial couplings on either the SM side or the DM side, and non-zero axial couplings in both sectors. Our results can be summarized as follows: 1. \[it:VV\] [**Vector(SM)–Vector(DM):**]{} In this case no additional new physics is needed to guarantee perturbative unitarity and the mass of the $Z'$ can be generated via the Stueckelberg mechanism. This model is however highly constrained phenomenologically and a thermal DM is excluded for large parts of the parameter space due to strong limits on the spin-independent DM-nucleon scattering cross section. Generally, if at least one of the axial couplings is non-zero one needs new physics to unitarize the longitudinal component of the $Z'$. As a simple example we consider a SM-singlet Higgs breaking the dark $U(1)'$. Unitarity then requires the mass of the new Higgs to be comparable to the $Z'$ mass. Models with non-zero axial couplings are therefore expected to have a rich phenomenology with promising experimental signatures in DM direct detection experiments and invisible Higgs decays as well as additional DM annihilation channels. 2. \[it:AA\] [**Axial(SM)–Axial(DM):**]{} The crucial observation in this case is that gauge invariance of the SM Yukawa terms requires that the SM Higgs has to be charged under the $U(1)'$. This requirement has important implications for phenomenology: - Electroweak symmetry breaking leads to mass mixing between the $Z'$ and the SM $Z$-boson, which is strongly constrained by EWPT. - The axial couplings of SM fermions to the $Z'$ are necessarily flavour universal and equal for quarks and leptons. Hence, it is not possible to couple the DM particle to quarks without also inducing couplings of the $Z'$ to leptons. Since the LHC is very sensitive to dilepton resonances, the resulting bounds severely constrain the model (dominating over constraints from monojet and dijet searches). 3. \[it:AV\] [**Axial(SM)–Vector(DM):**]{} The constraints from EWPT and dilepton resonance searches are largely independent of the coupling between the $Z'$ and DM and therefore also apply in the case of purely vectorial couplings on the DM side. However, gauge invariance of the Yukawa couplings implies that it is impossible for the $Z'$ to have purely axial couplings to quarks. Consequently, as soon as there is a vectorial coupling on the DM side, one necessarily obtains a vector-vector component inducing unsuppressed spin-independent DM-nucleus scattering, which is strongly constrained by direct detection (see item \[it:VV\] above). 4. \[it:VA\] [**Vector(SM)–Axial(DM):**]{} In contrast to the couplings between the $Z'$ and quarks it is possible for the DM-$Z'$ coupling to be purely axial. Indeed, this situation arises naturally in the case that the DM particle is a Majorana fermion such that the vector current vanishes. If the couplings on the SM side are purely vectorial (i.e. left- and right-handed SM fields have the same charge), the SM Higgs is uncharged under the $U(1)'$ and consequently the constraints discussed in item \[it:AA\] do not apply. Furthermore, the tree-level direct detection cross section is velocity suppressed, leading to much weaker constraints on this particular scenario. Nevertheless, sizeable spin-independent DM-nucleus scattering can be induced at loop level. In addition, kinetic mixing between the $Z'$ and SM gauge bosons (at tree level or loop-induced) can be potentially important for EWPT and dilepton signatures. Assuming $\epsilon = 0$ at $\Lambda=10$ TeV, we find that bounds from searches for dilepton resonances due to loop-induced kinetic mixing can still be relevant and give constraints that are complementary to the ones obtained from monojet and dijet searches. All in all we find that imposing gauge invariance and conservation of perturbative unitarity has important implications for the phenomenology of DM interacting via a vector mediator and that relevant experimental signatures are not captured by considering only the interactions of the vector mediator with DM and quarks. This observation is relevant for the interpretation of various recent analyses of $Z'$-based simplified models, e.g. [@Lebedev:2014bba; @Hooper:2014fda; @Chala:2015ama; @Alves:2015dya; @Blennow:2015gta; @Heisig:2015ira]. Indeed, the $Z'$ model considered here is severely constrained by EWPT and dilepton resonance searches, either due to tree-level effects or loop-induced kinetic mixing. Moreover, the general expectation is that the mixing between the dark Higgs and the SM Higgs is sizeable and that as a result Higgs portal interactions are present in addition to the interactions mediated by the $Z'$. The weakest constraints are obtained in the case of purely vectorial couplings on the SM side and purely axial couplings on the DM side. Indeed, this is the only case where LHC monojet and dijet searches are potentially competitive with other kinds of constraints. In all cases that we have considered we find the hypothesis of thermal DM production to be under significant pressure. In large regions of the parameter space which is still allowed by experiments additional annihilation channels (beyond the $Z'$-mediated interactions) are necessary to avoid DM overabundance. A more systematic parameter scan of the model will be performed in a forthcoming publication [@upcoming]. Two final comments are in order. First, in this work we have not taken into account gauge anomalies. In general new fermions are needed to cancel the anomalous triangle diagrams, potentially leading to additional signatures and further constraints on the model. However, due to the constraints implied by gauge invariance of the SM Yukawa terms, the gluon-gluon-$Z'$ anomaly vanishes automatically, so that the new fermions need not be charged under colour, making them difficult to probe at the LHC. Second, the requirement of universality of all axial fermion charges (including leptons) follows from the gauge invariance of the SM Yukawa term. It relies on the fact that in the SM all fermion masses are generated by the same Higgs doublet. If the Higgs sector is more complicated, for example in a two-Higgs-doublet model, this condition is relaxed and it is possible to have different axial couplings to up- and down-type quarks or different axial couplings to quarks and to leptons. In any case, such extensions of the SM go significantly beyond the simplified model approach and would most likely have a number of implications for Higgs physics, EWPT and other searches for new physics. In conclusion we would like to emphasise that one of the most intriguing implications of our study is that a model with a vector mediator should generically also contain a scalar mediator, corresponding to the dark Higgs that generates the vector mass. In the limit that the mass of the vector mediator is much larger than the mass of the scalar, our $Z'$ model can also be used to study simplified models with a scalar mediator, where gauge invariance and perturbative unitarity can be similarly problematic. Indeed, our findings suggest that a strict distinction between simplified models with scalar and vector mediators is unnatural and many of the issues with these models may be best addressed in a more realistic set-up combining the two. Future direct detection experiments together with the upcoming runs of the LHC will be able to thoroughly explore the parameter space of such a realistic simplified model and test the hypothesis of DM as a thermal relic. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Sonia El Hedri, Juan Herrero-Garcia, Matthew McCullough and Oscar Stål for helpful discussions, and Michael Duerr and Jure Zupan for valuable comments on the manuscript. FK would like to thank the Oskar Klein Center and Stockholm University for hospitality. This work is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe as well as the ERC Starting Grant ‘NewAve’ (638528). Coupling structure from mixing {#ap:mixing} ============================== Gauge boson mixing {#app:Z} ------------------ In this appendix we discuss the mixing of a gauge boson $\hat{Z}'$ of a new $U(1)'$ gauge group with the SM $U(1)_Y$ gauge field $\hat B$ and the neutral component $\hat W^3$ of the $SU(2)_\mathrm{L}$ weak fields, where we use hats to denote the interaction eigenstates in the original basis. The mixing will then lead to the mass eigenstates $Z'$, $Z$ and $A$. Following the discussion in [@Frandsen:2011cg], we consider an effective Lagrangian including both kinetic mixing and mass mixing (see also [@Babu:1997st]) $$\begin{aligned} {\cal L} =& \; {\cal L}_{SM} -\frac{1}{4}\hat{X}^{\mu\nu}\hat{X}_{\mu\nu} + {\frac{1}{2}} m_{\hat Z'}^2 \hat{Z'}_\mu \hat{Z'}^\mu - {\frac{1}{2}} \sin \epsilon\, \hat{B}_{\mu\nu} \hat{X}^{\mu\nu} +\delta m^2 \hat{Z}_\mu \hat{Z'}^\mu \;, \label{eq:Lappendix}\end{aligned}$$ where $\hat{X}^{\mu \nu} \equiv \partial^\mu \hat{Z'}^\nu - \partial^\nu \hat{Z'}^\mu$. Furthermore, we have defined $\hat{Z}\equiv \hat{c}_\mathrm{W} \hat{W}^3- \hat{s}_\mathrm{W} \hat{B}$, where $\hat{s}_\mathrm{W} \, (\hat{c}_\mathrm{W})$ is the sine (cosine) of the Weinberg angle and $\hat g', \, \hat g$ are the corresponding gauge couplings. The field strengths are diagonalised and canonically normalised via the following two consecutive transformations [@Babu:1997st; @Chun:2010ve; @Frandsen:2011cg] $$\begin{aligned} \label{eq:Zpmixing} \left(\begin{array}{c} \hat B_\mu \\ \hat W_\mu^3 \\ \hat Z'_\mu \end{array}\right) & = \left(\begin{array}{ccc} 1 & 0 & -t_\epsilon \\ 0 & 1 & 0 \\ 0 & 0 & 1/c_\epsilon \end{array}\right) \left(\begin{array}{c} B_\mu \\ W_\mu^3 \\ Z'_\mu \end{array}\right) \ , \\ \left(\begin{array}{c} B_\mu \\ W_\mu^3 \\ Z'_\mu \end{array}\right) & = \left(\begin{array}{ccc} \hat c_\mathrm{W} & -\hat s_\mathrm{W} c_\xi & \hat s_\mathrm{W} s_\xi \\ \hat s_\mathrm{W} & \hat c_\mathrm{W} c_\xi & - \hat c_\mathrm{W} s_\xi \\ 0 & s_\xi & c_\xi \end{array} \right) \left(\begin{array}{c} A_\mu \\ Z_\mu \\ R_\mu \end{array}\right) \; ,\end{aligned}$$ where $$\begin{aligned} t_{2\xi}=\frac{-2c_\epsilon(\delta m^2+m_{\hat Z}^2 \hat s_\mathrm{W} s_\epsilon)} {m_{\hat Z'}^2-m_{\hat Z}^2 c_\epsilon^2 +m_{\hat Z}^2\hat s_\mathrm{W}^2 s_\epsilon^2 +2\,\delta m^2\,\hat s_\mathrm{W} s_\epsilon} \; . \label{eq:xi}\end{aligned}$$ For $\epsilon \ll 1$ and $\delta m^2 \ll m_{\hat Z}^2, m_{\hat Z'}^2$, this equation can be approximated by $$\label{eq:xi_def} \xi = \frac{\delta m^2 + m_{\hat Z}^2 \hat s_\mathrm{W} \epsilon}{m_{\hat Z}^2 - m_{\hat Z'}^2} \; .$$ The mass eigenvalues $m_Z$ and $m_{Z'}$ are given by $$\begin{aligned} m_Z^2 & = m_{\hat Z}^2 (1+{\hat s}_\mathrm{W} \, t_\xi \, t_\epsilon)+\frac{\delta m^2 \, t_\xi}{c_\epsilon} \nonumber \\ & \approx m_{\hat Z}^2 + (m_{\hat Z}^2 - m_{\hat Z'}^2) \xi^2 \; , \\ m_{Z'}^2 & = \frac{m_{\hat Z'}^2 + \delta m^2 ({\hat s}_\mathrm{W} \, s_\epsilon-c_\epsilon \, t_\xi)}{c_\epsilon^2 \, (1+{\hat s}_\mathrm{W} \, t_\xi \, t_\epsilon)} \nonumber \\ & \approx m_{\hat Z'}^2 + m_{\hat Z'}^2 \xi (\xi - {\hat s_\mathrm{W}} \epsilon) - m_{\hat Z}^2 (\xi - {\hat s_\mathrm{W}} \epsilon)^2\; .\end{aligned}$$ We define the ‘physical’ weak angle via $$\begin{aligned} s_\mathrm{W}^2 \, c_\mathrm{W}^2=\frac{\pi \, \alpha(m_{Z})}{\sqrt{2} \, G_\mathrm{F} \, m_{Z}^2} \; , \label{eq:swcw}\end{aligned}$$ where $\alpha = e^2 / (4\pi)$. Eq. (\[eq:swcw\]) also holds with the replacements $s_\mathrm{W}\to\hat s_\mathrm{W}$, $c_\mathrm{W}\to\hat c_\mathrm{W}$ and $m_{Z}\to m_{\hat Z}$, leading to the identity $s_\mathrm{W} \, c_\mathrm{W} \, m_{Z}=\hat s_\mathrm{W} \, \hat c_\mathrm{W} \, m_{\hat Z}$. This equation implies $$s_\mathrm{W}^2 = \hat s_\mathrm{W}^2 - \frac{\hat s_\mathrm{W}^2 \, \hat c_\mathrm{W}^2}{\hat c_\mathrm{W}^2 - \hat s_\mathrm{W}^2} \left(1 - \frac{m_{\hat Z'}^2}{m_{\hat Z}^2}\right) \xi^2 \; .$$ These equations allow us to fix $\hat s_\mathrm{W}$ and $m_{\hat Z}$ in such a way that we reproduce the experimentally well-measured quantities $s_\mathrm{W}$ and $m_{Z}$. The couplings of the $Z'$ to SM fermions induced via mixing can e.g. be found in [@Frandsen:2012rk]. Of particular interest to our current analysis are the couplings to leptons which are strongly constrained. In terms of the mixing parameters they can be written as $$\begin{aligned} g_{\ell}^\mathrm{V} &= \frac{1}{4}(3 {\hat g'} (\hat s_\mathrm{W} s_\xi-c_\xi t_\epsilon)- {\hat g}\hat c_\mathrm{W} s_\xi) \ , & g_{\ell}^\mathrm{A} &= -\frac{1}{4} ({\hat g'} (\hat s_\mathrm{W} s_\xi-c_\xi t_\epsilon) + {\hat g} \hat c_\mathrm{W} s_\xi) \; , \label{eq:dilepton}\end{aligned}$$ with $\hat{g}$ and $\hat{g}'$ the fundamental gauge couplings of $SU(2)_\text{L}$ and $U(1)_Y$. Scalar mixing {#app:scalar} ------------- Considering the SM Higgs $h$ plus the dark Higgs $s$, the most general scalar potential after electroweak and dark symmetry breaking can be written as $$V(s, h) = - \frac{\mu_s^2}{2} (s+w)^2 - \frac{\mu_h^2}{2} (h+v)^2 + \frac{\lambda_h}{4} (h+v)^4 + \frac{\lambda_s}{4} (s+w)^4 + \frac{\lambda_{hs}}{4} (h+v)^2(s+w)^2 \; .$$ For $\lambda_{hs} = 0$, we obtain the usual formulas $$\begin{aligned} v^2 & = \frac{\mu_h^2}{\lambda_h} \; , \quad m_h^2 = 2 \, \lambda_h \, v^2 \; ,\\ w^2 & = \frac{\mu_s^2}{\lambda_s} \; , \quad m_s^2 = 2 \, \lambda_s \, w^2 \; .\end{aligned}$$ In this case, there is no mixing between the two Higgs fields even at one-loop level. Nevertheless, there is no reason why $\lambda_{hs}$ should be negligible and therefore the two fields will in general mix. One then obtains for the minimum (assuming $4 \, \lambda_h \, \lambda_s > \lambda_{hs}^2$) $$\begin{aligned} v^2 & = 2 \frac{2 \, \lambda_s \, \mu_h^2 - \lambda_{hs} \, \mu_s^2}{4 \, \lambda_s \, \lambda_h - \lambda_{hs}^2} \; ,\\ w^2 & = 2 \frac{2 \, \lambda_h \, \mu_s^2 - \lambda_{hs} \, \mu_h^2}{4 \, \lambda_s \, \lambda_h - \lambda_{hs}^2} \; ,\;\end{aligned}$$ and for the mass squared eigenvalues $$m_{1,2}^2 = \lambda_h \, v^2 + \lambda_s w^2 \mp \sqrt{(\lambda_s w^2 - \lambda_h v^2)^2 + \lambda_{hs}^2 w^2 v^2} \; .$$ The corresponding mass eigenstates are $$\begin{aligned} H_1 = s \sin \theta + h \cos \theta \nonumber \\ H_2 = s \cos \theta - h \sin \theta \end{aligned}$$ with $$\tan 2\theta = \frac{\lambda_{hs} \, v \, w}{\lambda_h \, v^2 - \lambda_s \, w^2} \; .$$ For small $\lambda_{hs}$ we find $m_1^2 \approx 2 \, \lambda_h \, v^2 \equiv m_h^2$ and $m_2^2 \approx 2 \, \lambda_s \, w^2 \equiv m_s^2$. 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[^2]: It turns out that for certain processes the transversal part of the propagator leads to a logarithmic divergence for $m_{Z'}^2 \ll s$. This divergence is not related to the UV completeness of the theory, but signals breakdown of perturbativity in the IR, see also [@Hedri:2014mua]. By restricting to the longitudinal components of the $Z'$ [@Chanowitz:1978mv] we can avoid the occurence of those IR divergences. [^3]: Note that this process corresponds to an off-diagonal element of $\mathcal{M}_{if}$, with $i \neq f$, whereas the bounds from eq.  apply for diagonal elements. In order to apply the unitarity constraint we consider the $2\times 2$ submatrix of $\mathcal{M}_{if}$ spanned by the states $\psi \bar{\psi}$ and $Z'_L Z'_L$. For $s\to \infty$ only the off-diagonal element survives, and hence the eigenvalues of the matrix become equal to the off-diagonal element, and we can apply eq. . [^4]: Our result differs from the one in [@Shu:2007wg] by a factor $1/\sqrt{2}$. [^5]: As discussed below there can be anomalies which require additional fermions. [^6]: This will no longer be true if we allow for an explicit mass term for $\psi$. In this case the relative phase between $y_\text{DM}$ and the mass term is physical (see e.g. [@LopezHonorez:2012kv]). Here we do not allow for an explicit mass term and we assume that the vev of the singlet is the only source of $U(1)'$ symmetry breaking. [^7]: If right-handed neutrinos exist their charge $q_{\nu_R}$ would be constrained by $q_H = q_{\ell_L} - q_{\nu_R}$ to allow for a Yukawa term with the lepton doublet. In the following we assume that if right-handed neutrinos exist they are heavy enough to decouple from all relevant phenomenology. [^8]: This conclusion is in disagreement with the observations made in [@Hooper:2014fda]. [^9]: Note that since $\xi$ can be large in some regions of parameter space, it is not a good approximation to expand the annihilation cross section in $\xi$. We therefore use the exact expression for the mixing between the neutral gauge bosons in terms of $\epsilon$ and $\delta m^2$ as derived in the appendix. [^10]: To interpret the CMS results in the context of our model, we implement our model in `Feynrules v2` [@Alloul:2013bka] and simulate the monojet signal with `MadGraph v5` [@Alwall:2014hca] and `Pythia v6` [@Sjostrand:2006za]. Imposing a cut on the missing transverse energy of $\slashed{E}_T > 450\:\text{GeV}$, we exclude all parameter points that predict a contribution to the monojet cross section larger than $7.8\:\text{fb}$. We find good agreement between this procedure and an analogous implementation using `CalcHEP v3` [@Belyaev:2012qa] and `DELPHES v3` [@deFavereau:2013fsa]. [^11]: Note that as long as the mediator is produced on-shell, the production cross section is proportional to $(g^V_q)^2 + (g^A_q)^2$ and hence it is a good approximation to apply the bounds obtained for $g^V_q = 0$ and $g^A_q \neq 0$ also to the case $g^A_q = 0$ and $g^V_q \neq 0$. However, ref. [@Chala:2015ama] assumes a Dirac DM particle, while we focus on Majorana DM. As a result, the invisible branching fraction will be somewhat smaller in our case and bounds from dijet resonance searches will be strengthened. The dijet bounds we show are therefore conservative.

ArXiv

--- abstract: 'Collimated supersonic flows in laboratory experiments behave in a similar manner to astrophysical jets provided that radiation, viscosity, and thermal conductivity are unimportant in the laboratory jets, and that the experimental and astrophysical jets share similar dimensionless parameters such as the Mach number and the ratio of the density between the jet and the ambient medium. When these conditions apply, laboratory jets provide a means to study their astrophysical counterparts for a variety of initial conditions, arbitrary viewing angles, and different times, attributes especially helpful for interpreting astronomical images where the viewing angle and initial conditions are fixed and the time domain is limited. Experiments are also a powerful way to test numerical fluid codes in a parameter range where the codes must perform well. In this paper we combine images from a series of laboratory experiments of deflected supersonic jets with numerical simulations and new spectral observations of an astrophysical example, the young stellar jet HH 110. The experiments provide key insights into how deflected jets evolve in 3-D, particularly within working surfaces where multiple subsonic shells and filaments form, and along the interface where shocked jet material penetrates into and destroys the obstacle along its path. The experiments also underscore the importance of the viewing angle in determining what an observer will see. The simulations match the experiments so well that we can use the simulated velocity maps to compare the dynamics in the experiment with those implied by the astronomical spectra. The experiments support a model where the observed shock structures in HH 110 form as a result of a pulsed driving source rather than from weak shocks that may arise in the supersonic shear layer between the Mach disk and bow shock of the jet’s working surface.' author: - 'P. Hartigan , J. M. Foster B. H. Wilde R. F. Coker P. A. Rosen J. F. Hansen B. E. Blue R. J. R. Williams R. Carver A. Frank' title: 'Laboratory Experiments, Numerical Simulations, and Astronomical Observations of Deflected Supersonic Jets: Application to HH 110' --- Introduction ============ Collimated supersonic jets originate from a variety of astronomical sources, including active galactic nuclei [@agnref], several kinds of interacting binaries [@binaryjetref], young stars [@ysojetref], and even planetary nebulae [@pnjetref]. Most current jet research focuses on how accretion disks accelerate and collimate jets, or on understanding the dynamics of the jet as it generates shocks along its beam and in the surrounding medium. Both areas of research have broad implications for astrophysics. Models of accretion disks typically employ magnetized jets to remove the angular momentum of the accreting material. The distribution and transport of the angular momentum in an accretion disk affects its mass accretion rate, mixing, temperature profile, and density structure, and in the case of young stars, also helps to define the characteristics of the protoplanetary disk that remains after accretion ceases. At larger distances from the source, shock waves in jets clear material from the surrounding medium, provide insights into the nature of density and velocity perturbations in the flow, and enable dynamical studies of mixing, turbulence and shear. Jets from young stars are particularly good testbeds for investigating all aspects of the physics within collimated supersonic flows [see @ray07 for a review]. Shock velocities within stellar jets are low enough that the gas cools by radiating emission lines rather than by expanding. Relative fluxes of the emission lines determine the density, temperature, and ionization of the postshock gas, while the observed Doppler shifts and emission line profiles define the radial velocities and nonthermal motions within the jet [see @hartigan08 for a review]. Moreover, many stellar jets are located relatively close to the Earth, so that one can observe proper motions in the plane of the sky from observations separated by several years [@heathcote92]. Combining this information with radial velocity measurements gives the orientation of the flow to the line of sight. Using the Hubble Space Telescope, one can observe morphological changes of knots within jets, and follow how these changes evolve in real time [@hartigan01]. Results from these studies show that internal shock waves, driven by velocity variations in the flow, sweep material in jets into a series of dense knots. Typical internal shock velocities are $\sim$ 40 km$\,$s$^{-1}$, or $\sim$ 20% of the flow speed. In several cases it is easy to identify both the bow shock and the Mach disk from emission line images. Because stellar jets are mostly neutral, strong H$\alpha$ emission occurs at the shock front where neutral H is collisionally excited [@heathcote96]. Forbidden line radiation occurs in a spatially-extended cooling zone in the postshock material. Temperatures immediately behind the internal shocks can exceed $10^5$K, but the gas in the forbidden-line-emitting cooling zone is typically 8000 K. Young stars show a strong correlation between accretion and outflow, leading to the idea that accretion disks power the outflows [@heg95; @cabrit07]. Most current models use magnetic fields in the disk to launch a fraction of the accreting material from the disk into a collimated magnetized jet [@ferreira06]. Stellar jets often precess, and there is some evidence that they rotate [@ray07], although rotation signatures are difficult to measure because the rotational velocities are typically only a few percent of the flow speeds and precession can mimic rotational signatures [@cerquieriaref]. In all cases proper motion measurements show that jets move radially away from the source. Jets show no dynamical evidence for kink instabilities, and in fact while magnetic fields may dominate in the acceleration regions of jets, they appear to play a minor role in the dynamics at the distances where most jet knots are observed [@hartigan07]. In the cooling zones behind the shock waves the plasma $\beta$ can drop below unity, so that magnetic fields dominate thermal pressure in those areas. However, the magnetic pressure is small compared with the ram pressure of the jet. While more unusual than internal working surfaces produced by velocity perturbations, shocks also occur when jets collide with dense obstacles such as a molecular cloud. When the obstacle is smaller than the jet radius, it becomes entrained by the jet, and a reverse bow shock or ‘cloudlet shock’ forms around the obstacle [@schwartz78; @l1551]. Alternatively, when a large obstacle like a molecular cloud deflects the jet, a quasi-stationary deflection shock at the impact point forms, followed by a spray of shocked jet material downstream. The classic example of such a jet is HH 110 [@riera03; @lopez05]. Though the observations summarized above provide a great deal of information about stellar jets, several important questions remain unanswered. The basic mechanism by which disks load material onto field lines (assuming the MHD disk scenario is correct), and the overall geometry of this wind is unknown, and the roles of reconnection and ambipolar diffusion in heating the jet close to the star are unclear. At larger distances, the magnetic geometry and its importance in shaping the internal working surfaces is poorly-constrained, as are the time scales and spatial scales associated with mixing in supersonic shear layers and working surfaces. The inherently clumpy nature of jets also affects the flow dynamics and observed properties of jets in uncertain ways, and the degree to which fragmentation and turbulence influence the morphologies of jets is unknown. Developing laboratory analogs of stellar jets could help significantly in addressing the questions above. Observations of a specific astronomical jet are restricted to a small range of times and to a particular observing angle, while laboratory experiments have no such restrictions. In principle, one could explore a wide range of initial collimations, velocity and density structures within the jet, as well as densities, geometries, and magnetic field configurations with laboratory experiments. The experiments also provide a powerful and flexible way to test 3-D numerical fluid codes, and to investigate how real flows develop complex morphologies in 3-D. The challenge is to design an experiment that is relevant to the astrophysical case of interest. Laboratory experiments differ by 15 $-$ 20 orders of magnitude in size, density and timescale from stellar jets, but because the Euler equations that govern fluid dynamics involve only three variables, time, density, and velocity, the behavior of the fluid is determined primarily by dimensionless numbers such as the Mach number (supersonic or subsonic), Reynolds number (viscous or inertial), and Peclet number (importance of thermal conduction). If the experiments behave as a fluid and have similar dimensionless fluid numbers as those of stellar jets, then the experiments should scale well to the astrophysical case [@ryutov99]. Other parameters, such as magnetic fields and radiational cooling are more difficult to match, and it is impossible to study the non-LTE excitation physics in the lab because the critical density for collisional deexcitation is not scalable. In any case, the materials are markedly different between the experiments and stellar jets, so it is not possible to study emission line ratios in any meaningful way with current laboratory capabilities. Hence, at present the main utility of laboratory experiments of jets is to clarify how complex supersonic flows evolve with time. Laboratory work relevant to stellar jets is an emerging area of research, and several papers have appeared recently which address various aspects of supersonic flows in the appropriate regime [see @remington06 for a review]. @hansen07 observed how a strong planar shock wave disrupts a spherical obstacle and tested numerical models of the process, and @loupias generated a laboratory jet with a Mach number similar to that of a stellar jet. In a different approach, @lebedev04 and @ampleford07 used a conical array of wires at the Magpie facility to drive a magnetized jet, and explored the geometry of the deflection shock formed as a jet impacts a crosswind. Laboratory experiments have also recently studied the physics associated with instabilities along supernova blast waves [@snrref; @drake09], and the dynamics within supernovae explosions [@snrcoreref]. In this paper we present the results of a suite of experiments which deflect a supersonic jet from a spherical obstacle, where the dimensionless fluid parameters are similar to those present in stellar jets. In section 2, we describe our experimental design, consider how the experiments scale to astrophysical jets, demonstrate that the experiments are reproducible, and report how the observed flows change as one varies several parameters, including the distance between the axis of the jet and the obstacle, the time delay, the density probed with different backlighters, and the viewing angle. The numerical work is summarized in section 3. Detailed calculations with the 3-D RAGE code reproduce all of the major observed morphologies well. In section 4, we present new high-resolution optical spectra of the shocked wake of the HH 110 protostellar jet, and a new wide-field H$_2$ image of the region. These observations quantify how the internal dynamics of the gas behave as material flows away from the deflection shock and show how the jet entrains material from the molecular cloud core. Finally, in section 5 we consider how the experiments and the simulations from RAGE provide new insights into the internal dynamics of deflected supersonic jets, especially in the regions of the working surface and at the interface where jet material entrains and accelerates the obstacle. Laser Experiments of Deflected Supersonic Jets ============================================== Experimental Design ------------------- Figure 1 shows the experimental assembly we used to collide a supersonic jet into a spherical obstacle. The design consists of a 125 $\mu$m thickness titanium disk in direct contact with a 700 $\mu$m thickness titanium washer with a central, 300 $\mu$m diameter hole. The surface of the disk is heated by the thermal (soft x-ray) radiation from a hohlraum laser target, which itself is heated by 12 beams of the Omega laser at the University of Rochester [@soures96]. X-ray driven ablation of the surface of the disk creates a near-planar shock within the disk and washer assembly, and the subsequent breakout of this shock from the inner surface of the disk results in the directed outflow of a plug of dense, shock-heated titanium plasma through the cylindrical hole in the washer. This outflow is further collimated by the hole in the washer, and directed into an adjacent block of low-density (0.1 g$\,$cm$^{-3}$) polymer foam within which it propagates to a distance of $\sim$ 2 mm in $\sim$ 200 ns. After the primary jet forms, the ablation-driven shock continues to progress through the titanium target assembly and along the sides of the hole in the washer. As the hole collapses inward a secondary jet of material forms by a process analogous to that occurring in a shaped-charge explosive. The secondary jet forms after the primary jet, and propagates into the high-pressure cocoon of material already within the polymer foam. At later times, the shock propagates from the surrounding titanium washer into the foam, and causes the interface between the titanium washer and the hydrocarbon foam to move. A bow shock runs ahead of the jets into the foam. Both jets form as a result of hydrodynamic phenomena alone; there is no significant magnetic field and associated magneto-hydrodynamics, and the temperature of the jet is sufficiently low for thermal-conduction and radiative energy losses to be insignificant. Hence, the hydrodynamic phenomena which determine how the jet forms and evolves are scalable (see Section 2.2) to jets of very different dimensions that evolve over very different timescales. The experimental assembly described above is identical in many respects to that used in experiments we have reported previously [@foster05], but with the following two significant differences [see @rosen06; @coker07 for further details]: (1) in the present case we use indirect (hohlraum) drive, instead of the direct laser drive used in our earlier work; and (2) the foam medium through which the jet propagates contains an obstacle (a ball of CH). The hohlraum drive enables us to obtain greater spatial uniformity of the ablation of the titanium surface, and thereby generates a jet of improved cylindrical symmetry. The (optional) addition of an obstacle in the foam along the path of the jet medium makes it possible to set up, and in a controlled manner test, how the flow behaves when the two-dimensional cylindrical symmetry is broken. A pinhole-apertured, laser-produced-plasma, x-ray backlighting source projects an image of the jet onto radiographic film for later, detailed analysis. Details of the experimental setup are as follows. A 1600 $\mu$m diameter, 1200 $\mu$m length (internal dimensions) cylindrical gold hohlraum target with a single 1200 $\mu$m diameter laser-entry hole [see also @foster02] generates the radiation drive. The experimental package mounts over an 800 $\mu$m diameter hole in the end wall of the hohlraum, immediately opposite the laser entry hole. This axisymmetric configuration enables us to model the assembly at high resolution using two-dimensional radiation hydrocodes during the stages of formation and early-time evolution of the jet; the later stages of three-dimensional hydrodynamics are modeled by linking to a three-dimensional hydrocode, albeit at lower spatial resolution. The hohlraum is heated by 12 beams of the Omega laser with a total energy of 6 kJ in a 1 ns duration, constant power laser pulse of 0.35 $\mu$m wavelength. The resulting peak radiation temperature in the hohlraum measured with a filtered x-ray diode diagnostic (Dante) is 190 $-$ 200 eV [@foster02]. The titanium experimental assembly is made of an alloy with 90% titanium, 6% aluminium, and 4% vanadium. The diamond-polished surfaces of the hole and the planar surfaces of the components each have a 0.05 $-$ 0.3 $\mu$m peak-valley, and 0.01 $-$ 0.03 $\mu$m RMS, surface finish. We place a 100 $\mu$m thickness, 500 $\mu$m diameter gold ‘cookie-cutter’ disk between the hohlraum and the titanium disk to control the area of the titanium disk illuminated by x-rays, and to control the time it takes shocks to propagate into the titanium washer. By this means we are able to adjust, to some extent, the relative importance to the overall hydrodynamics of the primary and secondary jets, and the late-time motion of the titanium/foam interface that surround both jets. The medium through which the jets propagate is a 4 mm diameter, 6 mm length cylinder of resorcinol-formaldehyde (C$_{15}$H$_{12}$O$_4$; hereafter RF) foam, of 0.1 g$\,$cm$^{-3}$ density. We used RF foam because it has a very small ($<$ 1 $\mu$m) pore size. The obstacle in the foam is a 1 mm diameter, solid polystyrene (1.03 g$\,$cm$^{-3}$ density) sphere, supported by a small-diameter, silicon-carbide-coated tungsten stalk. The axial position and radial offset (impact parameter) of the sphere within the monomer material is set (approximately) before polymerisation of the foam, but is determined accurately after polymerisation by inspecting each experimental assembly radiographically. Typically, the axial position of the center of the ball, relative to the titanium-to-foam interface is 800 $-$ 1000 $\mu$m, and the impact parameter (perpendicular distance from the axis of the jet to the centre of the ball) is in the range 300 $-$ 500 $\mu$m. We could determine these quantities with an accuracy of approximately $\pm$ 10 $\mu$m, by analyzing pre-shot radiographs of the experimental assembly. The process of target fabrication is to some extent non-repeatable, and necessitates a separate hydrocode simulation of each experimental shot once the target dimensions are known. A sequence of experimental shots, diagnosed at different times, thus measures the hydrodynamic behavior of several very similar (but not strictly identical) target assemblies. A thin-foil, transition-metal, laser target illuminated with 2 $-$ 5 beams of the Omega laser creates the x-ray point backlighting source we use to diagnose the hydrodynamics. Each beam provides 400 J of energy in a 1 ns duration laser pulse, focused into a 600 $\mu$m diameter spot (spot size determined by use of a random-phase plate). X-ray emission from this laser-produced plasma passes through a laser-machined pinhole of typically 10 $-$ 20 $\mu$m diameter in 50 $\mu$m thickness tantalum foil, and generates a backlighting source of size comparable to the pinhole aperture. He-like resonance-line radiation of the backlighter target material dominates the spectrum of this x-ray backlighting source, and the choice of backlighter targets of typically titanium, iron or zinc results in He-like resonance line radiation of, respectively, 4.75, 6.7 and 9.0 keV. Radiation from the point x-ray backlighting source creates a point-projection (shadow) image of the experimental assembly on Kodak DEF x-ray film, with approximately 10-times magnification. Temporal resolution is determined by the duration of the x-ray backlighting source (very nearly equal to the laser pulse length). Motion blurring is insignificant for the 1 ns duration backlighting source. The time delay between the laser beams heating the hohlraum target, and the laser beams incident on the x-ray backlighting target is varied from shot to shot to build up a sequence of x-ray images of the jet hydrodynamics. Scaling the Experiments to Stellar Jets --------------------------------------- The size scales, times, densities and pressures within the Omega experiments differ markedly from those present in stellar jets. For the laboratory results to be meaningful the fluid dynamical variables must scale well, and the experiment should resemble the overall density and velocity structures present in astrophysical jets. We can connect the fluid dynamics of the astrophysical and experimental cases through the Euler equations for a polytropic gas [e.g. @ll87] $$\label{eq:mass} {{\partial \rho}\over{\partial t}} + \nabla \cdot \left(\rho\bf{v}\right) = 0$$ $$\label{eq:momentum} \rho\left({{\partial\bf{v}}\over{\partial t}} + \bf{v}\cdot\bf{\nabla v} \right) + \nabla {\rm P} = 0$$ $$\label{eq:energy} {{\partial {\rm P}}\over{\partial t}} + \gamma {\rm P} \nabla \cdot \bf{v} + \bf{v} \cdot \nabla {\rm P} = 0,$$ where $\rho$ is the density, $\bf{v}$ is the velocity, and P is the gas pressure. @ryutov99 showed that these equations are invariant to the rescaling $$r^\prime = ar ;\ \ \ \rho^\prime = b\rho ;\ \ \ P^\prime = cP$$ where a, b, and c are constants, provided one also rescales the time as $$t^\prime = a\sqrt{{b\over c}}t.$$ With this scaling, the velocity transforms as $$V^\prime = \sqrt{{c\over b}}V.$$ Solutions to the Euler equations will be identical in a dimensionless sense provided equation 6 holds. Hence, to verify that the experiment scales to the astrophysical case we need to estimate the pressures, temperatures, and velocities in both the experiment and in the HH 110 system. The parameters in the jet differ from those in the working surface where the jet deflects from the obstacle, and from those in the working surface where the bow shock impacts the ambient medium, so we must consider these regions separately. The density and temperature vary throughout stellar jets as material encounters weak shocks, heats, and cools, so we are interested in order of magnitude estimates of these quantites for the scaling estimates. As the discussion below will show, the two working surfaces scale well in our experiment but the jet scales less well. Consider the parameters in the jet first. Unlike most other jets from young stars, the HH 270 jet that collides with the molecular cloud to produce HH 110 is rather ill-defined, and consists of several faint wisps that resemble weak bow shocks [@choi06]. The electron density and ionization fraction in this part of the flow is poorly-constrained by observations, but an electron density of $10^3$ cm$^{-3}$ is typical for faint shocked structures of this kind. Taking a typical ionization fraction of 10% we obtain a total density of $10^4$ cm$^{-3}$, or about $2\times 10^{-20}$ g$\,$cm$^{-3}$. These values refer to the material in the radiating bow shocks; densities (and temperatures) between the bow shocks are likely to be lower. The temperature where \[S II\] radiates in the HH 270 bow shocks will be $\sim$ 7000 K. Material in the jet is mostly H, so using the ideal gas law we obtain 1$\times 10^{-8}$ dyne$\,$cm$^{-2}$ for the pressure. As shown in Table 1, a $\sim$ $5\times 10^{16}$, b $\sim$ $2\times 10^{-20}$, and c $\sim$ $3\times 10^{-19}$. With this scaling, 200 ns in the laboratory experiments corresponds to $\sim$ 80 years for an HH flow. Bow shocks in HH objects typically move their own diameter in $\sim$ 20 years, so the agreement with the observed timescales of HH objects and the experiment is ideal. However, the jet velocity transforms less well, with 10 km$\,$s$^{-1}$ in the experiment scaling to $\sim$ 40 km$\,$s$^{-1}$ in the stellar jet, where the actual velocity is a factor of four higher. This difference arises in part because the stellar jet cools radiatively, lowering the pressure and therefore the value of c. Another way to look at the velocity scaling is to consider the Euler number V/V$_E$, where V$_E$ = (P/$\rho$)$^{0.5}$ is the sound speed for $\gamma$=1. The Euler number in the HH 270 jet is about 20, and for other stellar jets may range up to 40. In the experiment this number is only $\sim$ 6. The main effect of the difference in Euler numbers is that the experimental jet has a wider opening angle than the stellar jet does. However, both numbers are significantly larger than unity, so both jets are highly supersonic. As we discuss in section 2.3, collapse and subsequent rebound of the washer along the axis of the flow determines to a large extent how the experimental jet is shaped, and has no obvious astrophysical analog. For this reason we will focus our analysis primarily on the leading bow shock and on entrainment of material in the obstacle rather than on the collimation of the jet. The working surfaces are generally modeled well by the experiment. In these regions the critical parameters are the timescales, which match very well, the Mach numbers in the shocks, and the density contrasts between the jet and the material ahead of the working surface. In stellar jets, the velocity of the bow shock into the preshock medium (equivalently, the velocity of the preshock medium into the bow shock) is similar to that of the jet, $\sim$ 200 km$\,$s$^{-1}$, because the jet is much denser than the ambient medium. The sound speed in the ambient medium can range from 1 $-$ 10 km$\,$s$^{-1}$, depending on how much ambient ultraviolet light heats the preshock gas. So the Mach number of the leading bow shock can range from about 20 $-$ 200. In our experiments, the sound speed ahead of the bow shock is low ($\sim$ 0.03 km$\,$s$^{-1}$), so the Mach number of the leading bow shock in the experiments is also very large, $\sim$ 200. When a stellar jet encounters a stationary obstacle like a molecular cloud, the preshock sound speed is that of the molecular cloud, $\lesssim$ 1 km$\,$s$^{-1}$, so the Mach number of material entering the shock into the molecular cloud is $\gtrsim$ 200. The Mach number of this shock in the experiments depends on the impact parameter, but is typically $\gtrsim$ 100. The ratio $\eta$ of the density in the jet to that in the ambient medium to a large extent determines the morphology of the working surface that accelerates the ambient medium. Overdense jets with $\eta$ $>$ 1 act like bullets, and have strong bow shocks and weak Mach disks, while the opposite case of $\eta$ $<$ 1 produces jets that ‘splatter’ from strong Mach disks (sometimes called hot spots) and create large backflowing cocoons [@krause]. Stellar jets are overdense, with $\eta$ $\sim$ 10 while extragalactic jets are underdense with $\eta$ $\lesssim$ $10^{-3}$ [@krause]. In our experiments, $\eta$ ranges from 8 $-$ 1 between 50 ns and 200 ns, respectively, in good agreement with the overdense stellar jet case. For the obstacle, the molecular cloud density increases from at least an order of magnitude less than that of the jet in the periphery of the cloud (essentially the ambient medium density), to $\gtrsim$ 2 orders of magnitude larger than the jet at the center of the core. So the equal densities of the jet and obstacle in the experiment cover a relevant astrophysical regime. The main difference between the experimental and astrophysical cases is the uniform density of the experimental obstacle as compared with the $\sim$ r$^{-2}$ density falloff in an isothermal molecular cloud core. Internal shock waves are by far the most common kind of shock in a stellar jet. Here, velocity perturbations of 30 $-$ 60 km$\,$s$^{-1}$ produce forward and reverse shock waves within a jet as it flows outward from the source at 150 $-$ 300 km$\,$s$^{-1}$ [e.g. @hartigan01]. Gas behind these shock waves cools rapidly to $\sim$ 4000 K by emitting permitted and forbidden line radiation [e.g. @hrh87], and then more slowly thereafter. Typically the temperature falls to $\sim$ 2000 K before the gas encounters another shock front, so the sound speed of the preshock gas is $\sim$ 5 km$\,$s$^{-1}$. Hence, the internal Mach number is $\sim$ 10 for these shock waves. Unlike the astrophysical case, our experiments do not produce multiple velocity pulses, and so are less relevant to studying the internal shocks within the beams of stellar jets. However, when jets are deflected obliquely from an obstacle the velocities in the deflected flow can remain supersonic, and may form weaker shocks in the complex working surface area that lies between the leading bow shock and the deflection shock (Mach disk analog) within the jet. Our experiments are very useful for studying the dynamics of this region, and for investigating how a jet penetrates and accelerates an obstacle along its path. For scaling to hold, several additional conditions must apply. First, dissipation mechanisms such as viscosity and heat conduction must be negligible. Hence, the Reynolds number $Re \sim VL/(\nu_{mat}+\nu_{rad})$ and Peclet number $Pe \sim VL/\chi$, need to be much larger than unity, where $\nu_{mat}$ and $\nu_{rad}$ are kinematic viscosities for matter and radiation, respectively, and $\chi$ is the thermal diffusivity. Using expressions for $\nu_{mat}, \nu_{rad}$, and $\chi$ from @ryutov99 and @drake06, we find thermal conductivity and viscosity are unimportant for both the experimental and astrophysical cases (Table 1). The Euler equations above implicitly assume that the gas behaves as a polytrope. In reality, stellar jets cool by radiating emission lines from shock-heated regions, and so are neither adiabatic ($\gamma$ = 5/3) nor isothermal ($\gamma$ = 1). Existing numerical simulations provide some insight into how jet morphologies change between the limiting cases of adiabatic and isothermal. In the adiabatic case, material heated by strong shocks cools by expanding, so working surfaces of jets tend to have rounder and more extended bow shocks, while working surfaces collapse into a dense plug for strongly-cooling jets [e.g. @blondin90]. The effect is less pronounced for weaker oblique shocks like those we are studying here in the wakes of the deflected flows. However, cooling should affect the morphologies of the stronger shocks as described above. In order to use the Euler equations, the laboratory jet must act as a fluid. Hence, the material mean free path ($\lambda_{mat} \sim v_{th}/\nu_c$ where $v_{th} \sim \sqrt(k_BT/m)$ is the particle thermal velocity and $\nu_c$ is the sum of ion and electron collisional frequencies) of the electrons and ions must be short compared with the size of the system ($\tau_{mat} \sim L/\lambda_{mat} \gg 1$). Table 1 shows that this condition is easily satisfied in both stellar jets and the laboratory experiments. The Euler equations do not include the radiative energy flux, and so we must verify that this number is small compared with the hydrodynamical energy flux. When, as in our experiments, the minimum radiation mean free path $\lambda_{rad}$ from Thomson scattering or thermal bremsstrahlung is short compared to $L$ (i.e., $\tau_{rad} \sim L/\lambda_{rad} \gg 1$), the appropriate dimensionless parameter for determining the importance of radiation is the Boltzmann number Bo\# $\sim \rho U^3 / f_{rad}$, where $U$ is a velocity scale and $f_{rad}$ is the radiative flux. In the Omega experiments, Bo\# $>>$ 1, implying that the radiative energy flux is unimportant in the flow dynamics. Radiative fluxes are also negligible in optically thin astrophysical shocks like those in stellar jets. Even in the brightest jets the observed radiative luminosity is $\lesssim$ $10^{-3}$ of the energy of the bulk flow [@bbm81], so the observations and experiments are consistent with regard to radiative energy flux. Finally, stellar jets have magnetic fields while our experiments do not. Field strengths within stellar jets are difficult to measure, but could dominate the flow dynamics close to the acceleration region [i.e., V$_{Alfven}$ $<$ V$_{flow}$ @hartigan07]. At the larger distances of interest to these experiments, fields are weaker and act mainly to reduce compression in the postshock regions of radiatively-cooling flows [@morse92]. In summary, the experiments are good analogs of jets from young stars. The Mach numbers of the jet relative to the ambient medium and to the obstacle are high in both cases, the density contrasts between the jet and the obstacle are similar, and the scaled times match almost exactly. Both systems behave like fluids, and viscosity, thermal conduction, and radiative fluxes are unimportant to the dynamics in both cases. The main differences are that stellar jets cool radiatively and the experimental jets do not, stellar jets may have magnetic fields that are not present in the experimental jets, and the density profile within the experimental jet is unlikely to have an astrophysical analog. Results ------- The complete suite of our experimental radiographs are available on the web (http:$//$ sparky.rice.edu$/\sim$hartigan$/$LLE\_shots.html). Because our experiment generates only a single image for each shot, if we wish to investigate how the flow changes with time or compare experiments that have different offsets of the ball from the axis of the jet (we refer to this distance as the ‘impact parameter’ in what follows), we must first quantify the degree to which target fabrication affects the radiographs of the flow. To this end, we obtained several groups of laser shots that had identical backlighters and delay times. One such set appears in Fig. 2, where the overall morphology of the flow and position of the bow shock is reproduced well between the shots, but irregularities appear in the bow shock shape that are specific to each target. In the case of the Ti backlighter image the bow shock resolves into a collection of nearly cospatial shells. The level of difference between the two images at the left of Fig. 2 indicates a typical variation caused by target fabrication and alignment. We varied the backlighter type, exploring V, Fe, Zn, and Ni. Each backlighter has a different opacity through the material, making it possible to probe the different depths within the structure of the flow at any given time. For example, the Ti backlighter image in Fig. 2 has significantly less opacity than that of the V images. We also tested four different types of foam, normal RF, large-pore RF, TPX (poly-4-methyl-1-pentene), and DVB (divinylbenzene), and found that the neither the foam pore size nor the foam type had any effect on the results. Fig. 3 summarizes the results from our experiments. The time sequence at the top shows how the jet evolves in a uniform medium without an obstacle. As described in section 2.1, the experiment first accelerates a plug of Ti into the foam, followed by a secondary jet of Ti that originates primarily from the collapse of the washer. At 50 ns, the radiograph shows a flat-topped profile that defines the shape of the plug, and as the plug proceeds into the foam the leading shock becomes bow-shaped. Irregularities in the shape of the bow shock are similar in size and shape to those caused by variations in the manufacture of the target (Fig. 2). By 100 ns the secondary jet has also formed, but it does not become well-collimated until about 200 ns. At 150 ns the end of the jet has a flute-shape, with a less-dense interior (see also Fig. 5, below). This shape is not a particularly good analog for a stellar jet, so we place less emphasis on this area in our analysis. The second row in Fig. 3 illustrates the principal shock structures that form when the jet encounters a sperical obstacle (labeled ‘Ball’ in the Figure) along its path. In general we expect two shocks to form when a continuous supersonic jet impacts ambient material $-$ a forward bow shock that accelerates the ambient medium, and a reverse shock, sometimes called a Mach disk, that decelerates the jet. The area between the forward and reverse shocks is known as the working surface, and within the working surface there is a boundary known as the contact discontinuity which separates shocked jet material from shocked ambient (or shocked ball) material. The two forward bow shocks, one into the ball and the other from the deflected jet into the foam, are clearly visible in the radiographs. However, the Mach disks for these bow shocks are more difficult to see in Fig. 3. At 150 ns the Ti plug is the primary driver of the deflected bow shock, and the RAGE simulations discussed below show that the shocked plug is located near the head of the bow shock at this time (Fig. 4). Fig. 5 shows that a disk-shaped area of high temperature material exists in the shocked plug and at the end of the flute, and it is tempting to associate these areas of hot gas with the Mach disk. The situation is complicated by the fact that the plug jet is impulsive, so one would expect the Mach disk to disappear once material in the plug has passed through it, though the temperature will remain elevated in this area. The last two rows of images in Fig. 3 depict two sequences of radiographs taken at common times after the deposition of the laser pulse, but with impact parameters that increase from left to right. As expected, the jet burrows into the ball more when the impact parameter is smaller, and is deflected more when the impact parameter is larger. In all cases the forward bow shock into the ball is quite smooth, with no evidence for any fluid dynamical instabilities. In contrast, the contact discontinuity between the shocked jet and shocked ball material is highly structured. As we discuss further in section 5, the irregularity of the contact discontinuity plays a major role in breaking up the ball. Images obtained with the Zn backlighter are best for revealing the morphology of the secondary jet, which at late times appears to fragment into a complex filamentary structure in the region of the flute. The ability to observe a flow from an arbitrary viewing angle is an attractive feature of the laboratory experiments that is not available to astrophysicists who study stellar jets. Fig. 6 shows two pairs of identical laser shots, one with a 150 ns time delay the other with a 200 ns delay, where the viewing angle changed by 90 degrees within each pair. The outline of the ball is clearly visible through the deflected jet in the symmetrical view ($\theta$=0) at 150 ns. Discerning the true nature of the deflected flow is much more difficult in the symmetrical view, where the radiograph resembles a single, less-collimated bow shock. Numerical Simulations of the Laboratory Experiments =================================================== The design of these experiments and their post-shot analysis was done with the RAGE (Radiation Adaptive Grid Eulerian) simulation code. We have also adapted the astrophysical MHD code AstroBEAR to model laser experiments by including a laser drive and real equation of state for different materials, but we summarize the results of this work elsewhere [@carver09]. RAGE Simulations ---------------- RAGE is a multi-dimensional, multi-material Eulerian radiation-hydrodynamics code developed by Los Alamos National Laboratory and Science Applications International (SAIC) [@gittings08]. RAGE uses a continuous (in space and time) adaptive-mesh-refinement (CAMR) algorithm to follow interfaces and shocks, and gradients of physical quantities such as material densities and temperatures. At each cycle, the code automatically determines whether to subdivide or recombine Eulerian cells. The user also has the option to de-zone (that is, reduce the resolution of the mesh) as a function of time, space, and material. Adjacent square cells may differ by only one level of resolution, that is, by a factor of 2 in cell size. The code has several interface-steepening options and easily follows contact discontinuities with fine zoning at the material interfaces. This CAMR method speeds calculations by as much as two orders of magnitude over straight Eulerian methods. RAGE uses a second-order-accurate Godunov hydrodynamics scheme similar to the Eulerian scheme of @colella85. Mixed cells are assumed to be in pressure and temperature equilibrium, with separate material and radiation temperatures. The radiation-transfer equation is solved in the grey, flux-limited-diffusion approximation. Given the placement of the ball with respect to the symmetry axis of the undeflected jet, these experiments are inherently three-dimensional. However, before the jet impacts the ball, the hohlraum and the jet are two-dimensional. This allows us to perform highly resolved, two-dimensional simulations in cylindrically-symmetric geometry to best capture the ablation of the titanium, the acceleration of the titanium plug through the vacuum free-run region in the washer, the collapse of the titanium hole onto the symmetry axis, and the subsequent jet formation. The two-dimensional simulations are initialized by imposing the measured radiation-drive temperature in a region that is inside the hohlraum. To save computer time, we have determined that it is sufficient to eliminate the hohlraum and just use the measured temperature profile as the source of energy that creates the ablation pressure to drive the jet. Neglecting asymmetries that arise from initial perturbations and unintentional misalignment of the hohlraum with the vacuum free-run region (the hole in the washer), the early-time experimental data show that the jet remains cylindrically symmetric until at least 50 ns. At approximately this time we link the two-dimensional, cylindrically-symmetric simulation into three dimensions and add the 1 mm diameter ball at the different impact parameters. While the two-dimensional simulations are run with a resolution of 1.5 $\mu$m to capture the radiation ablation of the titanium correctly, the three-dimensional simulations are typically run at lower resolution, especially during the design of the experiments. Fig. 4 shows the model density, composition, and expected radiograph for a deflected jet at 200 ns. Material labeled as blue (Ti plug) comes from the Ti disk, while orange (Ti sleeve) originates in the Ti washer and constitutes most of the secondary jet. The flute-shape of the secondary jet is clear in this figure. The areas of greatest interest are the region where the jet is creating a cavity in the ball, because this shows how jets destroy obstacles and entrain material, and the region of filaments in the working surface of the leading bow shock, because these filaments could in principle propagate downstream as weak shocks like those seen in HH 110. The processes of jet formation and propagation are illustrated in Fig. 5, which show ‘snapshots’ of the temperature and density distribution within the jet taken from a RAGE hydrocode simulation (single choice of axial position and impact parameter for the polystyrene sphere) at different times. The primary (plug) and secondary (‘shaped-charge’ hole collapse) jets are identified in Fig. 5, as well as the late-time motion at the titanium-to-foam interface that results in the formation of an opaque pedestal-like feature at the base of the jet in the synthetic radiographs. All these features are clearly discernible in the experimental data. Fig. 7 shows post-processed, zinc-backlit (9 keV x-ray backlighter energy) radiographic images from two perpendicular views that represent the time evolution of the jet, and how it subsequently deflects from the ball, in the RAGE simulation. Data Analysis and Comparisons with RAGE Simulations --------------------------------------------------- The experimental data recording the jet hydrodynamics are in the form of radiographic images recorded on x-ray-sensitive film. Fig. 3 shows a composite of experimental radiographs from several laser shots, to illustrate the formation and deflection of the jets, in a series of shots where the position and impact parameter of the polystyrene sphere inevitably vary somewhat from shot to shot. We digitized the film data with a Perkin-Elmer PDS scanning microdensitometer and converted the film density to exposure using calibration data for the Kodak DEF film that recorded the images [@henke86]. Ideally, the pinhole-apertured x-ray backlighting source would provide spatially uniform illumination of the experimental assembly, although in practice this is not the case because of laser intensity variations at the backlighting target and vignetting resulting from the specific size and shape of the pinhole aperture. Regions of the image resulting from x-ray transmission through the undisturbed foam (that is, outside the jet-driven bow shock) provide a means to determine the uniformity of backlighting intensity, after we allow for the known x-ray attenuation resulting from the undisturbed foam. Starting from this measured intensity distribution we use a polynomial fitting procedure to infer the unattenuated backlighter intensity that underlies the image of the jet and bow shock. We divide the backlit image data by the inferred (unattenuated) backlighter intensity to obtain a map of x-ray transmission through the experiment. We compare this x-ray transmission data with post-processed hydrocode calculations that simulate the experimental radiographs. Because the absolute variation of backlighter intensity across the image is small (typically, 10 $-$ 20% across the entire image), the polynomial fitting procedure enables the absolute x-ray transmission to be inferred with second-order accuracy. We use three principal metrics for comparing the experimental radiographs with synthetic x-ray images obtained by post-processing of our hydrocode simulations. These are: (1) the large-scale hydrodynamic motion (determined by comparison of the positions of the bow shock in experiment and simulation); (2) the spatial distribution of mass of the titanium jet material (obtained from the spatial integral of optical depth throughout the image, or from sub-regions of the image); and (3) the small-scale structure in the deflected jet (quantified by the two-dimensional discrete Fourier transform of sub-regions of the image, and its corresponding power spectral density function of spatial frequency). In making comparison of the experimental with hydrocode simulation, two specific points of detail require attention: (i) each laser-target assembly has its own specific location of the polystyrene sphere within the polymer foam cylinder (because of target-to-target variations arising in fabrication), and (ii) the angular orientation of the polymer-foam cylinder attached to the laser-heated hohlraum target determines the x-ray backlighting line of sight, relative to the plane in which the polystyrene sphere is displaced from the axis of the jet. Ideally all laser targets would be identical, and all backlit images would be recorded orthogonal to the axis of the target, and either in the plane of radial displacement of the polystyrene sphere or orthogonal to that plane. The geometry of the target chamber of the Omega laser facility determines the possibilities for backlighting orthogonal to the axis of the experimental assembly, and is well-characterized. However, to model a specific experiment fully, we must run three-dimensional hydrocode simulations specific to that experiment which include target-to-target differences of the fabrication assembly, followed by post-processing specific to the backlighter line of sight used in each experiment (which may differ by up to 10 degrees from the two preferred directions dictated by the symmetry of the experiment). The three-dimensional hydrocode simulations are expensive in their use of computing time and resources, and we therefore make detailed comparison of the experimental data with simulation for only a small number of representative cases. ### Large-Scale Hydrodynamics and Bow Shock Position Figs. 8 and 9 compare images of the deflected jet at 200 ns after the onset of radiation drive to the experimental assembly, and for backlighting lines of sight orthogonal to the plane of jet deflection (Fig. 8) and within the plane of jet deflection (Fig. 9). In each case, the data are compared with post-processed simulations from the RAGE hydrocode. The x-ray backlighting source was the 9.0 keV resonance line of He-like zinc (Fig. 8) or the 6.7 keV line radiation of He-like iron (Fig. 9). In each case, we corrected for spatial variations of incident backlighter intensity (as described above) and the images are therefore maps of absolute backlighter transmission through the experiment. The spatial resolution of both experiments was 15 $\mu$m. The bow shock in the hydrocarbon foam ahead of the jet is clearly visible, as is the late-time hydrodynamic behavior of the primary (outflow) jet (the mushroom-like feature lying off axis, resulting from deflection of the jet) and the secondary (hole-collapse) jet (the dense, near-axis stem apparently penetrating the initial position of the polystyrene sphere). Also evident in the radiographs is the mound-shaped pedestal that arises from motion of the titanium-to-foam interface following shock transit across this interface at late time. In the case of each experiment (the two images were obtained from different experimental shots) the radial offset of the center of the sphere from the axis (impact parameter) was close to 350 $\mu$m , and the axial position of the center of the sphere was close to 920 $\mu$m . A single RAGE simulation is shown for purposes of comparison, in which the impact parameter was 350 $\mu$m, and the axial position for the sphere 915 $\mu$m. The spatial resolution of this simulation was 3.1 $\mu$m. We make quantitative comparisons of experiment and simulation by comparison of lineouts of x-ray transmission in Fig. 10 and Fig. 11. The simulation reproduces the large-scale hydrodynamics of the experiment (bow-shock position, formation of the deflected jet, motion of the pedestal and the creation of a Mach-stem-like feature where it meets the bow shock). However, the simulation does less well with other features of the hydrodynamics, including the “clumpiness” of the deflected jet material and its proximity to the bow shock running ahead of the deflected jet, and the small-scale structure at the interface of the titanium washer and foam (apparently at the surface of the pedestal feature). The finely-resolved simulations capture more fine-scale structure $-$ the jet does indeed break up into structure similar to that seen in the experiment with simulations at 1 $\mu$m spatial resolution, although at a somewhat later time than observed experimentally. These small-scale structures tend to have larger local Mach numbers and are therefore closer to the bow shock. The filigree structure at the pedestal probably arises from Richtmyer-Meshkov growth of small-scale machining or polishing marks on the surface of the titanium washer, following shock transit across this interface. There are also multiple shock interactions that form Mach stems which are not captured by these simulations owing to the reduced computational resolution in the pedestal area. ### Spatial Distribution of Mass To proceed further with quantitative comparison with simulation, we consider the spatial distribution of mass of the materials present in the experiment. At each point in the experimental and synthetic images, the optical depth at the photon energy of the backlighter radiation is given by $$\tau = \kappa_1\sigma_1 + \kappa_2\sigma_2 + \kappa_3\sigma_3$$ where $\tau$ and $\sigma$ are the opacity and areal density (integral of density along the line of sight) and subscripts distinguish the titanium jet (1), the hydrocarbon foam (2) and the polystyrene sphere (3) materials, respectively. The opacity of the titanium jet is significantly greater than that of either the RF foam or the polystyrene sphere, and their temperatures are sufficiently low for the opacity of these materials to be essentially constant throughout the volume of the experiment. For example, at 9 keV (the photon energy of the zinc backlighting source) the opacities of titanium, RF foam and polystyrene are, respectively, 150, 4.12 and 2.83 cm$^2$g$^{-1}$. We divide the experimentally measured and simulated images, arbitrarily, into 500 $\mu$m square regions, and for each region we calculate the mean optical depth $\bar\tau$ using $\tau$ = $-$ln(I/I$_\circ$), and $$\bar\tau = {{\int\tau dA}\over{\int dA}}$$ where I/I$_\circ$ is the measured (or simulated) x-ray transmission, dA is the pixel area, and the integration extends over the area of each specific region of the image. A comparison of mean optical depth defined in this way, for both experiment and simulation, appears in Fig. 12. The grouping together of adjacent, square regions of the image enables the mean to be calculated for rather larger areas encompassing all of, or the majority of, the mass of the deflected jet. In particular, we consider two larger areas of the images shown in Fig. 12: the rectangular region composed of six adjacent 500 $\mu$m squares and labeled A, and the L-shaped region composed of three adjacent 500 $\mu$m squares and labeled B. These encompass essentially all of the mass of the jet that has interacted with the polystyrene sphere (region A), and all except the mass of the primary jet deflected by the sphere (in the case of the L-shaped region B). In each case, the simulation reproduces the experimentally measured optical depth to $\sim$ 10%: for the rectangular region A, mean optical depths are 0.94 for the experiment and 0.86 for the simulation; for the L-shaped region B, mean optical depths are 1.15 for the experiment and 1.10 for the simulation. The greatest difference between experiment and the simulation arises in the magnitude and distribution of mass of the primary (outflow) jet deflected by the obstacle. Material from this primary jet resides mainly in a further L-shaped region, labeled C in Fig. 12. Although the mean optical depths for region C differ by only 15% (0.72 in the case of experiment, 0.62 in the case of simulation), the distribution of mass shows significant variation (mean optical depths of 1.14 and 0.65 in the case of the 500 $\mu$m square region where the difference is most evident). Fig. 12 shows that the RF foam makes only a relatively small contribution to the measured optical depth, but we may assess the magnitude of this contribution simply by setting the opacity of the other components of the experiment (titanium jet and polystyrene sphere) to zero in the post-processing of the hydrocode simulation. We conclude that for the various 500 $\mu$m square regions of the images shown in Fig. 12, the optical depth of the foam lies in the range 0.10 $-$ 0.17 (this variation arises because of chord-length effects, and because density variations of the foam at the bow shock and within the cocoon). Hence, RF foam contributes a negligible optical depth in these experiments. ### Discrete Fourier Transform and Power Spectral Density To compare the experimental data with the small-scale structure of the jet in the simulations, we use the two-dimensional discrete Fourier transform (DFT) of optical depth. Starting from the experimental (or simulated) images of the experiment, we obtain maps of optical depth ($\tau$ = -ln(I/I$_\circ$)) and use the fast Fourier transform algorithm to obtain the DFT of selected regions of the experimental (or simulated) data and then proceed to calculate the power spectral density (PSD). Fig. 13 shows an image of the deflected jet within which we have identified two separate regions (green and dark red boxes), as well as an area of undisturbed hydrocarbon foam (bright red box), and part of the fiducial grid attached to the surface of the foam (blue box). For each region, we show the PSD of optical depth. We define the PSD as the sum of amplitude-squared of all Fourier components whose spatial frequency lies in the range k $-$ k + dk, where k = (k$_x^2$ + k$_y^2$)$^{0.5}$ and k$_x$ and k$_y$ are orthogonal spatial frequency components of the two-dimensional DFT. Fig. 13 clearly shows the fundamental spatial frequency of the grid and its harmonics, the flat spectrum of white noise of the backlighter transmission through the undisturbed foam, and a spectrum arising from the clumpy, perhaps near turbulent, structure arising within the deflected titanium jet. Fig. 14 shows an analogous PSD analysis, for three different RAGE simulations of the experiment. Our approach is the same for the simulation as it is for the experimental data: we obtain a map of optical depth from the simulated radiograph and then the PSD of regions whose size and position is identical to those chosen in analysis of the experimental data. In the case of Fig. 14, we show the result of RAGE simulations for three different levels of AMR calculational resolution: 12.5, 6.25, and 3.125 $\mu$m. Although the calculational resolution (dimension of the smallest Eulerian cell) differs in these three cases, we first obtain (by interpolating the post-processed data) a simulated radiograph with the same spatial resolution as the experimental data before proceeding to calculate the PSD. This procedure enables us to avoid any potential uncertainty of the scaling of PSD amplitude and frequency spacing when comparing with the experimental data. Astronomical Observations and Numerical Models of the Internal Dynamics within the Deflected Jet HH 110 ======================================================================================================= Spectral Maps of HH 110 ----------------------- We observed HH 110 on 9 Jan 2008 UT with the echelle spectrograph on the 4-m Mayall telescope at Kitt-Peak National Observatory in order to quantify the dynamics present in a shocked, deflected astrophysical jet. The 79-63 grating and 226-1 cross disperser combined with the T2KB CCD and 1.5 arcsecond slit gave a spectral resolution of 3.0 pixels, or 11.1 km$\,$s$^{-1}$ as measured from the FWHM of night sky emission lines. The CCD, binned by two along the spatial direction, produced a plate scale of 0.52 arcseconds per pixel. The position angle of the slit was 13 degrees, and the length of the slit was limited by vignetting to be about 140 arcseconds. Seeing was 1.3 arcseconds, and skies were stable with light cirrus. A plot of the slit position superposed on an archival HST image of HH 110 appears in Fig. 15. We employed an unusual mode of observation, long slit but with a wide order separating filter (GG 435). This setup causes orders to overlap at different spatial positions along the slit, but in the case of HH 110 there is no ambiguity because no continuum sources are present. The advantage of this setup is that one can obtain position-velocity diagrams simultaneously for all the bright emission lines, including \[S II\] 6716, \[S II\] 6731, \[N II\] 6583, \[N II\] 6548, and H$\alpha$. The \[O I\] 6300 and 6363 lines were also present, but the signal-to-noise of these faint lines was too low to warrant any profile analysis. Distortion in the spectrograph optics causes each emission line to be imaged in a curved arc whose shape varies between orders. Fortunately, there is strong background line emission in each of the emission lines from the HH 110 molecular cloud, and we used this emission to correct for distortion and to define zero radial velocity for the object. A third degree polynomial matched the distortion of the night sky emission lines within an rms of about 0.2 pixels (0.74 km$\,$s$^{-1}$). It is important to remove night sky emission lines as well as the background emission lines from the molecular cloud, as these lines contaminate multiple orders in our instrumental setup. To this end, we imaged blank sky near HH 110 frequently and subtracted this component from the object spectra after aligning the night sky emission between the object and sky frames to account for flexure in the spectrograph. Cosmic rays and hot pixels were removed using a routine described by @hartigan04, and flatfielding, bias correction, and trimming were accomplished using standard IRAF tasks[^1]. Alignment of individual exposures was accomplished in two ways. First, we compared the position of a star relative to the slit that was captured from the guide camera, whose plate scale of 0.16 arcseconds per pixel we determined by measuring the length of the slit image when a decker of known size truncated the slit. More precise positioning in the direction along the slit is possible by extracting the spatial H$\alpha$ trace and performing a spatial cross correlation between frames. The uncertainty in the positional measurements from the guide camera image inferred from the secondary corrections required by the cross-correlations is about 1.0 arcseconds. To align and compare different emission lines we must also register the position-velocity diagrams to account for the spatial positions of the orders and for the tilt of the spectrum across the CCD within each order, but this is easy to do with a continuum lamp exposure through a small decker to define the spectral trace. The guide camera images indicate, and the spectra confirm, that the last three object exposures drifted by 1.5 arcseconds relative to the position shown in Fig. 15, so these were not used in the final analysis. In all, the total exposure time for the spectra is 140 minutes. There were no differences between the \[S II\] 6716 and \[S II\] 6731 position-velocity diagrams, so we combined these to produce a single \[S II\] spectrum. The \[N II\] 6548 line is fainter by a factor of three than the companion line \[N II\] 6583, and the fainter line is also contaminated by faint residuals from a bright night sky emission line from an adjacent order, so we simply use the 6583 line for the \[N II\] line profiles. Internal Dynamics of HH 110 --------------------------- The image in Fig. 15 divides the position-velocity diagram into ten distinct emitting regions along the slit. Spectra for each of these positions appear in Fig. 17 for H$\alpha$ and for the sum of \[N II\] + \[S II\]. All regions have well-resolved emission line profiles in H$\alpha$, \[N II\], and in \[S II\]. Only object 9 showed any difference between the \[S II\] and \[N II\] profiles, with \[N II\] blueshifted by 12 km$\,$s$^{-1}$ relative to \[S II\]. In low-excitation shocks like HH 110, the forbidden line emission peaks when the gas temperature is $\sim$ 8000 K [@hartigan95], which corresponds to a spread in radial velocity owing to thermal motions of HWHM = ((2ln2)kT/m)$^{0.5}$, or 2.6 km$\,$s$^{-1}$ for N, and 1.7 km$\,$s$^{-1}$ for S. These thermal line widths will be unresolved with the Kitt-Peak observations, which have a spectral resolution of 11 km$\,$s$^{-1}$. Therefore, the observed widths of the forbidden lines measure nonthermal line broadening in the jet. This line broadening most likely arises from nonplanar shock geometry or clumpy morphologies on small scales, as the HST images show structure down to at least a tenth of an arcsecond. However, other forms of nonthermal line broadening, such as magnetic waves or turbulence, may also contribute to the line widths. Figs. 16 and 17 show that the H$\alpha$ emission line widths are larger than those of the forbidden lines. This behavior is expected because a component of H$\alpha$ occurs from collisional excitation immediately behind the shock front where the temperature is highest. The thermal FWHM of H$\alpha$ is given by $$V_{th} = \left(V_{OBS}^2 - V_{forb}^2\right)^{0.5}$$ where V$_{forb}$ = (V$_{NT}^2$ + V$_{INST}^2$)$^{0.5}$ is the observed FWHM of the forbidden lines, V$_{NT}$ the nonthermal FWHM and V$_{INST}$ the instrumental FWHM. We can use the thermal line width observation to measure the shock velocity in the gas. The temperature immediately behind a strong shock is given by $$T = {3\over 16}{\mu m_H V_S^2\over k}$$ where $\mu$ is the mean molecular weight of the gas and V$_S$ is the shock velocity. The mean molecular weight depends on the preshock ionization fraction of the gas, and the postshock gas will have different ion and electron temperatures immediately behind the shock until the two fluids equilibrate [@mckee74], but the equilibration distance should be unresolved for HH 110. For simplicity we take the preshock gas to be mostly neutral, so $\mu$ $\sim$ 1. For thermal motion along the line of sight, the FWHM of a hydrogen emission line profile is then $$V_{th} = 2.354\left({k T\over {m_H}}\right)^{0.5}$$ Combining these two equations we obtain $$V_S = 0.98 V_{th}$$ so the shock velocity is closely approximated by the observed thermal FWHM. Fig. 18 summarizes the kinematics and dynamics within HH 110. The radial velocity gradually becomes more negative at distances greater than about $2\times 10^{17}$ cm. The radial velocities in the different emission lines track one another well. However, the same is not true for the line widths: both Fig. 17 and Fig. 18 show clearly that H$\alpha$ is broader than the forbidden line profiles. This behavior is expected because when the preshock gas is neutral, much of the H$\alpha$ comes from collisional excitation immediately behind the shock front where the temperature is high. The low atomic mass of H also increases its line width relative to those of N and S. The graph shows that the nonthermal component of the line profiles stays approximately constant at $\sim$ 40 km$\,$s$^{-1}$, and the thermal component of H$\alpha$ and the shock velocity are $\sim$ 50 km$\,$s$^{-1}$. There are two epochs of HST images available (Reipurth PI), separated by about 22 months, and these data show intriguing and complex proper motions. Regions 1 through 4 have multiple shock fronts some of which move in the direction of the jet while others move along the deflected flow. Fig. 18 shows that this impact zone has higher nonthermal line widths than present in the flow downstream, consistent with the HST data. Throughout this rather broad region, denoted as such in Fig. 15, jet material impacts the molecular cloud. Hints of this behavior are evident in the ground based proper motion data of @lopez05, which show proper motion vectors directed midway between the direction of the jet and that of the deflected flow. Downstream from region 4, the material all moves in the direction of the deflected flow and gradually expands in size. The electron density of the shocked gas declines as the flow expands [@ro91]. The slow increase in the blueshifted radial velocity could arise if the observer sees a concave cavity that gradually redirects the deflected flow towards our line of sight. One gets the impression from the HST images of a series of weak bubbles that emerges from the impact zone. Published proper motion measurements [@lopez05] suggest tangential velocities of $\sim$ 150 km$\,$s$^{-1}$ along the deflected flow. Hence, the nonthermal broadening is $\sim$ 25% of the bulk flow speed in the deflected jet, while the internal shock speeds are typically 30% of the flow speed. The temperature immediately behind a 50 km$\,$s$^{-1}$ shock is $\sim$ $5.7\times 10^4$ K, and will drop to $\lesssim$ 5000 K in areas where forbidden lines have cooled. The corresponding sound speeds are $\sim$ 20 km$\,$s$^{-1}$ and 8 km$\,$s$^{-1}$, respectively, so the bulk flow speed is $\sim$ Mach 10 in the deflected flow, while the internal shocks there are $\sim$ Mach 3 $-$ 6. The magnetosonic Mach numbers will be lower, depending on the field strength. Wide-Field H$_2$ Images of the HH 110 Region -------------------------------------------- In order to better define how the HH 110 jet entrains material from the molecular cloud, and to verify that the deflected jet model is appropriate for this object, we obtained new wide-field near-infrared images of the region. In Figs. 19 and 20 we present a portion of an H$_2$ image taken 21 Sept., 2008 with the NEWFIRM infrared camera attached to the 4-m telescope at Kitt Peak National Observatory. The image was constructed from 20 individual dithers of 2 minutes apiece for a total exposure time of 40 minutes. The wide-field image in Fig. 19 shows VLA 1, the driving source of the HH 110 flow [@choi06]. The jet from this source, known as HH 270, does not radiate in H$_2$ until it strikes the molecular cloud, although the jet is visible in deep \[S II\] images [@choi06]. Our H$_2$ image clearly shows two other jets that emanate from sources embedded within the molecular cloud core. IRS 1 (aka IRAS 05487+0255) is a bright near-IR source, while IRS 2 appears to be obscured by a flared disk seen nearly edge-on (as in HH 30, @burrows96). These sources drive a molecular outflow along the direction of the jets we see in the H$_2$ image [@ro91]. Archival Spitzer images of the region at mid-infrared (3.6 $\mu$m $-$ 8 $\mu$m), and far-infrared (24 $\mu$m, 70 $\mu$m and 160 $\mu$m) wavelengths reveal three very bright sources that persist in all bands in the region, VLA 1, IRS 1, and IRS 2. The spectral energy distribution of IRS 1 is still rising at 100 $\mu$m, indicative of a heavily embedded source. The morphology of the HH 30 clone IRS 2 splits into two pieces at shorter wavelengths in the Spitzer images, consistent with an obscuring disk seen edge-on. Epoch 2000 coordinates for the midpoint of the HH 30-like disk in IRS 2 are 5:51:22.70 +2:56:05, and for IRS 1 are 5:51:22.60 +2:55:43. The ubiquity of jets in this region is a common occurrence, as most star-forming regions have multiple sources that drive jets. However, it does potentially bring into question whether or not what we and others [e.g. @rrh96] interpret as a deflected jet for HH 110 may simply be a distinct flow generated by some other embedded source to the northeast of the emission. The counter to this argument is that Fig. 19 does not show a source near the apex of HH 110, and the Spitzer images of the region also show nothing there at mid-infrared wavelengths. If the hot H$_2$ is dragged out from the molecular cloud core by the impact of the jet, there should be a spatial offset between the H$_2$ from the core and the H$\alpha$ in the jet, with the H$_2$ located on the side of the spray closest to the core. The color composite in Fig. 20 demonstrates this effect very well, as noted previously for a small portion of the jet [@nc96]. Connecting the Laboratory Experiments With Astrophysical Jets ============================================================= Our laboratory results highlight two aspects of the fluid dynamics that are particularly useful for interpreting astronomical images and spectra $-$ entrainment of ambient material and the dynamics within contact discontinuities. The experiments also serve as a reminder that viewing angle affects how a bow shock appears in an image. We discuss each of these ideas below. Interpretation of Astronomical Images ------------------------------------- Keeping in mind that magnetic fields may play some role in the dynamics, we can look to the experiments as a guide to how material from an obstacle like a molecular cloud core becomes entrained by a jet in a glancing collision. As noted above, in the astronomical images the H$_2$ emission in Fig. 20 must arise from the cloud core to be consistent with the observed spatial offset of the H$\alpha$ and H$_2$, and the lack of H$_2$ in the jet before it strikes the cloud. In the experiments, the jet entrains material in the ball in part because the flute penetrates into the ball and ‘scoops up’ whatever material falls within the flute (Fig. 4). This type of entrainment is probably of little interest astrophysically because its origin is unique to the relatively hollow density structure within the experimental jet. However, material from the ball is also lifted into the flow because the jet penetrates into the ball along the contact discontinuity. Both the radiographs and the RAGE simulations show this region to be highly structured (Figs. 3-5). The experiments indicate that once a part of the jet becomes deflected into the ball along the contact discontinuity it creates a small cavity where the jet material lifts small fragments of the ball into the flow. We see the same morphology in the astrophysical case. This type of entrainment is one that occurs as a natural outgrowth of the complex 3-D structure along a contact discontinuity. As Fig. 6 shows, the deflected bow shock appears much wider when the bow shock has a significant component along the line of sight. While this result is rather elementary, Fig. 6 provides a graphic example that is important to keep in mind when interpreting astronomical images of bow shocks when the shocked gas has a large redshift or blueshift. In such cases a flow may appear to be much less collimated than if one were to observe the bow shock perpendicular to its direction of motion. An example of a wide bow shock oriented at a large angle from the plane of the sky is HH 32A, which has been thoroughly studied at optical wavelengths [@beck04]. Filamentary Structures in the Experiment and the Astronomical Observations -------------------------------------------------------------------------- The working surface of the deflected bow shock in the experiment exhibits an intriguing filamentary structure in the observed radiographs and in the RAGE simulations (Figs. 3 and 21) that resembles the filamentary shock waves seen in HH 110 (Fig. 15). If differential motions between adjacent filaments in the working surface are supersonic, then weak shocks like those seen in HH 110 could form as the filaments interact at later times. To test this idea, we generated synthetic images of the Mach number and the velocity along a plane that contains the center of the ball in Fig. 21. Together, these two images indicate whether shock waves are likely to form within the working surface at later times. Neither image alone provides this information: a constant velocity flow with two adjecent regions of different temperature will show markedly different Mach numbers in close proximity but will not create a shock; similarly, a shock will only form between adjacent fluid elements with differing velocities and the same temperature if the difference between the Mach numbers exceeds unity. Velocity differences between the filaments in the working surface in Fig. 21 are typically 1 $-$ 2 km/s, or only about 10% of the initial jet velocity. In contrast, the shock waves in HH 110 have higher velocities, about 30% of the jet speed (section 4.2; Fig. 18). Moreover, Mach numbers in the filaments range from $\sim$ 1 $-$ 2, so differences between the filaments are $\lesssim$ 1, indicating that the relative motions between the filaments are subsonic. We conclude that the velocity differences between the filaments in the working surface are too low to form shock waves unless the filaments cool. Even if that were to occur, the shock velocities will be about a factor of three smaller relative to the jet speed than those seen in HH 110. Hence, the best explanation for the shocks observed in HH 110 is that the source is impulsive, where each pulse impacts the molecular cloud in a similar manner to our experimental setup. It is instructive to consider what one might observe in a position-velocity diagram in the experiment, were it possible to align a slit down the axis of the deflected flow and obtain a velocity-resolved spectrum, as is possible with astronomical observations. We show this exercise in Fig. 22, where we have simply assumed the emissivity of the gas to be proportional to its density. For a real emission line the emissivity depends on the temperature, density, and ionization state of the gas in a complex manner determined by the atomic physics of the line. However, it is still instructive to see what sort of morphologies appear in a p-v diagram of this sort. The synthetic p-v diagram in Fig. 22 contains a series of arcs that resemble those present in astronomical slit spectra of spatially-resolved bow shocks [@raga86; @hrm90]. These arcs occur when the slit crosses the jet or an interface such as the cavity evacuated by the jet. In the case of spatially resolved bow shocks, arcs in the p-v diagram result from the motion of a curved shell of material, where the orientation of the velocity vector relative to the observer changes along the slit. The lesson from the experiment seems to be that in highly structured flows like our deflected jet, curved cavities and filaments naturally produce p-v diagrams that contain multiple arcuate features. As in the case of resolved bow shocks, the velocity amplitude of the arcs in the p-v diagram is on the order of the shock velocity responsible for the arc. In an optically thin astrophysical nebula one can measure five out of the six phase space dimensions: x and y position on the sky from images, proper motion velocities V$_x$ and V$_y$ from images taken at two times, and the radial velocity V$_z$ along the line of sight from spectroscopy. As we have shown above, V$_z$ is an extremely useful diagnostic of the dynamics of a flow, but one that is currently not possible to measure in laboratory experiments. If such a diagnostic instrument could be developed it would open up a wide range of possibilities for new studies of supersonic flows. Summary and Future Work ======================= The combination of experimental, numerical and astronomical observational data from this study demonstrates the potential of the emerging field of laboratory astrophysics. In this paper we have studied how a supersonic jet behaves with time as it deflects from an obstacle situated at various distances from the axis of the jet. The laboratory analog of this phenomenon scales very well to the astrophysical case of a stellar jet which deflects from a molecular cloud core. An important component to our study was to expand the observational database of best astrophysical example, HH 110, by obtaining new spatially-resolved high-spectral resolution observations capable of distinguishing thermal motions from turbulent motions, and by acquiring a new deep infrared H$_2$ image that can be compared with existing optical emission line images from the Hubble Space Telescope. The laboratory experiments span a range of times, spatial offsets between the axis of the jet and the center of the ball (impact parameters), viewing angles, opacities (backlighters), and materials. The experiments are reproducible and do not depend on composition or structure of foam, or on the pinhole diameter of the backlighter (spatial resolution of the experiment). Synthetic radiographs of the experiment from RAGE match the experimental data extremely well, both qualitatively as images and quantiatively with Fourier analysis. In fact, the agreement is so good that we have used the synthetic velocity maps from RAGE to compare the internal dynamics of the experiment with those that we measure from the new spectral maps of HH 110. A new wide-field H$_2$ image supports a scenario where HH 110 represents the shocked ‘spray’ that results from a glancing collision of the HH 270 jet with a molecular cloud core. The H$_2$ in HH 110 is offset from the H$\alpha$ toward the side closest to the molecular core, consistent with the deflected jet model. The H$_2$ images also uncovered two sources within the core that drive collimated jets, one a bright near-infrared source, and the other a highly-obscured source that appears to be a dense protostellar disk observed nearly edge-on, as in HH 30. The experiments provide several important insights into how deflected supersonic jets like HH 110 behave. In the experiment, entrainment of material in the obstacle occurs in part because the morphology of the contact discontinuity between the shocked jet and shocked obstacle easily develops a complex 3-D structure of cavities that enables the jet to isolate clumps of obstacle material and entrain them into the flow. A similar process likely operates in HH 110. The experiments also reveal filamentary structure in the working surface area of the deflected bow shock, but the relative motion between these filaments is subsonic. Hence, while this dynamical process will generate density fluctuations in the outflowing gas, it cannot produce the filamentary structure and $\sim$ Mach 5 shocks shown by the new velocity maps of HH 110. For this reason the best model for HH 110 remains that of a pulsed jet which interacts with a molecular cloud core. Synthetic position-velocity maps along the deflected jet from the RAGE simulations of the experiments appear as a series of arcs, similar to those observed in astronomical observations of resolved bow shocks. A close examination of the experimental data shows that these arcs correspond to regions where the slit crosses different regions of the flow, such as cavities evacuated by the jet, the jet itself, or entrained material from the ball. This correspondance between the appearance of a p-v diagram and the actual morphology of a complex flow is an intuitive, although perhaps unexpected result of studying the dynamics within the experimental flow. Finally, observations of the deflected bow shock from different viewing angles emphasize that the observed morphology and collimation properties of bow shocks depend strongly upon the orientation of the flow with respect to the observer. As one would expect, a bow shock deflected toward the observer appears less collimated than one that is redirected into the plane of the sky. The impact parameter of the jet and obstacle determines how much the jet deflects from the obstacle and how rapidly the obstacle becomes disrupted by the jet. While experimental analogs of astrophysical jets are highly unlikely to ever reproduce accurate emission line maps, laser experiments can provide valuable insights into how the dynamics of complex flows behave. Our study of a deflected supersonic jet is only one example of how the fields of astrophysics, numerical computation, and laboratory laser experiments can compliment one another. We are currently embarking on a similar program to study the dynamics within supersonic flows that are highly clumpy, and other investigations are underway related to the launching and collimation of jets [@bellan09]. We are grateful to Dean Jorgensen, Optimation Inc., Burr-Free Micro Hole Division, 6803 South 400 West, Midvale, Utah 84047, USA, for supplying the precision-machined titanium-alloy components for these experiments. We thank K. Dannenberg for her assistance in manufacturing the initial targets for these experiments, the staff of General Atomics for their dedication in developing new targets and delivering them on time, the staff at Omega for their efficient operation of the laser facility, and an anonymous referee for useful comments regarding scaling. This research was made possible by a DOE grant from NNSA as part of the NLUF programs DE-PS52-08NA28649 and DE-FG52-07NA28056. 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Plasmas 3, 218 Schwartz, R. 1978, ApJ 223, 884 [lcc]{} = 0.08in =0em L (cm) \[jet size\] & 1e15 & 0.02\ V (km$\,$s$^{-1}$) \[jet velocity\] & 150 & 10\ $\rho$ (g$\,$cm$^{-3}$) \[jet density\] & 2e-20 & 1\ P (dyne cm$^{-2}$) \[jet pressure\] & 1e-8 & 3e10\ t$_{flow}$ (sec) \[flow timescale\] & 3e9 & 1e-7\ Composition & H & Ti\ T (eV) \[temperature of wake\] & 0.6 & 1\ Euler number in jet &22 & 6\ $\nu_{mat}$ (cm$^2$s$^{-1}$) \[viscosity\] & 9e13 & 2e-5\ $\nu_{rad}$ (cm$^2$s$^{-1}$) \[viscosity\] & 2e21 & 1e-12\ $\nu_c$ (s$^{-1}$) \[collision freq\] & 0.002 & 2e13\ $\chi$ (cm$^2$s$^{-1}$) \[diffusivity\] & 5e24 & 5e-18\ Re$_{mat}$ \[Reynolds number\] & 7e8 & 1e9\ Pe$_{mat}$ \[Peclet number\] & 1e7 & 3e4\ $\lambda_{mat}$ (cm) (mean-free path) & 3e8 & 8e-9\ $\lambda_{rad}$ (cm) (mean-free path) & $>>$L$^a$ & 3e-5\ $\tau_{mat}$ \[optical depth\] & 9e6 & 2e6\ $\tau_{rad}$ \[optical depth\] & $<<$1$^a$ & 7e2\ Bo\# & $>$1e3 & 2e4\ 6.9in 0.0in 0.0in 0.0in 0.0in [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy Inc., under cooperative agreement with the National Science Foundation.

ArXiv

--- abstract: 'We propose a method to improve image clustering using sparse text and the wisdom of the crowds. In particular, we present a method to fuse two different kinds of document features, image and text features, and use a common dictionary or “wisdom of the crowds” as the connection between the two different kinds of documents. With the proposed fusion matrix, we use topic modeling via non-negative matrix factorization to cluster documents.' author: - - - - bibliography: - 'asilomar\_imgtxt.bib' title: Improving Image Clustering using Sparse Text and the Wisdom of the Crowds --- Introduction ============ There has been substantial research in organizing large image databases. Often, these images have corresponding text, such as captions in textbooks and metatags. We investigate strategies to use this text information to improve clustering image documents into groups of similar images. Image Clustering is used for image database management, content based image searches, and image classification. In this paper, we present a method for improving image clusters using sparse text and freely obtainable information form the internet. The motivation behind our method stems from the idea that we can fuse image and text documents and use the “wisdom of the crowds” (WOC), the freely obtainable information, to connect the sparse text documents where WOC documents act as a representative of a single class. In Section 2, we breifly touch upon related material. In Section 3, we introduce our method of fusing text and image documents using the term frequency-inverse document frequency weighting scheme. We then describe how non-negative matrix factorization is used for the purpose of topic modeling in section 4. In Section 5, we present results from an application of our method. Related Works ============= There have been many studies on text document clustering and image clustering. A general joint image and text clustering strategy proceeds in two steps, first two different types of documents must be combined into a single document feature matrix. Then, a clustering technique is implemented. The term frequency-inverse document frequency (TF-IDF) is a technique to create a feature matrix from a collection, or corpus, of documents. TF-IDF is a weighting scheme that weighs features in documents based on how often the words occurs in an individual document compared with how often it occurs in other documents [@tf-idf_original]. TF-IDF has been used for text mining, near duplicate detection, and information retrieval. When dealing with text documents, the natural features to use are words (i.e. delimiting strings by white space to obtain features). We can represent each word by a unique integer. In order to use text processing techniques for image databases, we generate a collection of image words using two steps. First, we obtain a collection of image features, and then define a mapping from the image features to the integers. To obtain image features, we use the scale invariant feature transform (SIFT) [@lowe2004distinctive]. We then use k-means to cluster the image features into $K$ different clusters. The mapping from the image feature to the cluster is used to identify image words, and results in the image Bag-Of-Words model [@fei2005bayesian]. Topic modeling is used to uncover a hidden topical structure of a collection of documents. There have been studies on using large scale data collections to improve classification of sparse, short segments of text, which usually cluster inaccurately due to spareness of text [@phan2008learning]. Latent Dirichlet Allocation (LDA), singular value decomposition (SVD), and non-negative matrix factorization (NNMF) are just some of the models that have been used in topic modeling [@arora2012learning]. In our method, we integrate these techniques to combine and cluster different types of documents. We use SIFT to obtain image features and term frequency-inverse document frequency to generate a feature matrix in the fused collection of documents. Then, we use the non-negative matrix factorization to learn a representation of the corpus which is used to cluster the documents. Fusing Image and Text Documents =============================== We denote a collection of image documents $D = \{d_1, ... d_n\}$ and a collection of sparse text documents $S = \{s_1,...s_n\}$ and text document $s_i$ describes image document $d_i$ for i=1,...m. Some of the text documents may be empty, indicating the absence of any labeled text. **Image Documents** Using the scale invariant feature transform (SIFT) and k-means, we obtain $A \in \mathbb{R}^{n \times p}$ where $p$ is the number of image features and n is the number of image documents and element $A_{i,j}$ represents the number of times the image document $d_i$ contains the $j^{th}$ feature. **Wisdom of the Crowds** Due to the sparse nature of the text documents we are considering, the WOC is needed to link features that represent a single class. For example, if one wishes to obtain a class of documents and images about cats, text and images from a wikipedia page on cats can be used as the wisdom of the crowds. Using Wikipedia, we collect WOC documents $W = \{w_1, ... w_k\}$ where $k$ is the number of clusters we wish to cluster the images into. Each $w_i$ is a text document that contains features that collectively describe a single class. To create text features, we parse text documents by white space (i.e. break up text by words) and obtain a corpus $f = (f_1, ... f_q)$ of $q$ unique features. Let $C \in \mathbb{R}^{k \times q}$ Each $C_{i,j}$ is the number of times the feature $f_j$ appears in $w_i$. **Text Documents** In the same manner as with WOC documents, we parse text documents into features to obtain a corpus. In most cases, the features in this corpus have already appeared somewhere in the WOC documents so we use the same $f = (f_1,...f_q)$ from the previous step. If it is not the case, “missing" features can simply be appended to the list of features and the $C$ matrix extended to reflect the absence of the missing features. We calculate $B \in \mathbb{R}^{m \times q}$ where $m$ is the number of text documents, $q$ is the number of features in corpus $f$, and element $B_{i,j}$ is the number of times text document $s_i$ contains the feature $f_j$. We then extend $B \in \mathbb{R}^{m \times q}$ to $B \in \mathbb{R}^{n \times q}$ such that $B_{i,j}= 0$ for $i = m+1,...n, j=1,...q$. Intuitively, this means that none of the text features knowingly describes the $m+1, ... n$ image documents. We combine the image feature matrix $A$, the text feature matrix $B$, and the WOC matrix $C$ to initialize matrix $M$: $$M = \begin{bmatrix} A & B\\ 0 & C\\ \end{bmatrix},$$ where $M \in \mathbb{R}^{(m+k) \times (p+q)}$ and $0 \in \{0\}^{k \times p}$. We call $M$ our mixed document feature matrix. Each row represents a document and each column represents a feature (either an image feature or a text feature). Without the reweighing using IDF, it is difficult to use sparse text to aid in image classification. This is because the frequency of the image features outweigh any sort of effect the sparse text has in the classification of image documents. The inverse document frequency matrix $\text{IDF} \in \mathbb{R}^{p+q \times p+q}$ is defined as the diagonal matrix with nonzero elements: $$\text{IDF}_{j,j} = \log\dfrac{m+k}{|\{i : M_{i,j}>0\}|},$$ where ${|\{i : M_{i,j}>0\}|}$ is the number of documents containing the $j^{th}$ feature. We then re-evaluate M to be $M = M \times IDF$. ![Example of an M matrix with 4500 image documents, 9 WOC documents, and 450 text documents.[]{data-label="fig:mixmat"}](mixMat.jpg) Topic Modeling using Non-negative Matrix Factorization ====================================================== We use non-negative matrix factorization (NNMF) on the document feature matrix to cluster documents into topics. We consider the document feature matrix as a set of (m+k) points in a (p+q) dimensional space. Each document is a point and each feature is a dimension. We want to reduce the dimensionality of this space into $k^* << \min(m+k, p+q)$ dimensions [@Lsas]. NMF is a method that takes a non-negative matrix $M_+ \in \mathbb{R}^{(m+k) \times (p+q)}$ and factors it into two non-negative matrices $U_+ \in \mathbb{R}^{(m+k) \times k^*}$ and $V_+ \in \mathbb{R}^{k^* \times (p+q)}$ where $k^*$ is the rank of the desired lower dimensional approximation to $X$ [@LeeSeung]. We take the $(p+q)$-dimensional feature space and project it onto a $k^*$-dimensional topic space where $k^*$ is the number of desired classes. Denoting the Frobenius norm of $M$ as $||M||_F^2 = \sum_i\sum_j M_{i,j}^2$, we wish to obtain $U$ and $V$ by minimizing the following cost function: $$||M - UV||^2_F. \label{eq:nmf}$$ Intuitavely, $U_{i,j}$ tells us how well document $d_i$ fits into topic $j$ and $V_{i,j}$ tells us how well the $j^{th}$ feature describes the $i^{th}$ topic. In most applications of topic modeling using NNMF, a document $d_i$ belongs to topic $j$ if $$j = \operatorname*{arg\,max}_z U_{i,z}.$$ Because of the geometric nature of the NNMF topic modeling method, we also investigate the clusters that result from a k-means clustering on the rows of $U$, or the location of documents in the reduced-dimension topical space. Results ======= Evaluation Metrics ------------------ Purity and z-Rand scores are metrics used to evaluate cluster quality [@traud2008comparing], [@amigo2009comparison]. **Purity** Purity is a well known clustering measure that depends on some ground truth. This metric compares a cluster to the ground truth by comparing the intersection of the ground truth clustering with the new clustering. Purity can be computed by as follows: $$Purity(G,C) = \frac{1}{m} \sum_i \max_j |g_j \cap c_i|.$$ Here, $m$ is the number of documents, $G = \{g_1, ..., g_k\}$ is the ground truth or class assignment where each $g_j$ is a set of indices belonging to the $j^{th}$ class, and $C = \{c_1, ... c_t\}$ is the clustering from some method where each $c_i$ is the set of indices belonging to the $i^{th}$ cluster. It is important to note that purity is sensitive to the number of clusters. If every document had its own cluster, then the purity for this set of clusters is 1. To address this sensitivity, we also look at the $z$-rand metric. **Z-rand** To define the $z$-rand score we first define $p$ to be the number of pairs of documents that are in the same cluster as determined by our method and in the ground truth (i.e. the number of document pairs that are correctly clustered together). The $z$-rand score, $z_R$ is defined as: $$z_R = \dfrac{(p-\mu_p)}{\sigma_p},$$ where $\mu_p$ and $\sigma_p$ are the expected value the standard deviation of $p$ under a hypergeometric distribution with the same size of clusters. Intuitively, we are comparing the number of correctly identified pairings to the number of correctly identified pairings if the pairings were randomly selected. The higher the z-rand score, the better clusters as the clusters created are very different from randomly picked clusters.\ We apply our method to the Electro-Optical (EO) dataset provided by China Lake. This dataset consists of 9 classes of images where each class contains 500 images of a single vehicle from different angles. Because this dataset does not contain text data, we use wikipedia articles to create sparse text captions for a varying number of documents by randomly selecting 5 words from each wikipedia article to be an image caption. Using only the image documents and NNMF, the clusters produced score a mean purity of 0.6397 and mean z-rand of 1460.7. Matrix Purity Zrand ----------- -------------------- --------------------- A 0.6397 $\pm$ 0.012 1460.7 $\pm$ 52.18 $[A : B]$ 0.6597 $\pm$ 0.01 1538.6 $\pm$ 45.71 M 0.769 $\pm$ 0.0012 1909.5 $\pm$ 136.55 : Results from the EO data set. A is only using image features, $[A:B]$ is image features with sparse text, and M is image and text features with additional dicionary.[]{data-label="tab:mainresults"} For our first experiment, we investigate the usefulness of fusing image and text documents together and using the appropriate reweighting. In Table \[tab:mainresults\], we are comparing using only image features, using image and sparse text features, and using image features, sparse text features, and a dictionary. As one can see, using only image features, does the worst while using sparse text features helps only slightly. We attribute this slight improve to the fact that the text documents are sparse. When we use the WOC, we get a significant increase in purity and zrand. We also investigated the effect of varying the percentage of documents with both image and text features and found that in general, regardless of the number of image documents that contained sparse text, the purity stayed from 0.76-0.78, while the z-rand ranged from 1877.0-1938.2. To improve results, one may also remove stop words from the text features. Stop words are commonly used words such as ‘the’, ‘a’, and ‘is’. When we did this, we obtained a mean purity of 0.778. We found that each class can be broken down into three subclasses: front of vehicle, back of vehicle, and sides. So, using $k=27$, we greatly improve our results as shown in Table \[tab:27topics\]. % of documents with labels purity z-Rand ---------------------------- ----------------------- ----------------------- 0.2 0.88126$\pm$0.0035533 1579.9675$\pm$15.6499 0.4 0.87793$\pm$0.0046333 1571.9551$\pm$14.8006 0.6 0.88403$\pm$0.0043387 1566.322$\pm$13.8568 0.8 0.88341$\pm$0.0042595 1580.0351$\pm$13.7404 1 0.88071$\pm$0.0038168 1576.8284$\pm$16.8794 : Purity and z-Rand over different percentages of images documents with text documents where the number of text documents $m = \lfloor{np}\rfloor$ for EO data set using $k^* = 27$.[]{data-label="tab:27topics"} Conclusion ========== Fusing text documents and image documents makes it possible to improve image clusters. The results from the EO data set show that are method does make an improvement on the image clusters when comparing to using NNMF on only the image document feature matrix $A$. Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported in part by AFOSR MURI grant FA9550-10-1-0569. Arjuna Flenner was supported by ONR grants number N0001414WX20237 and N0001414WX20170. Deanna Needell was partially supported by Simons Foundation Collaboration grant $\#274305$.

ArXiv

--- abstract: 'Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}$ of discriminant $\Delta$ and some element $\lambda \in \mathcal{O}_{K}$. Our first main result says that if there is a prime $\mathfrak{p}$ of $K$ dividing $(f(\lambda))$ but not $(2\Delta)$, then the image of the natural $2$-adic Galois representation is open in ${\mathrm{GSp}}(T_{2}(J))$ and contains a certain congruence subgroup of ${\mathrm{Sp}}(T_{2}(J))$ depending on the maximal power of $\mathfrak{p}$ dividing $(f(\lambda))$. We also present and prove a variant of this result that applies when $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)(x - \lambda'')$ for distinct elements $\lambda, \lambda'' \in K$. We then show that the hypothesis in the former statement holds for almost all $\lambda \in \mathcal{O}_{K}$ and prove a quantitative form of a uniform boundedness result of Cadoret and Tamagawa.' author: - Jeffrey Yelton bibliography: - 'bibfile.bib' title: 'Boundedness results for $2$-adic Galois images associated to hyperelliptic Jacobians' --- Introduction {#S1} ============ Let $K$ be a number field with absolute Galois group $G_{K}$, and let $C$ be a hyperelliptic curve defined over $K$; i.e. $C$ is a smooth projective curve defined by an equation of the form $y^{2} = f(x)$ for some squarefree polynomial $f$ of degree $d \geq 3$. (Note that in the case of $d = 3$, $C$ is an elliptic curve.) It is well known that the genus of $C$ is given by $g = \lfloor (d + 1) / 2 \rfloor$. We denote the Jacobian variety of $C$ by $J$; it is an abelian variety of dimension $g$. For each prime $\ell$, we let $T_{\ell}(J)$ denote the $\ell$-adic Tate module of $J$, which is a free ${\mathbb{Z}}_{\ell}$-module of rank $2g$. We write $\rho_{\ell} : G_{K} \to {\mathrm{Aut}}(T_{\ell}(J))$ for the natural $\ell$-adic Galois action on this Tate module. The Tate module $T_{\ell}(J)$ is endowed with the Weil pairing defined with respect to the canonical principal polarization on $J$, which we write as $e_{\ell} : T_{\ell}(J) \times T_{\ell}(J) \to {\mathbb{Z}}_{\ell}$; it is a ${\mathbb{Z}}_{\ell}$-bilinear skew-symmetric pairing. Let ${\mathrm{Sp}}(T_{\ell}(J))$ denote the group of symplectic automorphisms of $T_{\ell}(J)$ with respect to the pairing $e_{\ell}$, and let $${\mathrm{GSp}}(T_{\ell}(J)) := \{\sigma \in \mathrm{Aut}_{{\mathbb{Z}}_{\ell}}(T_{\ell}(J))\ |\ e_{\ell}(P^{\sigma}, Q^{\sigma}) = e_{\ell}(P, Q)^{\chi_{\ell}(\sigma)}\ \forall P, Q \in T_{2}(J)\}$$ denote the group of symplectic similitudes, where $\displaystyle \chi_{\ell} : G_{K} \to {\mathbb{Z}}_{\ell}^{\times}$ is the $\ell$-adic cyclotomic character. It is well known that the image $G_{\ell}$ of $\rho_{\ell}$ is always a closed subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ and that in fact there is some hyperelliptic Jacobian $J$ of a given dimension $g$ such that the inclusion $G_{\ell} \subseteq {\mathrm{GSp}}(T_{\ell}(J))$ has finite index (or equivalently, that $G_{\ell}$ is an open subgroup of the $\ell$-adic Lie group ${\mathrm{GSp}}(T_{\ell}(J))$); see for instance [@yelton2015images Theorem 1.1]. Note that the subgroup $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J)) \subset G_{\ell}$ coincides with the image of the Galois subgroup which fixes the extension $K(\mu_{\ell}) / K$ obtained by adjoining all $\ell$-power roots of unity to $K$. Since $K$ is a number field, the extension $K(\mu_{\ell}) / K$ is infinite; it follows that $G_{\ell} \not\subset {\mathrm{Sp}}(T_{\ell}(J))$ and that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ if and only if $G_{\ell} \cap {\mathrm{Sp}}(T_{\ell}(J))$ has finite index in ${\mathrm{Sp}}(T_{\ell}(J))$. There have been many results stating that $G_{\ell}$ has finite index in ${\mathrm{GSp}}(T_{\ell}(J))$ under various hypotheses for the polynomial defining the hyperelliptic curve. For instance, Y. Zarhin has proven this for large enough genus in the case of hyperelliptic curves defined by equations of the form $y^{2} = f(x)$ or $y^{2} = f(x)(x - \lambda)$ with $\lambda \in K$, where the Galois group of $f$ is the full symmetric or alternating group ([@zarhin2002very Theorem 2.5] and [@zarhin2010families Theorem 8.3]; see also [@zarhin2013two Theorem 1.3] for a variant of this where the curve is defined using two parameters). A. Cadoret and A. Tamagawa have also proven ([@cadoret2012uniform Theorems 1.1 and 5.1]) that for any family of hyperelliptic Jacobians over a smooth, geometrically connected, separated curve over $K$, this openness condition will be satisfied for the $\ell$-adic Galois action associated to all but finitely many fibers, and that in fact the indices of the $\ell$-adic Galois images corresponding to these fibers are uniformly bounded. However, there have been very few results which give explicit bounds for the index of $G_{\ell}$ in ${\mathrm{GSp}}(T_{\ell}(J))$ in such cases. Our aim in this paper is to give some similar results on the openness of the $2$-adic Galois images in the group of symplectic similitudes associated to Jacobians of hyperelliptic curves whose defining polynomials satisfy certain hypotheses, and to provide formulas giving explicit bounds for the indices of the $2$-adic Galois images in these cases. (Unfortunately, our method currently cannot tell us anything about the $\ell$-adic Galois images for odd primes $\ell$. However, we are hopeful that it can be strengthened to show the openness of the $\ell$-adic Galois images as well under the same or similar hypotheses as is implied by the Mumford-Tate conjecture, and to show that the $\ell$-adic Galois images contain the full symplectic group for almost all $\ell$.) We state our main results below. In these statements as well as in the rest of the paper, we use the following notation. For any integer $N \geq 1$, we denote the level-$N$ congruence subgroup of ${\mathrm{Sp}}(T_{2}(J))$ by $\Gamma(N) := \{\sigma \in {\mathrm{Sp}}(T_{2}(J)) \ | \ \sigma \equiv 1 \ (\mathrm{mod} \ N)\}$. We denote the ring of integers of a number field $K$ by $\mathcal{O}_{K}$. Finally, we write $v_{2} : {\mathbb{Q}}^{\times} \to {\mathbb{Z}}$ for the (normalized) $2$-adic valuation on ${\mathbb{Q}}$. \[thm main1\] Let $K$ be a number field, and let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 2$ with discriminant $\Delta$. Let $J$ be the Jacobian of the hyperelliptic curve with defining equation $y^{2} = f(x)(x - \lambda)$ for some $\lambda \in \mathcal{O}_{K}$, and define the $2$-adic Galois image $G_{2}$ as above. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$. If in addition $d = 3$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + 1})$. \[thm main2\] Assume the same notation as in the statement of Theorem \[thm main1\], except that the defining equation of the hyperelliptic curve is $y^{2} = f(x)(x - \lambda)(x - \lambda')$ for distinct elements $\lambda, \lambda' \in \mathcal{O}_{K}$. Then if there is a prime $\mathfrak{p}$ of $\mathcal{O}_{K}$ which divides $(f(\lambda))$ but not $(\lambda - \lambda')$ or $(2\Delta)$ and a prime $\mathfrak{p}'$ which divides $(\lambda - \lambda')$ but not $(f(\lambda))$ or $(2\Delta)$, the Lie subgroup $G_{2} \subset {\mathrm{GSp}}(T_{2}(J))$ is open. In fact, we have $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{2v_{2}(m) + 2})$ if $v_{2}(m') \leq v_{2}(m)$ or $d = 2g$ and $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{v_{2}(m) + v_{2}(m') + 2})$ otherwise, where $m \geq 1$ is the greatest integer such that $\mathfrak{p}^{m} \mid (f(\lambda))$ and $m' \geq 1$ is the greatest integer such that $\mathfrak{p}'^{m'} \mid (\lambda - \lambda')$. If in addition $d = 2$, then $G_{2} \cap {\mathrm{Sp}}(T_{2}(J)) \supsetneq \Gamma(2^{\max \{v_{2}(m), v_{2}(m')\} + 1})$. \[rmk almost all fibers\] For a fixed irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$ with discriminant $\Delta$, it is not hard to show using Faltings’ Theorem that the hypotheses in Theorems \[thm main1\] is satisfied for almost all $\lambda \in \mathcal{O}_{K}$, as we will see in §\[S5\] (Remark \[rmk not PID\] below). It follows immediately that there are also infinitely many choices of $(\lambda, \lambda') \in K \times K$ satisfying the hypotheses in Theorem \[thm main2\] for this polynomial $f$. \[rmk end\] It is known that there is a finite algebraic extension $K'$ of $K$ over which every endomorphism of an abelian variety over a field $K$ is defined (see [@silverberg1992fields Theorem 2.4]), so that each endomorphism commutes with the action of ${\mathrm{Gal}}(\bar{K} / K')$ on torsion points. Note that the only endomorphisms in ${\mathrm{End}}(T_{\ell}(J))$ which commute with everything in an open subgroup of ${\mathrm{GSp}}(T_{\ell}(J))$ are scalars. It therefore follows from the above theorems that the endomorphism ring of any hyperelliptic Jacobian $J$ satisfying the hypotheses of Theorem \[thm main1\] or of Theorem \[thm main2\] coincides with ${\mathbb{Z}}$ and that such a $J$ is absolutely simple. The key ingredient used in proving the above theorems is a method of describing Galois actions on $\ell$-adic Tate modules of hyperelliptic Jacobians defined over strictly Henselian local fields of residue characteristic $p \neq 2, \ell$ by looking at the valuations of the differences between the roots of the defining polynomial, which is derived from results shown in joint work with H. Hasson ([@hasson2017prime]). This way of looking at $\ell$-adic Galois actions associated to hyperelliptic Jacobians over local fields is very similar to the “method of clusters" used by S. Anni and V. Dokchitser in [@anni2017constructing]. Our approach seems quite powerful and should lead to many similar boundedness results in a number of situations where one can compute valuations of the differences between the roots of the defining polynomial with respect to various primes of the ground field. Unfortunately, in practice, these valuations (or even the roots themselves) may be difficult to calculate, and so our main focus here is on obtaining results such as the ones stated above where the hypotheses are very easy to verify. The rest of this paper is organized as follows. In §\[S2\], we use the main results of [@hasson2017prime], which show that over a strictly Henselian local field of characteristic $p \neq 2$, for primes $\ell \neq p$, the $\ell$-adic Galois action factors through the tame quotient of the absolute Galois group and can be described in terms of Dehn twists with respect to certain loops on a complex hyperelliptic curve. In particular cases such as when exactly two roots of the defining polynomial coalesce in the reduction over the residue field, we will show (Proposition \[prop Galois action local\]) that such a Dehn twist induces a transvection in the symplectic group. We will later put local data together to show that over a number field $K$, the $2$-adic Galois image contains certain powers of several sufficiently “independent" transvections. In §\[S3\], we will demonstrate using elementary matrix algebra that the group generated by these powers of transvections contains a certain congruence subgroup. In §\[S4\], we will use what we have shown in §\[S2\] and §\[S3\] to prove Theorems \[thm main1\] and \[thm main2\] as well as to prove an auxiliary result that applies to a more general situation (Theorem \[thm several primes\]). Finally, in §\[S5\], we will assume that $K$ has class number $1$ and show using Theorem \[thm main1\] that for a given $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 3$, the $2$-adic Galois image associated to the hyperelliptic curves defined by $y^{2} = f(x)(x - \lambda)$ for all but finitely many $\lambda \in \mathcal{O}_{K}$ contains a principal congruence subgroup which depends only on $d$ (Theorem \[thm uniform bounds\]). In fact, for $d \geq 4$ even, in Theorem \[thm uniform bounds\](c) we will provide a uniform bound for indices of the $2$-adic Galois images associated to almost all fibers of such a one-parameter family over the $K$-line, as is guaranteed by [@cadoret2012uniform Theorems 1.1 and 5.1]. The author is grateful to a MathOverflow user whose comment on question 264281 helped to inspire the arguments for the results in §\[S5\]. Hyperelliptic Jacobians over local fields and tame Galois actions {#S2} ================================================================= We retain all notation introduced in the previous section. In this section, we write $\widehat{{\mathbb{Z}}}$ for the profinite completion of ${\mathbb{Z}}$ and use the symbol $\widehat{\pi}_{1}$ to denote the profinite completion of the fundamental group of a topological space. For any profinite group $G$, we write $G^{(p')}$ for its maximal prime-to-$p$ quotient. For any profinite group $G$, let $G^{(p')}$ denote its prime-to-$p$ quotient. Note that since $G^{(p')}$ is a characteristic quotient of $G$, any action on $G$ induces an action on $G^{(p')}$. Now we choose a prime $\mathfrak{p}$ of $K$ of residue characteristic $p \neq 2$. Fix a strict Henselization of the localization of $K$ at the prime $\mathfrak{p}$ and denote it by $\mathcal{R}_{\mathfrak{p}}$ and its fraction field by $\mathcal{K}_{\mathfrak{p}}$; this comes with an embedding $\mathcal{K}_{\mathfrak{p}} \hookrightarrow \bar{K}$. Let $\pi \in K$ be a uniformizer of the discrete valuation ring $\mathcal{R}_{\mathfrak{p}}$. We fix a compatible system of $N$th roots of unity $\zeta_{N} \in \bar{K}$ for $N = 1, 2, 3, ...$; that is, we require that $\zeta_{N'N}^{N'} = \zeta_{N}$ for any integers $N, N' \geq 1$. Note that since $R$ is strictly Henselian, $\zeta_{N} \in R \subset K$ for any $N$ not divisible by $p$. Let $G_{K, \mathfrak{p}}$ denote the absolute Galois group of $\mathcal{K}_{\mathfrak{p}}$, and let $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ denote its tame quotient. It follows from a special case of Abhyankar’s Lemma that the maximal tamely ramified extension $\mathcal{K}_{\mathfrak{p}}^{{\mathrm{tame}}}$ is given by $\mathcal{K}_{\mathfrak{p}}(\{\pi^{1/N}\}_{(N, p) = 1})$, where $\pi^{1/N}$ denotes an $N$th root of $\pi$, and that $G_{K, \mathfrak{p}}^{{\mathrm{tame}}} \cong \widehat{{\mathbb{Z}}}^{(p')}$ is topologically generated by the automorphism which acts on $K^{{\mathrm{tame}}}$ by fixing $K$ and sending each $\pi^{1/N}$ to $\zeta_{N}\pi^{1/N}$. We fix, once and for all, an embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ where $\zeta_{N}$ is sent to $e^{2 \pi \sqrt{-1} / N}$ for $N \geq 1$, so that we have an inclusion $\mathcal{K}_{\mathfrak{p}} \subset {\mathbb{C}}$. Let $d \geq 2$ be an integer and choose distinct integral elements $\alpha_{1}, ... , \alpha_{d} \in K$. Choose polynomials $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{d} \in {\mathbb{C}}[x]$ satisfying $\tilde{\alpha}_{i}(\pi) = \alpha_{i}$ for $1 \leq i \leq d$ and such that the $x$-adic valuation of $\tilde{\alpha}_{i}$ and $\tilde{\alpha}_{j}$ and the $\pi$-adic valuation of $\alpha_{i}$ and $\alpha_{j}$ are equal (such polynomials exist as is shown in the discussion in [@hasson2017prime §3.3]). Let $\varepsilon > 0$ be a real number small enough that $\tilde{\alpha}_{i}(z) \neq \tilde{\alpha}_{j}(z)$ for all $i \neq j$ and for all $z \in B_{\varepsilon}^{*} := \{z \in {\mathbb{C}}\ | \ |z| < \varepsilon\} \smallsetminus \{0\}$. We define a family $\mathcal{F} \to B_{\varepsilon}^{*}$ of $d$-times-punctured Riemann spheres by letting $$\mathcal{F} = {\mathbb{P}}_{{\mathbb{C}}}^{1} \times B_{\varepsilon}^{*} \smallsetminus \bigcup_{i = 1}^{d} \{(\tilde{\alpha}_{i}(z), z) \ | \ z \in B_{\varepsilon}^{*}\}.$$ Choose a basepoint $z_{0} \in B_{\varepsilon}^{*}$. The fundamental group $\pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts by monodromy on the fundamental group $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ of the fiber over $z_{0}$ with basepoint $\infty$. We write $\rho_{{\mathrm{top}}} : \pi_{1}(B_{\varepsilon}^{*}, z_{0}) \to {\mathrm{Aut}}(\pi_{1}(\mathcal{F}_{z_{0}}, \infty))$ for this action. The action $\rho_{{\mathrm{top}}}$ extends uniquely to a continuous action of the profinite completion $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ on the profinite completion $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ (see the discussion in [@hasson2017prime §1.1]), which we also denote by $\rho_{{\mathrm{top}}}$. We note that $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ is isomorphic to $\widehat{{\mathbb{Z}}}$ and is topologically generated by the element $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ represented by the loop given by $t \mapsto e^{2 \pi \sqrt{-1} t}z_{0}$ for $t \in [0, 1]$. Meanwhile, the absolute Galois group $G_{K, \mathfrak{p}}$ acts naturally on the étale fundamental group $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)$ via the $K_{\mathfrak{p}}$-point lying under the geometric point $\infty : {\mathrm{Spec}}({\mathbb{C}}) \to {\mathbb{P}}_{K_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$. After identifying $\pi_{1}^{{\mathrm{\acute{e}t}}}({\mathbb{P}}_{\bar{K}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ with $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ via Riemann’s Existence Theorem and the inclusion of algebraically closed fields $\bar{\mathcal{K}}_{\mathfrak{p}} \subset {\mathbb{C}}$, we write $\rho_{{\mathrm{alg}}} : G_{K, \mathfrak{p}} \to {\mathrm{Aut}}(\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')})$ for this action. We denote the actions on prime-to-$p$ quotients of étale fundamental groups induced by $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ by $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ respectively. For the statement of Theorem \[thm comparison punctured projective line\](a) below, we require the terminology of Dehn twists. Let $\gamma : [0, 1] \to M$ be a simple loop on any complex manifold $M$; we will often identify $\gamma$ with its image in $M$. We define the *Dehn twist* on $M$ with respect to the loop $\gamma$. It is an element of the mapping class group of $M$ represented by a self-homeomorphism of $M$ which can be visualized in terms of a small tubular neighborhood of $\gamma \subset M$, in the following way: the Dehn twist keeps the outer edge of the tubular neighborhood fixed while twisting the inner edge one full rotation counterclockwise and acts as the identity everywhere else on $M$. Since this Dehn twist depends only on the homology class $[\gamma] \in H_{1}(M, {\mathbb{Z}})$ of any loop $\gamma$, we will denote it by $D_{[\gamma]}$. (See [@farb2011primer Chapter 3] for more details.) The following theorem is a compilation of all the necessary results describing and comparing $\rho_{{\mathrm{top}}}$ and $\rho_{{\mathrm{alg}}}$ that are proven in [@hasson2017prime] (Theorems 1.2 and 2.3 and Remark 3.10 of that paper). \[thm comparison punctured projective line\] In the above situation, we have the following. a\) Let $\mathcal{I}$ be the set of all pairs $(I, n)$ where $I \subseteq \{1, ... , d\}$ is a subset and $n \geq 1$ is an integer such that $x^{n} \mid \tilde{\alpha}_{i} - \tilde{\alpha}_{j} \in {\mathbb{C}}[x]$ for all $i, j \in I$ and such that $I$ is maximal among intervals with this property. If $\varepsilon$ is small enough, there exist pairwise nonintersecting loops $\gamma_{I, n} : [0, 1] \to \mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ for each $(I, n) \in \mathcal{I}$ such that $\delta \in \pi_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\pi_{1}(\mathcal{F}_{z_{0}}, \infty)$ in the same way that the product $\prod_{(I, d) \in \mathcal{I}} D_{[\gamma_{I, n}]}$ of Dehn twists on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ does. These loops $\gamma_{I, n}$ each have the property of separating the subset $\{\tilde{\alpha}_{i}(z_{0})\}_{i \in I}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$, and two such loops $\gamma_{I, n}$ and $\gamma_{I', n'}$ are homologous if and only if $I = I'$. b\) The actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$ factor through $\pi_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')}$ and $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ respectively. c\) We have isomorphisms $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})^{(p')} \stackrel{\sim}{\to} G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ inducing an isomorphism of the actions $\rho_{{\mathrm{top}}}^{(p')}$ and $\rho_{{\mathrm{alg}}}^{(p')}$. Moreover, we can choose the isomorphism $\phi$ so that it takes any element represented by a loop on $\mathcal{F}_{z_{0}}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\tilde{\alpha}_{i}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{d} \cup \{\infty\}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d} \cup \{\infty\}$. In other words, the prime-to-$p$ monodromy action $\rho_{{\mathrm{top}}}^{(p')}$ can be described in terms of Dehn twists with respect to loops surrounding certain subsets of the removed points which coalesce at a certain rate as one approaches the center of $B_{\varepsilon}^{*}$; moreover, this is isomorphic to the algebraic action $\rho_{{\mathrm{alg}}}^{(p')}$ via an isomorphism of prime-to-$p$ étale fundamental groups which takes the image of a loop wrapping around a given $a_{i}(z_{0})$ to the image of a loop wrapping around $\alpha_{i}$. We now want to relate this to the action of $G_{K, \mathfrak{p}}$ on the prime-to-$p$ étale fundamental group of a smooth hyperelliptic curve over $K_{\mathfrak{p}}$. Let $\mathfrak{C}$ be a smooth, projective hyperelliptic curve over ${\mathbb{C}}$ of degree $d$ and genus $g$ defined by an equation of the form $y^{2} = \prod_{i = 1}^{d} (x - z_{i})$ for distinct roots $z_{i} \in \mathcal{K}_{\mathfrak{p}}$; if $d$ is odd (resp. even), then $d = 2g + 1$ (resp. $d = 2g + 2$). The hyperelliptic curve $\mathfrak{C}$ comes with a surjective degree-$2$ morphism $\mathfrak{C} \to {\mathbb{P}}_{{\mathbb{C}}}^{1}$ defined by projecting onto the $x$-coordinate. It is well known that this projection ramifies at $\infty$ if and only if $d = 2g + 1$; in this case, we write $z_{2g + 2} = \infty$. Then the projection is ramified at exactly the $(2g + 2)$-element set of $z_{i}$’s. Write $\mathfrak{B} \subset \mathfrak{C}({\mathbb{C}})$ for the set of inverse images of these ramification points in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ ($\mathfrak{B}$ is the set of *branch points* of $\mathfrak{C}$). Clearly the restriction to $\mathfrak{C} \smallsetminus \mathfrak{B}$ of the above projection map yields a finite degree-$2$ étale morphism $\mathfrak{C} \smallsetminus \mathfrak{B} \to {\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$. Choose a basepoint $P$ of the topological space ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{d}\}$ and a basepoint $Q$ of the topological space $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ such that $Q$ lies in the inverse image of $P$. Then after making identifications via Riemann’s Existence Theorem, we get an inclusion and surjections $$\label{eq maps of fundamental groups} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P) \rhd \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}), Q) \twoheadrightarrow H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$$ induced by the maps ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\} \leftarrow \mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B} \hookrightarrow \mathfrak{C}({\mathbb{C}})$ and by identifying the first singular homology group of $\mathfrak{C}({\mathbb{C}})$ with the abelianization of its fundamental group. The inclusion in (\[eq maps of fundamental groups\]) is an inclusion of a characteristic subgroup, so any automorphism of $\widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ induces an automorphism of $\widehat{\pi}_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, P)$. We write $\mathfrak{J}$ for the Jacobian of $\mathfrak{C}$. There is a well-known identification of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ with $H_{1}(\mathfrak{J}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ and in turn with $T_{\ell}(\mathfrak{J})$ for any prime $\ell$. Moreover, the intersection pairing on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ defined above carries over to the canonical Riemann form on the complex abelian variety $\mathfrak{J}({\mathbb{C}})$ and in turn to the Weil pairing $e_{\ell}$ on $T_{\ell}(\mathfrak{J})$ (see the results in [@lang2012introduction §IV.4 and §VIII.1] and in [@mumford1974abelian §24]). Given an element $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, we also write $c$ for the element $c \otimes 1 \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell} = T_{\ell}(\mathfrak{J})$. It is not difficult to show that the action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(\mathfrak{J})$ induced by $\rho_{{\mathrm{alg}}}^{(p')}$ via these identifications is the natural $\ell$-adic Galois action $\rho_{\ell}$: see, for instance, step 5 of the proof of [@yelton2015images Proposition 2.2]. \[dfn symplectic basis\] Given any complex hyperelliptic curve $\mathfrak{C}$ as above, we define the following objects. a\) For any $c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$, the *transvection* with respect to $c$, denoted $T_{c} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$, is the automorphism given by $v \mapsto v + \langle v, c \rangle c$. As above, we may identify $c$ with its image in $T_{\ell}(\mathfrak{J})$, and then $T_{c}$ is identified with the automorphism of $T_{\ell}(\mathfrak{J})$ given by $v \mapsto e_{\ell}(v, c) c$. b\) A *sympectic basis* of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is an ordered basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ satisfying the following properties:    (i) each $a_{i}'$ (resp. each $b_{i}'$) is represented by a loop on $\mathfrak{C}({\mathbb{C}})$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates $\{z_{2i - 1}, z_{2i}\}$ (resp. $\{z_{2i}, ... , z_{2g + 1}\}$) from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$; and    (ii) the (skew-symmetric) intersection pairing $\langle \cdot, \cdot \rangle$ on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is determined by $\langle a_{i}', b_{i}' \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}', a_{j}' \rangle = \langle b_{i}', b_{j}' \rangle = \langle a_{i}', b_{j}' \rangle = 0$ for $1 \leq i < j \leq g$. (See, for instance, the first figure in [@arnold1968remark], or [@farb2011primer Figure 6.1].) We note that a transvection in ${\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$ respects the intersection pairing $\langle \cdot, \cdot \rangle$; similarly, a transvection in ${\mathrm{Aut}}(T_{\ell}(\mathfrak{J}))$ respects the Weil pairing $e_{\ell}$ and thus lies in ${\mathrm{Sp}}(T_{\ell}(\mathfrak{J}))$. From now on, given a complex hyperelliptic curve $\mathfrak{C}$, we fix a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. In order to prove Proposition \[prop Galois action local\] below, we require a couple of lemmas, the first of which is purely topological. \[lemma lifts to Dehn twists\] Assume all of the above notation. a\) Let $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ be the image of a simple closed loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}, P\}$ which separates the set of $\alpha_{i}$’s into two even-cardinality subsets. Then the inverse image of $\gamma$ under the ramified degree-$2$ covering map $\mathfrak{C}({\mathbb{C}}) \to {\mathbb{P}}_{{\mathbb{C}}}$ consists of two connected components, each of which are simple closed loops whose homology classes $\pm c \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ differ by sign. As a particular case, if $\gamma_{i} \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ is the image of a loop which separates $\{z_{i}, z_{2g + 1}\}$ from its complement in $\{z_{j}\}_{j = 1}^{2g + 2}$ for some $i$, then the homology classes of these simple closed loops on $\mathfrak{C}({\mathbb{C}})$ lying above $\gamma_{i}$ are given by $\pm c_{i}'$ for some element $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ which is equivalent modulo $2$ to $$\label {eq c_i} \begin{cases} a_{(i + 1)/2}' + ... + a_{g}' + b_{(i + 1)/2}' & i \ \mathrm{odd}\\ a_{i/2 + 1}' + ... + a_{g}' + b_{i/2 + 1}' & i \ \mathrm{even} \end{cases}$$ b\) With $\gamma$ and $c$ as above, the Dehn twist $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the inclusion and quotient maps in (\[eq maps of fundamental groups\]), which is given by $T_{c}^{2} \in {\mathrm{Aut}}(H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}))$. The essential ideas of this argument are contained in the proof of [@mumford1984tata Lemma 8.12]. The maximal abelian exponent-$2$ quotient of $\pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ is isomorphic to $({\mathbb{Z}}/ 2{\mathbb{Z}})^{2g + 1}$ and is identified with the group of partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into two subsets (where the addition law is given by symmetric differences), by sending the homology class of any loop $\gamma \subset {\mathbb{P}}_{{\mathbb{C}}}^{1}$ to the subsets of $z_{i}$’s lying in each connected component of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \gamma$. We have a homomorphism of homology groups (composed with reduction modulo $2$) $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \to H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ coming from the inclusion of fundamental groups in (\[eq maps of fundamental groups\]). By [@arnold1968remark Lemma 1], this homomorphism factors through $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$. It is clear from this that we have an inclusion of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}/ 2{\mathbb{Z}})$ as the subgroup of $H_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, {\mathbb{Z}}/ 2{\mathbb{Z}})$ which is generated by the partitions $\{z_{2i - 1}, z_{2i}\} \cup \{z_{1}, ... , z_{2i - 2}, z_{2i + 1}, ... , z_{2g + 2}\}$ and $\{z_{2i}, ... , z_{2g + 1}\} \cup \{z_{1}, ... , z_{2i - 1}, z_{2g + 2}\}$ for $1 \leq i \leq g$. It follows from an easy combinatorial argument that this subgroup consists of the partitions of $\{z_{i}\}_{i = 1}^{2g + 2}$ into even-cardinality subsets. Thus, any loop $\gamma \in \pi_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}, P)$ whose image modulo $2$ is such a partition lifts to a loop $c \in \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$; the only other choice of loop on $\mathfrak{C}({\mathbb{C}})$ whose image is $\gamma$ must be based at $\iota(Q)$ and equal to the composition of the path $\gamma$ with $\iota$, where $\iota : \mathfrak{C}({\mathbb{C}}) \to \mathfrak{C}({\mathbb{C}})$ is the only nontrivial deck transformation. Since $\iota$ acts on the homology group by sign change (see [@farb2011primer §7.4]), this loop must be $-c$. If the image of $\gamma$ modulo $2$ is the partition of $\{z_{i}, z_{2g + 1}\}$ and its complement, then the desired statement in (a) follows by the straightforward verification that this partition is equal to the symmetric sum of the partitions corresponding to the basis elements $a_{i}'$ and $b_{i}'$ appearing in the formulas given in (\[eq c\_i\]). Thus, (a) is proved. Every self-homeomorphism of ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{z_{1}, ... , z_{2g + 2}\}$ fixing $P$ lifts uniquely to a self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ under which the image of a loop wrapping around a single point in $\mathfrak{B}$ is also a loop wrapping around a single point in $\mathfrak{B}$. Since the kernel of the quotient map $\pi_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}, Q) \twoheadrightarrow \pi_{1}(\mathfrak{C}({\mathbb{C}}), Q)$ is generated by squares of homotopy classes of such loops, it follows that $D_{[\gamma]}$ induces an automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ via the maps in (\[eq maps of fundamental groups\]). It is clear that the unique self-homeomorphism of $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ fixing $Q$ lifting a representative of $D_{[\gamma]}$ must represent the composition of Dehn twists on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$ with respect to the two lifts of $\gamma$ on $\mathfrak{C}({\mathbb{C}}) \smallsetminus (\mathfrak{B} \cup \{Q\})$, which are $\pm c$. Therefore, the induced automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) = H_{1}(\mathfrak{C}({\mathbb{C}}) \smallsetminus \{Q\}, {\mathbb{Z}})$ is determined by the product of $D_{c}$ and $D_{-c}$. Since Dehn twists do not depend on the orientation of loops, this product is $D_{c}^{2}$. Now (b) follows from the well known fact (see [@farb2011primer Proposition 6.3]) that the Dehn twist $D_{c}$ acts on $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ as the transvection $T_{c}$ since $\mathfrak{C}({\mathbb{C}})$ is a compact smooth manifold. \[lemma branch point infty\] Let $\alpha_{1}, ... , \alpha_{d}$ be distinct elements in $\mathcal{R}_{\mathfrak{p}}$. There exist elements $\alpha_{1}', ... , \alpha_{d + 1}' \in \mathcal{R}_{\mathfrak{p}}$ satisfying the following: \(i) The elements $\alpha_{i}' - \alpha_{j}'$ and $\alpha_{i} - \alpha_{j}$ have equal valuation for $1 \leq i < j \leq d$, and $\alpha_{d + 1}' - \alpha_{i}' \in \mathcal{R}_{\mathfrak{p}}^{\times}$ for $1 \leq i \leq d$. \(ii) Let $\alpha_{d + 1} = \infty \in {\mathbb{P}}_{\mathcal{K}_{\mathfrak{p}}}^{1}$. There is a $\mathcal{K}_{\mathfrak{p}}$-isomorphism $$\psi : {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\} \stackrel{\sim}{\to} {\mathbb{P}}_{\bar{\mathcal{K}}_{\mathfrak{p}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$$ such that the induced isomorphism $\psi_{*} : \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}, \psi^{-1}(\infty)) \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}, \infty)$ yields an isomorphism of $\rho_{{\mathrm{alg}}}$ with the analogously defined representation $\rho_{{\mathrm{alg}}}'$. The isomorphism $\psi_{*}$ takes any element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d + 1}\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates some singleton $\{\alpha_{i}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{d + 1}$ to an element represented by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}', ... , \alpha_{d + 1}'\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the singleton $\{\alpha_{i}'\}$ from its complement in $\{\alpha_{j}'\}_{j = 1}^{d + 1}$. Choose $\beta \in \mathcal{R}_{\mathfrak{p}}^{\times}$ satisfying $\beta \not\equiv \alpha_{j}$ (mod $\pi$) for $1 \leq j \leq d$ (this is always possible because the residue field $\mathcal{R}_{\mathfrak{p}} / (\pi)$ is infinite). Let $\alpha_{j}' = \alpha _{j}\beta / (\beta - \alpha_{j}) \in \mathcal{R}_{\mathfrak{p}}$ for $1 \leq j \leq d$ and $\alpha_{d + 1}' = \beta \in \mathcal{R}_{\mathfrak{p}}$, and let $\psi$ be the $\mathcal{K}_{\mathfrak{p}}$-morphism given by $x \mapsto x\beta / (\beta - x)$. Then property (i) follows from straightforward computation. Since $\psi$ is defined over $\mathcal{K}_{\mathfrak{p}}$, the isomorphism $\psi_{*}$ is equivariant with respect to the action of $G_{K, \mathfrak{p}} = {\mathrm{Gal}}(\bar{\mathcal{K}}_{\mathfrak{p}} / \mathcal{K}_{\mathfrak{p}})$. Moreover, after base change to ${\mathbb{C}}$, $\psi$ is a homeomorphism of punctured Riemann spheres, and the property given in (ii) follows. We are finally ready to state and prove the main result of this section, which is essentially a more concrete version of a particular case of Grothendieck’s criterion for semistable reduction ([@grothendieck1972modeles Proposition 3.5(iv)]). For the statement of the below proposition, we fix a symplectic basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$; the image $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\} \subset T_{\ell}(J)$ forms a symplectic basis of $T_{\ell}(J)$ with respect to the Weil pairing. \[prop Galois action local\] Let $C$ be a hyperelliptic curve of genus $g$ over $\mathcal{K}_{\mathfrak{p}}$ given by an equation of the form $y^{2} = h(x)$ for some squarefree polynomial $h$ of degree $d = 2g + 1$ or $d = 2g + 2$ with distinct roots $\alpha_{1}, ... , \alpha_{d} \in \mathcal{R}_{\mathfrak{p}}$, and let $J$ be its Jacobian. Suppose that exactly $2$ of the roots, $\alpha_{i}$ and $\alpha_{j}$, are equivalent modulo $\pi$, and let $m \geq 1$ be the maximal integer such that $\pi^{m} \mid (\alpha_{i} - \alpha_{j})$. Then for any prime $\ell \neq 2, p$, the image of the natural action of $G_{K, \mathfrak{p}}$ on $T_{\ell}(J)$ is topologically generated by the element $T_{c}^{2m} \in {\mathrm{Sp}}(T_{\ell}(J))$ for some $c \in T_{\ell}(J)$ determined by a loop on ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ whose image separates $\{\alpha_{i}, \alpha_{j}\}$ from the rest of the roots. As a particular case, if $i \leq 2g$ and $j = 2g + 1$, then this $\ell$-adic Galois image is topologically generated by $T_{c_{i}}^{2m}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ equivalent modulo $2$ to $a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2}$ (resp. $a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1}$) if $i$ is odd (resp. even). We first assume that $d = 2g + 2$. Let $\tilde{\alpha}_{1}, ... , \tilde{\alpha}_{2g + 2} \in {\mathbb{C}}[x]$ be the elements constructed from the $\alpha_{i}$’s as above, and define the family $\mathcal{F} \to B_{\varepsilon}^{*}$ as above. It is clear from the hypothesis on the roots $\alpha_{j}$ that the set $\mathcal{I}$ in the statement of Theorem \[thm comparison punctured projective line\] consists of only the elements $(\{i, 2g + 1\}, n)$ for $1 \leq n \leq m$. Theorem \[thm comparison punctured projective line\](a) then implies that a topological generator of $\widehat{\pi}_{1}(B_{\varepsilon}^{*}, z_{0})$ acts on $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ via the monodromy action $\rho_{{\mathrm{top}}}$ as $D_{[\gamma_{i}]}^{m}$, where $\gamma_{i}$ is a loop on $\mathcal{F}_{z_{0}} \smallsetminus \{\infty\}$ whose image in ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ separates the subset $\{\tilde{\alpha}_{i}(z_{0}), \tilde{\alpha}_{2g + 1}(z_{0})\}$ from its complement in $\{\tilde{\alpha}_{j}(z_{0})\}_{j = 1}^{2g + 2}$. Let $\mathfrak{C}$ be the complex hyperelliptic curve of degree $d = 2g + 2$ ramified over ${\mathbb{P}}_{{\mathbb{C}}}^{1}$ at the points $\tilde{\alpha}_{1}(z_{0}), ... , \tilde{\alpha}_{d}(z_{0}) \in {\mathbb{C}}$, which has a symplectic basis $\{a_{1}', ... , a_{g}', b_{1}', ... , b_{g}'\}$ of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$. By Lemma \[lemma lifts to Dehn twists\], the automorphism of $\widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)$ determined by $D_{[\gamma_{i}]}^{m}$ induces the automorphism of $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}$ given by $T_{c_{i}}^{2m} : v \mapsto v + 2m \langle v, c_{i}' \rangle c_{i}'$, where $c_{i}' \in H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}})$ is equivalent modulo $2$ to the formula in (\[eq c\_i\]). Now parts (b) and (c) of Theorem \[thm comparison punctured projective line\] say that there is an isomorphism $\phi : \widehat{\pi}_{1}(\mathcal{F}_{z_{0}}, \infty)^{(p')} \stackrel{\sim}{\to} \widehat{\pi}_{1}({\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{d}\}, \infty)^{(p')}$ making the action $\rho_{{\mathrm{top}}}$ isomorphic to the Galois action $\rho_{{\mathrm{alg}}}$. Since $p \neq 2$ and $\mathfrak{C}({\mathbb{C}}) \smallsetminus \mathfrak{B}$ and $C({\mathbb{C}}) \smallsetminus \{(\alpha_{i}, 0)\}_{i = 1}^{2g + 2}$ are the only degree-$2$ covers of $\mathcal{F}_{z_{0}}$ and ${\mathbb{P}}_{{\mathbb{C}}}^{1} \smallsetminus \{\alpha_{1}, ... , \alpha_{2g + 2}\}$ respectively, we see that the isomorphism $\phi$ induces an isomorphism $H_{1}(\mathfrak{C}({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')} \stackrel{\sim}{\to} H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which we also denote by $\phi$. It is clear from the property of $\phi$ given in Theorem \[thm comparison punctured projective line\](c) and from our characterization of homology groups with coefficients in ${\mathbb{Z}}/ 2{\mathbb{Z}}$ in the proof of Lemma \[lemma lifts to Dehn twists\] that $\phi(a_{j}') \equiv a_{j}$ and $\phi(b_{j}') \equiv b_{j}$ (mod $2$) for $1 \leq j \leq g$. In the case that $d = 2g + 1$, we get the same results by applying Lemma \[lemma branch point infty\], which allows us to replace $\alpha_{1}, ... , \alpha_{2g + 1}, \alpha_{2g + 2} := \infty$ with elements $\alpha_{1}', ... , \alpha_{2g + 2}' \in \mathcal{R}_{\mathfrak{p}}$ whose differences have the same valuations with respect to $\pi$. Putting this all together, we see that the tame Galois action on $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by $\rho_{{\mathrm{alg}}}$ sends a generator of $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ to the automorphism of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ given by $v \mapsto v + 2m \langle v, c_{i} \rangle_{\phi} c_{i}$ for some $c_{i} \in H_{1}(C({\mathbb{C}}), \infty) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ which is equivalent modulo $2$ to the formula given in the statement. Here $\langle \cdot, \cdot \rangle_{\phi}$ is the skew-symmetric pairing on $H_{1}(C({\mathbb{Z}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ induced by the interection pairing on $H_{1}(\mathfrak{C}, {\mathbb{Z}})$ via $\phi$; note that $\langle \cdot, \cdot \rangle_{\phi}$ is normalized so that $\langle H_{1}(C({\mathbb{C}}), {\mathbb{Z}}), H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \rangle_{\phi} = {\mathbb{Z}}$. As above, we identify the maximal pro-$\ell$ quotient of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes \widehat{{\mathbb{Z}}}^{(p')}$ with $T_{\ell}(J)$ and see that the natural $\ell$-adic Galois action $\rho_{\ell}$ factors through $G_{K, \mathfrak{p}}^{{\mathrm{tame}}}$ and takes a generator to the automorphism of $T_{\ell}(J)$ given by $v \mapsto v + 2m\langle v, c_{i} \rangle_{\phi} c_{i}$. But this automorphism must lie in ${\mathrm{Sp}}(T_{\ell}(J))$ by the Galois equivarience of the Weil pairing $e_{\ell}$ and the fact that $\mathcal{K}_{\mathfrak{p}}$ contains all $\ell$-power roots of unity. It is now an easy exercise to verify that this implies that $\langle v, c_{i} \rangle_{\phi} = \pm e_{\ell}(v, c_{i})$ for all $v \in T_{\ell}(J)$, and so the image of $\rho_{\ell}$ is generated by $T_{c_{i}}^{2m} : v \mapsto v + e_{\ell}(v, c_{i}) c_{i}$, as desired. In order to prove Theorems \[thm main1\] and \[thm main2\], we will put some local data together to show using Proposition \[prop Galois action local\] that the $\ell$-adic Galois image contains $\{T_{c_{1}}^{2m}, ... , T_{c_{2g}}^{2m}\}$ for homology classes $c_{i}$ as above and a certain integer $m$. Therefore, it is of interest to investigate the subgroup that this set generates. Unfortunately, since the $c_{i}$’s are only known modulo $2$, not much can be deduced except if $\ell = 2$. In this case, the subgroup of ${\mathrm{Sp}}(T_{2}(J))$ generated by the above set can be determined, which is the goal of the next section. Subgroups generated by powers of transvections {#S3} ============================================== For this section, we let $M$ be a free ${\mathbb{Z}}$-module of rank $2g$, equipped with a nondegenerate skew-symmetric ${\mathbb{Z}}$-bilinear pairing $\langle \cdot, \cdot \rangle : M \times M \to {\mathbb{Z}}$. For any ring $A$, this pairing induces in an obvious way a nondegenerate skew-symmetric $A$-bilinear pairing on the free $A$-module $M \otimes A$ which we also denote by $\langle \cdot, \cdot \rangle$. We write ${\mathrm{Sp}}(M \otimes A)$ for the symplectic group of $A$-automorphisms of $M \otimes A$ which respect this pairing; for any integer $N \geq 1$, we write $\Gamma(N)$ for the level-$N$ congruence subgroup of ${\mathrm{Sp}}(M \otimes A)$ as defined in §\[S1\]. For any element $c \in M \otimes A$, we write $T_{c} \in {\mathrm{Sp}}(M \otimes A)$ for the transvection with respect to $c$ and the pairing $\langle \cdot, \cdot \rangle$, as in Definition \[dfn symplectic basis\](a). We fix a basis $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ of $M$ and assume that the pairing is determined by $\langle a_{i}, b_{i} \rangle = -1$ for $1 \leq i \leq g$ and $\langle a_{i}, a_{j} \rangle = \langle b_{i}, b_{j} \rangle = \langle a_{i}, b_{j} \rangle = 0$ for $1 \leq i < j \leq g$. We also write $a_{i}, b_{i} \in M \otimes A$ for the elements $a_{i} \otimes 1$ and $b_{i} \otimes 1$ respectively for $1 \leq i \leq g$. Our main goal is to prove the following purely algebraic result. \[prop open subgroup\] Let $c_{1}, ... , c_{2g} \in M$ be elements such that $c_{i}$ is equivalent modulo $2$ to $$\begin{cases} a_{(i + 1)/2} + ... + a_{g} + b_{(i + 1)/2} & i \ \mathrm{odd}\\ a_{i/2 + 1} + ... + a_{g} + b_{i/2 + 1} & i \ \mathrm{even} \end{cases}$$ for $1 \leq i \leq 2g$. Then given integers $n, n' \geq 1$, the subgroup $G \subset \Gamma(2^{n})$ generated by the elements $T_{c_{1}}^{2^{n}}, ... , T_{c_{2g - 1}}^{2^{n}}, T_{c_{2g}}^{2^{n'}} \in \Gamma(2^{n})$ contains $\Gamma(2^{2n})$ (resp. $\Gamma(2^{n + n'})$) if $n' \leq n$ (resp. if $n' > n$). In particular, as a topological subgroup of ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, $G$ is open, and its associated Lie algebra $\mathfrak{g}$ coincides with the $2$-adic symplectic Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$. Note that the image of any transvection $T \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$ under the logarithm map is $T - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and that $\mathfrak{g}$ is generated as a Lie algebra by the logarithms of the transvections $T_{c_{i}}$. Therefore, in order to prove the second statement of this proposition, it suffices to show that $\{T_{c_{s}} - 1\}_{1 \leq s \leq 2g}$ generates the full Lie algebra $\mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$. However, in order to get both statements, we will prove something slightly stronger which is given by the following lemma. \[lemma transvection commutator basis\] Let $t_{i} = T_{c_{i}} - 1 \in \mathfrak{sp}(M \otimes {\mathbb{Q}}_{2})$ and let $\bar{t}_{i}$ denote the reduction of $t_{i}$ modulo $2$ for $1 \leq i \leq 2g$. Then an ${\mathbb{F}}_{2}$-basis for $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ is given by the set $\{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{1 \leq i < j \leq 2g}$. Let $\bar{c}_{i}$ denote the reductions modulo $2$ of the basis element $c_{i}$ for $1 \leq i \leq 2g$. We know from the characterization of the $c_{i}$’s modulo $2$ given in the statement that $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ is an ordered basis of $M \otimes {\mathbb{F}}_{2}$ with $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1 \in {\mathbb{F}}_{2}$ for $1 \leq i < j \leq 2g$. With respect to this basis, we may view $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ as the Lie algebra $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$. Now it is well known that $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ is a $(2g^{2} + g)$-dimensional vector space over ${\mathbb{F}}_{2}$. Since there are $2g^{2} + g$ elements listed in the set of $\bar{t}_{s}$’s and their commutators, it suffices to prove that this set is linearly independent over ${\mathbb{F}}_{2}$, using only the fact that $\langle \bar{c}_{i}, \bar{c}_{j} \rangle = \langle \bar{c}_{j}, \bar{c}_{i} \rangle = 1$ for $1 \leq i < j \leq 2g$. In order to do this, we first note that for any elements $a, b \in M \otimes {\mathbb{F}}_{2}$, the commutator of the logarithms of the transvections with respect to $a$ and to $b$ is the ${\mathbb{F}}_{2}$-linear operator in $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $v \mapsto \langle v, a \rangle \langle a, b \rangle b + \langle v, b \rangle \langle b, a \rangle a$ for all $v \in M \otimes {\mathbb{F}}_{2}$. Therefore, for $1 \leq i < j \leq 2g$, the linear operator $[\bar{t}_{i}, \bar{t}_{j}] = [\bar{t}_{j}, \bar{t}_{i}]$ is given by $$\label{eq commutator} v \mapsto \langle v, \bar{c}_{i} \rangle \bar{c}_{j} + \langle v, \bar{c}_{j} \rangle \bar{c}_{i}.$$ We compute using this formula that the upper left-hand $2 \times 2$ submatrices of $\bar{t}_{1}$, $\bar{t}_{2}$, and $[\bar{t}_{1}, \bar{t}_{2}]$ respectively are $$\label{eq base case} \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix},$$ and therefore these elements of $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ are linearly independent over ${\mathbb{F}}_{2}$. This in particular proves the statement for $g = 1$. Now assume inductively that $g \geq 2$ and that the statement is true for $g - 1$. Clearly, $\{\bar{c}_{3}, ... , \bar{c}_{2g}\}$ generates a $2(g - 1)$-dimensional subspace $\mathfrak{g}'$ of $M \otimes {\mathbb{F}}_{2}$ which it generates, and the intersection pairing of any two distinct elements of this set is $1$. Therefore, the inductive assumption implies that the $(2(g - 1)^{2} + (g - 1))$-element set $\{\bar{t}_{i}\}_{3 \leq i \leq g} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent; in fact, $\mathfrak{g}'$ is a copy of $\mathfrak{sp}_{2g - 2}({\mathbb{F}}_{2})$ lying inside $\mathfrak{sp}_{2g}({\mathbb{F}}_{2})$. We first claim that the subset $S_{1} := \{\bar{t}_{i}\}_{1 \leq i \leq 2g} \cup \{[\bar{t}_{1}, \bar{t}_{2}]\} \cup \{[\bar{t}_{i}, \bar{t}_{j}]\}_{3 \leq i < j \leq 2g}$ is linearly independent. To see this, note that we have already shown that $\{\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}]\}$ is linearly independent, and from the fact that the matrices in $\mathfrak{g}'$ have all $0$’s in their first and second rows, it is clear that $\bar{t}_{1}, \bar{t}_{2}, [\bar{t}_{1}, \bar{t}_{2}] \notin \mathfrak{g}'$. The elements of $S_{1}$ generating $\mathfrak{g}'$ are linearly independent by the inductive assumption, and so the claim follows. We next claim that the subset $S_{2} := \{[\bar{t}_{1}, \bar{t}_{i}]\}_{3 \leq i \leq 2g} \cup \{[\bar{t}_{2}, \bar{t}_{i}]\}_{3 \leq i \leq 2g}$ is linearly independent. In order to show this, consider a linear combination of elements of the set $S_{2}$ written as $\sum_{i = 3}^{2g} \beta_{i} [\bar{t}_{1}, \bar{t}_{i}] + \sum_{i = 3}^{2g} \gamma_{i} [\bar{t}_{2}, \bar{t}_{i}]$ with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Using the formula in (\[eq commutator\]), we see that this is the linear operator $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ given by $$u : v \mapsto \sum_{i = 3}^{2g} \beta_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \bar{c}_{i}) + \sum_{i = 3}^{2g} \gamma_{i} (\langle v, \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \bar{c}_{i})$$ $$\label{eq commutator2} = \langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle \bar{c}_{1} + \langle v, \bar{c}_{1} \rangle \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} + \langle v, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle \bar{c}_{2} + \langle v, \bar{c}_{2} \rangle \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i}.$$ Suppose that $u = 0$. Assume that $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \neq 0$. Then by the nondegeneracy of the symplectic pairing, we can choose $v \in M \otimes {\mathbb{F}}_{2}$ such that $\langle v, \sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} \rangle = 1$. Then $u(v)$ written as a linear combination using the basis $\{\bar{c}_{1}, ... , \bar{c}_{2g}\}$ has $\bar{c}_{1}$-coefficient equal to $1$, a contradiction because $u(v) = 0$. Therefore, $\sum_{i = 3}^{2g} \beta_{i} \bar{c}_{i} = 0$, which implies that $\beta_{3} = ... = \beta_{2g} = 0$. Now assume that $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \neq 0$. Then similarly, we can choose $w \in M \otimes {\mathbb{F}}_{2}$ such that $\langle w, \sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} \rangle = 1$ and get that $u(w)$ has $\bar{c}_{2}$-coefficient equal to $1$, a contradiction because $u(w) = 0$. Therefore, $\sum_{i = 3}^{2g} \gamma_{i} \bar{c}_{i} = 0$, which implies that $\gamma_{3} = ... = \gamma_{2g} = 0$, and so $S_{2}$ is linearly independent. We finally claim that if a linear combination of elements in $S_{1}$ is equal to a linear combination of elements in $S_{2}$, then these linear combinations must be trivial. Since the full set of $t_{i}$’s and their commutators coincides with $S_{1} \cup S_{2}$, this implies the statement of the proposition. Let $u \in \mathfrak{sp}_{2g}({\mathbb{F}}_{2})$ be a linear combination of elements in $S_{2}$, written as in (\[eq commutator2\]) with $\beta_{i}, \gamma_{i} \in {\mathbb{F}}_{2}$. Now it is clear from the formula there that for each of the matrices in $S_{2}$, the $(1, j)$th entries are all equal and the $(2, j)$th entries are all equal for $3 \leq j \leq 2g$. Then we see by putting $\bar{c}_{3}, ... , \bar{c}_{2g}$ into this formula that $\beta_{3} = ... = \beta_{2g}$ and $\gamma_{3} = ... = \gamma_{2g}$. But then we have $$\langle \bar{c}_{1}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{1}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \beta_{i}\bar{c}_{i} \rangle = \langle \bar{c}_{2}, \sum_{i = 3}^{2g} \gamma_{i}\bar{c}_{i} \rangle = 0,$$ and we get that the upper left-hand $2 \times 2$ submatrix of $u$ is the $0$ matrix. This is also true of any matrix in $\mathfrak{g}'$, so we know from what was given in (\[eq base case\]) that $t_{1}$, $t_{2}$, and $[t_{1}, t_{2}]$ do not appear when $u$ is written as a linear combination of elements in $S_{1}$, and therefore we have $u \in \mathfrak{g}'$. As was noted above, every matrix in $\mathfrak{g}'$ has all $0$’s in its first and second rows. Thus, for any $v \in M \otimes {\mathbb{F}}_{2}$, when $u(v)$ is written as a linear combination of $\bar{c}_{i}$’s, the $\bar{c}_{1}$-coefficient and the $\bar{c}_{2}$-coefficient are both $0$. Now by the same argument that was used for the previous claim, we have $\beta_{3} = ... = \beta_{2g} = \gamma_{3} = ... = \gamma_{2g} = 0$. Therefore $u = 0$, as desired. Let $N = 2n$ (resp. $N = n + n'$) if $n' \leq n$ (resp. if $n' > n$); that is, $N = n + \max \{n, n'\}$. In order to show that $G$ contains the congruence subgroup $\Gamma(2^{N})$, since $G$ is closed, it suffices to show that the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for each integer $m \geq 1$. We claim that in fact we only need to show this for $m = 1$. Indeed, for any $m \geq 1$, the restriction of the logarithm map to $\Gamma(2)$ sends each element $T \in \Gamma(2^{N + m - 1})$ to an element of $\mathfrak{sp}_{2g}({\mathbb{Z}}_{2})$ which is equivalent to $T - 1$ modulo $2^{N + m}$ since $(T - 1)^{2} \equiv 0$ modulo $2^{N + m}$. In this way, one verifies that there is an isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$, given by sending an element $t \in \mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $1 + 2^{N + m - 1}\tilde{t} \in {\mathrm{Sp}}(M \otimes {\mathbb{Z}}_{2})$, where $\tilde{t}$ is any operator in ${\mathrm{Sp}}(M \otimes {\mathbb{Z}}/ 2^{N + m}{\mathbb{Z}})$ whose image modulo $2$ is $t$. In particular, each $\Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$ is an elementary abelian group of exponent $2$ and rank $2g^{2} + g$ (this is also known from the proof of [@sato2010abelianization Corollary 2.2]). Moreover, it is easy to check that for each $m \geq 1$, the map sending each matrix in $\Gamma(2^{N})$ to its $2^{m - 1}$th power induces a group isomorphism $\Gamma(2^{N}) / \Gamma(2^{N + 1}) \stackrel{\sim}{\to} \Gamma(2^{N + m - 1}) / \Gamma(2^{N + m})$. It follows that if the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$, then the image of $G$ modulo $2^{N + m}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + m})$ for all $m \geq 1$. Thus, in order to prove the proposition, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains $\Gamma(2^{N}) / \Gamma(2^{N + 1})$. As above, let $t_{i}$ denote $T_{c_{i}} - 1$ for $1 \leq i \leq 2g$. Since Lemma \[lemma transvection commutator basis\] says that the image modulo $2$ of $\{t_{i}\}_{1 \leq i \leq 2g} \cup \{[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g}$ is a basis of $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, it suffices to show that the image of $G$ modulo $2^{N + 1}$ contains the images of the elements in $\{1 + 2^{N}t_{i}\}_{1 \leq i \leq 2g} \cup \{1 + 2^{N}[t_{i}, t_{j}]\}_{1 \leq i < j \leq 2g} \subset \Gamma(2^{2n})$. For ease of notation, we write $n'' = \max \{n, n'\}$. Clearly $G$ is generated by $\{1 + 2^{n}t_{i}\}_{1 \leq i \leq 2g - 1} \cup \{1 + 2^{n'}t_{2g}\}$. We verify using the property $t_{i}^{2} = 0$ that $(1 + 2^{n}t_{i})^{2^{n''}} \equiv 1 + 2^{n + n''}t_{i}$ (mod $2^{n + n'' + 1}$) for $1 \leq i \leq 2g - 1$; $(1 + 2^{n'}t_{2g})^{2^{n + n'' - n'}} \equiv 1 + 2^{n + n''}t_{2g}$ (mod $2^{n + n'' + 1}$); and $$(1 + 2^{n}t_{i})(1 + 2^{n''}t_{j})(1 + 2^{n}t_{i})^{-1}(1 + 2^{n''}t_{j})^{-1} \equiv 1 + 2^{n + n''}[t_{i}, t_{j}] \ (\mathrm{mod} \ 2^{n + n'' + 1})$$ for $1 \leq i < j \leq 2g$. Thus, $G \supset \Gamma(2^{n + n''})$, as desired. We now state and prove another proposition which will be needed only for the last statements of Theorems \[thm main1\] and \[thm main2\], which pertain to the case that the hyperelliptic curve has degree $4$. \[prop open subgroup degree 4\] Assume the notation of Proposition \[prop open subgroup\] and that $g = 1$ and $n' = n$, and let $c_{3} \in M$ be an element which is equivalent modulo $2$ to $a_{1}$. Then the subgroup of $\Gamma(2^{n})$ generated by $G$ and the element $T_{c_{3}}^{2^{n}}$ coincides with $\Gamma(2^{n})$. By the argument used at the beginning of the proof of Proposition \[prop open subgroup\], we only need to show that the images of $T_{c_{1}}^{2^{n}}, T_{c_{2}}^{2^{n}}, T_{c_{3}}^{2^{n}}$ modulo $2^{n + 1}$ generate $\Gamma(2^{n}) / \Gamma(2^{n + 1})$. Using the isomorphism from the additive group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$ to $\Gamma(2^{n}) / \Gamma(2^{n + 1})$ which was established in that proof, we see that it suffices to show that $\{\bar{t}_{1}, \bar{t}_{2}, \bar{t}_{3}\}$ is linearly independent and thus generates the rank-$3$ elementary abelian group $\mathfrak{sp}(M \otimes {\mathbb{F}}_{2})$, where $\bar{t}_{i}$ denotes the image modulo $2$ of $T_{c_{i}} - 1$ for $1 \leq i \leq 3$. But this is clear from noting that $c_{3} \equiv c_{1} + c_{2}$ (mod $2$) and writing out these linear operators as matrices with respect to an ordered basis of $M \otimes {\mathbb{F}}_{2}$ consisting of the images of $c_{1}$ and $c_{2}$. Proof of main theorems and a further result {#S4} =========================================== The main goal of this section is to prove Theorems \[thm main1\] and \[thm main2\]. Our strategy for this is to put together local results with respect to several primes of $K$ using Proposition \[prop Galois action local\] to get several elements in $G_{2}$ and then to use Proposition \[prop open subgroup\] to determine that the subgroup of $G_{2}$ generated by these elements contains a certain congruence subgroup of the symplectic group. This is realized by the following theorem, which can be used in a far more general situation than required for Theorems \[thm main1\] and \[thm main2\] and is therefore a useful result in its own right. \[thm several primes\] Let $J$ be the Jacobian of the hyperelliptic curve $C$ over $K$ of genus $g$ and degree $d'$ with defining equation $y^{2} = \prod_{i = 1}^{d'} (x - \alpha_{i})$ for some elements $\alpha_{i} \in \mathcal{O}_{K}$ for $1 \leq i \leq d'$, and define the $2$-adic Galois image $G_{2}$ as above. Suppose that there are distinct primes $\mathfrak{p}_{1}$, ... , $\mathfrak{p}_{d' - 1}$ of $K$ not lying over $(2)$ and such that for $1 \leq i \leq d' - 1$, the only two $\alpha_{j}$’s which are equivalent modulo $\mathfrak{p}_{i}$ are $\alpha_{i}$ and $\alpha_{d'}$; let $m_{i} \geq 1$ be the maximal integer such that $\mathfrak{p}_{i}^{m_{i}} \mid (\alpha_{d'} - \alpha_{i})$. Let $n = \max \{v_{2}(m_{i})\}_{i = 1}^{d' - 2}$. If $n' := v_{2}(m_{d' - 1}) \leq n$ or if $d' = 2g + 2$, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{2n + 2})$. Otherwise, we have $\Gamma(2) \supseteq G_{2} \supsetneq \Gamma(2^{n + n' + 2})$. Moreover, if $d' = 4$, then $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$. Using the embedding $\bar{K} \hookrightarrow {\mathbb{C}}$ fixed at the beginning of §\[S2\], we identify $T_{\ell}(J)$ with $H_{1}(C({\mathbb{C}}), {\mathbb{Z}}) \otimes {\mathbb{Z}}_{\ell}$ as we did in §\[S2\]. Let $\{a_{1}, ... , a_{g}, b_{1}, ... , b_{g}\}$ be a symplectic basis of $H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ as in Definition \[dfn symplectic basis\](b) with $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g + 1}, \infty)$ if $d'$ is odd and $(z_{1}, ... , z_{2g + 2}) = (\alpha_{1}, ... , \alpha_{2g}, \alpha_{2g + 2}, \alpha_{2g + 1})$ if $d'$ is even. Then Proposition \[prop Galois action local\] says that for $1 \leq i \leq 2g$, the image of $G_{K, \mathfrak{p}_{i}} \subset G_{K}$ in ${\mathrm{GSp}}(T_{2}(J))$ contains $T_{c_{i}}^{2m_{i}}$, where $T_{c_{i}} \in {\mathrm{Sp}}(T_{2}(J))$ is the transvection with respect to an element $c_{i} \in T_{2}(J)$ which is equivalent modulo $2$ to the formula given in (\[eq c\_i\]). Meanwhile, if $d'$ is even, the image of $G_{K, \mathfrak{p}_{2g + 1}}$ contains $T_{c_{d'}}^{2m_{2g + 1}}$ for some other element $c_{d'} \in T_{2}(J)$ corresponding to the lift of a loop separating $\{\alpha_{2g + 1}, \alpha_{2g + 2}\}$ from its complement in $\{\alpha_{j}\}_{j = 1}^{2g + 2}$. In any case, we have $T_{c_{1}}^{2m_{1}}, ... , T_{c_{2g}}^{2m_{2g}} \in G_{2}$; note that $m_{i} / 2^{v_{2}(m_{i})} \in {\mathbb{Z}}_{2}^{\times}$ and so by taking suitable powers we get that $T_{c_{1}}^{2^{v_{2}(m_{1}) + 1}}, ... , T_{c_{2g}}^{2^{v_{2}(m_{2g}) + 1}} \in G_{2}$. It then follows from Proposition \[prop open subgroup\] applied to $M := H_{1}(C({\mathbb{C}}), {\mathbb{Z}})$ that $G_{2} \supset \Gamma(2^{2n + 2})$ if $n' \leq n$ or if $d' = 2g + 2$ and $G_{2} \supset \Gamma(2^{n + n' + 2})$ otherwise. If $d' = 4$, then in particular, we have $T_{c_{1}}^{2^{\max\{n, n'\} + 1}}, T_{c_{2}}^{2^{\max\{n, n'\} + 1}}, T_{c_{3}}^{2^{\max\{n, n'\} + 1}} \in G_{2}$ with $c_{1}, c_{2}, c_{3}$ as described above. Clearly we have $c_{1} \equiv a_{1} + b_{1}$, $c_{2} \equiv b_{1}$, and $c_{3} \equiv a_{1}$ (mod $2$), and so Proposition \[prop open subgroup degree 4\] implies that $G_{2} \supseteq \Gamma(2^{\max \{n, n'\} + 1})$. \[rmk index bound\] We note that it is generally not difficult to compute the order of $G / \Gamma(2^{2n + 2})$ or $G / \Gamma(2^{n + n' + 2})$, where $G \subseteq G_{2}$ is the subgroup generated by the powers of transvections given in the proof above. Therefore, one may improve the upper bound for $[\Gamma(2) : G_{2}]$ which directly follows from the statement of Theorem \[thm several primes\]. For example, we have $[\Gamma(2) : G_{2}] \leq 2^{(2n + 1)(2g^{2} + g) - (n + 1)(d' - 1)}$ in the case that $n = n'$. [(Legendre curve)]{} \[ex Legendre\] For any $\lambda \in \mathcal{O}_{K} \smallsetminus \{0, 1\}$, let $E_{\lambda}$ be the elliptic curve over $K$ given by $y^{2} = x(x - 1)(x - \lambda)$. Suppose that there exist (necessarily distinct) primes $\mathfrak{p}_{1}$ and $\mathfrak{p}_{2}$ of $K$ not lying over $(2)$ and integers $m_{1}, m_{2}, \geq 1$ such that $\mathfrak{p}_{1}^{m_{1}}$ exactly divides $(\lambda)$ and $\mathfrak{p}_{2}^{m_{2}}$ exactly divides $(\lambda - 1)$. Then Theorem \[thm several primes\] tells us that the $2$-adic Galois image $G_{2}$ (strictly) contains $\Gamma(2^{v_{2}(m_{1}) + v_{2}(m_{2}) + 2})$. In the case that $m_{1} = m_{2} = 1$ (e.g. $K = {\mathbb{Q}}$, $\lambda = 6$, $\mathfrak{p}_{1} = (3)$, $\mathfrak{p}_{2} = (5)$), we get $\Gamma(2) \supset G_{2} \supsetneq \Gamma(4)$ and can therefore directly compute the precise subgroup $G_{2} \cap {\mathrm{Sp}}(T_{2}(E_{\lambda})) \subset \Gamma(2)$ using the well-known fact that the $4$-division field $K(E_{\lambda}[4])$ is generated over $K$ by $\{\sqrt{-1}, \sqrt{\lambda}, \sqrt{\lambda - 1}\}$. It is also possible to prove the statement of Proposition \[prop Galois action local\], and hence the subgroup $G_{2}$ for this example for the particular cases of $C = E_{\lambda}$ over $\mathcal{K}_{\mathfrak{p}_{1}}$ and over $\mathcal{K}_{\mathfrak{p}_{2}}$ using formulas for generators of $2$-power division fields of $E_{\lambda}$ over $K$ found in [@yelton2015dyadic], as the author has done in [@yelton2015hyperelliptic §3.4]. We now prove Theorems \[thm main1\] and \[thm main2\] together. First assume the notation and hypotheses of Theorem \[thm main1\]. Let $\alpha_{1}, ... , \alpha_{d}$ denote the roots of $f$, and write $L = K(\alpha_{1}, ... , \alpha_{d})$ for the splitting field of $f$ over $K$. Note that ${\mathrm{Gal}}(L / K)$ acts transitively on the $\alpha_{i}$’s since $f$ is irreducible. It then follows from the well-known description of the $2$-division field of a hyperelliptic Jacobian (see for instance [@mumford1984tata Corollary 2.11]) that $G_{K}$ does not fix the $2$-torsion points of $J$ and so $G_{2}$ is not contained in $\Gamma(2)$, while the image of ${\mathrm{Gal}}(\bar{K} / L)$ under $\rho_{2}$ coincides with $G_{2} \cap \Gamma(2)$. The fact that $\mathfrak{p} \nmid (2\Delta)$ implies that the extension $L / K$ is not ramified at $\mathfrak{p}$, and so $\mathfrak{p}$ splits into a product $\mathfrak{p}_{1} ... \mathfrak{p}_{r}$ of distinct primes in $L$, for some integer $r$ dividing $[L : K]$. Then since $\mathfrak{p}^{m}$ exactly divides $(f(\lambda)) = \prod_{i = 1}^{d}(\lambda - \alpha_{i})$, we have $\mathfrak{p}_{i} \mid (\lambda - \alpha_{i})$ for some $i$; we assume without loss of generality that $\mathfrak{p}_{1} \mid (\lambda - \alpha_{1})$. Then $\mathfrak{p}_{1}$ cannot divide $(\lambda - \alpha_{i})$ for any $i \in \{2, ... , d\}$, because otherwise for such an $i$ we would have $\mathfrak{p}_{1} \mid (\alpha_{i} - \alpha_{1}) \mid (\Delta)$, which contradicts the hypothesis that $\mathfrak{p} \nmid (2\Delta)$. It follows that $\mathfrak{p}_{1}^{m}$ exactly divides $(\lambda - \alpha_{1})$. Then by applying elements of ${\mathrm{Gal}}(L / K)$ that take $\alpha_{1}$ to each $\alpha_{i}$, we get other primes lying over $\mathfrak{p}$ whose $m$th powers exactly divide the ideals $(\lambda - \alpha_{i})$; we assume without loss of generality that $\mathfrak{p}_{i}^{m}$ exactly divides $(\lambda - \alpha_{i})$ for $1 \leq i \leq d$. Now since the $\mathfrak{p} \nmid (2\Delta)$ hypothesis implies that none of the $\mathfrak{p}_{i}$’s lie over $(2)$, we can apply Theorem \[thm several primes\] with $K$ replaced by $L$, $d' = d + 1$, $\alpha_{d'} = \lambda$, and $m_{1} = ... = m_{d' - 1} = m$ to get the statement of Theorem \[thm main1\]. Now assume the notation and hypotheses of Theorem \[thm main2\]. Then the argument for proving Theorem \[thm main2\] is the same except that when applying Theorem \[thm several primes\] we choose $\mathfrak{p}_{d' - 1}$ to be a prime of $L$ lying over $\mathfrak{p}'$, and we put $d' = d + 2$, $\alpha_{d'} = \lambda$, $\alpha_{d' - 1} = \lambda'$, $m = m_{1} = ... = m_{d' - 2}$, and $m' = m_{d' - 1}$. Realizing uniform boundedness along one-parameter families {#S5} ========================================================== Fix an irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$ of degree $d \geq 2$. Cadoret and Tamagawa have shown in [@cadoret2012uniform Theorems 1.1 and 5.1] that for all but finitely many $\lambda \in K$, the $\ell$-adic Galois image $G_{\ell, \lambda}$ associated to the Jacobian of the curve given by $y^{2} = f(x)(x - \lambda)$ is open in the $\ell$-adic Galois image $G_{\ell, \eta}$ associated to the generic fiber of the family parametrized by $\lambda$. These theorems also assert that there is some integer $B \geq 1$ depending only on $f$ and $\ell$ such that the index of $G_{\ell, \lambda}$ in $G_{\ell, \eta}$ is bounded by $B$ for all but finitely many $\lambda \in K$. The following theorem recovers the openness result for $\ell = 2$ when $d \geq 4$ and explicitly provides the aforementioned uniform bound when $d$ is even, under the assumption that $K$ has class number $1$. (Note that for the elliptic curve case, where $d \in \{ 2, 3\}$, such openness results are already known from the celebrated Open Image Theorem of Serre given by [@serre1989abelian IV-11], while uniform bounds are given by [@arai2008uniform Theorem 1.3].) It is interesting to note that Faltings’ Theorem is used both in the proof of [@cadoret2012uniform Theorem 1.1] and in our proof of the theorem below. \[thm uniform bounds\] Assume that $\mathcal{O}_{K}$ is a PID. Let $f \in \mathcal{O}_{K}[x]$ be an irreducible monic polynomial of degree $d \geq 3$ with discriminant $\Delta$. For each $\lambda \in K$, let $J_{\lambda}$ denote the Jacobian of the hyperelliptic curve $C_{\lambda}$ with defining equation $y^{2} = f(x)(x - \lambda)$, and write $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ for the image of the associated $2$-adic Galois representation. a\) If $d \geq 4$, then the Lie subgroup $G_{2, \lambda} \subseteq {\mathrm{GSp}}(T_{2}(J_{\lambda}))$ is open for all but finitely many $\lambda \in K$. b\) If $d = 3$ (resp. if $d \geq 5$), then $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{4}$ (resp. all but finitely many $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{2}$). c\) If $d = 4$, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(16)$ for all but finitely many $\lambda \in K$. If $d \geq 6$ is even, then we have $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ for all but finitely many $\lambda \in K$. Let $\Sigma \subset K^{\times}$ denote the multiplicative subgroup generated by the elements $\xi \in \mathcal{O}_{K}$ such that $\xi$ is divisible only by primes which divide $2\Delta$. We claim that $\Sigma$ is finitely generated. Indeed, there is an obvious map from $\Sigma$ to the free ${\mathbb{Z}}$-module formally generated by the (finite) set of prime ideals of $\mathcal{O}_{K}$ which divide $2\Delta$, and its kernel is the unit group $\mathcal{O}_{K}^{\times}$, which is also well known to be finitely generated. Choose any $\lambda \in K$, which we may write as $\mu / \nu$ for some coprime $\mu, \nu \in \mathcal{O}_{K}$, because $\mathcal{O}_{K}$ is a PID. Let $h(x) = \nu^{d}f(\nu^{-1} x)$, which is a monic polynomial in $\mathcal{O}_{K}[x]$; note that the discriminant of $h$ is equal to $\nu^{2d^{2} - d}\Delta$. Then there is a $K(\sqrt{\nu})$-isomorphism from $C_{\lambda}$ to the hyperelliptic curve $C_{\lambda}'$ whose defining equation is $y^{2} = \nu^{d}f(\nu^{-1}x)(x - \mu) \in \mathcal{O}_{K}[x]$, given by $(x, y) \mapsto (\nu x, \nu^{(d + 1) / 2}y)$. Thus, letting $J_{\lambda}'$ denote the Jacobian of $C_{\lambda}'$, the $2$-adic Tate modules $T_{2}(J_{\lambda})$ and $T_{2}(J_{\lambda}')$ are isomorphic as ${\mathrm{Gal}}(\bar{K} / K(\sqrt{\nu}))$-modules. In light of this, we replace $K$ with $K(\sqrt{\nu})$ and consider the $2$-adic Galois image $G_{2, \lambda}$ associated to $J_{\lambda}'$. Now Theorem \[thm main1\] says that if there is a prime element $\mathfrak{p}$ dividing $\nu^{d}f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$, then $G_{2, \lambda} \subset {\mathrm{GSp}}(T_{2}(J_{\lambda})) \cong {\mathrm{GSp}}(T_{2}(J_{\lambda}'))$ is open. It follows from the fact that $\mu$ and $\nu$ are coprime that $\nu^{d}f(\lambda)$ is not divisible by any prime element dividing $\nu$, so a prime $\mathfrak{p}$ satisfying the above condition does not divide $2\Delta$. The existence of such a prime is equivalent to the condition that $\nu^{d} f(\lambda) \notin \Sigma$, so to prove part (a) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{d}$ for only finitely many $\lambda \in K$. Note that any such $\lambda$ yields a solution $(x = \lambda, y) \in K \times K$ to an equation of the form $\xi y^{d} = f(x)$ with $\xi \in \Sigma'$, where $\Sigma' \subset \Sigma$ is a set of representatives of elements in $\Sigma / \Sigma^{d}$. If $d \geq 4$, then an application of the Riemann-Hurwitz formula shows that such an equation defines a smooth curve of genus $\geq 2$, and then Faltings’ Theorem implies that there are only finitely many solutions defined over $K$ to each such equation. Therefore, to prove (a) it suffices to show that there are only finitely many choices of $\xi$. But this follows from the fact that $\Sigma / \Sigma^{d}$ is finite because $\Sigma$ is finitely generated. Now assume that $d = 3$ or $d \geq 5$ and that $\lambda \in \mathcal{O}_{K}[(2\Delta)^{-1}] \cdot (K^{\times})^{s}$, with $s = 4$ if $d = 3$ and $s = 2$ otherwise. Then if we write $\lambda = \mu / \nu$ as above, we have $\nu \in \Sigma \cdot (K^{\times})^{s}$. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda})) \supsetneq \Gamma(4)$ both when $d = 3$ and when $d \geq 5$. Therefore, to prove (b) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$. This follows from the same argument as above, once we observe by Riemann-Hurwitz that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$. Finally, assume that $d \geq 4$ is even and choose any $\lambda = \mu / \nu \in K$ as before. Let $s = 4$ if $d = 4$ and let $s = 2$ otherwise. Suppose that $\nu^{d} f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$, which in both cases is equivalent to saying that $f(\lambda) \notin \Sigma \cdot (K^{\times})^{s}$. Then there is a prime element $\mathfrak{p}$ dividing $\nu^{d} f(\lambda)$ but not $2\nu^{d^{2} - d}\Delta$ (so $\mathfrak{p} \nmid 2\Delta$ as before) and such that the maximum integer $m \geq 1$ with $\mathfrak{p}^{m} \mid \nu^{d}f(\lambda)$ satisfies $v_{2}(m) \leq v_{2}(s) - 1$. Then Theorem \[thm main1\] implies that $G_{2, \lambda} \cap {\mathrm{Sp}}(T_{2}(J_{\lambda}))$ strictly contains $\Gamma(16)$ (resp. $\Gamma(4)$). Therefore, to prove (c) it suffices to show that $f(\lambda) \in \Sigma \cdot (K^{\times})^{s}$ for only finitely many $\lambda \in K$, which likewise follows from checking that the curves given by $\xi y^{s} = f(x)$ have genus $\geq 2$. \[rmk not PID\] a\) If we drop the assumption that $\mathcal{O}_{K}$ is a PID in the statement of Theorem \[thm uniform bounds\], then we observe from the proof above that we still get the statement of (a) when “all but finitely many $\lambda \in K$" is replaced by “all but finitely many $\lambda \in \mathcal{O}_{K}$" (and this statement holds for $d \in \{2, 3\}$ as well). In particular, this shows that the elements $\lambda \in \mathcal{O}_{K}$ which satisfy the hypothesis in Theorem \[thm main1\] account for all but finitely many of the elements $\lambda \in \mathcal{O}_{K}$ such that $G_{2, \lambda}$ is open. b\) In any case, the hypothesis “$\mathcal{O}_{K}$ is a PID" may be weakened to “the Hilbert class field tower of $K$ terminates". This follows from the fact that under this hypothesis, there is a finite extension of $K$ with class number $1$, and it clearly suffices to prove the assertions of Theorem \[thm uniform bounds\] when $K$ is replaced with a finite extension of $K$. \[rmk 4-torsion\] Parts (b) and (c) of Theorem \[thm uniform bounds\] say that for a given polynomial $f$ of degree $d \neq 4$, there are many elements $\lambda \in K$ such that $G_{2, \lambda} \supsetneq \Gamma(4)$. In these cases, it is always possible to compute the full structure of $G_{2}$ and determine its index in ${\mathrm{GSp}}(T_{2}(J_{\lambda}))$ by considering the Galois action on the $4$-torsion subgroup of $J$ and using formulas for the generators of the $4$-division field $K(J[4])$ over $K$. Such formulas are provided by [@yelton2015images Proposition 3.1] in the case that $d$ is odd and are found in [@yelton2015hyperelliptic §2.4] in the case that $d$ is even.

ArXiv

--- address: | Laboratori Nazionali dell”INFN, Via E.Fermi 40, I-00044 FRASCATI\ and CERN, EP Division\ E-mail: [email protected] author: - Monica Pepe Altarelli title: Higgs Searches and prospects from LEP2 --- Introduction ============ After reviewing the indirect information on the Higgs mass based on precise electroweak measurements performed at LEP1, SLD and at the TEVATRON, I will discuss the mechanisms of Higgs production and decay and the strategy adopted to search for the neutral Higgs boson (in the SM and in the MSSM) at LEP2 [@reviews]. I will summarise the results based on the analysis of approximately 170 ${\mbox{$\rm pb$} }^{-1}$ collected by each LEP experiment at ${\sqrt{s}}=189$ [ ]{}updated to the more recent Winter Conferences numbers [@felcini]. In the end I will briefly discuss the prospects for Higgs discovery at LEP2. Higgs mass from precision electroweak measurements and from theoretical arguments ================================================================================= The aim of precision electroweak tests is to prove the SM beyond the tree level plus pure QED and QCD corrections and to derive constraints on its fundamental parameters. Through loop corrections, the SM predictions for the electroweak observables depend on the top mass via terms of order $\rm{G_F}/{M_{\mathrm{t}}}^2$ and on the Higgs mass via logarithmic terms. Therefore from a comparison of the theoretical predictions [@pre_calc], computed to a sufficient precision to match the experimental capabilities and the data for the numerous observables which have been measured, the consistency of the theory is checked and constraints on ${M_{\mathrm{H }}}$ are placed, once the measurement of ${M_{\mathrm{t}}}$ from the TEVATRON is input. The present 95% C.L. upper limit on the Higgs mass in the SM is [@mh_smfits; @felcini] $$\label{mh_up} {M_{\mathrm{H }}}< 220\,{\mbox{${\rm {GeV}}/c^2$} }\,,$$ if one makes due allowance for unknown higher loop uncertainties in the analysis. The corresponding central value is still rather imprecise: $${M_{\mathrm{H }}}= 71^{+75}_{-42}\pm5\,{\mbox{${\rm {GeV}}/c^2$} }\,.$$ The range given by Eq.\[mh\_up\] may be compared with the one derived from theoretical arguments [@hambye]. It is well known that in the SM with only one Higgs doublet a lower limit on the Higgs mass ${M_{\mathrm{H }}}$ can be derived from the requirement of vacuum stability. This limit is a function of the energy scale $\Lambda$ where the model breaks down and new physics appears. Similarly an upper bound on ${M_{\mathrm{H }}}$ is obtained from the requirement that up to the scale $\Lambda$ no Landau pole appears. If, for example, the SM has to remain valid up to the scale $\Lambda\simeq{\rm M_{GUT}}$, then it is required that $135<{M_{\mathrm{H }}}<180~{\mbox{${\rm {GeV}}/c^2$} }$. In the MSSM two Higgs doublets are introduced, in order to give masses to the up-type quarks on the one hand and to the down-type quarks and charged leptons on the other. The Higgs particle spectrum therefore consists of five physical states: two CP-even neutral scalars (h,A), one CP-odd neutral pseudo-scalar (A) and a charged Higgs boson pair ($\rm{H}^{\pm}$). Of these, h and A could be detectable at LEP2 [@yellow]. In fact, at tree-level h is predicted to be lighter than the Z. However, radiative corrections to ${M_{\mathrm{h}}}$ [@ellis], which are proportional to the fourth power of the top mass, shift the upper limit of ${M_{\mathrm{h}}}$ to approximately 135 [ ]{}, depending on the MSSM parameters. Higgs production and decay ========================== At LEP2, the dominant mechanism for producing the standard model Higgs boson is the so-called Higgs-strahlung process ${\mathrm{e}^+\mathrm{e}^-}\to$ HZ [@khoze; @bjorken], with smaller contributions from the WW and ZZ fusion processes leading to H$\nu_{\rm{e}}\bar{\nu}_{\rm{e}}$ and H${\mathrm{e}^+\mathrm{e}^-}$ final states, respectively. A sizeable cross section (few 0.1 pb) is obtained up to ${M_{\mathrm{H }}}\sim {\sqrt{s}}- {M_{\mathrm{Z}}}$, so that an energy larger than 190 [ ]{}is needed to extend the search above ${M_{\mathrm{H }}}\simeq {M_{\mathrm{Z}}}$. For example the production cross section at ${\sqrt{s}}=189$ GeV for ${M_{\mathrm{H }}}=95$ [ ]{}is 0.18 pb, which for an integrated luminosity $\cal{L}$=170 ${\mbox{$\rm pb$} }^{-1}$/exp. gives 30 signal events per experiment. For the MSSM Higgs the main production mechanisms are the Higgs-strahlung process ${\mathrm{e}^+\mathrm{e}^-}\to$ hZ, as for the SM Higgs, and the associated pair production ${\mathrm{e}^+\mathrm{e}^-}\to$ hA [@ha-prod]. The corresponding cross sections may be written in terms of the SM Higgs-strahlung cross section, $\sigma^{\rm{SM}}$, and of the cross section $\sigma^{\rm{SM}}_{{{\nu}\overline{\nu}}}$ for the process $\rm{Z}^*\to{{\nu}\overline{\nu}}$ as $$\begin{aligned} \label{Zh-hA} \sigma({\mathrm{e}^+\mathrm{e}^-}\to\rm{Zh}) = & \rm{sin}^2(\beta-\alpha)\,\sigma^{\rm{SM}} \\ \sigma({\mathrm{e}^+\mathrm{e}^-}\to\rm{hA}) \propto & \rm{cos}^2(\beta-\alpha)\,\sigma^{\rm{SM}}_{{{\nu}\overline{\nu}}}. \nonumber \end{aligned}$$ The parameter $\rm{tan}\beta$ gives the ratio of the vacuum expectation values of the two Higgs doublets and $\alpha$ is a mixing angle in the CP-even sector. The Higgs-strahlung hZ process occurs at large $\rm{sin}^2(\beta-\alpha)$, i.e., at small $\rm{tan}\beta$. Conversely, at small $\rm{sin}^2(\beta-\alpha)$, i.e., at large $\rm{tan}\beta$, when hZ production dies out, the associated hA production becomes the dominant mechanism with rates similar to the previous case. In this region the masses of h and A are approximately equal. For masses below $\sim 110~$ [ ]{}, the SM Higgs decays into ${{\rm b}\overline{\rm b}}$ in approximately 85% of the cases and into ${\tau^+\tau^-}$ in approximately 8% of the cases. Similar branching ratios (BR) are expected for the MSSM Higgs bosons. Above ${M_{\mathrm{H }}}\sim 135$ [ ]{}, the BR into W and Z pairs becomes dominant. Searches at LEP2 ================ While at LEP1 energies the signal to noise ratio was as small as $10^{-6}$ due to the very high ${{\rm q}\overline{\rm q}}$ cross section, at LEP2 the signal to noise ratio is much more favourable, increasing to $\simeq1\%$. In order to reduce this background, mainly due to W pair production, ${{\rm q}\overline{\rm q}}$ (with two gluons or two additional photons in the final state) and ZZ events, use is made of b-tagging techniques which exploit the large BR of the Higgs into ${{\rm b}\overline{\rm b}}$. For ${M_{\mathrm{H }}}\simeq{M_{\mathrm{Z}}}$, as is the case for the expected experimental sensitivity, ZZ production represents an irreducible source of background since the Z decays into ${{\rm b}\overline{\rm b}}$ in 15% of the cases. The following event topologies are studied: - The leptonic channel (Z$\to {\mathrm{e}^+\mathrm{e}^-}, {\mu^+\mu^-}$, H$\to{{\rm b}\overline{\rm b}}$) which represents $7\%$ of the Higgs-strahlung cross section. These events are characterised by two energetic leptons with an invariant mass close to ${M_{\mathrm{Z}}}$ and a recoil mass equal to ${M_{\mathrm{H }}}$. Because of the clear experimental signature, no b-tag is necessary and therefore the signal efficiency is high, typically $\sim75\%$. - The missing energy channel (Z$\to {{\nu}\overline{\nu}}$, H$\to{{\rm b}\overline{\rm b}}$) comprising $\simeq20\%$ of the Higgs-strahlung cross section. This channel is characterised by a missing mass consistent with ${M_{\mathrm{Z}}}$ and two b-jets. The selection efficiency is $\simeq35\%$. - The four jet channel (Z$\to {{\rm q}\overline{\rm q}}$, H$\to{{\rm b}\overline{\rm b}}$) which is not as distinctive as the two previous topologies but compensates for this drawback with its large BR of $\simeq64\%$. The efficiency for this channel is typically $\simeq40\%$. - The ${\tau^+\tau^-}{{\rm q}\overline{\rm q}}$ channel (Z$\to {\tau^+\tau^-}$, H$\to{{\rm q}\overline{\rm q}}$ and vice-versa) with a $\simeq9\%$ BR. The event topology includes two hadronic jets and two oppositely-charged, low multiplicity jets due to neutrinos from the $\tau$ decays. The signal efficiency is of the order of 25%. The b-tagging algorithms are based on the long lifetime of weakly decaying b-hadrons, on jet shape variables such as charged multiplicity or boosted sphericity and on high $p_t$ leptons from semileptonic b decays. The b-jet identification is improved by combining information from the different b-tagging algorithms with tools like neural-networks and likelihoods. Typically, for a 60% signal efficiency, the WW background, which has no b-content, is suppressed by a factor over 100, and the ${{\rm q}\overline{\rm q}}$ and ZZ backgrounds by approximately a factor 10. With respect to the b-tagging algorithms developed for the measurement at LEP1 of $\rm{R_b}$, the b fraction of Z hadronic decays, the performances at LEP2 have improved by almost a factor of 2, due to vertex detectors with an extended solid angle coverage and to more efficient b-tagging techniques. All the analyses developed for the standard model Higgs produced via the Higgs-strahlung mechanism can be used with no modification for the supersymmetric case, provided that the Higgs decays to standard model particles (${{\rm b}\overline{\rm b}}$, ${\tau^+\tau^-}$). The results can then be reinterpreted in the MSSM context, by simply rescaling the number of expected events by the factor $\rm{sin}^2(\beta-\alpha)$. For the pair production process, the signal consists of events with four b-quark jets or a ${\tau^+\tau^-}$ pair recoiling against a pair of b-quark jets. Results and prospects ===================== Table \[tab:res\] shows the number of selected events in the data for the SM Higgs search, the expected number of background events and the expected numbers of signal events assuming ${M_{\mathrm{H }}}=95$ [ ]{} [@felcini; @al_moriond; @del_moriond; @l3_moriond; @op_moriond]. $n_{\rm obs}$ $n_{\rm back}$ $n_{\rm sig}$ --------------------------------------- --------------- ---------------- --------------- ALEPH 53 44.8 13.8 DELPHI 26 31.3 10.1 L3 30 30.3 9.9 OPAL 50 43.9 12.6 Total 159 150 46.4 $\Delta{M_{\mathrm{H }}}=92-96$ [ ]{} 47 37.5 24.6 : Standard Model Higgs search. Number of observed events in the data $n_{\rm obs}$, expected number of background events $n_{\rm back}$ and expected numbers of signal events $n_{\rm sig}$ assuming ${M_{\mathrm{H }}}=95$ [ ]{}for the four LEP experiments and for their combination. Also shown are the number of events observed and expected by the four experiments combined in the mass window $\Delta{M_{\mathrm{H }}}=92-96$ [ ]{}.[]{data-label="tab:res"} As can be observed from Table \[tab:res\], an excess of events is observed by ALEPH [@al_moriond] and OPAL [@op_moriond] which, in the case of OPAL, is concentrated in the mass region around ${M_{\mathrm{H }}}\simeq{M_{\mathrm{Z}}}$, while for ALEPH it is distributed over higher masses, typically $\geq95$ [ ]{}. These results translate into the lower limits shown in Table \[tab:lim\], together with the sensitivity (expected limit) of each experiment. -------- --------------- -------------- Observed Expected limit ([ ]{}) limit([ ]{}) ALEPH 90.2 95.7 DELPHI 95.2 94.8 L3 95.2 94.4 OPAL 91.0 94.9 -------- --------------- -------------- : Observed 95% C.L. lower limits on ${M_{\mathrm{H }}}$. Also shown are the limits predicted by the simulation if no signal were present. []{data-label="tab:lim"} Table \[tab:MSSM\_lim\] shows the preliminary 95% C.L. lower limits on ${M_{\mathrm{h}}}$ and ${M_{\mathrm{A}}}$ for the four LEP experiments [@felcini; @al_moriond; @del_moriond; @l3_moriond; @op_moriond], as well as the derived excluded ranges of $\tan\beta$ for both no mixing and maximal mixing in the scalar-top sector. -------- ---------------------------- ---------------------------- --------------------- ----------------------- ${M_{\mathrm{h}}}$ ([ ]{}) ${M_{\mathrm{A}}}$ ([ ]{}) $\tan\beta$ $\tan\beta$ max. mixing no mixing ALEPH 80.8 81.2 - $1<\tan\beta<2.2$ DELPHI 83.5 84.5 $0.9<\tan\beta<1.5$ $0.6<\tan\beta<2.6$ L3 77.0 78.0 $1.<\tan\beta<1.5$ $1.<\tan\beta<2.6$ OPAL 74.8 76.5 - $0.81<\tan\beta<2.19$ -------- ---------------------------- ---------------------------- --------------------- ----------------------- : Observed 95% C.L. lower limits on ${M_{\mathrm{h}}}$ and ${M_{\mathrm{A}}}$. Also shown are the derived excluded ranges of $\tan\beta$. The mass limits are given for $\tan\beta>1$, except for those of DELPHI, given for $\tan\beta>0.5$. []{data-label="tab:MSSM_lim"} In the years 1999 to 2000 LEP2 is expected to deliver a luminosity larger than 200 $\rm{pb}^{-1}$ per experiment at a centre-of-mass energy eventually as high as $\sim 200$ GeV. These data should allow to discover a SM Higgs of 107 [ ]{}or to exclude a Higgs lighter than $\sim$108 [ ]{} [@lellouch; @chamonix]. This is a particularly interesting region to explore, given the present indication for a light Higgs from the standard model fit of the electroweak precision data. The sensitivity to the Higgs in the MSSM will reach $\sim90$ [ ]{}for the high $\tan\beta$ region and $\sim108$ [ ]{}for $\tan\beta\simeq1$, therefore allowing good coverage of the MSSM plane. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank Cesareo Dominguez and Raul Viollier for their great hospitality and excellent organization of the Workshop and “Maestro” Patrick Janot for his precious advice and for carefully reading this manuscript. [99]{} For other more detailed reviews on the subject, see, [*e.g.*]{} P.Janot in [*Perspectives on Higgs Physics II*]{}, Advanced Series on Direction in High Energy Physic, Vol.17 (1997), ed. by G.L.Kane, 104;\ F.Richard, hep-ex/9810045, 28 Oct. 1998. M.Felcini, Rencontres de Moriond on ElectroWeak Interactions and Unified Theories, 13-20 March 1999, Les Arcs, France. , ed. by D.Bardin, W.Hollik and G.Passarino, CERN Yellow Report 95-03, March 1995 and references therein. LEP EW Working Group, available at\ http://www.cern.ch/LEPEWWG/plots/winter99/. See, [*e.g.*]{}, T.Hambye, K.Riesselmann, DESY-97-152, Aug. 1997. See, [*e.g.*]{} [*Physics at LEP2*]{}, ed. by G.Altarelli, T.Sjostrand and F.Zwirner, CERN 96-01, Vol.1 (1996) 351. J.Ellis, G.Ridolfi and F.Zwirner, [*Phys.Lett.*]{} [**B257**]{} (1991) 83; [*Phys.Lett.*]{} [**B262**]{} (1991) 477. B.L.Ioffe and V.A.Khoze, Sov.J.Part.Nucl.[****]{}9 (1978) 50. J.D.Bjorken, Proc. Summer Institute on particle Physics, SLAC Report 198 (1976). J.F.Gunion and H.E.Haber, [*Nucl.Phys.*]{} [**B272**]{} (1986) 1; [*Nucl.Phys.*]{} [**B278**]{} (1986) 449 and [*Nucl.Phys.*]{} [**B307**]{} (1988) 445. ALEPH Collaboration, Contribution to 1999 Winter Conferences, ALEPH 99-007, CONF 99-003, March 1999. DELPHI Collaboration, Contribution to 1999 Winter Conferences, DELPHI 99-8 CNF 208, March 1999. L3 Collaboration, Contributions to 1999 Winter Conferences, L3 Note 2382, 12 March 1999; L3 Note 2383, 15 March 1999. OPAL Collaboration, Contribution to 1999 Winter Conferences, OPAL PN382, March 12, 1999. E.Gross, A.L.Read and D.Lellouch, CERN-EP/98-094. P.Janot in [*Proceedings of the Workshop on LEP-SPS Performance*]{}, Chamonix IX, Jan. 1999, 222.

ArXiv

--- abstract: 'While the analysis of airborne laser scanning (ALS) data often provides reliable estimates for certain forest stand attributes – such as total volume or basal area – there is still room for improvement, especially in estimating species-specific attributes. Moreover, while information on the estimate uncertainty would be useful in various economic and environmental analyses on forests, a computationally feasible framework for uncertainty quantifying in ALS is still missing. In this article, the species-specific stand attribute estimation and uncertainty quantification (UQ) is approached using Gaussian process regression (GPR), which is a nonlinear and nonparametric machine learning method. Multiple species-specific stand attributes are estimated simultaneously: tree height, stem diameter, stem number, basal area, and stem volume. The cross-validation results show that GPR yields on average an improvement of 4.6% in estimate RMSE over a state-of-the-art k-nearest neighbors (kNN) implementation, negligible bias and well performing UQ (credible intervals), while being computationally fast. The performance advantage over kNN and the feasibility of credible intervals persists even when smaller training sets are used.' author: - 'Petri Varvia, Timo Lähivaara, Matti Maltamo, Petteri Packalen, Aku Seppänen[^1][^2][^3][^4]' bibliography: - 'IEEEabrv.bib' - 'bibliography.bib' title: Gaussian process regression for forest attribute estimation from airborne laser scanning data --- forest inventory, LiDAR, area based approach, machine learning, Gaussian process 1000 Copyright notice {#copyright-notice .unnumbered} ================ P. Varvia, T. Lähivaara, M. Maltamo, P. Packalen and A. Seppänen, “Gaussian Process Regression for Forest Attribute Estimation From Airborne Laser Scanning Data,” in IEEE Transactions on Geoscience and Remote Sensing. doi: 10.1109/TGRS.2018.2883495\ 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Introduction ============ Forest inventories based on airborne laser scanning (ALS) are becoming increasingly popular. Therefore, it is more and more important to have well performing methods for the estimation/prediction of stand attributes, such as basal area and tree height. Coupled with the prediction procedures, efficient methods for the quantification of prediction uncertainty are also urgently needed for forestry planning and assessment purposes [@kangas2018]. Operational forest inventories employing ALS data are most often implemented with the area based approach (ABA) [@naesset2002]. In ABA, metrics used as predictor variables are calculated from the ALS returns within a plot or grid cell. Using training plots with field-measured stand attributes, a model is formulated between the stand attributes and ABA metrics. This statistical model is then used to predict the stand attributes for each grid cell [@reutebuch2005; @maltamobook] and the predictions are finally aggregated to the desired area, e.g. to a stand. Although tree species is among the most important attributes of forest inventory, the ALS research does not particularly reflect this. One reason for this is that in many biomes the number of tree species is so high that it is practically impossible to separate them by remote sensing. In the Nordic countries, however, the majority of the growing stock comes from three economically valuable tree species. The species-specific prediction is approached two ways in Nordic countries: in Norway, stands are stratified according to tree species by visual interpretation of aerial images before the actual ALS inventory [@naesset2004], whereas in Finland, stand attributes are predicted by tree species using a combined set of metrics from ALS data and aerial images [@packalen2007]. In both approaches, aerial images are used to improve the discrimination of tree species. Uncertainty estimation is a key component in strategic inventories that cover large areas [@mandallaz2007]. ALS can be used in that context too. For example, ALS metrics can be used as auxiliary variables in model-based (e.g. [@staahl2010]) or model assisted (e.g. [@gregoire2010]) estimation of some forest parameter. Typically, sample mean and sample variance are estimated to the area of interest (e.g. 1000000 ha) using a certain number (e.g. 500) of sample plots and auxiliary variables covering all population elements. In the stand level forest management inventories, the situation is different: the point estimate and its confidence intervals are needed for each stand and there may not be any sample plots in most stands. Today, most ALS inventories can be considered as stand level management inventories. Commonly in ABA, when using prediction methods such as linear regression or , only point estimates without accompanying uncertainty metrics are computed. Plot or cell level prediction uncertainty has garnered some research interest in recent years and several methods of predicting plot/cell level variance have been proposed [@junttila2008a; @finley2013; @magnussen2016]. Recently, a Bayesian inference approach to quantify uncertainty within the framework of the ABA was proposed by Varvia *et al.* [@varvia]. The main shortcoming in the method proposed in [@varvia] is that it is computationally costly: wall-to-wall uncertainty quantification of a large forest area would require considerable computer resources. Gaussian process regression (GPR) [@rasmussenbook] is a machine learning method that provides an attractive alternative; compared with the more widely used machine learning methods, such as artificial neural networks [@niska2010neural; @alsdeeplearning], GPR also produces an uncertainty estimate for the prediction. Univariate GPR was tested for estimation of several total stand attributes by Zhao *et al.* [@alsgpr], where it was found to significantly outperform (log)linear regression. In this paper, we propose a multivariate GPR for simultaneous estimation of species-specific stand attributes within ABA. The estimation accuracy of GPR is compared with kNN and the uncertainty quantification performance with the Bayesian inference method of [@varvia]. Furthermore, the effect of training set size on its performance is evaluated. Materials ========= The same test data as in [@varvia] is used in this study. In this section, the data set is briefly summarized, for detailed description, see e.g. [@packalen2009; @Packalen2012]. The test area is a managed boreal forest located in Juuka, Finland. The area is dominated by Scots pine (*Pinus sylvestris* L.) and Norway spruce (*Picea abies* (L.) Karst.), with a minority of deciduous trees, mostly downy birch (*Betula pubescens* Ehrh.) and silver birch (*Betula pendula* Roth.). The deciduous trees are considered as a single group. The field measurements were done during the summers of 2005 and 2006. Total of 493 circular sample plots of radius 9 m are used in this study. The diameter at breast height (DBH), tree and storey class, and tree species were recorded for each tree with DBH larger than 5 cm and the height of one sample tree of each species in each storey class was measured. The heights of other trees on the plot were predicted using a fitted Näslund’s height model [@naslund]. The species-specific stand attributes were then calculated using the measured DBH and the predicted heights. The stand attributes considered in this study are tree height ($H_{\mathrm{gm}}$), diameter at breast height ($D_{\mathrm{gm}}$), stem number ($N$), basal area ($\mathit{BA}$), and stem volume ($V$). The ALS data and aerial images were captured in 13 July 2005 and 1 September 2005, respectively. The ALS data has a nominal sampling density of 0.6 returns per square meter, with a footprint of about 60 cm at ground level. The orthorectified aerial images contain four channels (red, green, blue, and near infrared). A total of $n_x=77$ metrics were computed from the ALS point cloud and aerial images and used in ABA. The metrics include canopy height percentiles, the corresponding proportional canopy densities, the mean and standard deviation of the ALS height distribution, the fraction of above ground returns (i.e. returns with $z>2$ m), and metrics computed from the LiDAR intensity. From the aerial images, the mean values of each channel were used along with two spectral vegetation indices [@packalen2009]. Methods {#sec:methods} ======= Let us denote a vector consisting of the stand attributes by $\mathbf{y}\in\mathbb{R}^{15}$; the vector $\mathbf{y}$ contains the species-specific (pine, spruce, deciduous) $H_{\mathrm{gm}}$, $D_{\mathrm{gm}}$, $N$, $\mathit{BA}$, and $V$, resulting in a total of $n_y=15$ variables. The vector of predictors (ALS and aerial image metrics) is denoted by $\mathbf{x}\in\mathbb{R}^{n_x}$. The general objective is to learn a nonlinear regression model $$\label{thefunc} \mathbf{y}=f(\mathbf{x})+\mathbf{e},$$ where $\mathbf{e}$ is an error term, from a set of $n_t$ training data $(\mathbf{Y}_t,\mathbf{X}_t)$. Let $\mathbf{Y}$ be a finite collection of points $\mathbf{y}^{(i)}=f(\mathbf{x}^{(i)})+\mathbf{e}^{(i)}$ concatenated in a long vector. In Gaussian process regression [@rasmussenbook], the joint probability distribution of these points $\mathbf{Y}$ is modeled as a multivariate normal distribution, with mean $\boldsymbol{\mu}_{\mathbf{y}}$ and covariance $\boldsymbol{\Gamma}_{\mathbf{y}}$ written as functions of $\mathbf{x}$: $$\begin{aligned} &\boldsymbol{\mu}_{\mathbf{y}} = \mathbf{m}(\mathbf{x}) = \mathbb{E}\{f(\mathbf{x})\},\\ &\boldsymbol{\Gamma}_{\mathbf{y}}=\mathbf{K}(\mathbf{x},\mathbf{x}^{\prime}) = \mathbb{E}\{(f(\mathbf{x})-\mathbf{m}(\mathbf{x}))(f(\mathbf{x}^{\prime})-\mathbf{m}(\mathbf{x}^{\prime}))^T\},\end{aligned}$$ where $(\:\cdot\:)^T$ is the matrix transpose. Let now $\mathbf{Y} = \begin{bmatrix} \mathbf{Y}_t & \mathbf{y}_*\end{bmatrix}^T$, that is, $\mathbf{Y}$ a vector consisting of the training data $\mathbf{Y}_t$, and a new point $\mathbf{y}_*$ which we want to estimate, using the corresponding measurement $\mathbf{x}_*$. For simplification, we set $\mathbf{m}(\mathbf{x})=0$. The mean term mostly affects the behavior when extrapolating far away from the space covered by the training data. The joint distribution of $\mathbf{Y}$ is then $$\label{joint} \begin{bmatrix} \mathbf{Y}_t \\ \mathbf{y}_* \end{bmatrix} \sim\mathcal{N}\left(0, \begin{bmatrix} \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)+\mathbf{E} & \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)^T \\ \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t) & \mathbf{K}(\mathbf{x}_*,\mathbf{x}_*)+\mathbf{E}_* \end{bmatrix}\right),$$ where $\mathbf{E}$ and $\mathbf{E}_*$ describe the covariance of the error $\mathbf{e}$, i.e. uncertainty of $\mathbf{y}$. In this work, we use $\mathbf{E}*=0.1\mathbf{D}$, where $\mathbf{D}$ is a diagonal matrix that contains the sample variances of the training data $\mathbf{Y}_t$ on the main diagonal. The error matrix $\mathbf{E}= 0.1\mathbf{D}\otimes \mathbf{I}$, where $\otimes$ is the Kronecker product and $\mathbf{I}\in\mathbb{R}^{n_t\times n_t}$ is an identity matrix. For brevity, following shorthand notations are introduced: $$\begin{aligned} &\mathbf{K} = \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)\in\mathbb{R}^{n_yn_t\times n_yn_t} \\ &\mathbf{K}_* = \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)\in\mathbb{R}^{n_y\times n_yn_t}.\end{aligned}$$ In GPR, the kernel matrices $\mathbf{K}$ and $\mathbf{K}_* $ are constructed based on a covariance function. In this study we use stationary Matérn covariance function with $\nu=3/2$, fixed length scale $l=10$, and $\sigma=1$: $$\label{matern} k(\mathbf{x},\mathbf{x^\prime}) = \left(1+\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right)\exp\left(-\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right),$$ where the distance metric $d(\mathbf{x},\mathbf{x^\prime})$ is the Euclidean distance. The covariance function $k(\mathbf{x},\mathbf{x^\prime})$ describes the covariance between the vectors $\mathbf{x}$ and $\mathbf{x^\prime}$ based on the distance between the vectors. The covariance function is the core component of GPR that specifies properties such as smoothness of the regressor. The covariance function is used to construct univariate kernel matrices $$\begin{aligned} &K(i,j) = k(\mathbf{x}_t^{(i)},\mathbf{x}_t^{(j)})\in\mathbb{R}^{n_t\times n_t} \\ &K_*(1,j) = k(\mathbf{x}^*,\mathbf{x}_t^{(j)})\in\mathbb{R}^{1\times n_t}.\end{aligned}$$ To get from the univariate kernels to multivariate kernels used in , the so-called separable kernel [@bonilla2008] is used: $$\begin{aligned} &\mathbf{K} = \boldsymbol{\Gamma}_{\mathbf{y}}\otimes K \\ &\mathbf{K}_* = \boldsymbol{\Gamma}_{\mathbf{y}}\otimes K_*,\end{aligned}$$ where $\boldsymbol{\Gamma}_{\mathbf{y}}\in\mathbb{R}^{n_y\times n_y}$ is a (prior) covariance for $\mathbf{y}$. In this work, $\boldsymbol{\Gamma}_{\mathbf{y}}$ is approximated by the sample covariance of $\mathbf{Y}_t$. It should be noted, that $\sigma=1$ is chosen in the kernel function , because the (prior) variances of $\mathbf{y}$ are added to the covariance kernel in this step through $\boldsymbol{\Gamma}_{\mathbf{y}}$. From the joint density , the conditional density of $\mathbf{y}_*$ given the training data and the measurement $\mathbf{x}_*$ is $$\mathbf{y}_*\:|\:\mathbf{Y}_t,\mathbf{X}_t,\mathbf{x}_*\sim\mathcal{N}(\mathbf{m}(\mathbf{y}_*),\boldsymbol{\Gamma}_{\mathbf{y}_*}),$$ where $$\begin{aligned} \label{predmean} &\boldsymbol{\mu}_{\mathbf{y}_*} = \mathbf{K}_*(\mathbf{K}+\mathbf{E})^{-1}\mathbf{y}_t \\ \label{predcov} &\boldsymbol{\Gamma}_{\mathbf{y}_*}= \boldsymbol{\Gamma}_{\mathbf{y}} + \mathbf{E}_* - \mathbf{K}_*(\mathbf{K}+\mathbf{E})^{-1}\mathbf{K}_*^T.\end{aligned}$$ The predictive mean $\boldsymbol{\mu}_{\mathbf{y}_*} $ is now the point estimate for the unknown vector of stand attributes $\mathbf{y}_*$ and $\boldsymbol{\Gamma}_{\mathbf{y}_*}$ provides the estimate covariance. As can be seen from the equation , the final prediction is a linear combination of the training data values $\mathbf{y}_t$. This mathematical connection to linear models is expected, because general linear model can be written as a special case of GPR [@rasmussenbook]. Correcting for negative predictions ----------------------------------- Unlike kNN and certain other machine learning methods, GPR extrapolates outside the training data. As an unwanted side effect of this extrapolation behavior, GPR can produce unrealistic negative predictions for the stand attributes. Several statistically rigorous methods for constraining the GPR predictions have been proposed [@DaVeiga2012; @jidling2017], but these methods increase the computational cost significantly and are nontrivial to implement. Here we adopt a simpler correction. For the point prediction, we compute a maximum a posteriori estimate by solving $$\hat{\mathbf{y}}_*=\mathrm{arg}\:\underset{\hat{\mathbf{y}}}{\mathrm{min}}\left\{(\hat{\mathbf{y}}-\boldsymbol{\mu}_{\mathbf{y}_*})^T\boldsymbol{\Gamma}_{\mathbf{y}_*}^{-1}(\hat{\mathbf{y}}-\boldsymbol{\mu}_{\mathbf{y}_*})\right\},\;\hat{\mathbf{y}}\geq 0.$$ The prediction $\hat{\mathbf{y}}_*$ is the mode of the truncated GPR predictive density. If the original GPR predictive mean is non-negative, $\hat{\mathbf{y}}_*$ is simply $\boldsymbol{\mu}_{\mathbf{y}_*}$. Due to the complicated structure of the marginal densities of a truncated multivariate Gaussian distribution [@Horrace2005], correcting the predictive intervals exactly is not computationally practical. Instead, the univariate Gaussian marginals of the GPR predictive density are truncated at zero. If the original 95% predictive interval for a stand attribute is $[a,b]$ and $a<0$, set $\hat{a}=0$ and calculate the new corrected upper bound $\hat{b}$ using the cumulative distribution of univariate truncated Gaussian by solving $$\Phi(\hat{b},\mu_{y_{*}},\sigma_{y_{*}}) = 0.95+0.05\Phi(0,\mu_{y_{*}},\sigma_{y_{*}}),$$ where $\Phi(\:\cdot\:,\mu_{y_{*}},\sigma_{y_{*}})$ is the cumulative distribution function of the univariate Gaussian distribution with the mean and standard deviation from the GPR predictive distribution. If $a\geq0$, the interval $[a,b]$ does not change. The corrected interval $[\hat{a},\hat{b}]$ is not a proper predictive interval of the truncated predictive distribution, unless $\boldsymbol{\Gamma}_{\mathbf{y}_*}$ is strictly diagonal. Reference methods ----------------- The GPR point estimates are compared with a state-of-the-art kNN algorithm. We select ten predictors from the (transformed) data using a simulated annealing -based optimization approach of [@Packalen2012] and use the most similar neighbor (MSN) method for selecting the neighbors. The number of neighbors is chosen to be $k=5$, as in [@packalen2009; @Packalen2012]. The predictor selection is done using the whole data set and leave-one-out cross-validation. The prediction credible intervals provided by GPR are compared with the Bayesian inference approach [@varvia]. In the Bayesian approach the posterior predictive density: $$\label{alsextposterior} \pi(\mathbf{y}_*|\mathbf{x}) \propto \begin{cases} \mathcal{N}(\mathbf{x}|\hat{\mathbf{A}}\boldsymbol{\phi}(\mathbf{y}_*)+\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}},\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}) &\\ \qquad\qquad\quad\;\cdot\:\mathcal{N}(\mathbf{y}_*|\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}},\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}), & \mathbf{y}_*\geq 0 \\ 0, & \mathbf{y}_*< 0, \end{cases}$$ is constructed based on the training data and the new measurement. The model matrix $\hat{\mathbf{A}}$, conditional (residual) error statistics $\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}$, and the prior statistics $\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}$ are learned from the training data. The density is then sampled using a Markov chain Monte Carlo method. The point estimate and 95% credible intervals are then calculated from the samples. {width="\textwidth"} Performance assessment ---------------------- The proposed GPR method is first evaluated using leave-one-out cross-validation (i.e. $n_t=492$). From the results, relative root mean square error (RMSE%), relative bias (bias%), and credible interval coverage (CI%) are calculated. Credible interval coverage is the percentage of the test plots where the field measured value of a stand attribute lies inside the computed 95% prediction interval; CI% thus has the ideal value of 95%. In addition to conducting a leave-one-out cross-validation, the effect of the number of training plots is evaluated. Training set sizes from $n_t=20$ to $n_t=400$ are tested with a stepping of 20. The cross-validation is performed by first randomly sampling $n_t$ plots to be used as a training set and then randomly selecting a single test plot from the remaining $493-n_t$ plots. This procedure is repeated 2000 times for each $n_t$ value. This way the number samples for each tested $n_t$ stays constant. The effect of training set size is only evaluated for GPR and kNN, due to the high computational cost of the reference Bayesian inference approach. Results and discussion {#sec.results} ====================== Species-specific attributes --------------------------- The RMSE% comparison between the GPR predictions, kNN and Bayesian inference is shown in the Figure \[fig:rmse\] for all estimated stand attributes. The numerical RMSE% value is shown above each bar. For pine, which is the dominant species in the study area, the GPR and kNN estimates have fairly equivalent performance: GPR is slightly better for all the stand attributes except height and basal area. The Bayesian linear inference estimates are notably worse. In the minority species (spruce and deciduous), the GPR estimates have consistently better RMSE% than kNN or Bayesian linear. On average, the relative improvement over kNN is 6.5% for the minority species and 4.6% for all species. Figure \[fig:bias\] shows a similar comparison of relative bias between the evaluated methods for all the estimated stand attributes. The numerical bias% value is printed for each bar. GPR estimates show smaller than 2% absolute bias for all the stand attributes, except the spruce basal area and volume. kNN shows small bias in the spruce attributes, but has a large bias in deciduous basal area and volume. The Bayesian linear results show notable bias in $N$, $\mathit{BA}$, and $V$. {width="\textwidth"} The CI coverages of GPR and the reference Bayesian inference method are compared in Figure \[fig:ci\]. The numerical CI% value is shown above each bar; the ideal value is here 95%. The CI% for the Bayesian linear estimates fall short of the 95% target, that is, the prediction intervals that are too narrow. The GPR prediction intervals perform well on basal area and stem volume, with good coverage also on stem number. The GPR CI% for these stand attributes is consistently better than the Bayesian linear. The GPR prediction intervals for height and diameter are overconfident, especially in the deciduous variables, and the performance is roughly similar to the Bayesian linear estimates. {width="140mm"} Total attributes ---------------- Point estimates and credible intervals for the total stem number, basal area, and stem volume were calculated from the species-specific results. The point estimates were computed by summing up the corresponding species-specific estimates. The GPR prediction interval for the total attributes is acquired from the prediction covariance $\boldsymbol{\Gamma}_{\mathbf{y}_*}$, because summation is a linear transformation. The results for the total attributes are shown in Figure \[fig:totvars\]. In RMSE%, GPR estimates show the best performance. Bayesian linear estimates have lower RMSE% than kNN in the total basal area and volume, while kNN is better in the stem number. In the relative bias, GPR has fairly low bias and performs worst in the total stem volume. kNN has consistent slight bias, while the Bayesian linear estimates show large bias in the stem number. In credible interval coverage, GPR produces too wide intervals for basal area and stem volume (CI% between 98-99%). The Bayesian linear intervals are, on the other hand, with CI% around roughly 80%. {width="140mm"} Effect of training set size --------------------------- RMSE% versus training set size is shown in Figure \[fig:rmsetrain\]. The dashed minimum line corresponds to the species-specific stand attribute with the lowest RMSE%, maximum to the stand attribute with the highest RMSE%, and mean is the average over the stand attributes. As expected, the RMSE% increases for both methods when the training set size decreases. GPR keeps the slight performance advantage over kNN even when using smaller training sets. The improvement in performance for training sets larger than c. 200 plots is fairly small. ![Lowest, average, and the highest relative RMSE as a function of training set size for GPR and kNN estimates.[]{data-label="fig:rmsetrain"}](rmse_nt.png){width="70mm"} Figure \[fig:biastrain\] shows the relative bias as a function of the training set size. When the training set size decreases, the estimated bias increases in both positive and negative directions, but with a general negative tendency. Smaller training sets are less likely to cover the full range of variation of the stand attributes in the population, which results in underestimation of large values: this would explain the observed tendency in bias. The largest negative biases produced by kNN are consistently larger than in GPR. ![Lowest, average, and the highest relative bias as a function of training set size for GPR and kNN estimates.[]{data-label="fig:biastrain"}](bias_nt.png){width="70mm"} Figure \[fig:citrain\] shows the CI% of the GPR estimates versus the training set size. The average CI% increases slightly as training set size is decreased, the lowest CI% increases considerably, while the highest CI% drops somewhat. The generally too narrow credible intervals signify overconfidence in the predictions, which implies that either the GP model is not optimal in its current formulation, or the stand attributes have not sufficiently explained the variation in the predictors. The latter explanation might cause that there are usually contradicting training data (i.e. training points that are close in the stand attribute space, but distant in the predictor space) in large training sets, which might partly explain the slight improvement of CI% when training set size decreases. With the Bayesian inference approach, on the contrary, a substantial drop in CI% in smaller training set sizes would be expected based on the results in [@varvia]. ![Lowest, average, and the highest CI% as a function of training set size for GPR estimates.[]{data-label="fig:citrain"}](ci_nt.png){width="70mm"} Discussion {#sec:disc} ---------- Conceptually, GPR is a non-parametric machine learning method that has similarities with kNN. Thus, many approaches proposed for improvement of kNN estimates within ABA could be also utilized to further improve GPR estimates. GPR seems to be insensitive to multicollinearity and quite large numbers of predictors can been used simultaneously [@alsgpr]. In this paper, fairly traditional ABA metrics were used, adding additional predictors, such as $\alpha$-shape [@alphashape] or composite metrics [@zhao2009], could potentially improve prediction performance. Additionally, dimension reduction, for example by using principal component analysis (PCA) [@junttila2015], would probably improve performance when using small training sets. Besides PCA, the deep belief network pretraining proposed in [@hinton2008] could be beneficial. The prediction step of GPR is not computationally much more costly than using kNN. The most computationally expensive part is the GPR model training, which requires computing the matrix inverse of a large matrix (see the equations and ). However, the matrix inverse can be precomputed for a given set of training data. After this, computing the prediction and the prediction interval only requires calculating matrix products. In the LOO case ($n_t=492$), computing the GPR prediction and intervals for a plot/cell took on average 345 ms in Matlab on a AMD Ryzen 1700X (3.4 GHz) processor, this is more expensive than kNN (1.5 ms), but still feasible for practice. For comparison, computing the Bayesian linear estimate took on average 18.5 s per plot. The training of GPR took 12.7 seconds. The present work used fixed length scale, covariance function, and error magnitude, because finding the optimal values for these (hyper)parameters automatically is generally a nonconvex and computationally difficult optimization problem. The values, $l=10$ for the correlation length, and 10% variance for the error $e$, were found by manual testing. The sub-optimal choice of these parameters might explain some of the tendency to produce too narrow prediction intervals for some stand attributes. Additionally, several commonly used covariance functions were tested; Matérn $3/2$ covariance function was found to be the best performing. More advanced covariance functions, such as nonstationary covariance functions [@paciorek2004] or spectral mixture covariance functions [@wilson2013; @wilson2016], could potentially improve prediction accuracy. Further research is still needed on finding the optimal model formulation. Due to extrapolation, GPR can produce unrealistic negative predictions. In this study, the negative predictions were corrected in a post-processing step. In the LOO cross-validation, total of 628 negative predictions occurred on 215 plots. Of these, 386 (61.5%) occurred in cases where the corresponding field-measured value was zero (i.e. a missing tree species). Furthermore, 93% of the negative predictions happened in cases where the corresponding field measurement was less than half of the average value of the stand attribute in the data set. The negative predictions thus occurred most commonly when predicting small stand attribute values. Additionally, 1039 cases where the field-measured stand attribute was zero were predicted to have a positive, non-zero value. Due to the more probable occurrence when predicting small values and the relatively symmetric distribution of the prediction error, the behavior of the negative predictions seems to be in line with the Gaussianity assumption in the GPR. In this study, GPR showed better reliability in all considered stand attribute predictions except the mean height of pine (the RMSE of basal area predictions of pine were practically the same for GPR and kNN) and the relative improvement of GPR predictions over the state-of-the-art method kNN were rather large being on average 4.6%. This is contrary to the earlier studies where the reliability of ALS based forest inventory system has been examined by comparing different estimation methods. For example, Maltamo *et al.* [@Maltamo2015] used visual pre-classification of aerial images to divide the study data into strata according to the main tree species and stand development stages. The aim was to improve species-specific estimates by applying more homogeneous reference data in kNN but the results were contradictory. The pre-classification did improve the accuracy of some species-specific stand attributes compared to the kNN estimates which applied whole study data as reference, but for some species-specific estimates the accuracy decreased. It is also notable that usually the accuracy of minor tree species did not improve, whereas in the present study the improvement was substantial especially for the minor species. Similar contradictory results have been obtained when comparing different statistical methods, such as neural networks or Bayesian approach [@niska2010neural; @varvia]. For example, Niska *et al.* [@niska2010neural] obtained more accurate species-specific volume estimates using neural networks at plot level than kNN but on the other hand kNN was more accurate on the stand level. R[ä]{}ty *et al.* [@raty2018] compared kNN estimates in which the species-specific estimates were obtained either by simultaneous imputation for all the species (as in this study) or by separate imputation for each species. The results concerning separate imputations were promising, but again, the results were contradictory. Conclusions =========== In this article, the feasibility of Gaussian process regression for the estimation of species-specific stand attributes within the area based approach was evaluated. In addition to testing the prediction performance, the prediction credible intervals were also evaluated. GPR estimates were compared with a state-of-the-art kNN-based algorithm and a linear Bayesian inference based method. The effect of training set size on the performance was also examined. The GPR estimates showed on average a 4.6% relative improvement in RMSE over the reference kNN method in the leave-one-out cross-validation, generally smaller bias, and credible interval performance on par with the linear Bayesian inference. The GPR estimates kept the advantage even when tested using smaller training set sizes. Especially the credible interval performance proved robust with respect to the training set size. The promising performance of GPR, the feasible computational cost, and that it provides prediction intervals make GPR an attractive method to use in forestry applications. Especially the plot level prediction uncertainty information provides many potential improvements in forest planning. [Petri Varvia]{} was born in Karttula, Finland, in 1988. He received M.Sc. and Ph.D. degrees in Applied Physics from the University of Eastern Finland in 2013 and 2018, respectively. He is currently a postdoctoral researcher at the Laboratory of Mathematics in the Tampere University of Technology. His scholarly interests include statistical inverse problems, Bayesian statistics and remote sensing. [Timo Lähivaara]{} received the M.Sc. and Ph.D. degrees from the University of Kuopio, Finland, and University of Eastern Finland in 2006 and 2010, respectively. Currently, he is a senior researcher at the Department of Applied Physics in the University of Eastern Finland. His research interests are in computational wave problems and remote sensing. [Matti Maltamo]{} was born in Jyväskylä, Finland in 1965. He received the M.Sc., Lic.Sc., and D.Sc. degrees (with honors) in forestry from the University of Joensuu, Joensuu, Finland, in 1988, 1992, and 1998, respectively. He is currently the Professor of Forest Mensuration Science with the Faculty of Science and Forestry, University of Eastern Finland. He has also worked as a visiting professor at the research group of professor Erik Naesset at the Norwegian University of Life Sciences. He was together with Naesset and Jari Vauhkonen the editor of the textbook “Forestry Applications of Airborne Laser Scanning – concepts and case studies” published in 2014. He has published about 165 scientifically refereed papers. His specific research topic is Forestry Applications of ALS. He is an Associate Editor of the journal Canadian Journal of Forest Research. Prof. Maltamo won together with professor Juha Hyyppä the First Innovation prize of the Finnish Society of Forest Science in 2010 about “Bringing airborne laser scanning to Finland. Maltamo also obtained bronze A.K. Cajander medal of the Finnish Society of Forest Science, 2012 [Petteri Packalen]{} was born in Rauma, Finland, in 1973. He received the M.Sc, Lic.Sc., and D.Sc. degrees in Forestry from the University of Joensuu, Joensuu, Finland, in 2002, 2007, and 2009, respectively. Currently, he is an Associate Professor in optimization of multi-functional forest management (Tenure Track) with the School of Forest Sciences, Faculty of Science and Forestry, University of Eastern Finland, Joensuu, Finland. Previously, he has been an Assistant, Senior Assistant, and Professor with the Faculty of Forestry, University of Joensuu. From August 2011 to July 2012, he was a Visiting Research Scientist at the Oregon State University, Corvallis, OR, USA. He has authored over 80 peer-reviewed research articles. Recently, his focus has been on time series, nearest neighbor imputation, combined use of ALS and spectral data in forest inventory, and the use of ALS in wildlife management. Since 2007, he has also been a Consultant for remote sensing-based forest inventory. His research interests include both practical and theoretical aspects of utilizing remote-sensing data in the monitoring and assessment of the forest environment. [Aku Seppänen]{} is an Associate Professor in the Department of Applied Physics at University of Eastern Finland, Kuopio. He received the M.Sc. and Ph.D. degrees from the University of Kuopio, Finland, in 2000 and 2006, respectively, and has authored 50 journal articles, 36 conference papers and 3 book chapters. His research interests are in statistical and computational inverse problems. He is one of the PIs in the Centre of Excellence in Inverse Modelling and Imaging (2018-2025) appointed by the Academy of Finland. The applications of his research include, e.g., industrial process imaging, non-destructive material testing and remote sensing of forest. [^1]: This work was supported by the Finnish Cultural Foundation, North Savo Regional fund, the Academy of Finland (Project numbers 270174, 295341, 295489, and 303801, and Finnish Centre of Excellence of Inverse Modelling and Imaging 2018-2025), and the FORBIO project (The Strategic Research Council, Grant No. 293380). [^2]: P. Varvia was with the Department of Applied Physics, University of Eastern Finland, FI-70211 Kuopio, Finland. He is now with the Laboratory of Mathematics, Tampere University of Technology, FI-33101 Tampere, Finland (e-mail: [email protected]). [^3]: T. Lähivaara and A. Seppänen are with the Department of Applied Physics, University of Eastern Finland, FI-70211 Kuopio, Finland. [^4]: M. Maltamo and P. Packalen are with the School of Forest Sciences, University of Eastern Finland, FI-80101 Joensuu, Finland.

ArXiv

--- abstract: 'A perturbative QCD treatment of the pion wave function is applied to computing the scattering amplitude for coherent high relative momentum di-jet production from a nucleon.' address: - | School of Physics and Astronomy,\ Tel Aviv University, 69978 Tel Aviv, Israel - | Department of Physics, Box 351560\ University of Washington\ Seattle, WA 98195-1560\ U.S.A. - | Department of Physics, Pennsylvania State University,\ University Park, PA 16802, USA author: - 'L. Frankfurt' - 'G. A. Miller' - 'M. Strikman' title: ' Perturbative Pion Wave function in Coherent Pion-Nucleon Di-Jet Production' --- \#1[[$\backslash$\#1]{}]{} Introduction ============ Consider a process in which a high momentum ($\sim$ 500 GeV/c) pion undergoes a coherent interaction with a nucleus in such a way that the final state consists of two jets (JJ) moving at high transverse relative momentum ($\kappa_\perp>1\sim2 $ GeV/c). In this coherent process, the final nucleus is in its ground state. This process is very rare, but it has remarkable properties[@fms93]. The selection of the final state to be a $q\bar q$ pair plus the nuclear ground state causes the $q\bar q$ component of the pion dominate the reaction process. At very high beam momenta, the pion breaks up into a $q\bar q$ pair well before hitting the nucleus. Since the momentum transfer to the nucleus is very small (almost zero for forward scattering), the only source of high momentum is the gluonic interactions between the quark and the anti-quark. Because $\kappa_\perp$ is large, the quark and anti-quark must be at small separations–the virtual state of the pion is a point-like-configuration[@fmsrev]. But the coherent interactions of a color neutral point-like configuration is suppressed by the cancellation of gluonic emission from the quark and anti-quark[@bb; @fmsrev]. Thus the interaction with the nucleus is very rare, and the pion is most likely to interact with only one nucleon. For this coherent process, the forward scattering amplitude is almost (since the momentum transfer is not exactly zero) proportional to the number of nucleons, $A$ and the cross section varies as $A^2$. This reaction, in which there are no initial or final state interactions, is an example of (color singlet) color transparency [@mueller; @fmsrev]. This is the name given to a high momentum transfer process in which the normal strongly absorbing interactions are absent, and the nucleus is transparent. The term “suppression of a color coherent process" could also be used, because it is the quantum mechanical destructive interference of amplitudes caused by the different color charges of a color singlet that is responsible for the reduced nuclear interaction. The forward angular distribution is difficult to observe, so one integrates the angular distribution, and the $A^2$ variation becomes $\approx A^{4/3}$. But the inclusion of the leading correction to this process, which arises from multiple scatterng of the point-like configuration causes a further increase in the $A$-dependence[@fms93]. Actually at sufficiently small $x_N={2\kappa_t^2\over s}\le {1\over 2m_NR_A}$, the situation changes since the quark-antiquark system scatters off the collective gluon field of the nucleus. Since this field is expected to be shadowed, one expects a gradual disappearance of color transparency for $ x\le 0.01$ - this is the onset of perturbative color opacity [@fms93]. Within the kinematical region of applicability of the QCD factorization theorem, the A dependence of this process is given by the factor: $A^{4/3}\left[G_A(x,Q^2)/G_N(x,Q^2)\right]^2$ [@fms93] Our interest in this curious process has been renewed recently by experimental progress[@danny]. The preliminary result from experiments comparing Pt and C targets is a dependence $\sim A^{1.55\pm0.05}$, qualitatively similar to our 1993 prediction. It is much stronger than the one observed for the soft diffraction of pions off nuclei (for a review and references see [@fmsrev]) , and it is qualitatively different from the behaviour $\sim A^{1/3}$ suggested in [@bb]. Since 1993 many workers have been able to make considerable progress in the theory related to the application of QCD to experimentally relevant observables, and we wish to incorporate that progress and improve our calculation. Our particular aim here is to use perturbative QCD to compute the relevant high-$\kappa_t, q\bar q$ component of wavefunction of the incident pion. We show here that QCD factorization holds for the leading term which dominates at large enough values of $\kappa_\perp$. In the following we discuss the different contributions to the scattering amplitudes as obtained in perturbative QCD. Amplitude for $\pi N\to N JJ$ =============================== Consider the forward ($t=t_{min}\approx 0$) amplitude, ${\cal M}$, for coherent di-jet production on a nucleon $\pi N\to N JJ$[@fms93]: (N)=f,\_,x , \[matel\]where $\widehat{f}$ represents the soft interaction with the target nucleon. The initial $\mid \pi\rangle$ and final $\mid f, \kappa_\perp x \rangle$ states represent the physical states, which generally involve all manner of multi-quark and gluon components. Our notation is that $x$ is the fraction of the total longitudinal momentum of the incident pion, and $1-x$ is the fraction carried by the anti-quark. The transverse momenta are given by $\vec{\kappa}_\perp$ and $-\vec{\kappa}_\perp$. As discussed in the introduction, for large enough values of $\kappa_\perp$, only the $q\bar q$ components of the initial pion and final state wave functions are relevant in Eq. (\[matel\]). This is because we are considering a coherent nuclear process which leads to a final state consisting of a quark and anti-quark moving at high relative transverse momentum. The quark and anti-quark ultimately hadronize at distances far behind the target, and this part of the process is analyzed by the experimentalists using a well-known algorithm[@danny]. We continue by letting the $q\bar q$ part of the Fock space be represented by $\mid \pi\rangle_{q\bar q}$, then \_[q|q]{} =G\_0() V\_[eff]{}\^\_[q|q]{}, \[pieq\] where $G_0(\pi)$ is the non-interacting $q\bar q$ Green’s function evaluated at the pion mass: p\_, yG\_0 ()p’\_,y’= [\^[(2)]{}(p\_-p\_’)(y-y’)m\_\^2- [p\_\^2 +m\_q\^2y(1-y)]{}]{}, where $m_q$ represents the quark mass, $y$ and $y'$ represent the fraction of the longitudinal momentum carried by the quark; and the relative transverse momentum between the quark and anti-quark is $p_\perp$ and $ V_{eff}^\pi$ is the complete effective interaction, which includes the effects of all Fock-space configurations. A similar equation holds for the final state: f,\_,x\_[q|q]{} =\_,x+G\_0(f) V\_[eff]{}\^f f,\_,x \_[q|q]{},\[fstate\] p\_, yG\_0 (f)p’\_,y’= [\^[(2)]{}(p\_-p\_’)(y-y’)m\_f\^2- [p\_\^2 +m\_q\^2y(1-y)]{}]{}, \[gf\] m\_f\^2, in which the first term on the right-hand-side of (\[fstate\]) is the plane-wave part of the wave function. The use of the wave functions (\[pieq\]) and (\[fstate\]) in the equation (\[matel\]) for the scattering amplitude yields $$\begin{aligned} {\cal M}(N)&=&{1\over 2}(T_1+T_2),\nonumber\\ T_1&\equiv& \langle \kappa_\perp,x \mid \widehat{f}\mid \pi\rangle,\quad T_2\equiv _{q\bar q}\langle f,\kappa_\perp,x \mid V_{eff}^f G_0(f)\widehat{f} \mid \pi\rangle_{q\bar q}.\label{tdef}\end{aligned}$$ The term $T_2$ includes the effect of the final state $q\bar q$ interaction; this was not included in our 1993 calculation[@fms93], but its importance was stressed in [@jm] . We shall first evaluate $T_1$, and then turn to $T_2$. Evaluation of $T_1$ =================== The wave function $\mid \pi\rangle_{q\bar q}$ is dominated by components in which the separation between the constituents is of the order of the diameter of the physical pion, but there is a perturbative tail which accounts for short distance part of the pion wave function. This perturbative tail is relevant here because we need to take the overlap with the final state which is constructed from constituents moving at high relative momentum. If we concentrate on those aspects it is reasonable to consider only the one gluon exchange contribution $V^g$ to $V_{eff}^\pi$ and pursue the Brodsky-Lepage analysis [@BL] for the evaluation of this particular component. Their use of the light cone gauge $A^+=0$, simplifies the calculation. We also use their normalization and phase-space conventions. We want to draw attention to the issue of gauge invariance. The pion wave function is not gauge invariant, but the sum of two-gluon exchange diagrams for the pion transition to $q\bar q$ is gauge invariant. This is because only the imaginary part of the scattering amplitude survives in the sum of diagrams, and because two exchanged gluons are vector particles in a color singlet state as a consequence of Bose statistics. So, in this case, conservation of color current has the same form as conservation of electric current in QED. We also note, that in the calculation of hard high-momentum transfer processes, the $q\bar q$ pair in the non-perturbative pion wave function should be considered on energy shell. Corrections to this enter as an additional factor of $1\over \kappa_t^2$ in the amplitude. We define the non-perturbative part of the momentum space wave function as ( l\_,y)l\_, y\_[q|q]{}. We use the one-gluon exchange approximation to the exact wave function of Eq. (\[pieq\]) to obtain an approximate wave funtion, $\chi$, valid for large values of $k_\perp$. $$\begin{aligned} \chi(k_\perp,x)={-4\pi C_F} {1\over m_\pi^2-{ k_\perp^2 +m_q^2\over x(1-x)}} \int_0^1 dy\int {d^2 l_\perp\over(2\pi)^3} V^g(k_\perp,x;l_\perp,y) %%%need to put in wave fucntion \psi(l_\perp,y)\end{aligned}$$ with $$\begin{aligned} % V^g(q_\perp,x;l_\perp,y)=-4\pi C_F\alpha_s %{\bar u (x,q_\perp)\over\sqrt{x}}\gamma_\mu {u(y,l_\perp)\over \sqrt{y}} %{\bar v (x,-q_\perp)\over\sqrt{1-x}}\gamma_\nu % {1\over m_\pi^2-{q_\perp^2-m_q^2\over x} -{l_\perp^2+m_q^2\over 1-y} % -{(q_\perp-l_\perp)^2\over y-x}} +(x\to 1-x, y\to % -{(k_\perp-l_\perp)^2\over y-x}} +(x\to 1-x, y\to %%%\nonumber %\psi(l_\perp,y) V^g(k_\perp,x;l_\perp,y)=-4\pi C_F\alpha_s {\bar u (x,k_\perp)\over\sqrt{x}}\gamma_\mu {u(y,l_\perp)\over \sqrt{y}} {\bar v (x,-k_\perp)\over\sqrt{1-x}}\gamma_\nu {v(1-y,-l_\perp)\over \sqrt{1-y}}d^{\mu\nu} \nonumber\\ \times \left[ {\theta(y-x)\over y-x} {1\over m_\pi^2-{k_\perp^2-m_q^2\over x} -{l_\perp^2+m_q^2\over 1-y}} % LF check 1-y) \right], \end{aligned}$$ and $C_F={n_c^2-1\over 2n_c}={4\over 3}$. The range of integration over $l_\perp$ is restricted by the non-perturbative pion wave function $\psi$. Then we set $l_\perp$ to 0 everywhere in the spinors and energy denominators and evaluate the strong coupling constant at $k_\perp^2$: \_s(k\_\^2)=[4]{} ,where $\beta=11-{2\over 3}n_f$. Then $$\begin{aligned} V^g(k_\perp,x;l_\perp,y)\approx {-4\pi C_F\alpha_s(k_\perp^2)\over x(1-x) y(1-y)}V^{BL}(x,y)\end{aligned}$$ where $V^{BL}(x,y)$ is the Brodsky-Lepage kernal: V\^[BL]{}(x,y)=2,with the operator $\Delta$ defined by ${\Delta \over x-y}\phi(x)={\phi(x)-\phi(y)\over x-y}$. This kernal includes the effects of vertex and quark mass renormalization. The net result for the high $k_\perp$ component of the pion wave function is then (k\_)=[4C\_F \_s(k)k\_\^2]{} \_0\^1 dy V\^[BL]{}(x,y)[(y,k\_\^2)y(1-y)]{} \[pieq1\], where $$\phi(y,q_\perp^2)\equiv \int {d^2 l_\perp\over(2\pi)^3} \theta(q_\perp^2-l_\perp^2)\psi(l_\perp,y).$$ The quark distribution amplitude $\phi$ can be obtained using QCD evolution[@BL]. Furthermore, the analysis of experimental data for virtual Compton scattering and the pion form factor performed in [@tolya; @kroll] shows that this amplitude is not far from the asymptotic one for $k^2_\perp\ge 2-3$ GeV$^2$ (x)=a\_0x(1-x), \[phi\]where $a_0=\sqrt{3}f_\pi$ with $f_\pi\approx 93$ MeV. Equation (\[pieq1\]) represents the high relative momentum part of the pion wave function. Using the asymptotic function (\[phi\]) in Eq. (\[pieq1\]) leads to an expression for $\psi(k_\perp,x)\propto x(1-x)/k^2_\perp$ which is of the factorized form used in Ref. [@fms93]. To compute the amplitude $T_1$, it is necessary to specify the scattering operator $\widehat f.$ For high energy scattering the operator $\widehat {f}$ changes only the transverse momentum and therefore in the coordinate space representation $\widehat f$ depends on $b^2$. The transverse distance operator $\vec b = (\vec{b}_{q}-\vec{b}_{\bar{q}})$ is canonically conjugate to $\vec{\kappa}_\perp$. At sufficiently small values of $b$, the leading twist effect and the dominant term at large $s$ arises from the diagrams when pion fragments into two jets as a result of interactions with the two-gluon component of gluon field of a target, see Figure 1. The perturbative QCD determination of this interaction involves a diagram similar to the gluon fusion contribution to the nucleon sea-quark content observed in deep inelastic scattering. One calculates the box diagram for large values of $\kappa_\perp$ using the wave function of the pion instead of the vertex for $\gamma^*\to q\bar q$. The application of QCD factorization theorem leads [@fmsrev; @bbfs; @frs] to $$%\sigma^{q \bar{q}}_{T} \widehat f(b^2)=i s \frac{\pi^2}{3} b^2 \left[ x_N G_N(x_N, \lambda/b^2) \right] \alpha_{s}(\lambda/b^2), \label{eq:1.27}$$ in which $x_N=2\kappa_\perp^2/s$ where $G_N$ is the gluon distribution function of the nucleon, and $\lambda(x=10^{-3})=9$ according to Frankfurt, Koepf, and Strikman, [@fks]. Accounting for the difference between the pion mass and mass of two jet system requires us to replace the target gluon distribution by the the skewed gluon distribution. The difference between both distributions is calculable in QCD using the QCD evolution equation for the skewed parton distributions[@ffs; @fg]. The most important effect shown in Eq. (\[eq:1.27\]) is the $b^2$ dependence which shows the diminishing strength of the interaction for small values of $b$. In the leading order approximation it is legitimate to rewrite $\sigma$ in the form: (b\^2)=is \_0 [b\^2b\_0\^2 ]{} \[fb2\] in which the logarithmic dependence on $b^2$ is neglected. Our notation is that $ \langle b_0^2 \rangle$ represents the pionic average of the square of the transverse separation, and \_s(\_\^2) \[x\_NG\_N\^[(skewed)]{} (x\_N,\_\^2)\] . The use of Eq. (\[fb2\]) allows a simple evaluation of the scattering amplitude $T_1$ because the $b^2$ operator acts as $-\nabla_{\kappa_\perp}^2$. Using Eqs. (\[pieq\]) and (\[fb2\]) in Eq. (\[tdef\]), leads to the result: T\_1=-4i[\_0b\^2 ]{}[4C\_F\_s(\_\^2) \_\^4]{} (1+[1]{})a\_0x(1-x). \[t1\] This is, except for the small $1\over \ln{\kappa_\perp^2\over \Lambda^2}$ correction ($\kappa_\perp\approx 2$ GeV and $\Lambda\approx 0.2$) arising from taking $-\nabla_{\kappa_\perp}^2$ on $\alpha_s$, is of the same form as the corresponding result of our 1993 paper. The amplitude of Ref. [@bb] varies as a Gaussian in $\kappa_\perp$. The $\kappa_\perp$ dependence: ${d\sigma(\kappa_\perp)\over d\kappa_\perp^2}\propto {1\over \kappa_\perp^8}$ follows from simple reasoning. The probability to find a pion at $b\le {1\over \kappa_\perp}$ is $ \propto b^2$, while the square of the total cross section for small dipole-nucleon interactions is $\propto b^4$. Hence the cross section of productions of jets with sufficiently large values of $\kappa_\perp $ is $\propto {1\over \kappa_\perp^6}$ leading to a differential cross section $\propto {1\over \kappa_\perp{^8}}$. Similar counting can be applied to estimate the $\kappa_\perp$ dependence for diffraction of a nucleon into three jets. Other amplitudes ================ So far we have emphasized that the amplitude we computed in 1993 is calculable using perturbative QCD. However there are four different contributions which occur at the same order of $\alpha_s$. The previous term in which the interaction with the target gluons follows the gluon-exchange represented by the potential $V^g$ in the pion wave function has been denoted by $T_1$. But there is also a term, in which the interaction with the target gluons occurs before the action of $V^g$ is denoted as $T_2$, see Figure 2. However, the two gluons from the nuclear target can also be annihilated by the exchanged gluon (color current of the pion wave function). This amplitude, denoted as $T_3$, is shown in Figure 3. The sum of diagrams when one target gluon is attached before the potential $V^g$ and a second after the potential $V^g$, see E.g. Figure  4, corresponds to an amplitude, $T_4$. We briefly discuss each of the remaining terms $T_2,T_3,T_4$. Their detailed evaluation will appear in a later publication. However we state at the outset that each of these amplitudes is suppressed by color coherent effects, and that each has the same $\kappa_\perp^{-4}$ dependence. The existence of the term $T_2$, which uses an interaction that varies as $b^2$, caused Jennings & Miller[@jm] to worry that the value of ${\cal M}_N$ might be severely reduced due to a nearly complete cancellation. Our preliminary and incomplete estimate obtained by neglecting the term arising from differentiation of the potential, $V^g$ finds instead enhancement. The $T_3$ or meson-color-flow term arises from the attachment of both target gluons to the gluon appearing in $V^g$ as well as the sum of diagrams where one target gluon is attached to potential $V^g$ in the pion wave function and another gluon is attached to a quark. This term is suppressed by color coherent destructive interference caused by the color neutrality of the $q\bar{q} g$ intermediate state. Thus this term has a form which is very similar to that of $T_1$ and $T_2$. The $T_4$ term arises from the sum of diagrams when one target gluon interacts with a quark in the pion wf before exchange by potential $V^g$ and second gluon interacts after that. The sum of these diagrams seems to be O because, for our kinematics, the $q$ and $\bar q$ in the initial and final states are not causally connected in these diagrams. The mathematical origin of this near 0 arises from the sum of diagrams having the form of contour integral: $\int d\nu \frac{1}{(\nu-a-i\epsilon)(\nu-b-i\epsilon)},$ which vanishes because one can integrate using a closed contour in the lower half complex $\nu$-plane. Summary Discussion ================== The purpose of this paper has been to show how to apply leading-order perturbative QCD to computing the scattering amplitude for the process: $\pi N\to JJ$. The high momentum component of the pion wave function, computable in perturbation theory is an essential element of the amplitude. Another essential feature of our result (20) is the $\sim {1\over \kappa_\perp^4}$ dependence of the amplitude manifest in a $\kappa_\perp^{-8}$ behavior of the cross section. This feature needs to be observed experimentally before one can be certain that the experiment [@danny] has verified the prediction of Ref. [@fms93]. Aknowledgements =============== It is a pleasure to dedicate this work to Prof. Kurt Haller, who has been recently interested in color transparency[@kh1], and who has long been interested in the fundamentals of QCD as applied to light-cone physics[@kh2]. Happy birthday, Kurt, and our best wishes for many more to come. This work has been supported in part by the USDOE. L. Frankfurt, G.A. Miller, M.Strikman, Phys. Lett. [**B304**]{} 1, (1993) See the review: L.L. Frankfurt, G.A. Miller and M. Strikman, Ann. Rev. Nucl. Part. Sci. [**44**]{}, 501 (1994) hep-ph/9407274. G.F. Bertsch, S.J. Brodsky, A.S. Goldhaber and J.F. Gunion, Phys. Rev. Lett. [**47**]{}, 297 (1981). A.H. Mueller, in Proceedings of 17’th Rencontre de Moriond, MOriond, 1982, ed. J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, France, 1982) p.13 Fermilab experiment E791, R. Weiss-Babai, talk at Hadron ’97 and D. Ashery, talk at International Workshop of Diffractive physics, Rio de Janeiro, Feb. 1998. B.K. Jennings and G.A. Miller Phys.Rev. [**C50**]{}, 3018 (1994). S.J. Brodsky G.P. Lepage, Phys. Rev. [**D22**]{}, 2157 (1982) B.Blattel, G.Baym, L.L.Frankfurt, and M.Strikman, Phys. Rev. Lett. [**71**]{} 896 (1993). L. Frankfurt, A. Radyushkin, and M. Strikman, Phys. Rev. [**D55**]{} ,98 (1997) L. Frankfurt, W. Koepf, M. Strikman, Phys. Rev. [**D54**]{}, 3194 (1996) L. Frankfurt, A. Freund, V. Guzey, M. Strikman Phys. Lett. [**B418**]{} 345,1998, Erratum-ibid.B429:414,1998. A. Freund, V. Guzey, hep-ph - 9801388; hep-ph - 9806267. A.V. Radyushkin, hep-ph/9707335; A. Szczepaniak, A. Radyushkin and C. Ji, Phys. Rev. [**D57**]{}, 2813 (1998) hep-ph/9708237; I.V. Musatov and A.V. Radyushkin, Phys. Rev. [**D56**]{}, 2713 (1997) hep-ph/9702443. P. Kroll and M. Raulfs, Phys. Lett. [**B387**]{}, 848 (1996) hep-ph/9605264. L. Chen and K. Haller, “Quark confinement and color transparency in a gauge invariant formulation of QCD," hep-th/9803250. K.. Haller, “Gauge Theories In The Light Cone Gauge," Phys. Rev. [**D42**]{}, 2095 (1990). [Figure 1. Contribution to $T_1$. The high momentum component of the pion interacts with the two-gluon field of the target. Only a single diagram of the four that contribute is shown.]{} [Figure 2. Contribution to $T_2$. The high momentum component of the final $q\bar q$ pair interacts with the two-gluon field of the target. Only a single diagram of the four that contribute is shown.]{} [Figure 3. Contribution to $T_3$. The exchanged gluon interacts with the two-gluon field of the target. Only a single diagram of the several that contribute is shown.]{} [Figure 4. Contribution to $T_4$. A gluon interacts with a quark and another with the exchanged gluon. Only a single diagram of the several that contribute is shown.]{}

ArXiv

--- abstract: 'One-dimensional structures of non-Hermitian plasmonic metallic nanospheres are studied in this paper. For a single sphere, solving Maxwell’s equations results in quasi-stationary eigenmodes with complex quantized frequencies. Coupled mode theory is employed in order to study more complex structures. The similarity between the coupled mode equations and the effective non-Hermitian Hamiltonians governing open quantum systems allows us to translate a series of collective phenomenon emerging in condensed matter and nuclear physics to the system of plasmonic spheres. A nontrivial physics emerges as a result of strong non-radiative near field coupling between adjacent spheres. For a system of two identical spheres, this occurs when the width of the plasmonic resonance of the uncoupled spheres is twice the imaginary component of the coupling constant. The two spheres then become coupled through a single continuum channel and the effect of coherent interaction between the spheres becomes noticeable. The eigenmodes of the system fall into two distinct categories: superradiant states with enhanced radiation and dark states with no radiation. The transmission through one-dimensional chains with an arbitrary number of spheres is also considered within the effective Hamiltonian framework which allows us to calculate observables such as the scattering and transmission amplitudes. This nano-scale waveguide can undergo an additional superradiance phase transition through its coupling to the external world. It is shown that perfect transmission takes place when the superradiance condition is satisfied.' author: - Amin Tayebi - Scott Rice bibliography: - 'Refs.bib' title: 'Superradiant and Dark States in Non-Hermitian Plasmonic Antennas and Waveguides' --- [^1] Introduction ============ Manipulation of light in nanometer scales via surface plasmonic resonances of metallic structures has attracted a great deal of attention over the past two decades [@intro1]. Optical antennas capable of localizing light in sub-wavelength regions have resulted in a new generation of photonic devices with applications ranging from imaging [@intro2; @intro2p5; @intro3] to biosensing [@intro4; @intro5] and emission enhancement of photon sources [@intro6; @intro7; @intro8]. In addition, plasmonics ought to play an important role in the efficient reception and transport of optical energy in light harvesting devices [@intro15]. Therefore, various waveguide structures are being investigated in order to control and further improve the propagation of light in micrometer length scales [@intro13; @intro14; @Plasmonics.9.925; @PhysRevB.82.035434]. More recently, it has been shown that surface plasmons can exhibit quantum interference [@intro9; @intro10]. This has sparked a great interest in studying the quantum properties of surface plasmons and exploiting plasmonic devices as potential building blocks of quantum computers and quantum circuits [@intro11; @intro12]. It was previously suggested that plasmonic structures could be mapped to quantum systems governed by non-Hermitian Hamiltonians [@NonhemitianPlasmonic1; @NonhemitianPlasmonic2]. In [@NonhemitianPlasmonic2], the radiation properties of an array of optical dipole antennas are manipulated by altering the anti-Hermitian coupling strength between the elements of the array. However, due to the complexity of nano-dipole antennas and the lack of closed form expressions for the fields, the mapping to a non-Hermitian Hamiltonian was achieved via numerical simulation and curve fitting. In this paper we consider systems of plasmonic nano-spheres using the effective non-Hermitian Hamiltonian framework. This framework provides a general platform for studying different physical systems; it has been previously utilized in various problems ranging from quantum signal transmission in nano-structures [@celardo09; @Greenberg_transport] to solid state quantum computing [@Tayebi_qc; @PhysRevA.78.062116] and nuclear reactions [@VZ777; @AZ07; @PhysRevLett.115.052501; @DrZ_New; @PhysRevC.86.044602]. The description of plasmonic structures via the effective Hamiltonian is achieved due to the correspondence between the Feshbach formalism and the coupled mode theory of optical resonators. The effective non-Hermitian Hamiltonian approach allows us to translate phenomena already known in condensed matter and nuclear systems to the plasmonic system under study. One such example is the existence of superradiant and subradiant, or dark, states. In addition, the effective Hamiltonian framework can be readily used in order to calculate observables, such as the transmission coefficient through a plasmonic waveguide. In Section \[secII\] we discuss the effective non-Hermitian Hamiltonian formalism used for studying *open* system in condensed matter [@Greenberg_transport; @Tayebi_qTrans; @Amin_Scott_Conf; @YakovG3], nuclear physics [@Volya_NucP; @Volya2014_new; @Volya_new2; @AUERBACH200445] and quantum optics [@YakovG1; @YakovG2; @YakovG4]. We then discuss the coupled mode formalism used for studying optical and plasmonic systems. The connection between the effective Hamiltonian and coupled mode theory is illustrated through simple two-level examples. We then consider a single plasmonic metallic nanosphere. Section \[secIII\] considers a single plasmonic metallic nanosphere. The wave equation is solved in order to find the natural resonant frequencies of the single sphere, and discuss the intrinsic radiative nature of nanospheres. This provides a basis for the consideration of two spheres via coupled mode theory. The signature of superradiance emerges when the interaction between adjacent optical nano antennas occurs through a single continuum channel, resulting in states with enhanced radiation and confined dark modes. This is discussed in Section \[secIV\]. The effect of these states on energy transmission through a one-dimensional chain of spheres is considered in Section \[secV\], with applications to optical frequency nanoscale antennas and waveguide-like structures. Section \[secVI\] includes the summary, concluding remarks and future work. The effective non-hermitian hamiltonian and coupled mode theory {#secII} =============================================================== In this section we briefly discuss the effective non-Hermitian Hamiltonian approach and the coupled mode theory and show how both theories result in matrices of similar structures. The Effective Hamiltonian ------------------------- Consider a quantum system described by the Hamiltonian $H_0$ and its discrete set of eigenvectors $\ket{i}$ that interacts with its surrounding environment. The environment is thermodynamically large and is characterized by infinitely many channels with continuous energy spectrum $\ket{c;E}$. The Hilbert space of this problem can be divided into two subspaces with the help of projection operators $\mathcal{Q}$ and $\mathcal{P}$. The operator $\mathcal{Q}$ only acts on the subspace of the closed system and the operator $\mathcal{P}$ acts on the environment only. Without loss of generality, one can further assume that the projections are orthogonal, therefore $\mathcal{P}+\mathcal{Q}=1$ and $\mathcal{P}\mathcal{Q}=\mathcal{Q}\mathcal{P}=0$. The total Hamiltonian $H$ i.e. the Hamiltonian of the system, the environment and their interactions, is then decomposed into $H=H_{\mathcal{Q}\mathcal{Q}}+H_{\mathcal{Q}\mathcal{P}}+H_{\mathcal{P}\mathcal{Q}}+H_{\mathcal{P}\mathcal{P}}$, where $H_{\mathcal{Q}\mathcal{Q}}=\mathcal{Q}H\mathcal{Q}$, $H_{\mathcal{Q}\mathcal{P}}=\mathcal{Q}H\mathcal{P}$, $H_{\mathcal{P}\mathcal{Q}}=\mathcal{P}H\mathcal{Q}$ and $H_{\mathcal{P}\mathcal{P}}=\mathcal{P}H\mathcal{P}$. The effective Hamiltonian is achieved by projecting the stationary wave function in the Schrödinger equation $H\Psi=E\Psi$ into the subspace of the closed system $$\label{effectiveHamiltonian1} \mathscr{H}_{\text{eff}} (E) \mathcal{Q} \Psi = E\mathcal{Q} \Psi,$$ where $$\label{effectiveHamiltonian2} \mathscr{H}_{\text{eff}} (E) = H_{\mathcal{Q}\mathcal{Q}} + H_{\mathcal{Q}\mathcal{P}} \frac{1}{E-H_{\mathcal{P}\mathcal{P}} } H_{\mathcal{P}\mathcal{Q}} .$$ The first term in the left handside of (\[effectiveHamiltonian2\]) is the Hamiltonian of the closed system $\mathcal{Q} H \mathcal{Q}=H_0$. The second term in (\[effectiveHamiltonian2\]), can be further simplified by calculating the matrix element between two intrinsic states of the closed system $\ket{i}$ and $\ket{j}$ $$\label{effectiveHamiltonian3} \langle i| H_{\mathcal{Q}\mathcal{P}} \frac{1}{E-H_{\mathcal{P}\mathcal{P}} } H_{\mathcal{P}\mathcal{Q}} \ket{j}= \sum_{c}\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}},$$ where $A_{i}^{c}(E)$ are transition amplitudes from continuum channel $\ket{c;E}$ to internal state $\ket{i}$; $A_{i}^{c}(E)=\langle i|H_{\mathcal{Q}\mathcal{P}}\ket{c;E}$. Usually continuum channels couple to the system only if the energy is above a certain threshold in which case the channel is open, otherwise the channel is closed and the transition amplitude vanishes. Using the Sokhotski-Plemelj theorem, the integral in (\[effectiveHamiltonian3\]) can be decomposed into Hermitian and anti-Hermitian parts $$\label{effectiveHamiltonian4} \vspace*{-16mm} \sum_{c} \int{dE'}\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'} = \\$$ $$\sum_{c}\mathcal{P}.\mathcal{V}.\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}} - i \pi \sum_{c_{open}} A_{i}^{c}(E){A_{j}^{c}}^{*}(E), \nonumber$$ where P.V. denotes the Cauchy principal value. Accordingly, two operators are defined; $\Delta(E)$, corresponding to the Hermitian component in (\[effectiveHamiltonian4\]) with matrix elements $$\label{realpartHeff} \Delta_{i,j}(E) = \sum_{c}\mathcal{P}.\mathcal{V}.\int{dE'\frac{A_{i}^{c}(E'){A_{j}^{c}}^{*}(E')}{E-E'}},$$ and $W(E)$ corresponding to the anti-Hermitian component in (\[effectiveHamiltonian4\]), with matrix elements $$\label{imagpartofHeff} W_{ij}(E)= 2\pi \sum_{c_{open}} A_{i}^{c}(E){A_{j}^{c}}^{*}(E).$$ Thus, in operator form, the *energy-dependent* effective Hamiltonian is $$\label{effectiveHamiltonian_complete} \mathscr{H}_{\text{eff}} (E)= H_0+\Delta(E)-\frac{i}{2}\,W(E).$$ The two terms, $\Delta(E)$ and $W(E)$, also known as the *self energy*, completely take the effect of the interaction with the environment into account. The Hermitian part, $\Delta(E)$, renormalizes the energies of the closed system. Notice that the summation in (\[realpartHeff\]) runs over all continuum channels, open and close. This is because even when the running energy, $E$, is below the threshold such that $A_{i}^{c}(E)$ vanishes, the integral is still non-vanishing and thus takes the *virtual* coupling to the continuum into account. On the other hand, $W(E)$ which is responsible for the decay width of the energy states of the effective Hamiltonian (\[effectiveHamiltonian\_complete\]), arises only due to real interaction processes with the environment. In many situations, including the case we are concerned with in this paper, the energy window of interest is relatively narrow and the transition amplitudes $A_{i}^{c}(E)$ are smooth functions of energies. These amplitudes can therefore be considered as energy-independent quantities. Consequently, the integral in the Hermitian component of the self energy (\[realpartHeff\]) vanishes and the effective Hamiltonian reduces to $$\label{ReducedHeff} \mathscr{H}_{\text{eff}}=H_{0}-\frac{i}{2}\,W.$$ It is interesting to look at the statistics of the *quasi-stationary* eigenenergies of the effective Hamiltonian $$\label{Heff_energies} E_{n}=\mathrm{E}_{n}-\frac{i}{2}\Gamma_n,$$ where $\mathrm{E}_{n}$ is the real component of the energy and $\Gamma_n$ is the width of the state and related to the lifetime by $\tau_n=\hbar/\Gamma_n$. The positiveness of $\Gamma_n$ is guaranteed by the Cholesky factorized form of $W$ in (\[imagpartofHeff\]) which makes $W$ a positive definite matrix. We define the quantity $\xi$ that parameterizes the strength of interaction with the external world, thus $$\label{ReducedHeff_xi} \mathscr{H}_{\text{eff}}=H_{0}-\frac{i}{2}\xi\,W.$$ Because the framework is exact and no approximation was used, $\xi$ can take arbitrarily small values representing weak interactions or extremely large values representing strong interactions with the external world. For weak interactions, when $\xi$ is small, the anti-Hermitian component, $W$ is a perturbation to $H_0$. The complex eigenenergies are then narrow resonances with almost uniform width distribution [@SOKOLOV1; @Volya2016]. In the opposite limit of strong interactions, the anti-Hermitian component becomes the dominant term and $H_0$ is the perturbation. It is clear from (\[imagpartofHeff\]) that the rank of $W$ is equal to the number of open channels which is normally much smaller than the number of intrinsic states of the closed system. Therefore a few states become dominant resonances that consume the entire width and the remaining states become long-lived states that decouple from the environment. Due to the resemblance of this phenomenon to the Dicke superradiance in quantum optics, we term the broad short-lived resonances as superradiant states and the narrow short-live resonances as subradiant states. It was shown [@Tayebi_qc; @Amin_Thesis] that if the system is connected to the environment through its physical boundaries then the superradiant states are localized to the boundaries of the system, and the subradiant states are pushed away from the boundaries and trapped within the interior of the system. The superradiance *phase transition* is discussed more rigorously in [@SOKOLOV1; @Auerbach]. It was shown that the important parameter to consider is the ratio of $\langle \Gamma \rangle$, the average widths of the energies in (\[Heff\_energies\]), to $D$, the mean level spacing of the close system. The transition occurs when $\langle \Gamma \rangle / D \approx 1$, this is when the resonances are maximally overlapped and the superradiant states emerge. The effective non-Hermitian Hamiltonian framework also provides us with useful expressions corresponding to observables such as the scattering and transmission amplitudes [@SOKOLOV1; @Auerbach; @SOKOLOV2; @Amin_Thesis]. Here we are particularly interested in transmission through a one dimensional chain of metallic nano-spheres. The transmission amplitude from a continuum channel $a$ to channel $b$ through the system is given by $$\label{Transmission_amp} Z^{ab}(E)=\sum_{i,j} A_{i}^{a} \bigg( \frac{1}{E-\mathscr{H}_{\text{eff}}} \bigg)_{i,j} {A_{j}^{b}}^*.$$ The transmission probability, $T^{ab}(E)$ is therefore $$\label{ProcessAmplitude} T^{ab}(E)=|Z^{ab}(E)|^2.$$ The amplitude in (\[Transmission\_amp\]) has a simple interpretation: the particle enters into state $\ket{j}$ through channel $b$. Next the propagator takes the particle from state $\ket{j}$ to $\ket{i}$ considering all possible *paths*. Finally the particle escapes the system from site $\ket{i}$ through continuum channel $a$. Coupled Mode Theory ------------------- In this paper, we employ coupled mode theory in order to study systems of interacting plasmonic spheres. This method is reminiscent of the time-dependent perturbation theory in quantum mechanics; it has been used for the investigation of coupled resonators in optical systems [@CMT_optic1; @CMT_optic2; @CMT_optic3], wireless energy transfer loop antennas [@CMT_antennas1], and plasmonic structures including antennas [@NonhemitianPlasmonic2; @CMT_antennas1] and waveguides [@CMT_waveguide1; @CMT_waveguide2; @CMT_waveguide3]. The coupled mode approach significantly simplifies the complexity of the problem: instead of solving the wave equation one needs to solve a system of linear algebraic equations. In addition, it provides a clear and intuitive picture of how interactions between the constituents of the system can dramatically change the dynamics. The formulation provided in this paper, similar to [@cmt_qnm_new], is rather general. No specific boundary conditions are assumed and hence it is applicable to the system of coupled plasmonic nanoantennas discussed in the future sections. Consider two non-magnetic dielectric resonators with relative dielectric constants $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$. The resonators occupy a volume in space, $V_1$ and $V_2$, respectively. In addition, consider that the relative dielectric constants $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$ are equal to unity for points outside of the resonators. In a time harmonic scenario each resonator, when isolated, satisfies the wave equation $$\label{wave_eqn_cmt} \vec{\nabla} \times \vec{\nabla} \times \vec{\mathcal{E}}^{\alpha}_n(\vec{r})-\bigg(\frac{ \omega_{\alpha , n}}{c}\bigg)^2 \epsilon_{\alpha} (\vec{r}) \vec{\mathcal{E}}^{\alpha}_n(\vec{r})=0,$$ where $\alpha=1, 2$ denotes the resonator number and $c$ is the speed of light in the background medium which is assumed to be the free space for simplicity. Due to the sharp discontinuity between the resonator and the background at the resonator boundaries, the modes are quantized and characterized by the integer number $n=1, 2, 3, ...$ and their eigenfrequency $\omega_{\alpha,n}$. The modes of isolated resonators are normalized according to $$\label{normalization_isolated_mode_gen} \int_{V} \epsilon_{\alpha}(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\alpha}_n(\vec{r}) d^3r = \delta_{mn},$$ where $V$ is the total volume in which fields are present and $m$ and $n$ are the mode indices and $\delta_{mn}$ is the Kronecker delta. The normalization expression is of crucial importance and its form is dictated by the boundary conditions of the problem. For instance in [@cmt_qnm_new] the normalization is similar to (\[normalization\_isolated\_mode\_gen\]), however with no complex conjugation. As we will see, the normalization expression has to be modified when discussing spherical plasmonic particles in the following sections, in order for the normalization expression to remain finite when integrated over all space. For now, we assume that the normalization rule is given by the general dot product definition provided in (\[normalization\_isolated\_mode\_gen\]) with the integration volume being all space, as this is usually the case in electromagnetic textbooks. We will revisit the normalization definition later in this paper. Next we assume that, for a system of two coupled resonators, the total electric field, $\vec{\mathcal{E}}(\vec{r})$, can be written as a superposition of a finite number of individual modes of the two resonators: $$\label{ansatz_cmt} \vec{\mathcal{E}}(\vec{r})=\sum_{n=1}^{N} \Big[ a_1(n) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \vec{\mathcal{E}}^{2}_n(\vec{r})\Big],$$ where $N$ is the total number of modes. The total electric field satisfies the wave equation $$\label{wave_eqn_cmt_total_field} \vec{\nabla} \times \vec{\nabla} \times \vec{\mathcal{E}}_n(\vec{r})-\Big(\frac{\omega_n}{c}\Big)^2 \epsilon(\vec{r}) \ \vec{\mathcal{E}}_n(\vec{r})=0,$$ where $\omega_n$ are the eigenfrequencies of the coupled system and $\epsilon(\vec{r})$ is the dielectric constant at a given point, $\vec{r}$, when both resonators are simultaneously present. The function $\epsilon(\vec{r})$ is equal to $\epsilon_1(\vec{r})$ and $\epsilon_2(\vec{r})$ for points inside the first and second resonator, respectively, and is equal to unity otherwise. Plugging the ansatz (\[ansatz\_cmt\]) into the wave equation (\[wave\_eqn\_cmt\_total\_field\]) and using its linearity and (\[wave\_eqn\_cmt\]) we arrive at $$\begin{aligned} \label{series_eqn} \sum_{n=1}^N \Big[ & a_1(n) \big(\omega_{1,n}\big)^2 \epsilon_{1} (\vec{r}) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \big(\omega_{2,n}\big)^2 \epsilon_{2} (\vec{r}) \vec{\mathcal{E}}^{2}_n(\vec{r}) \Big] \nonumber \\ & =\omega_n^2 \epsilon (\vec{r}) \sum_{n=1}^N \Big[ a_1(n) \vec{\mathcal{E}}^{1}_n(\vec{r}) + a_2(n) \vec{\mathcal{E}}^{2}_n(\vec{r}) \Big].\end{aligned}$$ Using the normalization rule (\[normalization\_isolated\_mode\_gen\]) to project (\[series\_eqn\]) onto $\vec{\mathcal{E}}^{1}_m(\vec{r})$ and $ \vec{\mathcal{E}}^{2}_m(\vec{r})$ for all values of $m$: $m=1,2,..,N$, we obtain a system of $2N$ linear equations. In the matrix form $$\label{cmt_equation_complete_form} \begin{pmatrix} \bm{T^{11}} & \bm{T^{12}} \\ \bm{T^{21}} & \bm{T^{22}} \end{pmatrix} \begin{pmatrix} \bm{\Omega_{1}}^2 & \bm{0} \\ \bm{0} & \bm{\Omega_{2}}^2 \end{pmatrix} \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix}= \omega^2 \begin{pmatrix} \bm{L^{11}} & \bm{L^{12}} \\ \bm{L^{21}} & \bm{L^{22}} \end{pmatrix} \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix},$$ where $\vec{A}_1$ and $\vec{A}_2$ are $N \times 1$ vectors of the coefficients $a_1(n)$ and $a_2(n)$ in the ansatz (\[ansatz\_cmt\]), respectively. $\bm{\Omega_{1}}$ and $\bm{\Omega_{2}}$ are $N \times N$ diagonal matrices containing the eigenfrequencies of the isolated resonators with matrix elements $$\big(\Omega_{\alpha}\big)_{mn}= \omega_{\alpha,n} \ \delta_{mn}$$ where as previously $\alpha=1, 2$. The matrix elements of the four square $N \times N$ matrices $\bm{T^{\alpha \beta}}$, where $\alpha, \beta=1, 2$, are given by $$T^{\alpha\beta}_{mn}= \int_{V} \epsilon_{\beta}(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\beta}_n(\vec{r}) d^3r.$$ According to (\[normalization\_isolated\_mode\_gen\]), $\bm{T^{11}}$ and $\bm{T^{22}}$ are equal to the identity matrix $\bm{1}$. Finally the elements of the matrices $\bm{L^{\alpha, \beta}}$ are given by $$\label{L_matrix_cmt} L^{\alpha\beta}_{mn}=\int_{V} \epsilon(\vec{r}) {\vec{\mathcal{E}}^{\alpha^*}_m}(\vec{r}). \vec{\mathcal{E}}^{\beta}_n(\vec{r}) d^3r.$$ Because the dielectric function $\epsilon(\vec{r})$ is the sum of the two dielectric function, the matrix elements $L^{12}_{m,n}$ and $T^{12}_{m,n}$ are related via $$\label{T_matrix_cmt} L^{12}_{mn}=T^{12}_{mn}+\int_{V_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{2}_n(\vec{r}) d^3r,$$ where the integration is carried out over the volume of the first resonator, $V_1$ only. Accordingly, we define the matrix $\bm{K^{12}}$ with matrix elements $$\label{coupling_coeff_K1} K^{12}_{mn}=\int_{V_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{2}_n(\vec{r}) d^3r.$$ Therefore $$\label{kappa_matrix1} \bm{L^{12}}=\bm{T^{12}}+\bm{K^{12}}.$$ Similarly $L^{21}_{mn}$ is related to $T^{21}_{mn}$ as $$L^{21}_{mn}=T^{21}_{mn}+\int_{V_2} \big(\epsilon_2(\vec{r})-1 \big) \vec{\mathcal{E}}^{2^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{1}_n(\vec{r}) d^3r.$$ Correspondingly $\bm{K^{21}}$ is defined with matrix elements $$\label{coupling_coeff_K2} K^{21}_{mn}=\int_{V_2} \big(\epsilon_2(\vec{r})-1 \big) \vec{\mathcal{E}}^{2^{*}}_m(\vec{r}).\vec{\mathcal{E}}^{1}_n(\vec{r}) d^3r.$$ Hence $$\label{kappa_matrix2} \bm{L^{21}}=\bm{T^{21}}+\bm{K^{21}}.$$ In order to simplify (\[cmt\_equation\_complete\_form\]), we accept a number of approximations that are commonly used in studying systems of weakly coupled resonators [@Elnaggar_app; @Elnaggar_CMT1; @Elnaggar_CMT2]. We assume that the diagonal matrix elements in (\[L\_matrix\_cmt\]) are approximately equal to unity, i.e. $L^{11}_{m,n}=L^{22}_{m,n} \approx 1$ and therefore $\bm{L^{11}}=\bm{L^{22}}\approx \bm{1}$. This is justified due to the strong field confinement within the dielectric regions. Furthermore, we assume that the coupling is weak and therefore the coupling elements in (\[T\_matrix\_cmt\]) satisfy the condition $T^{12}_{mn} T^{21}_{m'n'} \ll 1$. Using these approximations along with (\[kappa\_matrix1\]) and (\[kappa\_matrix2\]), the coupled mode equation (\[cmt\_equation\_complete\_form\]) reduces to $$\label{cmt_equation_reduced_form1} \begin{pmatrix} \bm{1} & -\bm{K^{12}} \\ -\bm{K^{21}} & \bm{1} \end{pmatrix} \begin{pmatrix} \bm{\Omega_{11}^2} & \bm{0} \\ \bm{0} & \bm{\Omega_{22}^2} \end{pmatrix} \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix}= \omega^2 \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix}.$$ It is also helpful to linearize the system of equations (\[cmt\_equation\_reduced\_form1\]). This can be done by noting that the eigenmodes of the isolated resonators are not far apart and are clustered around their mean value [@Haus_CMT], i.e. $\omega \approx \omega_{\alpha,n}$. Under this approximation $\omega$ and $\omega_{\alpha,n}$ satisfy the following $$\omega^2-\big(\omega_{\alpha,n} \big)^2 \approx 2\omega_{\alpha,n}(\omega-\omega_{\alpha,n} ).$$ This brings us to the final form of the coupled mode equations $$\label{cmt_equation_reduced_form2} \begin{pmatrix} \bm{1} & -\frac{1}{2}\bm{K^{12}} \\ -\frac{1}{2}\bm{K^{21}} & \bm{1} \end{pmatrix} \begin{pmatrix} \bm{\Omega_{11}} & \bm{0} \\ \bm{0} & \bm{\Omega_{22}} \end{pmatrix} \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix}= \omega \begin{bmatrix} \vec{A}_1 \\ \vec{A}_2 \end{bmatrix}.$$ In the simplest situation when the two resonators are identical and only one mode of an isolated resonator is considered, the electric field of the coupled system can be expressed as $\vec{\mathcal{E}}(\vec{r})= a_1 \vec{\mathcal{E}}^1(\vec{r}) + a_2 \vec{\mathcal{E}}^1(\vec{r})$. According to (\[cmt\_equation\_reduced\_form2\]) the coupled mode equations are then given by $$\begin{aligned} \label{CMT2res_timeDomain} \omega_0 a_1 + \kappa \ a_2=\omega a_1, \nonumber \\ \omega_0 a_2 + \kappa^*a_1=\omega a_2, \end{aligned}$$ where $\omega_0$ is the eigenfrequency of the isolated resonators. Using (\[coupling\_coeff\_K1\]) and (\[cmt\_equation\_reduced\_form2\]), $\kappa$ is given by $$\label{coupling_coefficient} \kappa=-\frac{1}{2}\omega_0 \int_{\text{V}_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1^*}(\vec{r}). \vec{\mathcal{E}}^2(\vec{r}) d^3r.$$ The coupling coefficient $\kappa$ has a simple interpretation: it is the interaction energy between the field generated by the second resonator and the dipole moment of the first resonator averaged over one period. The complex conjugation of the coupling coefficient in (\[CMT2res\_timeDomain\]) is dictated by the energy conservation, assuming there is no loss or gain in the system [@Haus_CMT]. The eigenfrequencies of the coupled system, which are guaranteed to be real due to the Hermitian form of the equations in (\[CMT2res\_timeDomain\]), are $$\omega_{\pm}=\omega_0 \pm |\kappa|,$$ where the frequencies of the coupled system, $\omega_{+}$ and $\omega_{-}$, correspond to the symmetric eigenstate with $a_1=a_2=1/\sqrt{2}$ and the anti-symmetric eigenstate with $a_1=-a_2=1/\sqrt{2}$, respectively. One can readily see the similarity between the coupled mode theory and the quantum theory as both are a theory of waves. Equation (\[CMT2res\_timeDomain\]) is the Schrödinger equation for a two-level system (a qubit). In quantum mechanical language, $\omega_0$ is the energy of the *unperturbed* states and the off-diagonal matrix element $\kappa$ represents the interaction strength between the two states which is responsible for the level repulsion and avoided crossing of the final *mixed* states. An interesting dynamic of the two-level system is the so-called Rabi oscillation. Let the system at time $t=0$ be prepared in the unperturbed state with energy $\omega_1$. Then the probability $P(t)$ to find the system in the same state at time $t$ is [@zelevinskybook] $$P(t)=1-\text{sin}^2\Big(\frac{\omega_R t}{2} \Big),$$ where $\omega_R=\omega_{+}-\omega_{-}=2|\kappa|$ is the Rabi frequency of the excitation oscillating back and forth between the two levels. Because the unperturbed energies of the two states are equal, the probability goes through the minimum, $P=0$, which indicates that the excitation can be completely transferred, leaving no residue in the initial state. The Rabi oscillation was predicted in systems of optical waveguides [@Rabi_waveguide] and in coupled ring resonators [@Rabi_ringRes]. We now focus on a more realistic case: two coupled identical dielectric resonators where in general, due to damping and leakage of the resonators, the energy is no longer conserved. Therefore the governing equations need not be Hermitian. In the case of open systems one has to modify the normalization expression (\[normalization\_isolated\_mode\_gen\]) which leads to an altered expression for $\kappa$. The coupling coefficient in this case is similar to (\[coupling\_coefficient\]) but with no complex conjugation (see [@cmt_qnm_new] for detail). The coupled equations (\[CMT2res\_timeDomain\]) are modified to a more general form $$\begin{aligned} \label{CMT2res_freqDomian} \omega_0 a_1 + \kappa a_2 &=\omega a_1 \nonumber \\ \omega_0 a_2 + \kappa a_1 &=\omega a_2, \end{aligned}$$ where $\omega_0$ can now be complex: $\omega_0=\eta_0-i\gamma_0/2$ representing loss and radiation. Similarly, $\kappa$ is in general complex as well: $\kappa=\kappa'-i\kappa''$ where $\kappa'$ and $\kappa''$ are real numbers. The left hand side of the coupled equations (\[CMT2res\_freqDomian\]) can then be written as the summation of two matrices, a Hermitian matrix, $H'_0$, and an anti-Hermitian matrix $W'$: $$\label{CMT_to_eff_Hamiltonian} \begin{pmatrix} \omega_0 & \kappa \\ \kappa & \omega_0 \end{pmatrix}= H'_0-\frac{i}{2}W',$$ where $$\label{CMT_Matrix_hermitian} H'_0= \begin{pmatrix} \eta_0 & \kappa' \\ \kappa' & \eta_0 \end{pmatrix},$$ and $$\label{CMT_Matrix_nonhermitian} W'= \begin{pmatrix} \gamma_0 & 2\kappa'' \\ 2\kappa'' & \gamma_0 \end{pmatrix}.$$ The similarity between the coupled mode theory and the effective non-Hermitian Hamiltonian formalism becomes apparent by comparing (\[ReducedHeff\]) and (\[CMT\_to\_eff\_Hamiltonian\]). The problem of two coupled dielectric resonators is mapped to a two level quantum system where in general each level is coupled to an independent continuum channel. This is because in general the rank of $W'$ is 2, therefore, according to (\[imagpartofHeff\]), one requires two independent open channels to construct the anti-Hermitian matrix $W'$. In the particular case when $\gamma_0=\pm 2\kappa''$, the rank of $W'$ is equal to unity, therefore the superradiance condition is fulfilled and only one open channel is required to construct the matrix $W'$. In this case, the effective Hamiltonian has two distinct eigenvalues, a purely real eigenvalue or the subradiant state , reminiscent of dark modes in open quantum systems, with eigenfrequency $\eta_0-\kappa'$, and a complex eigenvalue with enhanced radiation properties and eigenfrequency $\eta_0+\kappa'-i\gamma_0$, which is the superradiant state. A Single Metallic Sphere {#secIII} ======================== In this section we consider a single isolated metallic sphere embedded in a homogeneous background dielectric material. The problem is treated classically by solving Maxwell equations. In the absence of external sources and assuming a harmonic time dependence of the form $$\label{Phase_convention} \vec{\mathbb{E}}(\vec{r},t)=e^{i\omega t} \vec{\mathcal{E}}(\vec{r}),$$ the governing equation is the well known Helmholtz equation $$\label{Helmholtz_eqn} \nabla^2 \vec{\mathcal{E}}(\vec{r})-k^2\vec{\mathcal{E}}(\vec{r})=0.$$ The wave number $k$ is defined for the interior and exterior regions of the plasmonic sphere according to $$\begin{aligned} k = \begin{cases} k_{\text{in}}=\frac{\omega}{c} \sqrt{\epsilon_{\text{in}}} & r\leq a,\\ k_{\text{out}}=\frac{\omega}{c} \sqrt{\epsilon_{\text{out}}} & r> a, \end{cases}\end{aligned}$$ where $c$ is the speed of light in vacuum and $a$ is the radius of the sphere which is located at the origin of the coordinate system. The relative dielectric constants of the metallic sphere and the background medium are denoted by $\epsilon_{\text{in}}$ and $\epsilon_{\text{out}}$, respectively. Plasmonic structures are usually made of noble metals, such as gold and silver, with face-centered cubic lattice, or alkali metals, such as sodium and potassium, with body-centered cubic lattice. Due to their symmetric crystal lattice types, they are isotropic to light and their relative permittivity is characterized by a scalar. This dielectric constant is well described by the Drude-Sommerfeld model $$\begin{aligned} \label{Drude_dielectric_func} \epsilon_{\text{in}}(\omega)=\epsilon_{\infty}-\frac{\omega_p^2}{\omega^2-i\omega\gamma_s},\end{aligned}$$ where $\omega_p$ is the plasma frequency which is defined by the electron effective mass $m^*$, the vacuum permittivity $\epsilon_0$, electron charge $e$, and electron density $n$; $\omega_p^{2}= ne^2/\epsilon_0m^*$. The loss within the dielectric material due to various processes, such as electron-phonon interaction, impurities and scattering, is incorporated into the relaxation rate $\gamma_s$. The negative sign of this term in the denominator is dictated by the phase convention adopted in (\[Phase\_convention\]). The phenomenological parameter $\epsilon_{\infty}$ accounts for the contribution of the bound electrons to the polarization of the dielectric material. For typical metals the plasma frequency, $\omega_p$, ranges from 3 to 15 eV (700-3600 THz) which mainly falls into the ultraviolet spectrum [@Drude_model_plasma_1; @Drude_model_plasma_2; @Drude_model_plasma_3; @Drude_model_plasma_4; @Drude_model_plasma_5]. The damping rate $\gamma_s$ is much smaller than the plasma frequency, $\gamma_s \ll \omega_p$, being of the order $10^{-2}-10^{-1}$ eV (2.4-24 THz). Finally, the correction term $\epsilon_{\infty}$ typically ranges from 1 to 10 [@Silver_drude]. In an ideal electron gas, $\epsilon_{\infty}=1$ and $\gamma_s=0$, therefore the dielectric function (\[Drude\_dielectric\_func\]) reduces to $\epsilon_{\text{in}}(\omega)=1-\omega_p^2/\omega^2$. Below the plasma frequency the dielectric function is negative and the field can not penetrate inside the metal. For frequencies larger than the plasma frequency however, the dielectric constant becomes positive and the fields can penetrate the metal i.e. the metal becomes transparent. Using the spherical coordinate system, the solutions of the Helmholtz equation (\[Helmholtz\_eqn\]) for the plasmonic sphere can be divided into two categories: transverse magnetic (TM) modes with no radial magnetic field and transverse electric (TE) modes with no radial electric field component. In this work we consider the TM modes only. The components of the electric field are given by [@harringtonbook]: $$\begin{aligned} \label{field_exprs} \mathcal{E}_r&=\zeta \ C(r;a) \ell(\ell+1) \frac{f_{\ell}(kr)}{\epsilon(r)kr} Y_\ell^m(\theta,\phi), \nonumber \\ \mathcal{E}_\theta&=\zeta \ C(r;a)\frac{1}{\epsilon(r)kr} \frac{\partial}{\partial(kr)}\Big(krf_\ell(kr)\Big)\frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi), \\ \mathcal{E}_\phi&=\zeta \ C(r;a)\frac{1}{\epsilon(r)kr} \frac{\partial}{\partial(kr)}\Big(krf_\ell(kr)\Big)\frac{1}{\sin\theta}\frac{\partial}{\partial \phi}Y_\ell^m(\theta,\phi), \nonumber\end{aligned}$$ where $\zeta$ is the normalization constant discussed in detail in the next section. The dielectric constant is equal to $\epsilon_{\text{in}}$ and $\epsilon_{\text{out}}$ for $r\leq a$ and $r>a$, respectively. $Y_\ell^m(\theta,\phi)$ are the spherical harmonics with $\ell=0,1,2,...$ and $m=0,\pm 1,\pm 2,...,\pm \ell$. The case of $\ell=0$ results in the trivial solution. The first non-trivial solution corresponds to the dipole mode, $\ell=1$. The function $f_{\ell}(kr)$ is equal to the spherical Bessel function of the first kind and the spherical Hankel function of the second kind, for $r\leq a$ and $r>a$, respectively. $$\begin{aligned} f_{\ell}(kr) = \begin{cases} j_{\ell}(k_{\text{in}}r) & r\leq a,\\ h_{\ell}^{(2)}(k_{\text{out}}r) & r> a. \end{cases}\end{aligned}$$ The Bessel function $j_{\ell}(k_{\text{in}}r)$ represent standing waves within the plasmonic sphere while, noting the phase convention (\[Phase\_convention\]), the Hankel function $h_{\ell}^{(2)}(k_{\text{out}}r)$ describes radially outward traveling waves which satisfy the Sommerfeld radiation boundary condition. The coefficient $C(r;a)$ guarantees that the boundary conditions are satisfied at the boundary of the sphere (see [@SingleSph1] for details) $$\begin{aligned} \label{general_constant} C(r;a) = \begin{cases} \big[ \ j_{\ell}(k_{\text{in}}a)\big]^{-1} & r\leq a,\\ \big[ \ h_{\ell}^{(2)}(k_{\text{out}}a)\big]^{-1} & r> a. \end{cases}\end{aligned}$$ Matching the interior and the exterior fields leads to the characteristic equation of the discrete eigenfrequencies of the system: $$\label{charac_eqn_single_sphere} \epsilon_{\text{in}}\bigg[1+k_{\text{out}}a \ \frac{h_{\ell}^{(2)'}(k_{\text{out}}a)}{h_{\ell}^{(2)}(k_{\text{out}}a)} \bigg]=\epsilon_{\text{out}}\bigg[1+k_{\text{in}}a \ \frac{j'_{\ell}(k_{\text{in}}a)}{j_{\ell}(k_{\text{in}}a)} \bigg].$$ Here, the prime denotes differentiation with respect to the argument of the function, i.e. $j'_{\ell}(k_{\text{in}}r)= \partial j_{\ell}(k_{\text{in}}r)/ \partial (k_{\text{in}}r )$. For a given radius, different modes can be labeled by $\ell$: $\omega_{0,\ell}$, where the subscript $0$ denotes isolated single spheres. In case of small spherical particles, when $ka \ll 1$, considering the dipole mode $\ell=1$, the spherical Bessel and Hankel functions can be approximated by their leading order terms: $j_{1}(k_{\text{in}}a) \sim k_{\text{in}}a/3$ and $h_{1}^{(2)}(k_{\text{out}}a) \sim i(k_{\text{out}}a)^{-2}$. Therefore the characteristic equation (\[charac\_eqn\_single\_sphere\]) reduces to $\epsilon_{\text{in}}=-2\epsilon_{\text{out}}$ which leads to the well known resonance frequency of $\omega=\omega_{p}/\sqrt{3}$ for an ideal electron gas with $\epsilon_{\infty}=1$ and $\gamma_s=0$. Next, we numerically solve eq. (\[charac\_eqn\_single\_sphere\]) for a silver sphere with a free space background. The parameters of the Drude-Sommerfeld model for silver are [@Silver_drude]: the plasma frequency $\omega_p=8.9$ eV, the damping rate $\gamma_s=0.1$ eV, and $\epsilon_{\infty}=5$. The eigenfrequencies are always complex, which indicates the radiative nature of the nanospheres [@SingleSphDAMPING]. The real and imaginary components of the eigenfrequencies as a function of radius and for various values of $\ell=1, 2, 3, 4$, are shown in Fig. \[SilverResonance\]. The real part is the frequency required to excite a mode, for instance with a laser, and the imaginary component is the associated width of the mode. For all the modes, as expected, the real component decreases monotonically as the radius increases. To see the capability of the plasmonic sphere to manipulate light in sub-wavelength dimensions consider the dipole resonance ($\ell=1$) for a 50 nm sphere. The resonant frequency is about 3 eV corresponding to a free space wavelength of approximately 413 nm which is an order of magnitude larger than the radius of the sphere. The imaginary part of the eigenfrequencies consists of both non-radiative and radiative components, $\text{Im}(\omega_{0,\ell})=\gamma^{\text{nrad}}+\gamma_{\ell}^{\text{rad}}$. The non-radiative damping is associated with the loss within the plasmonic sphere. It was discussed in [@SingleSphDAMPING] that the non-radiative component can approximately be considered size-independent and is of equal value for all different modes, $\gamma^{\text{nrad}}=(1/2) \gamma_{s}$. It is therefore clear from Fig. \[SilverResonance\](b) that the dipole is the most radiative mode. For $\ell=1$, initially the sphere becomes more radiative as the radius increases. However, larger spheres have less pronounced radiation properties. For all higher order modes, the imaginary part grows as the radius increases. The electric field patterns of a silver sphere with a radius of 40 nm and various values of $\ell$ are shown in Fig. \[Fieldplots\]. The field values are displayed logarithmically, and normalized to the maximum of the electric field. As the $\ell$ value increases the fields become more tightly bound to the surface of the sphere. This becomes important when we consider the coupling between two spheres in the next section. It is also instructive to look at the electric field pattern for the dipole mode. Fig. \[Field2Dplots\] represents the electric field pattern in polar coordinates for various radial distances from the center of the sphere. The black arrow represents the dipole orientation. In all six figures, $\rho$ is the radial distance, perpendicular to the dipole axis and $z$ is the direction along the dipole. The blue line represents the relative field strength at a given polar angle. Furthermore, the field strength is normalized to the maximum value of the electric field. Fig. \[Field2Dplots\](a) shows the pattern at the surface of the plasmonic sphere, $r/a=1$. At locations close to the surface of the sphere, the fields reach a maximum along the direction of the dipole. At $r/a=3$ and $r/a=5$ the patterns are almost omnidirectional (see Figs. \[Field2Dplots\](b) and (c)). The well known torus shape radiation pattern of the dipole only emerges in the far field. This is shown in Figs. \[Field2Dplots\](d), (e) and (f). The eigenmodes and plasmonic resonances of single metallic nanospheres of various sizes and the effect of different background dielectrics have been considered in detail in [@SingleSph3] for gold nanoparticles and in [@SingleSph2] for alkali metals such as sodium, lithium and Cesium; see [@SingleSphDAMPING] for a more detail description of the size dependency properties of nanospheres. Below we consider the case of two identical plasmonic spheres and discuss the coupling between them and the eigenmodes of the system. Superradiant and Dark States in The System of Two Coupled Spheres {#secIV} ================================================================= Given the importance of two level systems in physics [@Amin_Holstein], we now consider the case of two coupled metallic spheres within the coupled mode theory framework discussed earlier. The interaction between the spheres can greatly alter the radiation properties of the system and result in resonance frequencies profoundly different from those for the isolated spheres. Here, we limit our consideration to the dipole modes only. This is justified by the earlier discussion that the dipole mode is the most radiative mode and also has the longest range compared to higher multipolarities. The isolated frequencies discussed in the previous section serve as the diagonal elements in the coupled mode matrix (\[CMT\_to\_eff\_Hamiltonian\]). The real and imaginary components of the dipole eigenmode are the diagonal elements of the Hermitian (\[CMT\_Matrix\_hermitian\]) and the anti-Hermitian (\[CMT\_Matrix\_nonhermitian\]) matrices, respectively. The coupling between two modes, $\kappa$, makes up the off-diagonal matrix elements of the final matrix (\[CMT\_to\_eff\_Hamiltonian\]). However, a difficulty arises due to the normalization of the single sphere modes. In the far field region, $k_{\text{out}}r \gg 1$, the spherical Hankel functions behave asymptotically as $$\label{asymptotic_fields} h_{\ell}^{(2)}(k_{\text{out}}r) \approx (i)^{\ell+1} \frac{e^{-ik_{\text{out}}r}}{k_{\text{out}}r}.$$ Because of the complex nature of the eigenfrequencies, the asymptotic form of the fields given by (\[asymptotic\_fields\]) grows exponentially in space as $r \rightarrow \infty$. This growth is however compensated by the exponential decay in time in the complete expression of the field (\[Phase\_convention\]) when the time dependency is considered. As a result the amplitude of the wave front of the total field (\[Phase\_convention\]) reaching any point in the asymptotic region is proportional to $1/r$, as it is expected. Nevertheless, the modes (\[field\_exprs\]) should be properly normalized. The correct normalization of such modes was discussed in [@normalization_1D2; @normalization_1D1; @normalization_1D3; @Amin_Thesis] for one-dimensional problems. The generalization to three dimensions by three different methods is discussed in [@normalization_3D1], [@normalization_3D2], and [@normalization_3D3]. However, in [@normalization_comparison], it was shown that all three expressions are compatible. The normalization condition is given by [@normalization_comparison]: $$\label{normalization_int} \int_V \sigma(\vec{r},\omega) \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r + \frac{i\epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r=1$$ where $V$ is the integration volume and $\partial V$ is its surface. The integration volume is assumed to be sufficiently large, so the fields at its surface are accurately approximated by asymptotic expressions of the spherical Hankel function provided in (\[asymptotic\_fields\]). The modified dielectric function $\sigma(\vec{r},\omega)$ which incorporates the dispersiveness of the medium is given, according to [@DispersiveMedia] as $$\label{modified_dielectric} \sigma(\vec{r},\omega)=\frac{1}{2\omega} \frac{\partial}{\partial \omega} \Big(\omega^2 \epsilon(\vec{r},\omega)\Big).$$ Contrary to the normalization discussed earlier, the dot-product in (\[normalization\_int\]) does not require any complex conjugation of the fields. It is therefore easier to use the so-called tesseral harmonics instead of the conventional spherical harmonics in the field expression (\[field\_exprs\]). The tesseral harmonics (sometimes also called real spherical harmonics) are nothing but even and odd superpositions of the traditional spherical harmonics, see Appendix \[THandIs\]. The normalization condition (\[normalization\_int\]) defines the constant $\zeta$ in the field expressions (\[field\_exprs\]) up to a phase. It is shown in Appendix \[QNMNAp\] that assuming the volume of integration as a sphere, the volume and surface terms in the normalization expression can be evaluated explicitly. It is furthermore proved in the same appendix that the condition (\[normalization\_int\]) reduces to: $$I \big[ j_{\ell}(k_{\text{in}}a) \big] - I \big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]=1,$$ where the functional $I \big[f_{\ell}(kr) \big]$ is given by (\[volumeterm4\]) $$\begin{aligned} I \big[f_{\ell}(kr)\big]=\sigma(\vec{r},\omega) \ \zeta^2 \ C^2(r;a)\frac{\ell(\ell+1)}{k^2} \bigg[ r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr) +\frac{k^2r^3}{2}\Big(f_{\ell}^2(kr)-f_{\ell-1}(kr)f_{\ell+1}(kr) \Big) \bigg].\end{aligned}$$ Once the normalization constant $\zeta$ is found the coupling between two modes can be calculated according to $$\label{coupling_coefficient_modified} \kappa=-\frac{1}{2}\omega_0 \int_{\text{V}_1} \big(\epsilon_1(\vec{r})-1 \big) \vec{\mathcal{E}}^{1}(\vec{r}). \vec{\mathcal{E}}^2(\vec{r}) d^3r,$$ where $\text{V}_1$ is again the volume of sphere 1. Contrary to (\[coupling\_coefficient\]), the fields from both spheres are treated on equal footing and no complex conjugation is required due to the normalization definition (\[normalization\_int\]). In what follows we show the eigenfrequencies of a system of two coupled spheres with different sizes and different separation distances. The coupling $\kappa$ is calculated numerically and the coupled mode matrix (\[CMT\_to\_eff\_Hamiltonian\]) is constructed and diagonalized. In the first case two identical silver spheres with radii of 10 nm are considered and the eigenfrequencies of the system are calculated for two different dipole orientations and as a function of the separation distance between the spheres. It is important to note that due to the symmetry of the dipole modes, dipoles with perpendicular orientations do not couple. Therefore we consider two parallel orientations only. As a check of the coupled mode theory approximation, we have also computed the resonant frequencies of two coupled metallic spheres via a modal solution. We refer to this modal solution as exact even though we only retain a finite number of modes that adequately resolves the resonant frequencies. The modal solutions are rigorously based upon solving Maxwell’s equations, and as such we treat them as exact, in contrast to the approximations used to generate the coupled mode theory results. The modal solution is derived by considering the discrete modes for an isolated sphere, which have electric fields as given by (\[field\_exprs\]), and corresponding magnetic fields determined via Maxwell’s equations. If we numerically truncate the modes at a maximum $\ell$ value, then this results in a finite set of possible $\ell$ and $m$ values. Each of these ($\ell$,$m$) combinations can be considered as defining a basis function, with distinct basis functions defined for the region interior and exterior to the sphere. The weighted summation of these basis functions can then approximate the fields supported everywhere by the metallic sphere. We now consider two spheres with these allowed modal solutions, and our goal is to find which particular combination of modal coefficients satisfies the boundary condition of the tangential electric and magnetic fields being continuous at the surfaces of both spheres. This is accomplished by choosing $N$ points spaced approximately equally over the surface of each sphere, where $N$ exceeds the number $M$ of modal coefficients we wish to determine. Each point on the surface of a sphere defines a constraint equation, with $4N$ total constraint equations due to there being four field components to match at each surface point: two tangential electric field components, and two magnetic field components. If we were able to exactly represent the fields, then at resonance the sum of the mode contributions at each point would equal zero, resulting in a matrix equation $Ax=0$ involving a $4N$-by-$M$ matrix $A$ and a length-$M$ vector $x$ representing the mode coefficients. However, because we are working with a truncated set of modes, we cannot solve the boundary condition exactly. We instead seek complex frequencies at which the smallest singular value of $A$ achieves a local minimum. These frequencies are approximately equal to the true resonant frequencies, and for the results plotted in this paper, frequency convergence was observed for maximum $\ell$ values ranging from 2 to 8, with a larger $\ell$ value being required as the sphere separation decreases. Fig. \[coupling\_10nm\_sphs\](a) shows the case where the two dipoles are parallel (vertical orientation). Both the real and imaginary components of the two eigenfrequencies of the system are plotted as a function of the separation $d$ between the two spheres normalized by the sphere radius $a$. The red solid lines and the black diamonds correspond to calculations via coupled mode theory (CMT) and modal expansion (Exact), respectively. There is good agreement between the two methods across the entire separation distances, even for small separations or strong interaction regime. This indicates that the dipole mode has the largest contribution in the coupling strength between the spheres and therefore coupled mode theory provides an acceptable approximation of the solution. The maximum coupling occurs when the two spheres are in contact with one another which in turn results in the largest deviation from the unperturbed frequency of a single sphere (dotted black lines). As the separation increases, the coupling decreases and the two eigenfrequencies approach the unperturbed eigenmode. A similar phenomenon is seen for the case of two dipoles in Fig. \[coupling\_10nm\_sphs\](b) (horizontal orientation). However, for a small separation distance between the spheres, the splitting of the two eigenmodes is larger for horizontal (colinear) dipole orientation than for vertical (parallel) dipole orientation. This is because, as shown in the previous section, the electric field is maximum along the dipole direction in the near field (see Fig. \[Field2Dplots\]). Consequently, this orientation results in larger coupling coefficients between the spheres. According to our findings in the previous section, silver spheres with the larger 40 nm radii are more radiative than the 10 nm radii spheres just considered. It is therefore desirable to look at the case of coupling between larger spheres since the coupling is stronger. Fig. \[coupling\_40nm\] shows the eigenfrequencies of a system of two silver spheres with radii of 40 nm. Because the coupling coefficient between two horizontal dipoles is greater than that of two vertical dipoles, we only consider the horizontal case. The difference between the superradiant and subradiant states is more pronounced in this case. At $d/a \approx 3$, coupled mode theory predicts a maximal difference between the imaginary components of the two eigenmodes. Similar to the previous case, the two eigenfrequencies approach the unperturbed resonance as the separation distance increases. At lower separations however, when the spheres are strongly interacting, coupled mode theory solution deviates more from the exact solution compared to the 10 nm spheres shown in Fig. \[coupling\_10nm\_sphs\](b). This indicates the importance of higher order modes in the interaction strength of larger spheres, and that the coupled mode results can improve if these modes are included in the calculations. ![Real and imaginary components of the eigenfrequencies of a system of two coupled identical silver spheres with horizontal dipole orientation calculated via coupled mode theory shown in red (solid lines) and modal expansion shown in black (diamonds). The radii of the spheres are 40 nm. $d$ is the center-to-center separation and $a$ is the radius of the spheres. The black dotted lines represent the unperturbed eigenfrequency of a single sphere.[]{data-label="coupling_40nm"}](coupling40nmh_silver_NEW2.pdf) According to our calculations, an exact dark mode does not exist for a system of two silver plasmonics dipoles. Therefore a numerical search over the parameters of the Drude-Sommerfeld dielectric function (\[Drude\_dielectric\_func\]) was performed in order to find material properties for which two plasmonic spheres can support a dark mode. Fig. \[coupling\_darkmode\] shows the eigenfrequencies of a system of two spheres with Drude-Sommerfeld parameters of $\epsilon_{\infty}=1$, $\omega_p=10.918$ eV and $\gamma_s=0$. At $d/a \approx 3.5$, the rank of the anti-Hermitian part of the coupled mode matrix is almost unity indicating that the interaction between the two spheres occurs through a single continuum channel. Consequently, the imaginary component of the subradiant mode is extremely small, in the order of 10$^{-3}$ eV. This is indicated with a black circle in the figure. ![Real and imaginary components of the eigenfrequencies of a system of two coupled identical spheres with horizontal dipole orientation. The material properties of the spheres are: $\epsilon_{\infty}=1$, $\omega_p=10.918$ eV and $\gamma_s=0$. The radii of the spheres are 20 nm. $d$ is the center to center separation and $a$ is the radii of the spheres. The black dotted lines represent the unperturbed eigenfrequency of a single sphere.[]{data-label="coupling_darkmode"}](couplingDARKMODE_NEW2.pdf) Plasmonic Waveguide {#secV} =================== We now consider the signal transmission through a plasmonic waveguide, namely a one-dimensional chain of identical spheres. The idea that such a structure acts as a waveguide due to interparticle coupling was proposed in [@Quinten98] and experimentally verified in [@waveguide_exp]. Here, it is assumed that the two edges of the waveguide are connected to an instrument capable of exciting the system of spheres with frequency $\omega_e$ and measuring the electric field intensity. Fig. \[plasmonic\_waveguide\_schematics\] depicts the schematic of the plasmonic waveguide and the two probes symmetrically coupled to the edges of the chain with coupling constants $\gamma_e$. Similar to the tight binding model of crystals in condensed matter physics, it is further assumed that each sphere in the chain only interacts with its nearest neighbor. This system can be modeled with the effective non-Hermitian Hamiltonian (\[ReducedHeff\]) $$\label{effective_hamiltonian_plasmonic_waveguide} \mathscr{H}_{\text{eff}}= \begin{bmatrix} \frac{i}{2}\gamma_e+\omega_0 & \kappa & 0 & \dots & 0 & 0 \\ \kappa & \omega_0 & \kappa & \dots & 0 & 0 \\ 0 & \kappa & \omega_0 & \kappa & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & \kappa & \frac{i}{2}\gamma_e+\omega_0 \end{bmatrix} ,$$ where $\omega_0$ is the unperturbed dipole frequency of an isolated sphere, and $\kappa$ is the coupling coefficient (\[coupling\_coefficient\_modified\]) between adjacent dipoles. It is important to mention that the addition of the anti-Hermitian matrix elements, $\frac{i}{2}\gamma_e$, with a positive sign is due to the phase convention adopted earlier in (\[Phase\_convention\]). ![Schematics of a plasmonic waveguide; a one-dimensional chain of five silver spheres with nearest neighbor coupling $\kappa$. The two edges are symmetrically coupled to continuum, the excitation source with frequency $\omega_e$, with coupling coefficient $\gamma_e$.[]{data-label="plasmonic_waveguide_schematics"}](TransportSchematic.pdf) Through its coupling to the two probes, the system can undergo an additional superradiance phase transition, other than that discussed in the previous section. This is illustrated by considering two different plasmonic waveguides. In both cases, according to our findings in the last section, in order to maximize the coupling between the neighboring sites the spheres are in contact with one another and the dipole orientation of the spheres is considered to be along the waveguide (horizontal orientation). In the first case, a chain of five silver spheres with radii of 10 nm is considered (Fig. \[plasmonic\_waveguide\_schematics\]). The resulting effective Hamiltonian (\[effective\_hamiltonian\_plasmonic\_waveguide\]) describing the system is a 5x5 square matrix with diagonal elements, $\omega_0=3.3468 + i0.0519$ eV, and off-diagonal matrix elements $\kappa=-0.2459 + i0.0029$ eV. The continuum coupling coefficient $\gamma_e$ is treated as a variable that changes from small, $\gamma_e=0.01$ eV, to extreme values $\gamma_e=10$ eV. The evolution of the complex eigenvalues of the effective Hamiltonian as the coupling to the continuum varies is shown in Fig. \[complex\_eigenvals\_sphrs\](a). At small values of $\gamma_e$ all the eigenvalues acquire a small width through the coupling to the continuum. The widths of the complex eigenmodes almost uniformly increase as the system is more strongly coupled to the continuum, up until $\gamma_e \approx 1$. At this point, the eigenvalues have reached their maximum width and, with further increasing $\gamma_e$, the system undergoes a phase transition (superradiance transition) when the eigenmodes become segregated into two distinct categories: superradiant and subradiant states. At strong coupling, the two superradiant states, their number being equal to the number of continuum channels (two probes), steal the entire available width of the system and leave the remaining states as narrow resonances. The second waveguide differs only in that the size of the spheres now have radii of 40 nm. In this case, the diagonal unperturbed frequencies are $\omega_0=3.1172 + i0.1910$ eV and the off-diagonal coupling coefficients are $\kappa=-0.2606 + i0.0475$ eV. The continuum coupling $\gamma_e$ is again varied from $\gamma_e=0.01$ eV to $\gamma=10$ eV, and the complex eigenvalues are plotted in Fig. \[complex\_eigenvals\_sphrs\](b). In general the picture is similar to the previous case. The superradiant transition can be clearly seen as the coupling $\gamma_e$ increases to extreme values. We now study the propagation of a signal through the two waveguides by calculating the transmission coefficient. Using (\[ProcessAmplitude\]) and (\[Transmission\_amp\]) we arrive at the following expression for the transmission coefficient $$T(\hbar \omega_e) = \Bigg| \frac{\gamma_e/ \kappa}{ \prod_{r=1}^{N}\big[(\hbar \omega_e - \hbar \omega_r)/ \kappa \big]} \Bigg|^2,$$ where $\omega_r$ are the complex frequencies of the effective Hamiltonian (\[effective\_hamiltonian\_plasmonic\_waveguide\]) and $N$ is its dimension. Transmission as a function of the excitation frequency, $\omega_e$, is shown in Fig. \[PlasmonicTransport10nm\] for the waveguide with 10 nm spheres. At weak coupling to the continuum, $\gamma_e=0.03$ eV, Fig. \[PlasmonicTransport10nm\](a), the five resonances are distinguishable. However, the resonances are not well separated due to the complex coupling coefficient between spheres, $\kappa$, which provide the eigenvalues of the effective Hamiltonian an initial width even for the closed system ($\gamma_e=0$). The case of intermediate coupling, when $\gamma_e=0.55$ eV, is shown in Fig. \[PlasmonicTransport10nm\](b). This is when the system is on the road to superradiance transition and all the eigenvalues of the Hamiltonian have large widths. Consequently, the resonances overlap and the transmission is dramatically enhanced. The case of strong couplings, Fig. \[PlasmonicTransport10nm\](c), has a picture similar to that of the weak coupling case. However, only three resonances remain. The two giant superradiant states do not participate in signal transmission and transmission is greatly suppressed due to the small width of the remaining subradiant states. We follow the same steps of weak, intermediate and strong coupling to continuum in order to study transmission through the waveguide with 40 nm spheres. Due to larger coupling, $\kappa$, between adjacent spheres the eigenvalues of the effective Hamiltonian poses a relatively large initial width even for small coupling to the continuum. Therefore, contrary to the previous case, the resonances overlap and are not separated even at weak coupling, Fig. \[PlasmonicTransport40nm\](a). Similar to before, the transmission is greatly enhanced at the superradiance transition, and at extreme couplings, we are back to suppressed transmission. Conclusion {#secVI} ========== We studied the resonant frequencies of plasmonic spherical nanoantennas by solving the full wave equation. These eigenfrequencies are always complex due to radiation and damping. Utilizing the effective non-Hermitian Hamiltonian framework, it was shown that a system of coupled two spheres can have modes with distinct properties; a superradiant mode with enhanced radiation and a dark mode with extremely damped radiation. Signal transmission through one dimensional chains was also considered. The coupling of the edge spheres to the continuum can drastically change transport properties of the system. A different superradiant transition arises through this interaction. Transmission is greatly enhanced at this transition. A possible direction to improve the accuracy of the results is to modify the Drude-Sommerfeld model by including terms that take into account surface scattering effects. It would be interesting to study the contribution of surface scattering to the total damping and radiation of the nanospheres. Another possibility is to consider higher order modes and their effect on the eigenfrequencies of the coupled system. These are left for future work. A. T. is grateful to his PhD advisor, Vladimir Zelevinsky for his guidance and many helpful discussions and thanks A. Stain for her support and assistance. S. R. thanks his doctoral advisor J. Verboncoeur for his mentorship. Quasi Normal Modes Normalization {#QNMNAp} ================================ In this appendix we explicitly normalize the fields of a plasmonic sphere by evaluating the normalization expression (\[normalization\_int\]) $$\label{normalization_int2} \int_V \sigma(\vec{r},\omega) \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r + \frac{i\epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r=1.$$ Due to the homogeneity of the sphere dielectric function $\epsilon_{\text{in}}$, and the surrounding background $\epsilon_{\text{out}}$, the modified dielectric function $\sigma(\omega)$ given in (\[modified\_dielectric\]) is only a function of frequency and can be taken out of the integral. In what follows we first evaluate the volume term in (\[normalization\_int2\]) assuming the normalization volume itself is a sphere with a radius $R$ where $a \ll R$. We first consider the volume term. Using the field expressions given in (\[field\_exprs\]) the volume term of the normalization (\[normalization\_int\]) is expressed as $$\begin{aligned} \label{volumeterm1} & \sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r =\sigma(\omega) \ \zeta^2 \ \int_0^{R} dr r^2 \ C^2(r;a) \Bigg\{ \big(\ell(\ell+1)\big)^2 \bigg( \frac{f_{\ell}(kr)}{kr} \bigg)^2 \int d\Omega \bigg(Y^m_\ell(\theta,\phi)\bigg)^2 \nonumber \\ &+ \bigg( \frac{1}{kr} \frac{\partial}{\partial(kr)} \big( krf_{\ell}(kr) \big) \bigg)^2 \int d\Omega \bigg[ \bigg( \frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi) \bigg)^2 +\frac{1}{\text{sin}^2\theta} \bigg( \frac{\partial}{\partial \phi} Y_\ell^m(\theta,\phi) \bigg)^2 \bigg] \Bigg\},\end{aligned}$$ where $d\Omega=\sin \theta d\theta d \phi$ is the solid angle differential in spherical coordinates. The integrals involving spherical harmonics can be evaluated by using the orthogonality relation of the tesseral harmonics and identity (\[SphericalHarmonic\_angular\_integral\_identity\]) in Appendix \[THandIs\]. Thus, (\[volumeterm1\]) reduces to $$\begin{aligned} \label{volumeterm2} & \sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r= \nonumber \\ & \sigma(\omega) \ \zeta^2 \ \ell(\ell+1) \int_0^R dr r^2 \ C^2(r;a) \Bigg\{ \ell(\ell+1) \bigg( \frac{f_{\ell}(kr)}{kr} \bigg)^2 + \bigg( \frac{1}{kr} \frac{\partial}{\partial(kr)} \big( krf_{\ell}(kr) \big) \bigg)^2 \Bigg\}.\end{aligned}$$ Due to the discontinuity of the function $f_{\ell}(kr)$ and the coefficients $C(r;a)$ at the surface of the plasmonic sphere \[see eqn. (\[general\_constant\])\], the radial integral in (\[volumeterm2\]) has to be divided into two terms: $\int_0^R=\int_0^a + \int_a^R$. Each term can be evaluated with the help of (\[SphericalHarmonic\_integral\_identity2\]) and (\[SphericalHarmonic\_integral\_identity3\]). This brings us to the final expression for the volume term of the normalization $$\label{volumeterm3} \sigma(\omega) \int_V \ \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^3r = I \big[ j_{\ell}(k_{\text{in}}a) \big] - I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big] + I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big],$$ where the functional $I\big[f_{\ell}(kr)\big]$ is defined as $$\begin{aligned} \label{volumeterm4} & I\big[f_{\ell}(kr)\big]= \sigma(\omega) \ \zeta^2 \ C^2(r;a)\frac{\ell(\ell+1)}{k^2} \bigg[ r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr)+\frac{k^2r^3}{2}\Big(f_{\ell}^2(kr)-f_{\ell-1}(kr)f_{\ell+1}(kr) \Big) \bigg].\end{aligned}$$ As before, $f_{\ell}^{'}(kr)$ implies differentiation with respect to the argument i.e. $\frac{\partial}{\partial (kr)} f_{\ell}(kr)$. Note that in evaluating the first term of the right hand side of (\[volumeterm3\]), $I \big[ j_{\ell}(k_{\text{in}}a) \big] $, one has to use all the parameters corresponding to the region interior to the plasmonic sphere. i.e. $\epsilon_{\text{in}}(\omega)$, $k_{\text{in}}$ and $C(r;a)$ for $r \leq a$ as it is defined in (\[general\_constant\]). Accordingly, the same applies to the second and third terms, $I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]$ and $I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big]$, for which one has to use the parameters corresponding to the background material. Next we show that the last term in the right hand side of (\[volumeterm3\]), $I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big]$, exactly cancels out with the surface term in (\[normalization\_int\]). Therefore, as expected, the normalization condition becomes independent of the integration volume. Because the integration sphere was assumed to be sufficiently large, asymptotic expressions can be used to evaluate both these terms. The spherical Hankel functions at large radial distances have the following asymptotic form $$\label{Asymptotic_hankel} h_{\ell}^{(2)}(kr)\sim i^{(\ell+1)} \frac{e^{-ikr}}{kr} \Bigg(1-i \frac{\ell(\ell+1)}{2kr} \Bigg).$$ The last term in the right hand side of (\[volumeterm4\]) can therefore be approximated as $$\label{VolumeLargeContr} I\big[h_{\ell}^{(2)}(k_{\text{out}}R)\big] \approx \frac{-i \epsilon_{\text{out}}}{2k} C^2(r;a) (-)^{\ell+1}e^{-2ik_{\text{out}}R}.$$ We now consider the surface term of the normalization condition (\[normalization\_int\]). Considering that in the far field the dominant terms are $\mathcal{E}_\theta$ and $\mathcal{E}_\phi$ and using the asymptotic form of the spherical Hankel function (\[Asymptotic\_hankel\]), the leading order of the surface term is $$\label{Surfaceterm_final} \frac{i \epsilon_{\text{out}}}{2k} \int_{\partial V} \vec{\mathcal{E}}(\vec{r}).\vec{\mathcal{E}}(\vec{r}) d^2r \approx \frac{i \epsilon_{\text{out}}}{2k^3} C^2(r;a) (-)^{\ell+1} e^{-2ik_{\text{out}}R}$$ It is clear that the surface term (\[Surfaceterm\_final\]) and (\[VolumeLargeContr\]) of the volume term cancel out. With this, the normalization condition (\[normalization\_int2\]) reduces to $$\label{normalization_reduced} I \big[ j_{\ell}(k_{\text{in}}a) \big] - I\big[h_{\ell}^{(2)}(k_{\text{out}}a)\big]=1,$$ which provide us the coefficient $\zeta$ in (\[field\_exprs\]). Tesseral Harmonics and Spherical Functions Identities {#THandIs} ===================================================== This appendix contains the definition of tesseral harmonics and a list of identities that are used throughout the paper. The tesseral harmonic are linear superpositions of the complex spherical harmonics with same $\ell$ and opposite sign $m$ values. Therefore the azimuthal dependency of the functions are in the form of $\text{sin}(m\phi)$ and $\text{cos}(m\phi)$ instead of the usual exponential form $e^{im\phi}$. They are defined as [@THarmonics]: $$\begin{aligned} Y_{\ell}^m(\theta,\phi) = \begin{cases} \sqrt{\frac{2\ell+1}{2 \pi}\frac{(\ell-|m|)!}{(\ell+|m|)!}} P_{\ell}^{|m|}(\text{cos} \theta) \text{sin}(|m|\phi) & m<0 \\ \sqrt{\frac{2\ell+1}{4 \pi}} P_{\ell}^{m} & m=0\\ \sqrt{\frac{2\ell+1}{2 \pi}\frac{(\ell-m)!}{(\ell+m)!}} P_{\ell}^{m}(\text{cos} \theta) \text{cos}(m\phi) & m>0 \end{cases}\end{aligned}$$ where $P_{\ell}^{m}$ are the associated Legendre polynomials. The tesseral harmonics satisfy the same orthogonality relation as the complex spherical harmonics. The tesseral harmonics also satisfy the following identity $$\begin{aligned} \label{SphericalHarmonic_angular_integral_identity} & \int d\Omega \Bigg\{ \bigg( \frac{\partial}{\partial \theta} Y_\ell^m(\theta,\phi) \bigg)^2 +\frac{1}{\sin^2\theta} \bigg( \frac{\partial}{\partial \phi} Y_\ell^m(\theta,\phi) \bigg)^2 \Bigg\} = \nonumber \\ & \ell (\ell+1).\end{aligned}$$ This can be proven by starting with the fact that the harmonics fulfill the identity $r^2 \nabla^2 Y_{\ell}^m(\theta,\phi)=-\ell(\ell+1)Y_{\ell}^m(\theta,\phi)$. One can then get to (\[SphericalHarmonic\_angular\_integral\_identity\]), by calculating the matrix element of the operator $r^2 \nabla^2$ and using integration by parts. A useful identity of the spherical Bessel and Hankel functions is the following $$\begin{aligned} \label{SphericalHarmonic_integral_identity2} \int dr \Bigg\{ \ell (\ell+1) f_{\ell}^2(kr) + \bigg( \frac{\partial}{\partial (kr)} krf_{\ell}(kr) \bigg)^2 \Bigg\}= \nonumber \\ r f_{\ell}^2(kr)+kr^2 f_{\ell}(kr) f_{\ell}^{'}(kr)+k^2 \int dr r^2 f_{\ell}^2(kr).\end{aligned}$$ where as previously $f_{\ell}^{'}(kr)$ implies differentiation with respect to the argument. The equality can be proven by using integration by parts and the spherical Bessel differential equation to get $$\begin{aligned} & \ell (\ell+1) f_{\ell}^2(kr) = \nonumber \\ & k^2r^2 f_{\ell}(kr) f_{\ell}^{''}(kr)+2krf_{\ell}(kr)f_{\ell}^{'}(kr)+k^2r^2f_{\ell}^2(kr),\end{aligned}$$ and $$\label{SphericalHarmonic_integral_identity3} \int dr r^2 f_{\ell}^2(kr) = \frac{r^3}{2} \bigg( f_{\ell}^2(kr)- f_{\ell-1}(kr) f_{\ell+1}(kr) \bigg).$$ [^1]: [email protected]

ArXiv

--- author: - 'Michele Leone, Sumedha, and Martin Weigt' title: | Unsupervised and semi-supervised clustering by message passing:\ Soft-constraint affinity propagation --- Introduction ============ Clustering is a very important problem in data analysis [@JAIN; @DUDA]. Starting from a set of data points, one tries to group data such that points in one cluster are more similar in between each other than points in different clusters. The hope is that such a grouping unveils common functional characteristics. As an example, one of the currently most important application fields for clustering is the informatical analysis of biological high-throughput data, as given e.g. by gene expression data. Different cell states result in different expression patterns. If data are organized in a well-separated way, one can use one of the many unsupervised clustering methods to divide them into classes [@JAIN; @DUDA]; but if clusters overlap at their borders or if they have involved shapes, these algorithms in general face problems. However, clustering can still be achieved using a small fraction of previously labeled data (training set), making the clustering [*semi-supervised*]{} [@BOOK; @DOMANY2]. While designing algorithms for semi-supervised clustering, one has to be careful: They should efficiently use both types of information provided by the geometrical organization of the data points as well as the already assigned labels. In general there is not only one possible clustering. If one goes to a very fine scale, each single data point can be considered its own cluster. On a very rough scale, the whole data set becomes a single cluster. These two extreme cases may be connected by a full hierarchy of cluster-merging events. This idea is the basis of the oldest clustering method, which still is amongst the most popular one: [*hierarchical agglomerative clustering*]{} [@SOKAL; @JOHNSON]. It starts with clusters being isolated points, and in each algorithmic step the two closest clusters are merged (with the cluster distance given, e.g., by the minimal distance between pairs of cluster elements), until only one big cluster appears. This process can be visualized by the so-called dendrogram, which shows clearly possible hierarchical structures. The strong point of this algorithm is its conceptual clarity connected to an easy numerical implementation. Its major problem is that it is a greedy and local algorithm, no decision can be reversed. A second traditional and broadly used clustering method is [*K-means clustering*]{} [@MCQUEEN]. In this algorithm, one starts with a random assignment of data points to $K$ clusters, calculates the center of mass of each cluster, reassigns points to the closest cluster center, recalculates cluster centers etc., until the cluster assignment is converged. This method is a very efficiently implementable method, but it shows a strong dependence on the initial condition, getting trapped by local optima. So the algorithm has to be rerun many times to produce reliable clusterings, and the algorithmic efficiency is decreased. Further on $K$-means clustering assumes spherical clusters, elongated clusters tend to be divided artificially in sub-clusters. A first statistical-physics based method is [*super-paramagnetic clustering*]{} [@DOMANY1; @DOMANY2]. The idea is the following: First the network of pairwise similarities becomes preprocessed, only links to the closest neighbors are kept. On this sparsified network a ferromagnetic Potts model is defined. In between the paramagnetic high-temperature and the ferromagnetic low-temperature phase a super-paramagnetic phase can be found, where already large clusters tend to be aligned. Using Monte-Carlo simulations, one measures the pairwise probability for any two points to take the same value of their Potts variables. If this probability is large enough, these points are identified to be in the same cluster. This algorithm is very elegant since it does not assume any cluster number of structure, nor uses greedy methods. Due to the slow equilibration dynamics in the super-paramagnetic regime it needs, however, the implementation of sophisticated cluster Monte-Carlo algorithms. Note that also super-paramagnetic clusterings can be obtained by message passing techniques, but these require an explicit breaking of the symmetry between the values of the Potts variables to give non-trivial results. Also in the last years, many new clustering methods are being proposed. One particularly elegant and powerful method is [*affinity propagation*]{} (AP) [@FREY], which gave also the inspiration to our algorithm. The approach is slightly different: Each data point has to select an exemplar in between all other data points. This shall be done in a way to maximize the overall similarity between data points and exemplars. The selection is, however, restricted by a hard constraint: Whenever a point is chosen as an exemplar by somebody else, it is forced to be also its own self-exemplar. Clusters are consequently given as all points with a common exemplar. The number of clusters is regulated by a chemical potential (given in form of a self-similarity of data points), and good clusterings are identified via their robustness with respect to changes in this chemical potential. The computational hard task to optimize the overall similarity under the hard constraints is solved via message passing [@YEDIDIA; @MAXSUM], more precisely via belief propagation, which are equivalent to the Bethe-Peierls approximation / the cavity method in statistical physics [@MezardParisi; @MyBook]. Despite the very good performance on test data, also AP has some drawbacks: It assumes again more or less spherical clusters, which can be characterized by a single cluster exemplar. It does not allow for higher order pointing processes. A last concern is the robustness: Due to the hard constraint, the change of one single exemplar may result in a large avalanche of other changes. The aim of [*soft-constraint affinity propagation*]{} (SCAP) is to use the strong points and ideas of affinity propagation – the exemplar choice fulfilling a global optimization principle, the computationally efficient implementation via message-passing techniques – but curing the problems arising from the hard constraints. In [@SCAP] we have proposed a first version of this algorithm, and have shown that on gene-expression data it is very powerful. In this article, we propose a simplified version which is more efficient. Finally we show that SCAP also allows for a particularly elegant generalization to the semi-supervised case, [*i.e.*]{} to the inclusion of partially labeled data. As shown in some artificial and biological benchmark data, the partial labeling allows to extract the correct clustering even in cases where the unsupervised algorithm fails. The plan of the paper is the following: After this Introduction, we present in Sec. \[sec:scap\] the clustering problem and the derivation of SCAP, and we discuss time- and memory-efficient implementations which become important in the case of huge data sets. In Sec. \[sec:data\] we test the performance of SCAP on artificial data with clustered and hierarchical structures. Sec. \[sec:semi\] is dedicated to the generalization to semi-supervised clustering, and we conclude in the final Sec. \[sec:conclusion\]. The algorithm {#sec:scap} ============= Formulation of the problem -------------------------- The basic input to SCAP are pairwise similarities $S(\mu,\nu)$ between any two data points $\mu,\nu\in \{1,...,N\}$. In many cases, these similarities are given by the negative (squared) Euclidean distances between data points or by some correlation measure (as Pearson correlations) between data points. In principle they need not even to be symmetric in $\mu$ and $\nu$, as they might represent conditional dependencies between data points. The choice of the correct similarity measure will for sure influence the quality and the details of the clusterings found by SCAP, it depends on the nature of the data which shall be clustered. Here we assume therefore the similarities to be given. The main idea of SCAP is that each data point $\mu$ selects some other data point $\nu$ as its [*exemplar*]{}, i.e. as some reference point for itself. The exemplar choice is therefore given by a mapping $$\label{eq:c_map} {\mathbf c}:\ \ \{1,...,N\} \ \mapsto\ \{1,...,N\}$$ where, in difference to the original AP and the previous version of SCAP, no self-exemplars are allowed: $$\label{eq:no_self_exemplar} \forall \mu\in\{1,...,N\}:\ \ c_\mu \neq \mu\ .$$ The mapping ${\mathbf c}$ defines a directed graph with links going from data points to their exemplars, and clusters in this approach correspond to the connected components of (an undirected version) this graph. The aim in constructing ${\mathbf c}$ is to minimize the Hamiltonian, or cost function, $$\label{eq:H} {\cal H} ({\mathbf c}) = - \sum_{\mu=1}^N S(\mu,c_\mu)\ +\ p\ {\cal N}_c\ ,$$ with ${\cal N}_c$ being the number of distinct selected exemplars. This Hamiltonian consists of two parts: The first one is the negative sum of the similarities of all data points to their exemplars, so the algorithm tries to maximize this accumulated similarity. However, this term alone would lead to a local greedy clustering where each data point chooses its closest neighbor as an exemplar. The resulting clustering would contain ${\cal O}(N)$ clusters, so increasing the amount of data would lead to more instead of better defined clusters. The second term serves to [*compactify*]{} the clusters: $\chi_\mu$ is one iff $\mu$ is an exemplar, so each exemplar has to pay a [*penalty*]{} $p$. Since this penalty does not depend on how many data points actually choose $\mu$ as their exemplar (the in-degree of $\mu$), mappings ${\bf c}$ with few exemplars of high in-degree are favored, leading to more compact clusters. In this way, the parameter $p$ controls the cluster number, robust clusterings are recognized due to their stability under changing $p$. Since the cluster number is not fixed a priori, SCAP also recognizes successfully a hierarchical cluster organization. For later convenience we express the exemplar number as $${\cal N}_c = \sum_{\mu=1}^N \chi_\mu({\mathbf c})\ ,$$ using an indicator function $$\chi_\mu({\mathbf c}) = \left\{ \begin{array}{ccl} 1 && {\rm if}\ \exists \nu:\ c_\nu=\mu\\ 0 && {\rm else} \end{array} \right.$$ which denotes the [*soft local constraint*]{} acting on each data point. Note that this problem setting is slightly different from the one used in the first derivation of SCAP in [@SCAP]. There self-exemplars were allowed, and only selecting an exemplar which was not a self-exemplar led to the application of the penalty $ p$. The number of self-exemplars itself was coupled to a second parameter, the self-similarity. In [@SCAP] we already found that the best results were obtained for very small self-similarities. Actually the algorithm presented here can be obtained from the previous formulation by explicitly sending all self-similarities $S(\mu,\mu)\to -\infty$. The resulting formulation is easier both in implementation and interpretation since it does not include self-messages. Derivation of the algorithm --------------------------- The exact minimization of this Hamiltonian is a computationally hard problem: There are $(N-1)^N$ possible configurations $c$ to be tested, resulting in a potentially super-exponential running time of any exact algorithm. We therefore need efficient heuristic approaches which, even if not guaranteeing to find the true optimum, are algorithmically feasible. An approach related to the statistical physics of disordered systems is the implementation of message-passing techniques, more precisely of the belief propagation algorithm [@YEDIDIA; @MAXSUM]. The latter is equivalent to an algorithmic interpretation of the Bethe-Peierls approximation in statistical physics: Instead of solving exactly the thermodynamics of the problem, we use a refined mean-field method. To do so, we first introduce a formal inverse temperature $\beta$ and the corresponding Gibbs weight $$\label{eq:gibbs} {\mathbf P} ({\mathbf c}) \sim \exp\{-\beta {\cal H}({\mathbf c}) \}\ .$$ The temperature will be sent to zero at the end of the calculations, to obtain a weight concentrated completely in the ground states of ${\cal H}$. In principle one should optimize ${\mathbf P}({\mathbf c})$ with respect to the joint choice of all exemplars, we will replace this by the independent optimization of all marginal single-variable probabilities. We thus need to estimate the probabilities $$\label{eq:marginal} P_\mu(c_\mu) = \sum_{\{c_\nu;\nu\neq\mu\}} {\mathbf P} ({\mathbf c})$$ which in principle contain a sum over the $(N-1)^{N-1}$ configurations of all other variables. From this marginal probability we can define an exemplar choice as $$\label{eq:argmax} c_\mu^\star = \underset{c_\mu}{\rm argmax} \lim_{\beta\to\infty} P_\mu(c_\mu)\ .$$ Note that this becomes the correct global minimum of ${\mathbf P}$ if the latter is non-degenerate which is a reasonable assumption in the case of real-valued similarities $S(\mu,\nu)$. We want to estimate these marginal distributions using belief propagation, or equivalently the Bethe-Peierls approximation. For doing so, we first represent the problem by its factor graph as given in Fig. \[fig:factor\_graph\_scap\]. The variables are represented by circular variable nodes, the constraints $\chi_\mu$ by square factor nodes. Due to the special structure of the problem, every variable node corresponds to exactly one factor node. Each factor node is connected to all variable nodes which are contained in the constraint (which are all but the one corresponding to the factor node). The similarities act locally on variable nodes, they can be interpreted as $(N-1)$-dimensional local vector fields. ![Factor graph for SCAP: Circles denote variable nodes, related to the variables $c_\mu$, whereas squares denote the constraints $\chi_\mu$. A link is drawn whenever a variable compares in a constraint, i.e. all variable nodes $\nu\neq\mu$ are connected to factor node $\chi_\mu$. Similarities act as external $(N-1)$-dimensional fields on the variables. The figure also displays the two message types send from variables to constraints and back.[]{data-label="fig:factor_graph_scap"}](factor_graph_scap){width="\columnwidth"} Belief propagation works via the exchange of messages between variable and factor nodes. Let us denote first $A_{\mu\to\nu}(c_\nu)$ the message sent from constraint $\mu$ to variable $\nu$, measuring the probability that $\mu$ forces $\nu$ to select $c_\nu$ as its exemplar. Second we introduce $B_{\nu\to\mu}(c_\nu)$ as the probability that variable $\nu$ would choose $c_\nu$ as its exemplar without the presence of constraint $\mu$. Than we can write down closed iterative equations, called belief-propagation equations, $$\begin{aligned} \label{eq:BP} A_{\mu\to\nu}(c_\nu) & \propto & \prod_{\lambda\neq\mu,\nu} \left[\sum_{c_\lambda} B_{\lambda\to\mu}(c_\lambda) \right] \exp\{-\beta\, p\, \chi_\mu({\mathbf c})\} \nonumber\\ B_{\mu\to\nu}(c_\mu) & \propto & \prod_{\lambda\neq\mu,\nu} A_{\lambda\to\mu}(c_\mu) \exp\{ \beta\, S(\mu,c_\mu)\} \ .\end{aligned}$$ We see that the message $A_{\mu\to\nu}$ from constraint $\mu$ to variable $\nu$ depends on the choices all other variables would take without constraint $\mu$, times the Gibbs weight of constraint $\chi_\mu$. The message $B_{\mu\to\nu}$ from variable $\mu$ to constraint $\nu$ depends on the messages from all other constraints to $\mu$, and the local field $\vec S$ on $\mu$. The approximate character of belief propagation stems from the fact that the joint distributions over all neighboring variables is taken to be factorized into single variable quantities. Having solved these equations we can easily estimate the true marginal distributions $$\label{eq:p_mu} P_{\mu}(c_\mu) \propto \prod_{\lambda\neq\mu} A_{\lambda\to\mu}(c_\mu) \exp\{ \beta\, S(\mu,c_\mu)\}$$ which are the central quantities we are looking for. However, looking at the first of Eqs. (\[eq:BP\]), we realize that it still contains the super-exponential sum. Further on, we need a memory space of ${\cal O}(N^3)$ to store all these messages, which is practical only for small and intermediate data sets. This problem can be resolved exactly by realizing that $A_{\mu\to\nu}(c_\mu)$ takes only two values for fixed $\mu$ and $\nu$, namely $A_{\mu\to\nu}(\mu)$ and $A_{\mu\to\nu}(c\neq\mu)$ [^1]. We therefore introduce the reduced messages $$\begin{aligned} \tilde A_{\mu\to\nu} &=& \frac{A_{\mu\to\nu}(\mu)}{A_{\mu\to\nu}(c\neq\mu)} \nonumber\\ \tilde B_{\mu\to\nu} &=& B_{\mu\to\nu}(\nu)\ .\end{aligned}$$ After a little book-keeping work to consider all possible cases, the sums in Eqs. (\[eq:BP\],\[eq:p\_mu\]) can be performed analytically resulting in a set of equivalent relations $$\begin{aligned} \label{eq:scap_T} \tilde A_{\mu\to\nu} & =& \left[ 1+(e^{\beta p}-1) \prod_{\lambda\neq\mu,\nu} (1-\tilde B_{\lambda\to\mu}) \right]^{-1} \nonumber\\ \tilde B_{\mu\to\nu} & =& \left[ 1+ \sum_{\lambda\neq\mu,\nu} e^{ \beta S(\mu,\lambda)-\beta S(\mu,\nu) } \tilde A_{\lambda\to\mu} \right]^{-1} \nonumber\\ P_\mu(c) & =& \frac{e^{\beta\,S(\mu,c)} \tilde A_{c\to\mu}} {\sum_{\lambda\neq\mu}e^{\beta\,S(\mu,\lambda)} \tilde A_{\lambda\to\mu}}\ .\end{aligned}$$ These equations are the [*finite-temperature SCAP equations*]{}. Note that the complexity of evaluating the first line is decreased from ${\cal O}(N^N)$ to ${\cal O}(N)$ and therefore feasible even for very large data sets. Also the memory requirements are decreased to ${\cal O}(N^2)$. As we will see later on, a clever implementation will, in particular in the zero-temperature limit, further decrease time- and space-complexity. SCAP in the zero-temperature limit ---------------------------------- Even if Eqs. (\[eq:scap\_T\]) are already relatively simple, the zero temperature limit of these equations becomes even simpler and bears a very intuitive interpretation. To achieve this limit, we have to transform the variables in the equations from probabilities to local fields, and introduce $$\begin{aligned} a_{\mu\to\nu} &=& \frac 1\beta \ln \tilde A_{\mu\to\nu} \nonumber\\ r_{\mu\to\nu} &=& \frac 1\beta \ln \frac {\tilde B_{\mu\to\nu}} {1-\tilde B_{\mu\to\nu}}\ .\end{aligned}$$ We call $a_{\mu\to\nu}$ the [*availability*]{} of $\mu$ to be an exemplar for $\nu$, whereas $r_{\mu\to\nu}$ measures the [*request*]{} of $\mu$ to point $\nu$ to be its exemplar. Using the fact that sums over various exponential terms in $\beta$ are dominated by the maximum term, we readily conclude $$\begin{aligned} \label{eq:SCAP1} r_{\mu\to\nu} &=& S(\mu,\nu) - {\rm max}_{\lambda\neq\mu,\nu} \left[ S(\mu,\lambda) + a_{\lambda\to\mu} \right] \nonumber\\ a_{\mu\to\nu} &=& {\rm min} \left[ 0,\, - p + \sum_{\lambda\neq\mu,\nu} {\rm max} (0,\, r_{\lambda\to\mu}) \right] \end{aligned}$$ to hold for these two fields. These equations have a very nice and intuitive interpretation in terms of a social dynamics of exemplar selection. The system tries to maximize its overall similarity (or gain) which is the sum over all similarities between data points and their exemplars, but each exemplar has to pay a penalty $ p$. Therefore each data point $\mu$ sends requests to all their neighbors $\nu$, which are composed by two contributions: The similarity to the neighbor itself, minus the maximum over all similarities to the other points $\lambda\neq\mu,\nu$ - the latter already being corrected for by the availability of the other points to be an exemplar. Now, data points $\mu$ communicate their availability to be an exemplar for any other data point $\nu$. For doing so, they sum up all positive requests from further points $\lambda\neq\mu,\nu$, and compare it to the penalty they have to pay in case they accept to be an exemplar. If the accumulated positive requests are bigger than the penalty, $\mu$ agrees right away to be the exemplar for $\nu$. If on the other hand the penalty is larger than the requests, $\mu$ communicates to $\nu$ the difference - so the answer is not a simple “no” but is weighed. Point $\nu$ should overcome this difference with its similarity. Consequently the exemplar choice of $\mu$ happens via the selection of the neighbor $\nu$ who has the highest value of the similarity corrected by the availability of $\nu$ for $\mu$, i.e. we have $$\label{eq:SCAP2} c_\mu^\star = \underset{\nu}{\rm argmax} \left[ S(\mu,\nu)+a_{\nu\to\mu} \right]\ .$$ Eqs. (\[eq:SCAP1\],\[eq:SCAP2\]) are called [*soft-constraint affinity propagation*]{}. They can be solved by first iteratively solving (\[eq:SCAP1\]), and then plugging the solution into (\[eq:SCAP2\]). The next two sub-sections will show how this can be done in a time- and memory efficient way. Time-efficient implementation ----------------------------- The iterative solution of Eqs. (\[eq:SCAP1\]) can be implemented in the following way: 1. Define the similarity $S(\mu,\nu)$ for each set of data points. Choose the values of the self-similarity $\sigma$ and of the constraint strength $ p$. Initialize all $a(\mu,\nu)=r(\mu,\nu)=0$ 2. For all $\mu\in\{1,...,N\}$, first update the $N$ [*requests*]{} $r_{\mu\to\nu}$ and then the $N$ [*availabilities*]{} $a_{\mu\to\nu}$, using Eqs. (\[eq:SCAP1\]). 3. Identify the exemplars $c_\mu^\star$ by looking at the maximum value of $S(\mu,\nu)+a_{\nu\to\mu}$ for given $\mu$, according to Eq. (\[eq:SCAP2\]). 4. Repeat steps 2-3 till there is no change in exemplars for a large number of iterations (we used 10-100 iterations). If not converged after $T_{max}$ iterations (typically 100-1000), stop the algorithm. Three notes are necessary at this point: - Step 3 is formulated as a sequential update: For each data point $\mu$, all outgoing responsibilities and then all incoming availabilities are updated before moving to the next data point. In numerical experiments this was found to converge faster and in a larger parameter range than the damped parallel update suggested by Frey and Dueck in [@FREY]. The actual implementation uses a random sequential update, i.e. each time step 3 is performed, we generate a random permutation of the order of the $\mu\in\{1,...,N\}$. - The naive implementation of the update equations (\[eq:SCAP1\]) requires ${\cal O}(N^2)$ updates, each one of computational complexity ${\cal O}(N)$. A factor $N$ can be gained by first computing the unrestricted max and sum once for a given $\mu$, and then implying the restriction only inside the internal loop over $\nu$. Like this, the total complexity of a global update is ${\cal O}(N^2)$ and thus feasible even for very large data sets. - Belief propagation on loopy graphs is not guaranteed to converge. We observe, that even in cases where the messages do not converge to a fixed point but go on fluctuating, the exemplar choice converges. In our algorithm, we therefore apply frequently the stationarity of ${\mathbf c}^\star$ as a weaker convergence criterion than message convergence. Memory-efficient implementation ------------------------------- Another problem of SCAP can be its memory size, Eqs. (\[eq:SCAP1\]) require the storage of three arrays of size $N^2$. This can be a problem if we consider very large data sets. A particularly important example are gene-expression data, which may contain more than 30,000 genes. If one wants to cluster these genes to identify coexpressed gene groups, the required memory size becomes fastly much larger than the working memory of a standard desktop computer, restricting the size of data sets to approximately $N<10^4$. However, this problem can be resolved in the zero-temperature equations by not storing messages and similarities (which are indexed by two numbers) but only site quantities (which are indexed by a single number) reducing thus the memory requirements to ${\cal O}(N)$. This allows to treat even the largest available data sets efficiently with SCAP. As a first step, we note that in most cases data are multi-dimensional. For example in gene expression data, a typical data sets contains about 100 micro-arrays measuring simultaneously 5,000-30,000 genes. If we want to cluster arrays, for sure a direct implementation of Eqs. (\[eq:SCAP1\],\[eq:SCAP2\]) is best. In particular only the similarities are needed actively instead of the initial data points. If, on the other hand, we want to clusterize genes, it is more efficient to calculate similarities whenever needed from the original data, instead of memorizing the huge similarity matrix. Once this is implemented, we can also get rid of the messages $a_{\mu\to\nu}$ and $r_{\mu\to\nu}$. First we introduce $$\begin{aligned} h_\mu^{(1)} &=& \underset{\lambda\neq\mu}{\rm max} \left[ S(\mu,\lambda) + a_{\lambda\to\mu} \right] \nonumber\\ c_\mu^{(1)} &=& \underset{\lambda\neq\mu}{\rm argmax} \left[ S(\mu,\lambda) + a_{\lambda\to\mu} \right] \nonumber\\ h_\mu^{(2)} &=& \underset{\lambda\neq\mu,c_\mu^{(1)}}{\rm max} \left[ S(\mu,\lambda) + a_{\lambda\to\mu} \right]\ .\end{aligned}$$ These quantities, together with the similarities (directly calculated from the original data) are sufficient to express all requests, $$r_{\mu\to\nu} = S(\mu,\nu) - h_\mu^{(1)} + \left(h_\mu^{(1)}-h_\mu^{(2)}\right) \, \delta_{\nu,c_\mu^{(1)}}$$ with $\delta_{\cdot,\cdot}$ being the Kronecker-symbol. A similar step can be done for the availabilities. We introduce $$u_\mu = \sum_{\lambda\neq\mu} {\rm max} (0,\, r_{\lambda\to\mu})$$ and express the availability as $$\begin{aligned} a_{\mu\to\nu} &=& {\rm min} \Big[ 0,\, - p + u_\mu \\ & & \left. - {\rm max}\left\{0, \, S(\nu,\mu) - h_\nu^{(1)} + \left(h_\nu^{(1)}-h_\nu^{(2)}\right) \, \delta_{\mu,c_\nu^{(1)}} \right\} \right] \nonumber\end{aligned}$$ Note that after convergence we have trivially $$c_\mu^\star = c_\mu^{(1)}$$ for all $\mu\in\{1,...,N\}$. In this way, instead of storing $S(\mu,\nu)$, $a_{\mu\to\nu}$ and $r_{\mu\to\nu}$ we have to store only the data, $h_\mu^{(1,2)},\ c_\mu^{(1)}$ and $u_\mu$. The largest array is the data set itself, all other memorized quantities require much less size. For large data sets, in this way the memory usage becomes much more efficient. Even if the algorithm requires more steps to be executed (similarities and messages have to be computed whenever they are needed, instead of a single time in each update step), the more efficient memory usage leads to strongly decreased running times. Artificial data {#sec:data} =============== In [@SCAP] we have shown that SCAP is able to successfully cluster biological data coming from gene-expression arrays. This is true also for the simplified version derived in the present work. Here we aim, however, at a more theoretical analysis on artificial data which will bring light into some characteristics of SCAP, and which will allow for a more detailed comparison to the performance of AP as defined originally in [@FREY]. To start with, we first consider numerically data having only one level of clustering, later on we extend this study to more than one level of clusters, i.e. to a situation where clusters of data points itself are organized in larger clusters. One cluster level ----------------- The first step is very simple: We define an artificial data set having only one level of clustering. We therefore start with $N$ data points which are divided into $q$ equally sized subsets. For each pair inside such a subset we draw randomly and independently a similarity from a Gaussian of mean $\alpha$ and variance one, whereas pair similarities of data points in different clusters are drawn as independent Gaussian numbers of zero mean and variance one, cf. Fig. \[fig:clusters\] for an illustration. The parameter $\alpha$ controls the separability of the clusters, for small $\alpha<1$ clusters are highly overlapping, and SCAP is expected to be unable to separate the $q$ subsets, whereas for large values $\alpha > 3$ a good separability is expected. Alternative definitions of the similarities where data points are defined via high-dimensional data with higher intra-cluster correlations, lead to similar results and are not discussed here. ![Artificial data set for testing SCAP: $N$ data points (crosses) are organized into $q$ clusters (full circles), similarities for pair of points in the same cluster are drawn independently from a Gaussian $N_{\alpha,1}(S)$ of mean $\alpha$ and variance 1, between clusters from $N_{0,1}(S)$. The parameter $\alpha>0$ determines the separability of the clusters.[]{data-label="fig:clusters"}](clusters){width="\columnwidth"} First we study the dependence of the SCAP results on the parameter $\alpha$, see Fig. \[fig:scap\_alpha\]. For $\alpha=1$, we see that there is no signal at all at five clusters, and the error number (measured as the number of points having exemplars in a different cluster) grows starting from a high value. Data are completely mixed, which is clear since $N_{0,1}$ and $N_{1,1}$ are strongly overlapping. For $\alpha=3$, a clear plateau at five clusters appears, and the error rate until this plateau is low. Only when we force the system to form less than five clusters, the error rate starts to grow considerably. This picture becomes even more pronounced for larger $\alpha$; the distributions of intra- and inter-cluster similarities are perfectly separated, SCAP makes basically no errors until it is forced to do so since it forms less than five clusters. The error rate is not found to go beyond five errors, which is very small considering the fact that at least four errors are needed to interconnect the five clusters. ![Results of SCAP as a function of $ p$ for various values of $\alpha$. Displayed are the number of clusters (black lines) and errors (red lines). Results are for $N=100$, averaged over 1000 samples.[]{data-label="fig:scap_alpha"}](scap_alpha){width="\columnwidth"} Fig. \[fig:scap\_N\] shows the $N$-dependence of the SCAP results. The parameter $ p$ has to be rescaled by $N$ to re-balance the increased number of contributions to the overall similarity in the model’s Hamiltonian. One sees that the initial cluster number for $ p = 0$ is linear in $N$, but the penalty successfully forces the system to show a collective behavior with macroscopic clusters. The plateau length for different $N$ values is comparable, even if for larger $N$ the decay from the plateau to 1-2 clusters is much more abrupt. ![Dependence of the SCAP results for different values of $N$. Curves result from averages over 1000 random samples.[]{data-label="fig:scap_N"}](scap_N){width="\columnwidth"} Fig. \[fig:scap\_T\] studies the influence of the formal temperature on SCAP. In some cases finite-temperature SCAP shows more efficient convergence, so it is interesting to see how much information is lost by increasing the formal temperature. The left panel of Fig. \[fig:scap\_T\] represents again the cluster number (resp. error number) as a function of $ p$. We see that for very low temperature ($T=0.25$ in the example) results are hardly distinguishable from the zero-temperature results. If we further increase the temperature we observe that the plateau at five clusters becomes less pronounced and shifted to larger $ p$. To get rid of this shift, we show in the right panel a parametric plot of the two most interesting quantities: The error number as a function of the cluster number. This plot shows again that the errors start to grow considerably (with decreasing cluster number) as soon as we go below five clusters. For low enough temperatures, the curves practically collapse, so very few of the clustering information is lost. Only for higher temperatures the error number starts to grow already at higher cluster numbers. The pronounced change when we cross the number of clusters is lost. Therefore, as long as the plateau is pronounced in the left panel, also the error number remains almost as low as in zero temperature on the plateau. ![Temperature dependence of the SCAP results for $N=100,\, \alpha=3$. The left figure shows the $ p$ dependence of the cluster number (full lines) and of the error number (dashed lines). The right figure shows a parametric plot of the numbers of errors vs. clusters. Curves result from averages over 1000 random samples.[]{data-label="fig:scap_T"}](scap_T){width="\columnwidth"} Last but not least, we compare the performance of SCAP to the original AP proposed in [@FREY]. AP shows a slightly different behavior than SCAP. The latter has only one plateau at the correct cluster number, whereas AP shows a long plateau at five clusters, but also less pronounced shoulders at multiples of this number. Both algorithms can be compared directly when plotting the number of errors against the cluster number, see Fig. \[fig:scap\_ap\]. Note that in principle this test is a bit easier for AP since a part of the data points are self-exemplars, which are not counted as errors. Nevertheless SCAP shows much less errors, in particular also on the plateau of five clusters. The hard constraint in AP forbidding higher order pointing processes is too strong even for a simple data set as the one considered here, simply because the random generation of the similarities makes all points on statistically equivalent, not preferring one as a cluster center. The more flexible structure of SCAP is able to cope with this fact and is therefore results in a more precise clustering. Note that this difference increases with growing size $N$ of the data set: Whereas the error number of SCAP at five clusters slightly decreases with $N$, the corresponding number for AP grows. This is again due to the hard constraint which forces inside a cluster more and more data points to refer to the cluster exemplar. ![SCAP vs. AP: The number of errors (divided by $N$) is plotted against the cluster number, for $\alpha=3$ and various values of $N$. Curves result from averages over 1000 samples.[]{data-label="fig:scap_ap"}](scap_ap){width="\columnwidth"} Hierarchical cluster organization --------------------------------- ![Artificial data with two-level hierarchical organization. Data (crosses) are organized in clusters (full circles), which themselves are collected in larger clusters (dashed circles). Similarities are drawn from Gaussians as shown in the figure, with $0<\alpha_0<\alpha_1$.[]{data-label="fig:clusters_2level"}](clusters_2level){width="\columnwidth"} To test if SCAP is also able to detect a hierarchical cluster organization we have slightly modified the generator, as shown in Fig. \[fig:clusters\_2level\]. We divide the set of $N$ data points into $q_0$ superclusters, and each of these into $q_1$ clusters (in the Fig. $q_0=q_1=3$). Similarities are drawn independently for each pair of points. If points are in the same cluster, we use a Gaussian $N_{\alpha_1,1}(S)$ of mean $\alpha_1$ and variance 1, if they are in the same supercluster but not in the same cluster, we use $N_{\alpha_0,1}(S)$, and for all pairs coming from different superclusters we draw similarities from $N_{0,1}(S)$. The means fulfill $0<\alpha_0<\alpha_2$. ![SCAP for a systems with two hierarchical levels of clustering, for $N=180,\ q_0=q_1=3,\ \alpha_0=3,\ \alpha_1=6$, averages are performed over 2000 samples. The black line shows the cluster number, two clear plateaus at 9 clusters resp. 3 super-clusters are observed. The red line gives the number of data points selecting an exemplar in a different cluster, the green line even in a different super-cluster. Both quantities are divided by 6 to put them on the same scale as the cluster number. []{data-label="fig:scap_2level"}](scap_2level){width="\columnwidth"} Fig. \[fig:scap\_2level\] shows the findings for $N=180,\ q_0=q_1=3,\ \alpha_0=3,\ \alpha_1=6$. We clearly see that SCAP is able to uncover both cluster levels, pronounced plateaus appear at 3 and 9 clusters. The plot also shows two different error measures: The number of points which choose an exemplar which is not in the same cluster (red line in the figure), and the number of points choosing even an exemplar in a different supercluster (green line). As long as we have more than 9 clusters, there are very few of both error types (increasing $\alpha_0$ further decreases this number). Once we force clusters at the finest level to merge, the first type of error starts to grow. The second grows if we observe some merging of superclusters, i.e. if the cluster number found by SCAP is around or below 3. Note the little bump in the errors at the beginning of the three-cluster plateau: There even some links between different superclusters appear. In fact, in this region the algorithm does not converge in messages in many cases, leading to many errors. In the middle of the plateau, however, convergence is much more stable and error rates are small. To summarize this section, SCAP is able to infer the cluster structure of artificial data, even if the latter are organized in a hierarchical way. Results are very robust and show less errors than the AP with its hard constraints. Extension to semi-supervised clustering {#sec:semi} ======================================= In case labels are provided for some data points, they can be exploited to enhance the algorithmic performance. We propose the following way: Identically labeled data are collected in [*macro-nodes*]{}, one for each label. Since macro-nodes are labeled, they do not need an exemplar, but they may serve as exemplars for other data. If there are $N$ [*unlabeled*]{} points and $m$ known labels, the exemplar mapping thus gets generalized to ${\bf c}: \{1,...,N\} \mapsto \{1,...,N,N+1,...,N+m\}$ where indexes $N+1,...,N+m$ correspond to macro-nodes. We define the similarity of an arbitrary unlabeled point to a macro-node as the maximum of similarities between the point and all elements of the macro-node [^2]. The Hamiltonian now becomes: $${\cal H}_2[{\bf c}] = -\sum_{\mu=1}^{N} S(\mu,c_\mu) +p_1\sum_{\mu=1}^N\chi_{\mu}[{\bf c}]+p_2\sum_{\nu=N+1}^{N+m} \chi_{\nu}[{\bf c}] \label{eq:cost2}$$ Note that neither the sizes of the training set nor of the macro-nodes appear explicitly. They are implicitly present via the determination of the similarities between data and macro-nodes. In principle, we can choose different values of $p_1$ and $p_2$, more precisely $p_1>p_2$, to reduce the cost of choosing macro-nodes as exemplars as compared to normal data points. However, this usually forces data to choose the closest macro-node instead of making a collective choice using the geometrical information contained in the data set. We found $p_1=p_2=p$ to work best. ![Factor graph and message direction: Circles (variable nodes) are unlabeled data points, squares (factor nodes) constraints due to unlabeled (light) and macro-nodes (dark). Similarities act as $(N+m-1)$-dimensional external fields on the unlabeled data points. Messages are exchanged between all connected pairs of data points and constraints.[]{data-label="AP-FG"}](AP-FG){width="\columnwidth"} Compared to Fig. \[fig:factor\_graph\_scap\], the factor graph becomes slightly more complicated. As is shown in Fig. \[AP-FG\], $m$ new factor nodes are added to the graph representing the constraints constituted by the macro-nodes. This modification allows, however, to follow exactly the same route from the Hamiltonian to the final SCAP equations: $$\begin{aligned} \label{eq:semiSCAP1} a_{\mu\to\nu} &=& {\rm min} [ 0,\, -p + \sum_{\lambda\neq\mu,\nu} {\rm max} (0,\, r_{\lambda\to\mu}) ] \\ r_{\nu\to\mu} &=& S(\nu,\mu) - {\rm max}_{\lambda\neq\mu,\nu} \left[ S(\nu,\lambda) + a_{\lambda\to\nu} \right] \nonumber\end{aligned}$$ Remember that $\mu\in\{1,...,N\}$ corresponds to the unlabeled data points, whereas $\nu\in\{1,...,N+m\}$ enumerates the constraints and thus the possible exemplars. At infinite $\beta$, the exemplar choice becomes polarized to one solution (for non-degenerate similarities) and reads $$\label{eq:semiSCAP2} c_\nu^\star = {\rm argmax}_{\mu\in\{1,...,N+m\},\mu\neq\nu} \left[ S(\nu,\mu)+a_{\mu\to\nu} \right]\ .$$ Compared to Eqs. (\[eq:SCAP1\],\[eq:SCAP2\]) only the number of constraints becomes modified. The introduction of macro-nodes actually allows for a very elegant generalization of SCAP from the unsupervised to the semi-supervised case. Artificial data {#artificial-data} --------------- To test the performance of unsupervised vs. semi-supervised SCAP, we turned first to some artificial cases. [*Data set 1:*]{} We randomly selected points in two dimensions clustered in a way clearly visible to human eye (Fig. \[small1\]). The similarity between data points is measured by the negative Euclidean distance. The clusters are so close that the distance between points on the borders of two clusters is sometimes comparable to the distance between points inside one single cluster. This makes the clustering by unsupervised methods harder. For example, look at Fig. \[small1\], upper row: In this case, the best unsupervised SCAP clustering makes a significant fraction of errors, and does not recognize the two clusters. The best results with unsupervised SCAP are actually obtained when we allow it to divide the data into four clusters. ![Upper row: 3 best clusterings seen by unsupervised SCAP. $N=600$, $300$ in each cluster. Lower row: same data set with $t$ trainers (larger circles) for each cluster.[]{data-label="small1"}](first-fig1){width="\columnwidth"} ![Histogram of the number of errors for 10 000 random choices of $t=5,10,20$ labeled data points. For better visibility, bars are reduced in width and shifted relative to each other for different training-set sizes (bin size 5).[]{data-label="hist"}](hist){width="\columnwidth"} On the other hand semi-supervised SCAP recognizes two clusters very fast. When we introduce some labeled points, we find a significant improvement of the output, cf. Fig. \[small1\], lower row. Already as few as 5 labeled points per cluster increase the performance substantially. Larger training sets lead typically to less errors. In the semi-supervised SCAP, clustering is very stable and does not change when we increase $p$. In Fig. \[small1\] we show the clusters for one random choice of labeled set. In general one can argue that the clustering would change with the way the labeled set is distributed inside a cluster. In Fig. \[hist\] we show a histogram for 10000 random selections of the training set, for training set size $t=5,\ 10,\ 20$. We observe that a majority of clusterings found makes only few errors (the peak for less then 5 errors is cut in height for better visibility), but a small number of samples lead to a substantial error number. These samples are found to have labeled exemplars which are concentrated in regions mostly far form the regions where clusters are close, so a relatively large part of these regions is assigned erroneously to the wrong label. The probability of occurrence of such unfavorable situations goes down exponentially with the size of the training data set. [*Data set 2:*]{} With partial labeling, there are often cases where no information is available on some of the classes. Semi-supervised SCAP is able to deal with this situation because it can output clusters without macro-nodes, i.e. clusters without reference to any of the trainers’ labels. As an example, we add to the artificial data set a third cluster of similar size and shape, without adding any new trainer. As shown in Fig. \[small4\] and \[small5\], the algorithm detects correctly both the labeled and the unlabeled clusters for a wide range of parameters. ![Upper row: $N=600$, with 200 data points in each cluster. In all three cases we choose $p=0.5$. In the semi-supervised case $t=10$ each of the lower clusters; $t=0$ for the upper one. The two semi-supervised results are for different training sets (TS1 and TS2). Lower row: same data with $p=1$.[]{data-label="small4"}](second-fig1){width="\columnwidth"} ![$N=600$, with 200 data points in each cluster. First and second rows contains clustering for unsupervised and semi-supervised learning for $p=2,6,10$ (left to right). Semi-supervised: 10 trainers each for lower clusters, 0 for the upper one. One can see how increasing $p$ leads to an artificial merging of the labeled clusters ($p=6$). However, in the Semi-supervised case a stable region of p arises where the third cluster is well discerned while the labeled ones are still naturally separated.[]{data-label="small5"}](last-fig1){width="\columnwidth"} Iris data --------- This is a classic data set used as a bench mark for testing clustering algorithm [@IRIS]. The data consist of measurements of sepal length, sepal width, petal length and petal width, performed for 150 flowers, chosen from three species of the flower Iris. Unsupervised SCAP already works well making only 9 errors. Introducing $t$ trainers per class, the error number further decreases as shown in table \[iris\_data\_set\]. t 3 4-10 15-30 40 -------- --- ------ ------- ---- errors 7 6 2 1 : Errors in labeling Iris data, in dependence on the number $t$ of labeled data points.[]{data-label="iris_data_set"} We also performed semi-supervised clustering where we provided labels for only two out of the three data sets. Depending on the number and distribution of labeled points the algorithm produced 5-9 errors. Semi-supervised SCAP worked better when we provided information on the clusters corresponding to [*versicolor*]{} and [*virginica*]{} species. This is not surprising as these two are known to be closer to each other than to [*setosa*]{}, whose points set is well discerned even in the unsupervised case. Summary and outlook {#sec:conclusion} =================== In this paper, a further simplification of soft-constraint affinity propagation, a message-passing algorithm for data clustering, was proposed. We have presented a detailed derivation, and have discussed time- and memory-efficient implementations. The latter are important in particular for the clustering of huge data sets of more than $10^4$ data points, an example would be gene which shall be clustered according to their expression profiles in genome-wide micro-array experiments. Using artificial data we have shown that SCAP can be applied successfully to hierarchical cluster structures, a model parameter (the penalty $p$ for exemplars) allows to tune the clustering to different resolution scales. The algorithm is computationally very efficient since it involves updating ${\cal O}(N^2)$ messages, and it converges very fast. SCAP can be extended to semi-supervised clustering in a straightforward way. Semi-supervised SCAP shares the algorithmic simplicity and stability properties of its unsupervised counter part, and can be seen as a natural extension. The algorithm allows to assign labels to previously unlabeled data, or to identify additional classes of unlabeled data. This generalization allows to cluster data even in situations where cluster shapes are involved, and some additional information is needed to distinguish different clusters. In its present version, SCAP does not yet fully exploit the information contained in the messages, only the maximal excess similarity is used to determine the most probable exemplar. In the case where labels are not exclusive, one can also use the information provided by the second, third etc. best exemplar. This could be interesting in particular in cases, where similarity information is sparse, a popular example being the community search in complex networks. In a future work we will explore these directions in parallel to a theoretical analysis of the algorithmic performance on artificial data, which will provide a profound understanding of the strength and also the limitations of (semi-supervised) SCAP. [*Acknowledgments*]{}: We acknowledge useful discussions with Alfredo Braunstein and Andrea Pagnani. The work of S. and M.W. is supported by the EC via the STREP GENNETEC (“Genetic networks: emergence and complexity”). [99]{} M. Leone, Sumedha, and M. Weigt, Bioinformatics [**23**]{}, 2708 (2007). A.K. Jain, M.N. Murthy, and P.J. Flynn, ACM Computing Surveys [**31**]{}, 264 (1999). R.O. Duda, P.E. Hart, and D.G. Stork, [*Pattern Classification*]{}, 2nd. ed. (Wiley-Interscience, 2000). O. Chapelle, B. Sch[ö]{}lkopf, and A. Zien (eds.), [*Semi-Supervised Learning*]{} (MIT Press, Cambridge MA 2006). G. Getz, N. Sehntal and E.Domany, Proceedings of “Learning with Partially Classified Training Data” ICML 2005, p.37. R.R. Sokal, C.D. Michener, University of Kansas Scientific Bulletin (1958). S.C. Johnson, Psychometrika [**2**]{}, 241 (1967). J. McQueen, in [*Proc. 5th Berkeley Symp. on Math. Stat. and Prob.*]{}, L. Le Cam, J. Neyman (ed.) (Uni. of California Press 1967). M. Blatt, S. Wiseman, and E. Domany, Phys. Rev. Lett. [**76**]{},3251 (1996). J.S. Yedidia, W.F.Freeman, and Y.Weiss, IEEE Trans. Inform. Theory [**47**]{}, 1 (2005). F.R. Kschischang, B.J. Frey, and H.A. Loeliger, IEEE Trans. Inform. Theory [**47**]{}, 1 (2001). B.J. Frey and D. Dueck, Science [**315**]{}, 972 (2007). M. Mézard and G. Parisi, Eur. Phys. J. B [**20**]{}, 217 (2001). A.K. Hartmann and M. Weigt, [*Phase Transitions in Combinatorial Optimization Problems*]{} (Wiley-VCH, Berlin 2005) R.O. Duda, P.E. Hart, [*Classification and Scene Analysis*]{} (Wiley, New York 1973). [^1]: This observation was first done in the case of original AP in [@FREY], and can be simply extended to our model [^2]: Other choices, such as taking the average or center of mass distance, have been tried, but lead to worse results.

ArXiv

--- abstract: 'Since AlN has emerged as an important piezoelectric material for a wide variety of applications, efforts have been made to increase its piezoelectric response via alloying with transition metals that can substitute for Al in the wurtzite lattice. Herein, we report density functional theory calculations of structure and properties of the Cr-AlN system for Cr concentrations ranging from zero to beyond the wurtzite-rocksalt transition point. By studying the different contributions to the longitudinal piezoelectric coefficient, we propose that the physical origin of the enhanced piezoelectricity in Cr$_x$Al$_{1-x}$N alloys is the increase of the internal parameter $u$ of the wurtzite structure upon substitution of Al with the larger Cr ions. Among a set of wurtzite-structured materials, we have found that Cr-AlN has the most sensitive piezoelectric coefficient with respect to alloying concentration. Based on these results, we propose that Cr-AlN is a viable piezoelectric material whose properties can be tuned via Cr composition. We support this proposal by combinatorial synthesis experiments, which show that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus $d_{33}$ is approximately four times larger than that of pure AlN. This finding, combined with the relative ease of synthesis under non-equilibrium conditions, may propel Cr-AlN as a prime piezoelectric material for applications such as resonators and acoustic wave generators.' author: - 'Sukriti Manna$^1$, Kevin R. Talley$^{2,3}$, Prashun Gorai$^{2,3}$, John Mangum$^2$, Andriy Zakutayev$^3$, Geoff L. Brennecka$^2$, Vladan Stevanović$^{2,3}$, and Cristian V. Ciobanu$^1$[^1]' title: Enhanced piezoelectric response of AlN via CrN alloying --- Introduction ============ Aluminum nitride has emerged as an important material for micro-electromechanical (MEMS) based systems[@fu2017advances; @muralt2008recent] such as surface and bulk acoustic resonators,[@fu2017advances; @loebl2003piezoelectric] atomic force microscopy (AFM) cantilevers,[@fu2017advances] accelerometers,[@gerfers2007sub; @wang2017mems] oscillators,[@zuo20101] resonators for energy harvesting,[@wang2017aln; @elfrink2009vibration] and band-pass filters.[@yang2003highly] The advantages of using AlN in MEMS devices include metal$-$oxide$-$semiconductor (CMOS) compatibility, high thermal conductivity, and high temperature stability. In addition, its low permittivity and high mechanical stiffness are particularly important for resonantor applications.[@fu2017advances; @muralt2017aln] However, the piezoelectric constants of AlN thin films are lower than those of other commonly used piezoelectric materials. For example, the out-of-plane piezoelectric strain modulus[@nomenclature] $d_{33}$ of reactively sputtered AlN films is reported to be 5.5 pC/N, whereas $d_{33}$ for ZnO can be at least twice as large,[@kang2017enhanced] and PZT films can be over 100 pC/N.[@muralt2008recent] It is therefore desirable to find ways to increase the piezoelectric response of AlN in order to integrate AlN-based devices into existing and new systems. A common way to engineer piezoelectric properties of AlN is by alloying with transition metal nitrides (Sc, Y, others), which can lead to a several-fold increase in the field-induced strain via increases in the longitudinal piezoelectric coefficient $e_{33}$ and simultaneous decreases in the longitudinal elastic stiffness $C_{33}$.[@akiyama2009enhancement; @caro2015piezoelectric; @manna2017tuning; @tasnadi2010origin] In the case of ScN alloying, the origins of this response have been studied,[@tasnadi2010origin] and it is presumed that other such systems which also involve AlN alloyed with rocksalt-structured end members are similar: as the content of the rocksalt end member in the alloy increases, the accompanying structural frustration enables a greater piezoelectric response. This structural frustration, however, is also accompanied by thermodynamic driving forces for phase separation[@hoglund2010wurtzite] which, with increased alloy concentration, lead to the destruction of the piezoelectric response upon transition to the (centrosymmetric, cubic) rocksalt structure. The experimental realization of large alloy contents without phase separation or severe degradation of film texture and crystalline quality can be quite difficult,[@hoglund2010wurtzite; @mayrhofer2015microstructure] even when using non-equilibrium deposition processes such as sputtering. Thus, it is desirable to find alloy systems for which the structural transition from wurtzite to rocksalt occurs at low alloying concentrations since these may be more easily synthesized and more stable, while also (hypothetically) providing comparable property enhancements as those observed in the more-studied Sc-AlN alloy system. Among the AlN-based systems presently accessible experimentally, Cr-AlN has the lowest transition composition between the wurtzite and rocksalt structures, occurring at approximately 25% CrN concentration.[@mayrhofer2008structure; @holec2010pressure] This motivates the investigation of the piezoelectric properties of the Cr$_x$Al$_{1-x}$N system, which we also refer to, for simplicity, in terms of Cr substitution for Al. In this article, we study Cr-substituted AlN using density functional theory (DFT) calculations of structural, mechanical, and piezoelectric properties. Given that Cr has unpaired $d$ electrons, a challenge to overcome in these calculations is the simulation of a truly representative random distribution of the spins of Cr ions, whose placement in the AlN lattice involves not only chemical disordering, but spin disordering as well. Among a set of wurtzite-based materials, we have found that Cr-doped AlN is the alloy whose piezoelectric stress coefficient $e_{33}$ is the most sensitive to alloying concentration and also has the lowest wurtzite-to-rocksalt transition composition. The key factor leading to the enhanced piezoelectricity in Cr$_x$Al$_{1-x}$N alloys is the ionic contribution to the coefficient $e_{33}$; this ionic contribution is increased through the internal $u$ parameter of the wurtzite structure when alloyed with the (larger) Cr ions. Therefore, we propose Cr$_x$Al$_{1-x}$N as a viable piezoelectric material with properties that can be tuned via Cr composition. To further support this proposal, we have performed combinatorial synthesis and subsequent characterization of Cr$_x$Al$_{1-x}$N films, and have showed that Cr can be incorporated in the AlN lattice up to 30% before a detectable transition to rocksalt occurs. At this Cr content, the piezoelectric modulus $d_{33}$ is four times larger than that of AlN. Pending future device fabrication and accurate measurements of properties and device performance, this significant increase in $d_{33}$ can propel Cr-AlN to be the choice material for applications such as resonators, GHz telecommunications, or acoustic wave generators. Methods ======= Paramagnetic Representation of Cr-AlN Alloys -------------------------------------------- Starting with a computational supercell of wurtzite AlN, any desired Cr concentration is realized by substituting a corresponding number of Al ions with Cr ions in the cation sub-lattice. In order to realistically simulate the chemical disorder of actual Cr-AlN alloys while maintaining a tractable size for the computational cell, we use special quasirandom structures (SQS).[@zunger1990special; @van2009multicomponent; @van2013efficient] The Cr$^{3+}$ ions have unpaired $d$ electrons, which require spin-polarized DFT calculations. Another important aspect of the calculations is that the Cr$_x$Al$_{1-x}$N alloys are paramagnetic,[@mayrhofer2008structure; @endo2007crystal; @endo2005magnetic] and this state has to be captured explicitly in the DFT calculations. Therefore, in addition to the configurational disorder simulated via SQS, the paramagnetic state requires truly random configurations for the spins associated with the Cr$^{3+}$ ions.[@alling2010effect; @abrikosov2016recent] However, as shown by Abrikosov [*et al.*]{},[@abrikosov2016recent] the paramagnetic state can be approximated by using disordered, collinear, static spins because such state yields zero spin-spin correlation functions. To represent the paramagnetic state of Cr$_x$Al$_{1-x}$N, for a given alloy structure with $n$ Cr sites, we performed a minimum of $n \choose 2$ and maximum 20 calculations. In these calculations, the spins on Cr sites are randomly initialized subject to the restriction of zero total spin for each concentration and each SQS structure. An example of such a random distribution of initial spins is illustrated in Figure \[schematics-spin\] for $x$ = 25% Cr concentration. ![Schematics of cation sublattice of Cr$_{x}$Al$_{1-x}$N alloy. Al (Cr) sites are shown as gray (green) spheres. At a given Cr concentration, the Cr sites of each configuration have a different and random spin initialization with zero total spin in order to capture the paramagnetic state.[]{data-label="schematics-spin"}](fig1-spindisorder){width="7cm"} Details of the DFT Calculations ------------------------------- Structural optimizations and calculations of piezoelectric and elastic constants were carried out using the Vienna Ab-initio Simulation Package (VASP),[@kresse1996efficiency] with projector augmented waves (PAW) in the generalized gradient approximation using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation function[@perdew1996generalized] and an on-site Hubbard term[@dudarev1998electron] $U$ for the Cr $3d$ states. The plane wave cutoff energy was set to 540 eV in all calculations. For the wurtzite structures, we have used $4\times4\times 2$ (128 atoms) and $2\times2\times2$ (32 atoms) SQS supercells; for the rocksalt structures, the computations were carried out on $2\times 2 \times 2$ (64 atoms) SQS supercells. Brillouin zone sampling was performed by employing $1\times 1\times 1$ and $2 \times2\times 2$ Monkhorst-Pack[@monkhorst1976special] $k$-point meshes for the wurzite and rocksalt structures, respectively, with the origin set at the $\Gamma$ point in each case. Piezoelectric coefficients were calculated using density functional perturbation theory, and the elastic constants were computed by finite differences.[@gonze1997dynamical; @wu2005systematic] The on-site Coulomb interaction for Cr atoms was set at 3 eV, through a Dudarev approach.[@dudarev1998electron] Before performing the calculations for elastic and piezoelectric constants, we performed cell shape, volume, and ionic relaxations in order to obtain the equilibrium lattice parameters and ionic positions at each particular Cr concentration and SQS alloy. Experimental Procedures ----------------------- Combinatorial synthesis of Cr$_x$Al$_{1-x}$N films was performed through reactive physical vapor deposition (PVD). Two inch diameter circular aluminum (99.9999%) and chromium (99.999%) metallic targets were arranged at 45$^{\rm o}$ angles measured from the normal to a plasma-cleaned Si(100) substrate inside a custom vacuum system with a base pressure of $5\times 10^{-6}$ torr. Magnetron RF sputtering with a power of 60 W for aluminum targets and 40 W for the chromium targets was performed at a deposition pressure of $3\times 10^{-3}$ torr, with 8 sccm of argon and 4 sccm of nitrogen, and a substrate temperature of 400 $^{\rm o}$C. Aluminum glow discharges were oriented opposite to each other, with the chromium target perpendicular to both, resulting in a film library with a compositional range in one direction.[@config1; @config2] Each sample library was subdivided into eleven regions across the composition gradient, which were subsequently characterized by x-ray diffraction (XRD) and x-ray fluorescence (XRF), performed on a Bruker D8 Discovery diffractometer with a 2D area detector in a theta-2theta configuration and a Fischer XUV vacuum x-ray spectrometer, respectively. Results and Discussion ====================== Enthalpy of Mixing ------------------ The enthalpy of mixing as a function of the Cr concentration $x$, at zero pressure, is defined with respect to the pure wurzite-AlN and rocksalt-CrN phases via $$\Delta H_{\text{mix}}(x) = E_{\text{Cr}_{x}\text{Al}_{1-x}\text{N}} - xE_{\text{rs}\text{-CrN}} -(1-x)E_{\text{w}\text{-AlN}},$$ where $E_{\text{Cr}_{x}\text{Al}_{1-x}\text{N}}$, $E_{\text{rs}-\text{CrN}}$, and $E_{\text{w}-\text{AlN}}$ are the total energies per atom of the SQS alloy, pure AlN phase, and pure CrN phase, respectively . The DFT calculated mixing enthalpies for the wurtzite and rocksalt phases of Cr$_{x}$Al$_{1-x}$N are shown in Figure \[enthalpy\](a). The wurzite phase is found to be favorable up to $x=0.25$, beyond which rocksalt alloys are stable; this wurzite to rocksalt phase transition point is consistent with previous experimental observations and other theoretical predictions.[@mayrhofer2008structure; @holec2010pressure] We have compared the mixing enthalpy of the Cr$_x$Al$_{1-x}$N alloys with that of several other common wurzite-based nitrides,[@akiyama2009enhancement; @tholander2016ab; @vzukauskaite2012yxal1; @manna2017tuning; @mayrhofer2015microstructure] Sc$_{x}$Al$_{1-x}$N, Y$_{x}$Al$_{1-x}$N, and Y$_{x}$In$_{1-x}$N, with the results shown in Figure \[enthalpy\](b). The mixing enthalpies are positive for all cases, meaning that the alloying of AlN or InN with their respective end members is an endothermic process. In practice, these alloys are formed as disordered solid solutions obtained using physical vapor deposition techniques operating at relatively low substrate temperatures because of the energetic plasmas involved.[@hoglund2010wurtzite; @luo2009influence] Figure \[enthalpy\](b) shows that the mixing enthalpy in Cr$_x$Al$_{1-x}$N lies between values corresponding to other systems synthesized experimentally, hence Cr$_x$Al$_{1-x}$N is no more difficult to synthesize than the others. More importantly, the enthalpy calculations show that the transition to rocksalt occurs at the lowest alloy concentration across the wurtzite systems considered, which is important for achieving maximum piezoresponse-enhancing structural frustration with a minimum of dopant concentration in order to retain the single-phase wurtzite. ![(a) DFT-calculated mixing enthalpies of the wurtzite and rocksalt phases of Cr$_{x}$Al$_{1-x}$N as functions of Cr concentration. (b) Calculated mixing enthalpies for several wurzite-based nitride alloys grown experimentally.[]{data-label="enthalpy"}](fig2-enthalpy){width="7cm"} {width="13cm"} {width="10cm"} ![(a) Variation of internal parameter, u with Cr addition. (b) Crystal structure of wurzite AlN.[]{data-label="tetrahedra"}](fig5-u_plot){width="5cm"} Piezoelectric Stress Coefficients --------------------------------- The piezoelectric coefficients $e_{ij}$ for different spin configurations in SQS supercells with the same Cr content are shown in Figure \[all-e\]. For clarity, the panels in Figure \[all-e\] are arranged in the same fashion as the piezoelectric tensor when represented as a matrix in Voigt notation. The vertical scale is the same for all coefficients except $e_{33}, e_{31}$, and $e_{32}$. The scatter in the results corresponds to different SQS supercells at each Cr concentration; this is an effect of the finite size of the system, in which local distortions around Cr atoms lead to small variations of the lattice constants and angles. It is for this reason that we average the SQS results at each Cr concentration, thereby obtaining smoother variations of the piezoelectric coefficients. At 25% Cr, the value of $e_{33}$ becomes $\sim$1.7 times larger than that corresponding to pure AlN. The piezoelectric coefficient $e_{33}$ of wurzite Cr$_x$Al$_{1-x}$N is shown in Figure \[e33-details\](a) as a function of Cr concentration, and can be written as[@bernardini1997spontaneous] $$e_{33}(x) = e_{33}^{\text{clamped}}(x) + e_{33}^{\text{non-clamped}}(x), \label{eqe33partition}$$ in which $e_{33}^{\text{clamped}}(x)$ describes the electronic response to strain and is evaluated by freezing the internal atomic coordinates at their equilibrium positions. The term $e_{33}^{\text{non-clamped}}(x)$ is due to changes in internal coordinates, and is given by $$e_{33}^{\text{non-clamped}}(x) = \frac{4eZ_{33}^{*}(x)}{\sqrt{3} a(x)^{2}}\frac{du(x)}{d\epsilon} \label{nclamped}$$ where $e$ is the (positive) electron charge, $a(x)$ is the equilibrium lattice constant, $u(x)$ is the internal parameter of the wurtzite, $Z_{33}^{*}(x)$ is the dynamical Born charge in units of $e$, and $\epsilon$ is the macroscopic applied strain. $e_{33}^{\text{non-clamped}}(x)$ describes the piezoelectric response coming from the displacements of internal atomic coordinates produced by the macroscopic strain. Based on Eqs. (\[eqe33partition\]) and (\[nclamped\]), panels (b) through (f) in Figure \[e33-details\] show the different relevant quantities contributing to $e_{33}$ in order to identify the main factors responsible for the increase of piezoelectric response with Cr addition. Direct inspection of Figures \[e33-details\](a-c) indicates that the main contribution to the increase of $e_{33}$ comes from the non-clamped ionic part, Figure \[e33-details\](c). Since the Born charge $Z_{33}^{*}$ \[Figure \[e33-details\](d)\] is practically constant, the key factor that leads to increasing the piezoelectric coefficient is the strain sensitivity $du/d\epsilon$ of the internal parameter $u$ \[Figure \[e33-details\](e)\]. Although the internal parameter $u$ is an average value across the entire supercell, the individual average $u$ parameters can also be determined separately for AlN and CrN tetrahedra \[Figure \[tetrahedra\](a,b)\]. The internal parameter $u$ of AlN tetrahedra \[Figure \[tetrahedra\](b)\] does not change significantly, while that of the CrN tetrahedra grows approximately linearly with Cr concentration \[Figure \[tetrahedra\](a)\]. In an alloy system where AlN tetrahedra are the majority, this variation can be understood based on (i) the fact that the ionic radius of Cr is about 10% larger than that of Al, and (b) the increase in Cr concentration will lead to average $u$ parameters mimicking the variation of the $u$ parameter corresponding to CrN tetrahedra. Comparison with Other Wurtzite-Based Alloys ------------------------------------------- ![Comparison of change in $e_{33}$ with addition of different transition metals for $x\leq$ 25% regime.[]{data-label="e33-compare"}](fig6-e33-compare){width="7cm"} The results from calculations of the piezoelectric properties of Cr$_x$Al$_{1-x}$N with $x$ from 0 to 25% Cr are plotted in Figure \[e33-compare\], together with the calculated values for Sc$_x$Al$_{1-x}$N, Y$_x$Al$_{1-x}$N, and Y$_x$In$_{1-x}$N. In Cr$_x$Al$_{1-x}$N, $e_{33}$ increases rapidly from 1.46 to 2.40 C/m$^{2}$ for Cr concentrations from 0 to 25%. For all other alloys considered, the increase is smaller in the same interval of solute concentration: for Sc$_x$Al$_{1-x}$N, Y$_x$Al$_{1-x}$N, and Y$_x$In$_{1-x}$N, $e_{33}$ increases, respectively, from 1.55 to 1.9 C/m$^{2}$, 1.55 to 1.7 C/m$^{2}$, and 0.9 to 1.2 C/m$^{2}$. Within the $x\leq$ 25% range, Cr is more effective than any of the other studied transition elements in improving piezoelectric response of AlN-based alloys. ![Variation of (a) $C_{33}$ and (b) $d_{33}$ for several nitride-based wurtzite alloys.[]{data-label="C33d33"}](fig7-C33d33){width="6cm"} The experimentally measurable property is $d_{33}$, which is commonly known as piezoelectric strain modulus and relates the electric polarization vector with stress. The relationship between the piezoelectric strain and stress moduli is[@nye1985physical] $$d_{ij} = \sum_{k=1}^{6} e_{ik}(C^{-1})_{kj}, \label{eC}$$ where $C_{ij}$ are the elements of the stiffness tensor in Voigt notation. The variation of the elastic constant $C_{33}$ in Cr$_x$Al$_{1-x}$N with $x$ is shown in Figure \[C33d33\](a), along with the other systems considered here. For all of these wurtzite-based piezoelectrics, the increase in piezoelectric response with alloying element concentration is accompanied by mechanical softening (decrease in $C_{33}$). From Eq. (\[eC\]), it follows that the increase in $e_{33}$ (Figure \[e33-compare\]) and the mechanical softening \[Figure \[C33d33\](a)\] cooperate to lead to the increase of $d_{33}$ values with alloy concentration $x$. Our calculated $d_{33}$ values for Cr$_x$Al$_{1-x}$N are in good agreement with experimental data from Ref.  \[Figure \[C33d33\](b)\] for Cr concentrations up to 6.3%. Beyond this concentration, Luo et al.[@luo2009influence] report a drop in the $d_{33}$ values of their films, which is attributed to changes in film texture. We have also extended our calculations of piezoelectric coefficients beyond 25% Cr composition in wurtzite structures. Figure \[e33-25\] shows that $e_{33}$ continues to increase at least up to 37.5% Cr. The calculations done at 50% Cr, which start with wurtzite SQS configurations, evolve into rocksalt configurations during relaxation, which explains the decrease of $e_{33}$ to zero in Figure \[e33-25\]. ![Variation of $e_{33}$ as function of Cr concentration for wurtzite phase alloys up to 50%.[]{data-label="e33-25"}](fig8-e33-boundary){width="7.5cm"} Material $e_{33}$ (C/m$^2)$ $d_{33}$ (pC/N) Refs. ------------------ -------------------- ----------------- ------- -- AlN 1.55 4.5-5.3 Sc-AlN, 10% Sc 1.61 7.8 Y-AlN, 6% Y 1.5 4.0 Y-InN, 14% Y 1.1 5.1 [*this work:*]{} Cr-AlN, 12.5% Cr 1.84 9.86 Cr-AlN, 25.0% Cr 2.35 16.45 Cr-AlN, 30.0% Cr 2.59 19.52 : Piezoelectric properties of AlN and a few wurtzite alloys for piezoelectric device applications.[]{data-label="table:properties"} It is worthwhile to compare the performance of several AlN wurzite-based materials for their use in applications. These applications, which are mainly resonators, ultrasound wave generators, GHz telecommunications, FBAR devices, bulk or surface acoustic generators, and biosensors, lead to a multitude of application-specific figures of merit for different utilization modes of the piezoelectric material. However, most figures of merit rely on the piezoelectric properties $e_{33}$ and $d_{33}$, both of which in general should be as large as possible for increased piezoelectric device sensitivity. The most used wurtzite material for these applications is AlN, although there are several other options as well (refer to Table I). Alloying with ScN is promising in that it offers an increased $d_{33}$ for about 10% Sc concentration; larger Sc concentrations are possible, but the growth process becomes more costly and the material is likely to lose texture with increased Sc content. Options such as alloying with YN offer marginal improvement at 6% Y content, and YN-doped InN (14% Y content) fares similarly (Table I). Our results indicate that CrN alloying of AlN can reach superior values for the piezoelectric properties, nearly quadrupling the value of $d_{33}$ (Table I) with respect to AlN. The fact that the transition point is the lowest (Figure \[enthalpy\]) of all wurtzite-based materials relevant for the technologies mentioned above, makes the CrN alloying easier compared with the other materials (which require higher alloy content) and hence renders Cr-AlN a prime candidate for synthesis of new, CrN-alloyed piezoelectrics for resonators and acoustic generators. As we shall see in Sec. III.D, the non-equilibrium growth techniques can bring Cr content past the transition point without significant formation of the (non-piezoelectric) rocksalt phase. Consequently, the piezoelectric properties are expected to be significantly better than those of AlN, especially $d_{33}$ (refer to Table \[table:properties\]). Indeed, this is born out in experiments (data points in Figure \[C33d33\]b). Measurements of figures of merit for specific device configurations will be needed in the future, as those require not only combinations of elastic and piezoelectric properties, but dielectric properties as well.[@manna2017tuning] Experimental Results -------------------- ![ Representative transmission electron microscopy (TEM) image (a) and an energy dispersive spectroscopy (EDS) line scan (b) of an (Al$_{1-x}$Cr$_x$)N film cross section containing $\sim$7% Cr, confirming the incorporation of Cr into wurtzite solid solution. EDS data were collected along the dashed black line shown in panel (a).[]{data-label="TEM"}](fig9-TEMPresentation1){width="8.5cm"} ![X-ray diffraction patterns of the thin film combinatorial libraries plotted against the film composition, with comparison to the patterns for wurtzite (WZ) [@eddine1977etude] and rocksalt(RS)[@wyckoff1960crystal] structures. For alloying content $x<30\%$, the films grow predominantly with the wurtzite structure. At higher Cr concentrations, $x>30\%$, both rocksalt and wurtzite phases are detected, and the wurtzite exhibits degraded texture. No films were produced with compositions in regions where no intensity is shown. []{data-label="kevin"}](fig10-kevinXRD){width="8.5cm"} To bring experimental support to our proposal that the Cr-AlN system can become a key piezoelectric material to replace AlN and perhaps even ZnO for future applications, we have to ensure that the texture obtained during growth is stable for sufficiently high CrN concentrations. After synthesizing Cr-AlN alloys through reactive PVD, we have performed transmission electron microscopy (TEM) analysis of the films grown in order to check for textural integrity (i.e., grains oriented primarily with the $c$ axis close to the surface normal) and for the onset of the rocksalt phase. At CrN content below 25% \[the theoretical boundary shown in Figure \[enthalpy\](a)\], our films display no significant texture variations. For example, Figure 9(a) shows a typical TEM micrograph wherein texture is preserved over the film thickness. Additionally, our energy dispersive spectrocopy (EDS) characterization shows nearly constant Cr content through the sample \[Figure \[TEM\](b)\]. Further characterization by XRD was performed for all CrN compositions in the combinatorially synthesized films. Figure \[kevin\] shows the XRD results for the 88 discrete Cr$_x$Al$_{1-x}$N compositions produced in an effort to test the possibilities of synthesizing alloys in a wide range of concentrations, including alloys beyond the wurtzite-to-rocksalt transition point. At low alloying levels, the films grow exclusively with the wurtzite structure and a $(002)$ preferred orientation, as indicated by the dominant presence of the wurtzite $(002)$ diffraction peak (Figure \[kevin\], left side). Films grown by reaction PVD under the conditions used here accept chromium into the wurtzite lattice and grow primarily with the ideal $(002)$ orientation. With increased CrN content, the wurtzite $(012)$ and $(010)$ peaks appear, indicating some deviations from the original, and still predominant $(002)$ orientation of the film. The metastability of this alloy is overcome at an approximate composition of $x \simeq 30$%, where the polycrystalline rocksalt phase appears, as revealed by the rocksalt $(002)$ and $(111)$ peaks (Figure \[kevin\], right hand side). These experimental results show that wurtzite Cr$_x$Al$_{1-x}$N solid solutions can be synthesized without observable phase separation up to concentrations of 30% Cr. Wurtzite material still exists at global compositions beyond 30%, but in a wurtzite-rocksalt phase mixture, which will diminish the piezoelectric properties because of the presence of a significant amount of centrosymmetric rocksalt phase in the mixture. There are few studies of Cr alloyed into wurtzite AlN,[@luo2009influence; @felmetsger2011; @endo2005magnetic] reporting Cr-doped alloys grown by magnetron sputtering. The Cr concentration previously attained is below 10%, although the limits of Cr alloying were not actually tested in the previous reports.[@luo2009influence; @felmetsger2011; @endo2005magnetic] Our combinatorial synthesis results show that Cr can be doped into the wurtzite lattice up to 11% before the predominant (002) film texture starts to change, and up to 30% before the rocksalt phase appears. Concluding Remarks ================== By using a physical representation of the paramagnetic state of substitutional Cr in a wurtzite AlN matrix and performing the necessary averaging over spin configurations at each Cr concentration, we computed the structural, mechanical, and piezoelectric properties of Cr-AlN alloys. Our combinatorial synthesis experiments showed that Cr-AlN are relatively easy to synthesize, and also showed that the reactive PVD procedure resulted in Cr-AlN alloys retaining the wurtzite structure for alloying concentrations up to 30% Cr. Remarkably, our DFT calculations of piezoelectric properties revealed that for 12.5% Cr $d_{33}$ is twice that of pure AlN, and for 30% Cr this modulus is about four times larger than that of AlN. From a technological standpoint, this finding should make Cr-AlN the prime candidate to replace the current-wurtzite based materials in resonators and acoustic wave generators. The larger piezoelectric response (than AlN) may lead to smaller power consumption and perhaps even to avenues to further miniaturize various devices. While the substitutional alloying with Cr would improve the piezoelectric response for every type of device in which currently AlN is being used, one may wonder why not alloying with other trivalent metals, such as Y or Sc. In particular, Sc has been shown to significantly increase the piezoelectric modulus as well.[@tasnadi2010origin] Even though Sc-AlN has more exciting properties[@caro2015piezoelectric; @tasnadi2010origin] than Cr-AlN, the reason why ScN alloys have not taken over the resonator market so far is that the outstanding enhancements in piezoelectric properties occur at very high Sc concentration (Fig. 2, $x>55$%), at which the stability of the wurtzite phase is rather poor. Cr-AlN has a low wurtzite-to-rocksalt transition concentration, and therefore can offer certain piezoelectric enhancements at alloying levels that are easier to stabilize during the synthesis. In order to ensure significant impact of Cr-AlN alloys as materials to outperform and replace the established piezoelectrics AlN and ZnO, two avenues should be pursued in the near future. First, to benefit from the $300$% increase in $d_{33}$ at 30% Cr content, it is not sufficient that the rocksalt phase does not form up to that Cr concentration: we also have to avoid the formation of (012) and (010)-oriented grains during growth, which would downgrade (simply through directional averaging) the piezoelectric enhancements associated with the (002)-oriented grains. To that end, we envision changing substrates so as to enable better lattice matching with Cr-alloys with over 25% Cr. This can effectively prevent the (012) and (010) textures from emerging, therefore creating the conditions to take advantage of the large increase in $d_{33}$ reported here. Second, future experimental efforts should measure device performance especially to understand the additional aspect of how Cr content in wurtzite affects the bandgap and whether there would be deleterious leakage effects at larger Cr concentrations. Assuming a worst case scenario, these effects can be mitigated by co-alloying with a non-metalic atomic species (e.g. boron). Pursuing the two directions above can make Cr-AlN suitable for simultaneous optical and mechanical resonators,[@yale1; @yale2] which are relatively new applications that currently exploit multi-physics aspects of AlN. At present, the characterization of Cr-AlN for these multifunctional applications that require simultaneous engineering of the photonic and acoustic band structure is rather incipient, and only few relevant properties of the Cr-AlN alloys are known: for example, for a Cr concentration of about 2%, the bandgap is virtually unchanged, while the adsorption band decreases from 6 to 3.5 eV.[@highly2] Future theoretical and experimental work to investigate, e.g., photoelastic effect and optical attenuation, is necessary in order to fully uncover the potential of Cr-doped AlN for these applications. For now, we surmise that the technological reason for which one would replace AlN with Cr-AlN for use in multifunctional resonators is the trade-off between the increase in vibrational amplitude and decrease in frequency: while low amounts of Cr may lower the frequency somewhat, the oscillation amplitude would increase due to larger piezoelectric response. The decrease in frequency can be mitigated by co-doping with a small trivalent element (boron), as shown for other doped AlN alloys.[@manna2017tuning] Last but not least, it is worth noting that doping with Cr could enable magnetic polarizaton of the Cr ions in the wurtzite lattice and/or of the minority carriers: these effects are non-existent in pure AlN, and could be pursued for spintronic applications or for low-hysteresis magnets.[@endo2005magnetic] The significant increase of piezoelectric modulus reported here provides significant drive to pursue the two directions identified above, and overcome routine barriers towards establishing Cr-AlN as a replacement for AlN with large performance enhancements. [*Acknowledgments.*]{} The authors gratefully acknowledge the support of the National Science Foundation through Grant No. DMREF-1534503. 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ArXiv

--- author: - 'Kory M. Stiffler' title: 'A WALK THROUGH SUPERSTRING THEORY WITH AN APPLICATION TO YANG-MILLS THEORY: K-STRINGS AND D-BRANES AS GAUGE/GRAVITY DUAL OBJECTS' ---

ArXiv

--- abstract: 'While 3GPP has been developing NB-IoT, the market of Low Power Wide Area Networks has been mastered by cheap and simple Sigfox and LoRa/LoRaWAN technologies. Being positioned as having an open standard, LoRaWAN has attracted also much interest from the research community. Specifically, many papers address the efficiency of its PHY layer. However MAC is still underinvestigated. Existing studies of LoRaWAN do not take into account the acknowledgement and retransmission policy, which may lead to incorrect results. In this paper, we carefully take into account the peculiarities of LoRaWAN transmission retries and show that it is the weakest issue of this technology, which significantly increases failure probability for retries. The main contribution of the paper is a mathematical model which accurately estimates how packet error rate depends on the offered load. In contrast to other papers, which evaluate LoRaWAN capacity just as the maximal throughput, our model can be used to find the maximal load, which allows reliable packet delivery.' author: - bibliography: - 'biblio.bib' title: 'Mathematical Model of LoRaWAN Channel Access [^1]' --- at (current page.south) ; LoRa, LoRaWAN, LPWAN, Channel Access, Performance Evaluation, ALOHA Introduction ============ LoRaWAN is a relatively new protocol designed to provide cheap and reliable wireless connectivity in various Internet of Things scenarios. Being a Low Power Wide Area Network technology operating in the ISM band, it rapidly got popularity in both industry and academic communities. Literature review shows that in spite of numerous studies of its PHY layer [@centenaro2016long; @vangelista2015long; @goursaud2015dedicated], the MAC layer got little attention, even though it has multiple issues [@bankov2016limits; @mikhaylov2016analysis] that limit its performance. However, as LoRaWAN is designed to support networks of thousands of devices, it is crucial not only to consider the performance of this technology in point-to-point scenarios, but also to evaluate its applicability in case of highly-populated networks. To calculate throughput of LoRaWAN networks, in existing studies of the MAC layer (e.g., see [@adelantado2017understanding]), the authors typically use the classical approach for modeling ALOHA networks [@aloha]. The papers (e.g. [@augustin2016study]) also limit the study to unacknowledged mode, which has no control acknowledgements (ACKs). Thus, with no control traffic the throughput increases. However the reliability of transmission decreases. In this paper, we provide a mathematical model for a LoRaWAN network operating in the acknowledged mode. We explain why the usage of classical ALOHA-like approach underestimates the collision probability and develop an accurate mathematical model which takes into account LoRaWAN peculiarities related to retransmission policy. LoRaWAN Channel Access Description ================================== A typical LoRaWAN [@lorawan] network consists of end devices, called *motes*, gateways (GWs), and a server. Motes are connected to the GWs via wireless LoRa links. Gateways gather information from the motes, send it to the server via an IP network, and forward packets from the server to the motes. LoRaWAN devices operate in different ways. Depending on operation, the standard describes three classes of devices. The basic functionality for sporadic uplink data transmission is described as class A operation and is studied in this paper. A LoRaWAN network simultaneously works in several wireless channels. For example, in Europe they can use three main channels and one downlink channel. To transmit a data frame, each mote randomly selects one of the main channels (see Fig. \[fig:channel\_access\]). Having received the frame, the GW sends two ACKs. The first one is sent in the main channel, where the frame was received, $T_1$ after frame reception. The second ACK is sent in the downlink channel after timeout $T_2 = T_1 + \SI{1}{\s}$. If a mote receives no ACK, it makes a retransmission. The standard recommends making a retransmission in a random time drawn from $[1, 1 + W]$ seconds, where $W = 2$. Note that the recommended $W$ is too small and, as we show in the paper leads to the “avalanche effect”. At the PHY layer, LoRaWAN uses Chirp Spread Spectrum modulation. Its main feature is that signals with different spreading factors can be distinguished and received simultaneously, even if they are transmitted in the same time on the same channel. Spreading factor, together with the channel width and the coding rate, determines the data rate. Lower data rates extend transmission range and improve transmission reliability. For the first transmission attempt, the rate is determined by the GW. The standard also recommends decrementing data rate every two consequent transmission failures, limiting the number of retransmissions by $RL = 7$. The first ACK is sent at a data rate that is lower than the data rate for the frame transmission by a configurable offset (it can be zero). The second ACK should always be sent at a fixed data rate, by default the lowest one. (0,2.2) – (11.5,2.2); at (0, 2.9) ; at (0, 1.4) ; (0,0.8) – (11.5,0.8); at (11.5, 1.9) [$t$]{}; at (11.5, 0.5) [$t$]{}; (1, 2.2) rectangle (3.6, 2.8); at (2.3, 2.5) ; (6, 1.6) rectangle (8.4, 2.2); at (7.25, 1.9) ; (8.5, 0.8) rectangle (10.9, 1.4); at (9.7, 1.1) ; (6, 2) – (6, 3.4); (3.6, 0) – (3.6, 3.4); (8.5, 0) – (8.5, 1); (3.6,3.1) – (6,3.1); (3.6,0.2) – (8.5,0.2); at (4.75, 3.3) [$T_1$]{}; at (6, 0.5) [$T_2$]{}; Problem Statement {#sec:scenario} ================= Consider a LoRaWAN network that consists of a GW and $N$ motes and operates in $F$ main channels and one downlink channel. The motes use data rates $0, 1, ..., R$, set by the GW. Let $p_i$ be the probability that a mote uses data rate $i$. We consider that a frame collision occurs when two frames are transmitted in the same channel at the same data rate, and they intersect in time. The motes generate frames according to a Poisson process with total intensity $\lambda$ (the network load). All motes transmit frames with 51-byte Frame Payload which corresponds to the biggest payload that can fit a frame at the lowest data rate. The frames are transmitted in the acknowledged mode, and ACKs carry no frame payload. We consider a situation, when motes have no queue, i.e. if two messages are generated, a mote transmits the most recent one. For the described scenario, it is important not only to know the nominal channel capacity, but also to find the maximal load at which the network can provide reliable communications. In other words, we need *to find the packet error rate (PER) as a function of network load $\lambda$*. Mathematical Model ================== To solve the problem, we develop a mathematical model of the transmission process. As the first transmission attempts are described by the Poisson process, to find the PER in these assumptions, in Section \[first\], we consider the approach used to evaluate ALOHA networks[@aloha] and extended to take into account ACKs. This approach is however inapplicable for retransmissions, because they do not form a Poisson process, so in Section \[retries\] we propose another way to take them into account and thus to improve the accuracy of the model. The First Transmission Attempt {#first} ------------------------------ The first transmission attempt is successful with probability $$\label{eq:success1} P_{S,1} = \sum_{i = 0}^{R} p_{i} P^{Data}_i P^{Ack}_{i},$$ where $P^{Data}_i$ is the probability that the data frame is transmitted without collision at data rate $i$ and $P^{Ack}_i$ is the probability that at least one ACK out of two is received by the mote, provided that the data frame is successful. Since the packets transmitted in different channels and at different rates do not collide, we need to consider separately each combination of channel and data rate. Specifically for rate $i$ and one of $F$ channels, the load equals $r_i = \frac{\lambda p_i}{F}$. A data frame transmission is successful if it intersects with no transmission of another frame or an ACK sent by the GW as a response to previous frame. Let $T^{Data}_{i}$ and $T^{Ack}_i$ be the durations of a data frame and an ACK, respectively, at rate $i$. Intersection with a frame does not occur if no frames are generated in the interval $[-T^{Data}_{i}, T^{Data}_{i}]$, relative to the beginning of the considered frame. For a Poisson process of frame generation, such an event happens with probability $e^{-2 r_i T^{Data}_{i}}$. We consider that the GW cancels ACK transmission if it is receiving a data frame, so a collision can happen only if the ACK is generated in the interval $[-T^{Ack}_{i}, 0]$. The rate of ACK generation is $P^{Data}_i r_i$, so the probability to avoid collision with an ACK is $e^{-r_i P^{Data}_i T^{Ack}_{i}}$. Finally, $P^{Data}_i$ can be found from the following equation: $$P^{Data}_i = e^{-(2 T^{Data}_{i} + P^{Data}_i T^{Ack}_{i}) r_i}.$$ As for ACKs, the probability that at least one ACK arrives is calculated according to the inclusion-exclusion principle: $$P^{Ack}_i = P^{Ack1}_{i} + P^{Ack2}_i - P^{Ack1}_i P^{Ack2}_i,$$ where $P^{Ack1}_{i}$ and $P^{Ack2}_i$ are the probabilities that the first and the second ACK, respectively, is transmitted successfully, provided that data was transmitted at rate $i$. The first ACK is transmitted successfully if no data frame intersects it: $$P^{Ack1}_i = e^{-\left(\min\left(T_1, T^{Data}_{i}\right) + T^{Ack}_{i}\right) r_i}.$$ Here we take the minimum of $T^{Data}_{i}$ and $T_1$, because if a frame exceeds $T_1$, it breaks the acknowledged frame, but such an event is already taken into account by $P^{Data}_i$. The second ACK is transmitted successfully if no data frame is successful in any other channel or at any other data rate, such that its second ACK would intersect the considered one: $$P^{Ack2}_i = e^{-T^{Ack}_{0} \lambda \left(1 - \frac{p_i}{F}\right) \sum_{j = 0}^{R} P^{Data}_j p_j}.$$ Retransmissions {#retries} --------------- Consider a case, when two motes transmit frames with collision, as shown in Fig. \[fig:retransmission\]. Let 0 be the time when the frame of mote A begins, and $x$ be the offset for frame of mote B. Motes choose a channel for retransmission randomly. If they choose different channels, the collision is resolved. Otherwise, with probability $\frac{1}{F}$, they choose the same channel. In this case, let $y$ and $z$ be the times when motes A and B start their retransmission, respectively. The value of $y$ is distributed uniformly in the interval $[\tau, \tau + W]$, where $\tau$ is the frame duration $T$ plus the timeout for the ACK. The value of $z$ is distributed uniformly in the interval $[\tau + x, \tau + x + W]$. The retransmission results in a new collision, if $[z, z + T]$ intersects with $[y, y + T]$, which happens with the probability $$\begin{aligned} P_x &= \frac{\int\limits_{0}^{T} r_i e^{-r_i x} \int\limits_{0}^{W} \int\limits_{x}^{W + x} \frac{\mathbbm{1}\left(y \leq z \leq y + T\right) + \mathbbm{1}\left(z \leq y \leq z + T\right)}{W^2} dz dy dx}{\int\limits_{0}^{T} r_i e^{-r_i x} dx} =\\ &=\frac{T}{W^2} \left(2 W - \frac{3}{2} T - \frac{2}{T r_{i}^2} + \frac{1}{r_i \tanh(\frac{r_i T}{2})}\right),\end{aligned}$$ where $\mathbbm{1}(condition)$ is the indicator function which equals 1 if $condition$ is true and 0 otherwise. Motes have the same probability of being the first and the second one, so the probability that there is no collision equals $$P^{Data}_{i, Re} = 1 - 2 P_x / F.$$ (0,1) – (3.3,1); (4.1,1) – (11,1); at ( 11, 0.6) [$t$]{}; at (0.5, 0.6) [$0$]{}; at (1.3, 0.6) [$x$]{}; at (2.0, 0.6) [$T$]{}; at (3.6, 1) [$...$]{}; at (4.5, 0.6) [$\tau$]{}; at (5.7, 0.6) [$y$]{}; at (7.5, 0.6) [$z$]{}; at (9.5, 0.6) [$\tau + W$]{}; (0.5, 1) rectangle (2.0, 1.5); (1.3, 1) rectangle (2.8, 1.5); (0, 2.2) rectangle (1.5, 2.8); at (3.5, 2.6) [frame of mote A]{}; (6.0, 2.2) rectangle (7.5, 2.8); at (9.5, 2.6) [frame of mote B]{}; (4.5, 0.9) – (4.5, 1.8); (5.7, 1) rectangle (7.2, 1.5); (7.5, 1) rectangle (9.0, 1.5); (9.5, 0.9) – (9.5, 1.8); The average probability of a successful transmission $P_{S}$ is $$P_{S} = P_{1} P_{S, 1} + (1 - P_{1}) P_{S, Re},$$ where $P_{S, Re}$ is the probability of a successful retransmission, calculated as in eq. , using $P^{Data}_{i, Re}$ instead of $P^{Data}_i$, and $P_{1}$ is the probability that the transmission is the first one (not a retry). $P_{1}$ is reverse to the average number of transmission attempts per a frame: $$P_1 = \left(1 + \left(1 - P_{S, 1}\right) \sum\limits_{r = 0}^{RL} \left(1 - P_{S, Re}\right)^r P^{r + 1}_{N}\right)^{-1},$$ where $P_{N} = \sum_{i = 0}^{R} p_i e^{-\frac{\lambda}{N}(T^{Data}_i + T_2 + T^{Ack}_0 + \langle T_{wait} \rangle)}$ is the probability that a new frame does not arrive during the transmission and $\langle T_{wait} \rangle = 1+W/2$ is the average interval that a mote waits before a retransmission. The packet error rate is calculated as $PER = 1 - P_{S}$. The model estimates PER correctly up to such network load, that new frames arrive at the motes as quickly as the motes drop the frames due to inability to resolve collisions after $RL$ retransmission attempts. It means that the load equals $$\lambda^* = F \left(\sum_{i = 0}^{R} p_i \left( T^{Data}_{i} + T_2 + T^{Ack}_{0} + \langle T_{wait} \rangle\right) RL \right)^{-1}.$$ Numerical Results ================= Let us use the developed model to evaluate performance of a LoRaWAN network. As in [@adelantado2017understanding], we consider a scenario, when the motes are distributed uniformly in a circular area with radius of $\SI{1}{\km}$ around the GW, and the path-loss is described by Okumura-Hata model for urban environment. We consider EU 863-880 MHz ISM band. In this case, the data rates are distributed as follows: $p_0 = 0.28, p_1 = 0.2, p_2 = 0.14, p_3 = 0.1, p_4 = 0.08, p_5 = 0.2$. We simulate a network with 1000 motes and compare the average $\mathrm{PER}$ and $\mathrm{PER}_1$ for the first transmission attempt with those obtained with the developed mathematical model. The results are shown in Fig. \[fig:per\]. Because of inefficient retransmission parameters the real $\mathrm{PER}$ is by 50% greater than $\mathrm{PER}_1$. Thus, by taking into account retransmissions, we have significantly improved the accuracy of the model. From Fig. \[fig:per\] we also see that we correctly estimate $\lambda^*$ which is the highest load when we can neglect high-order collisions and the “avalanche effect” inherent to the default retransmission parameters. Non-adaptive and small retransmission window does not allow to resolve collisions with high number of packets, and involving new motes in collisions is faster than packet dropping or collision resolution. This significantly limits the capacity of a LoRaWAN network. While the network can transmit several packets per second, because of a poor retransmission policy the PER rapidly tends to 1, when the load exceeds $10^{-1}$ packets per second. Conclusion {#sec:conclusion} ========== In the paper, we develop the first accurate mathematical model of acknowledged uplink transmissions in LoRaWAN networks with class A devices. We have shown that leaving out of consideration retransmission process significantly overestimates efficiency of a LoRaWAN network. In contrast, our model takes into account peculiarities of the retransmission process and correctly estimates packet error rate when the load is lower than some threshold $\lambda^*$, which is found in the paper. However the area with the higher loads is not interesting from a practical point of view. Indeed, after the load exceeds the described threshold, PER rapidly grows to 1 because retransmissions form an “avalanche”. Thus in this area LoRaWAN cannot provide reliable communications. [^1]: The reported study was partially supported by RFBR, research project No. 15-37-70004 mol\_a\_mos.

ArXiv

--- abstract: 'We present a combined x-ray diffraction and infrared spectroscopy study on the phase behavior and molecular dynamics of n-hexadecanol in its bulk state and confined in an array of aligned nanochannels of 8 nm diameter in mesoporous silicon. Under confinement the transition temperatures between the liquid, the rotator R$_{II}$ and the crystalline C phase are lowered by approximately 20 K. While bulk n-hexadecanol exhibits at low temperatures a polycrystalline mixture of orthorhombic $\beta$- and monoclinic $\gamma$-forms, geometrical confinement favors the more simple $\beta$-form: only crystallites are formed, where the chain axis are parallel to the layer normal. However, the $\gamma$-form, in which the chain axis are tilted with respect to the layer normal, is entirely suppressed. The $\beta$-crystallites form bi-layers, that are not randomly orientated in the pores. The molecules are arranged with their long axis perpendicular to the long channel axis. With regard to the molecular dynamics, we were able to show that confinement does not affect the inner-molecular dynamics of the CH$_2$ scissor vibration and to evaluate the inter-molecular force constants in the C phase.' author: - | R. Berwanger$^1$, A. Henschel$^2$, K. Knorr$^2$, P. Huber$^2$, and R. Pelster$^1$\ Universit[ä]{}t des Saarlandes, $^1$FR 7.2 Experimentalphysik & $^2$FR 7.3 Technische Physik,\ 66041 Saarbr[ü]{}cken, Germany title: | Phase transitions and molecular dynamics of n-hexadecanol\ confined in silicon nanochannels --- Introduction ============ The physical properties of condensed matter spatially confined in pores or channels of a few nanometer in diameter can differ markedly from the behavior in the bulk state. In particular, phase transitions can be entirely suppressed or significantly altered in comparison to their bulk counterparts [@Gelb1999; @AlbaSim2006; @Christenson2001; @Knorr2008]. Also the dynamics of condensed matter confined in mesopores, most prominently in the vicinity of glass transitions [@Koppensteiner2008; @Scheidler2000; @Kremer1999; @Jackson1991; @Barut98; @Pelster99prb; @Daoukaki98prb; @Pissis98; @Schranz2007; @Frick2003], can be affected markedly. Intimately related to these changes in the phase transition phenomenology the architectural principles of molecular solids can substantially differ in the spatially confined state from the bulk state. This depends, however, sensitively on the complexity of the building blocks. For simple van-der-Waals systems, such as Ar and N$_2$, a remarkable robustness of the bulk structures has been found for the solid state under confinement [@Huber1998; @Wallacher2001; @Knorr2003]. By contrast, the structural properties of pore fillings built out of more complex building blocks, such as linear hydrocarbons [@Huber2006; @Henschel2007; @Montenegro2003; @Xie2008; @Valliulin2006] or liquid crystals [@Crawford1996; @Kityk2008] are very susceptible to confinement on the meso- and nanoscale. For example, a quenching of the lamellar ordering of molecular crystals of n-alkanes has been observed in tortuous silica mesopores of Vycor [@Huber2004]. However, in tubular channels of mesoporous silicon this building principle of hydrocarbon molecular crystals survives, albeit a peculiar texture has been observed for the pore confined solids [@Henschel2007]: The long axes of the molecules and thus the stacking direction of the lamellae are oriented perpendicular to the long axis of the pores. Here we present an experimental study on a medium-length, linear alcohol C$_{16}$H$_{33}$OH, a representative of the 1-alcohol series, imbibed in mesoporous silicon. We explore the phase behavior of the confined alcohol by a combination of x-ray diffraction and infrared spectroscopy measurements. As we shall demonstrate, we profit in those experiments both from the parallel alignment of the silicon channels and from the transparency of the silicon host in the infrared region. Experimental ============ The porous silicon samples used in this study were prepared by electrochemical etching of a heavily p-doped (100) silicon wafer [^1] with a current density of 13 $\frac{mA}{cm^2}$ in a solution composed of HF, ethanol and H$_{2}$O (1:3:1 per volume) [@Lehmann1991; @Zhang2000; @Cullis1997]. These conditions led to a parallel arrangement of non-interconnected channels oriented with their long axes along the $<$100$>$ crystallographic direction of silicon, which coincides with the normal of the wafer surface. After the porous layer had reached the desired thickness of 70 microns, the anodization current was increased by a factor of ten with the result that the porous layer was released from the bulk wafer underneath. Using nitrogen sorption isotherms at $T=77$ K, we determined a porosity of 60% and a mean channel diameter of 8 nm. The single crystalline character of the matrix was checked by x-ray diffraction. Transmission electron micrographs of channel cross sections indicate polygonal, rough channel perimeters rather than circular, smooth circumferences [@Gruener2008]. The matrix both for the infrared spectroscopy and the x-ray measurements were filled completely via capillary action (spontaneous imbibition) with liquefied C$_{16}$H$_{33}$OH [@Huber2007]. Bulk excess material at the surface was removed by paper tissues.\ Infrared spectra in a range of wavenumbers $\overline{\nu}$ from 4000 to 800 cm$^{-1}$ with a resolution of 1 cm$^{-1}$ were measured with a Fourier Transform Spectrometer (FTIR Perkin Elmer System 2000). This range corresponds to frequencies from $3\cdot10^{13}$ Hz to $1.2\cdot10^{14}$ Hz (wavelengths from 10 $\mu$m to 2.5 $\mu$m). For both the bulk material and the filled porous samples the same sample holder was used, i. e. a copper cell with two transparent KBr windows. In the confinement experiments the long channel axes were oriented parallel to the beam axis, i. e. perpendicular to the electric field vector. The sample holder was placed into a cryostat (a closed cycle refrigerator CTI cryogenics, Model 22) allowing us to vary the temperature from 50 to 340 K. The temperature was controlled with a LakeShore 340 temperature controller with a precision of $\pm 0.25$ K. All IR-spectra that we show in the following were measured during cooling [**(typical cooling rates were of the order of 0.5 K/min)**]{}. Heating scans show the same behavior except for the transition temperatures, which are some degrees higher (see below).\ For the x-ray measurements the sample was mounted on a frame in a sample cell consisting of a Peltier cooled base plate and a Be cap. The cell was filled with He gas for better thermal contact. The Be cap sits in a vacuum chamber, the outer jacket of which has Mylar windows allowing the passage of the x-rays over a wide range of scattering angles $\theta$ within the scattering plane (see Fig. \[realRaum1\]). But the set-up allowed practically no tilt with respect to the scattering plane. The temperature was controlled by a LakeShore 330 over an accessible range from 245 K up to 370 K. The measurements were carried out on a two-circle x-ray diffractometer with graphite monochromatized CuK$_{\rm \alpha}$ radiation emanating from a rotating anode. The porous sheet was mounted perpendicular to the scattering plane. The two angles that could be varied were the detector angle $2\theta$ and the rotation angle $\omega$ about the normal of the scattering plane. The samples were studied as a function of temperature by performing several $\Phi$-scans. In this paper we concentrate on radial $2\theta$-$\omega$-Scans in reflection geometry, i.e. along q$_{\rm p}$ with $\Phi$=0°, and in transmission geometry, i.e. along q$_{\rm s}$ with $\Phi$=90° (see Fig. \[realRaum1\]). ![\[realRaum1\] ](realraum5.eps) Structure of bulk n-hexadecanol {#sec:bulkstruct} ------------------------------- n-Hexadecanol, C$_{16}$H$_{33}$OH, is an almost rod-like molecule with a length of 22  and a width of 4 . The C-atoms of the backbone are in an all-trans-configuration so that they are located in a plane [@Huber2004]. At low temperatures n-alcohols form bi-layered crystals in two possible modifications: the so-called $\gamma$-form, i. e. a monoclinic structure as sketched in Fig. \[fig:bulkstructure\_cryst\] ($C_{2h}^{6}-A2/a$ [@Metivaud2005; @Abrahamsson1960]), or the so-called $\beta$-form, i. e. an orthorhombic structure as sketched in Fig. \[fig:confstructure\_cryst\] [@Tasumi1964]. In the $\gamma$-form, the molecules include an angle of $122$° with the layer plane. Within the layers, they are close-packed in a quasi-hexagonal 2D array, described by the rectangular in-plane lattice parameters $a$ and $b$ (according to Ref. [@Abrahamsson1960] $a=7.42$  and $b=4.93$  holds, so that $a/b=1.5$). There are two different alternating orientations for the C-C-plane of the backbone leading to a herringbone structure (see Fig. \[fig:bulkstructure\_cryst\]b). The $\beta$-form exhibits an identical orientational order of the backbone, but the molecules’ axes remain perpendicular to the layers as sketched in Fig. \[fig:confstructure\_cryst\] [@Tasumi1964]. In addition, gauche- and trans-conformation of the CO-bond alternate with molecules in this phase, while they are in an all-trans configuration in the $\gamma$-form. In general, the $\gamma$-form dominates at low temperatures for the even alcohols, while the $\beta$-form is more frequent in odd n-alcohols [@Ventola2002; @Tasumi1964]. For n-hexadecanol both the orthorhombic $\beta$-form [@Tasumi1964] and the monoclinic $\gamma$-form [@Metivaud2005; @Abrahamsson1960] are reported. Depending on the preparation conditions it is possible to obtain a polycristalline mixture of the monoclinic $\gamma$- and the orthorhombic $\beta$-form [@Ventola2002]. ![\[fig:bulkstructure\_cryst\] $\gamma$-form of the crystalline low temperature phase of bulk C$_{16}$H$_{33}$OH ($T \le 310$ K). The structure is monoclinic. The left sketch shows the orientation of the molecules with respect to the layer normal $n$, the right sketch the in-plane arrangement, i. e. a projection of the backbones into the a-b-plane. Compare with the $\beta$-form sketched in Fig. \[fig:confstructure\_cryst\]](kristallinmonoklin.eps) ![\[fig:confstructure\_cryst\] $\beta$-form of the crystalline low temperature phase of C$_{16}$H$_{33}$OH. In contrast to the $\gamma$-form (see Fig. \[fig:bulkstructure\_cryst\]a), the long chain axes are not tilted but parallel to the layer normal $n$, i. e. the structure is orthorhombic. Bulk C$_{16}$H$_{33}$OH can exhibit a polycrystalline mixture of $\gamma$- and $\beta$-form (see Sec. \[sec:bulkstruct\]). Confinement into nanopores leads to the $\beta$-form (see below, Sec. \[sec:confstruct\]).](kristallinortho.eps) Upon heating, the crystalline phase undergoes a transition into a so-called Rotator-(II)-phase $R_{II}$, which is schematically depicted in Fig. \[fig:bulkstructure\_rot\] [^2]. This phase has a hexagonal in-plane arrangement with the $c$-direction perpendicular to the cell base. The hexagonal arrangement can be indexed with an orthorhombic cell with a ratio of rectangular basal lattice parameters of $a/b=\sqrt{3}$  [@Sirota1996]. On a microscopic level the change in the center of mass lattice from the low-temperature crystalline phase to the rotator phase can be attributed to the onset of hindered rotations of the molecules about their long axes between six equivalent positions (the stars in Fig. \[fig:bulkstructure\_rot\]b). Further heating above 322 K leads to the liquid state [@Sirota1996]. ![Structure of bulk n-hexadecanol in the Rotator-(II)-phase (for $310 \le T \le 322$ K), a hexagonal arrangement. The right picture shows the perfect hexagonal lattice in the a-b-plane. Confined C$_{16}$H$_{33}$OH exhibits the same structure in its rotator phase, but in a different temperature range (see below, Table \[tab:MeltingPoints\]). \[fig:bulkstructure\_rot\]](rotatorortho.eps) Results ======= Structure of confined n-hexadecanol {#sec:confstruct} ----------------------------------- We have determined structures, phase sequences and transition temperatures of n-C$_{16}$H$_{33}$OH confined in mesoporous silicon by x-ray diffractometry. The upper panel in Fig. \[realRaum\] shows diffraction patterns along q$_{\rm p}$ at selected temperatures while cooling. The appearance of a broad Bragg peak at $2 \theta \simeq 21$° indicates solidifaction. Its position is compatible with the leading hexagonal in-plane reflection of the $R_{II}$ phase. Upon further cooling a second peak at $2 \theta \simeq 24$° shows up. This change in the diffraction pattern indicates an uniaxial deformation of the hexagonal lattice. Both reflections can be mapped on a 2D rectangular mesh characteristic of an uniaxially deformed hexagonal cell. The overall resulting pattern is, however, incompatible with the monoclinic structure of the low temperature bulk crystalline phase. Additionally to the q$_{\rm p}$-scans, we performed also scans for a variety of additional orientations of the scattering vector with regard to the long axis of the channels. These patterns differ markedly, which is indicative of a strong texture of the pore confined cystallized alcohol. It is no powder in the crystallographic sense. In particular, there are strong in-plane reflections and no layering reflections for scans along q$_{\rm p}$, while the q$_{\rm s}$-scans for the same sample show at least very weak reflections characteristic of a bi-layer stacking and only very weak leading in-plane reflections (see Fig. \[realRaum\]). An analysis of the width of the layering reflections yields a coherence length of 7($\pm1.5$)nm. As discussed in more detail in Refs. [@Henschel2007] and [@Henschel2008], the overall picture which emerges from these results can be summed up as follows: the alcohol molecules form orthorhombic structures with a bilayer-stacking direction along the $c$-direction. Within the bilayers (the a-b-plane), the molecules’ backbones are untilted with regard to the stacking direction and the backbones are orientationally either fully ordered (in a herringbone fashion) or partially ordered, as known from the R$_I$ phase of n-alkanes. The superlattice reflection characteristic of the full, herringbone type orientational ordering has been searched for and could weakly be detected at low temperatures. The degree of uniaxial deformation of the hexagonal center of mass cell, quantified by the deviation of the ratio $a/b$ from its value in the hexagonal phase ($\sqrt{3}$), also indicates a full orientational ordered state (see Table I, [@Abrahamsson1960]). Thus, the diffraction data are compatible with the bulk $\beta$ modification discussed above. This conclusion is also supported by an analysis of the infrared spectroscopy data sets presented below. More importantly, the peculiar dependency of the diffraction patterns on the orientation of the q-vector with regard to the silicon host indicate that the bi-layer stacking direction is perpendicular to the long axis of the channels and, consequently, that the long axis of the molecules is oriented perpendicular to the long axis of the channels (see Fig. \[Porenschnitt\]). At first glance, this finding may appear somewhat counter-intuitive. Albeit it can be understood as resulting from the crystallization process in a strongly anistropic, capillary-like confined liquid [@Henschel2007; @Steinhart2006]. It is a well established principle in single crystal growth that in narrow capillaries the fastest growing crystallization direction prevails over other directions and propagates along the long axes of capillaries [@Palibin1933]. For layered molecular crystals of rod-like building blocks this direction is an in-plane direction, which is perpendicular to the long axis of the rods. If this direction is aligned parallel to the silicon nanochannels due to the crystallization process, it dictates a perpendicular arrangement of the molecules’ long axes with regard to the long channel axis, in agreement with the diffraction results presented here. --------- -------- ------------ -------- cryst. R$_{II}$ cryst. a \[Å\] 7.42 8.35 7.33 b \[Å\] 4.93 4.82 5.04 a/b 1.51 $\sqrt{3}$ 1.45 d \[Å\] 8.91 9.64 8.90 --------- -------- ------------ -------- : \[tab:ab\] Lattice parameters a and b of bulk and confined C$_{16}$H$_{33}$OH and the diagonal $d=\sqrt{a^2+b^2}$ of the subcell (see Fig. \[fig:lattice\]). The confined data result from our x-ray measurements and the bulk data are taken from the literature [@Abrahamsson1960]. ![\[realRaum\] ](diffraktogramme.eps) The temperature dependent diffraction study allows us to gain additional information on the relative stability of the different nanochannel confined phases. In Table \[tab:MeltingPoints\] we display the phase transition temperatures of confined C$_{16}$H$_{33}$OH as inferred from the appearance or disappaerance of characteristic Bragg peaks. There is a hysteresis between heating and cooling for both the fluid-R$_{II}$- and the R$_{II}$-C-transition (8 K and 3 K, respectively). Compared to the bulk data (see also Tab. \[tab:MeltingPoints\]), the transition temperatures of pore confined C$_{16}$H$_{33}$OH are lowered. On cooling, the lowering is of the order of $\Delta T$= 18 K for the fluid-$R_{II}$-transition and $\Delta T$= 26 K for the $R_{II}$-C-transition. This observation is analogous to phase transitions shifts reported for other pore condensates [@Christenson2001; @AlbaSim2006]. Furthermore, the temperature range of the confined $R_{II}$ phase, 14 K upon cooling and 19 K upon heating, is larger than that of the bulk material (12 K). Obviously, confinement stabilizes the orientational disordered $R_{II}$ phase, similarly as has been found for n-alkanes [@Henschel2007] and for other orientational disordered, plastic phases under spatial confinement [@Knorr2008]. fluid - R$_{II}$ R$_{II}$ - C fluid - R$_{II}$ R$_{II}$ - C -------------------- ------------------ -------------- ------------------ -------------- confined (cooling) 304 291 confined (heating) 312 293 bulk 322 310 : \[tab:MeltingPoints\] ![\[Porenschnitt\] ](streugeometrie1.eps) ![(color online). Temperature dependence of the area per molecule $A$ for C$_{16}$H$_{33}$OH confined in porous silicon.[]{data-label="fig:GitterparameterFlaeche"}](flaeche.eps) Since the pores were completely filled at higher temperatures, when hexadecanol is in its liquid state, the pore filling at low temperatures does not consist only of bi-layer crystals: the change of volume at the R$_{II}$-C phase transition is about 10% (see Fig. \[fig:GitterparameterFlaeche\]), so that there are voids and/or molecules that are not part of a bi-layer crystal. However, our experiments do not give us information about their spatial arrangement. Molecular dynamics ------------------ The dynamics of bulk-C$_{16}$H$_{33}$OH has already been investigated in IR-measurements in the past [@Metivaud2005; @Tasumi1964]. In order to show later on how the molecular dynamics is affected by spatial confinement on the nm-scale, we display some of our bulk spectra in the following. Here we focus on two characteristic vibrations, the OH-stretching and the CH$_2$-scissoring vibration.\ Figs. \[fig:OHspectrum\]a) and \[fig:OHTemp\]a) show the bulk spectra of the OH-stretching-band in the respective phases (compare with Figs. \[fig:bulkstructure\_cryst\]-\[fig:bulkstructure\_rot\]). In the liquid state (above 322 K) the peak maximum is located at about 3345 cm$^{-1}$. A decrease of temperature below 321 K yields a shift of the peak position to about 3325 cm$^{-1}$ indicating the molecular rearrangement in the $R_{II}$ phase. A further decrease of temperature below 310 K results in a splitting into two peaks at approximately 3310 cm$^{-1}$ and 3220 cm$^{-1}$. Confined C$_{16}$H$_{33}$OH shows a different behavior. There is only one peak in the whole temperature range, the position of which changes reflecting the transition between liquid phase and $R_{II}$ phase as well as between $R_{II}$ phase and C phase (see Figs. \[fig:OHspectrum\]b and \[fig:OHTemp\]b). The fact that the OH-band of bulk C$_{16}$H$_{33}$OH splits at low temperatures while no splitting is observed under confinement confirms the structural differences already observed in the x-ray experiment. For example, Tasumi et. al have studied bulk alcohols C$_n$H$_{2n+1}$OH from $n=11-37$ using infrared spectroscopy [@Tasumi1964], Ventòla et al. alcohols with $n=17-20$ [@Ventola2002]. Those alcohols showing at low temperatures (C phase) the monoclinic $\gamma$-form, such as C$_{16}$H$_{33}$OH, exhibit the splitting of the OH-band, while those that take the orthorhombic $\beta$-form show a single peak. This is due to differences in the spatial arrangement of the hydrogen bonds as well as in the distances of neighboring O-atoms: in the crystalline $\gamma$-form, where the molecule axis are tilted (see Fig. \[fig:bulkstructure\_cryst\]), the molecules show an all trans conformation, and the intra-layer O-distance ($\simeq 2.74$ A) differs from the inter-layer O-distance ($\simeq 2.69$ A). However, in the orthorhombic $\beta$-form (Fig. \[fig:confstructure\_cryst\]) trans- and gauche-molecules alternate and the intra-layer O-distance ($2.73$ A) nearly equals the inter-layer O-distance ($2.72$ A), so that the splitting is suppressed [@Tasumi1964]. Therefore, the observed OH-band splitting for bulk C$_{16}$H$_{33}$OH shows the presence of the $\gamma$-form. Either the whole bulk material exhibits the $\gamma$-form or there is a mixture of $\gamma$- and $\beta$-crystallites. The latter case is frequently observed [@Tasumi1964; @Ventola2002]: in fact, in the range of wavenumbers from 1150 cm$^{-1}$ to 950 cm$^{-1}$, where C-C stretching vibrations are visible, we see indications for a superposition of both forms (not shown). On the other hand, pore confined C$_{16}$H$_{33}$OH shows no OH-band-splitting at low temperatures. This reflects that the molecular arrangement doesn’t transform in the monoclinic $\gamma$-form but remains in an orthorhombic structure, i. e. only the $\beta$-form is present (compare Figs. \[fig:bulkstructure\_cryst\] and \[fig:confstructure\_cryst\]). This result is in agreement with the x-ray data presented above. [**Upon cooling, both the bulk and the confined hexadecanol pass from an hexagonal $R_{II}$- phase into a crystalline phase. The bulk material undergoes a stronger structural change, i. e. there is a mixture of the orthorombic $\beta$- and the monoclinic $\gamma$-form. The latter one is suppressed under confinement, so that only the $\beta$-form remains, which is quite similar to the hexagonal structure of the R$_{II}$-phase:**]{} the fact that the crystallites have to fit into nanopores of irregular shape might favor the geometrically more simple $\beta$-form [@Christenson2001; @Morishige2000](see Fig. \[Porenschnitt\]). ![(a) IR spectrum in the OH - stretching range for bulk C$_{16}$H$_{33}$OH. At lower temperatures the peak shifts to lower wavenumbers and then splits into two peaks. (b) Spectrum for confined C$_{16}$H$_{33}$OH, where no splitting is visible. \[fig:OHspectrum\]](ohspektrum.eps) ![(a) Wavenumber $\omega/2\pi c$ of the OH - stretching peak vs temperature for bulk C$_{16}$H$_{33}$OH \[compare with Fig. \[fig:OHspectrum\]a)\]. Three different phases are visible: a) above 321 K, b) from 310 to 321 K, where the peak position appears at lower wavenumbers and c) below 310 K where the peak splits up into two peaks. (b) Wavenumber of the OH - stretching peak for confined C$_{16}$H$_{33}$OH (compare with Fig. \[fig:OHspectrum\]b). The transition between the C and the R$_{II}$ phase seems to be smeared in a range around $T=291 \pm 5$ K. The R$_{II}$-liquid transition does not affect the OH-stretching. \[fig:OHTemp\]](ohpeakpos2.eps) Now let us turn towards the scissor-vibration of the $\text{CH}_2$ groups (bending mode) that will give us information about inner-molecular and inter-molecular force constants. The spectra are shown in Fig. \[fig:CHspectrum\]. At first, we want to discuss the bulk material. At high temperatures (liquid state) a superposition of two peaks at 1467 cm$^{-1}$ and 1460 cm$^{-1}$ is observed. In the intermediate temperature range ($R_{II}$ phase; see Fig. \[fig:bulkstructure\_rot\]) the intensity of the peak labeld “1” increases strongly. At low temperatures (C phase; see Fig. \[fig:bulkstructure\_cryst\]) this band splits up into two peaks. The latter transition can be clearly seen in Figs. \[fig:CHTemp\]a) and \[fig:CHInt\]a), where we display the peak positions and intensities as a function of temperature. The results are similar to those obtained for the bulk state of n-paraffines, that apart from the missing OH-group are similar in their structure, i. e. that have the same CH$_2$-backbone [@Snyder1961]. In IR-spectra only one CH$_2$-scissoring-band is observed at high temperatures, i. e.intra-molecular interactions of the CH$_2$-groups are too small to lead to a series of distinct peaks. The band splitting at low temperatures has been attributed to inter-molecular interactions (see Ref. [@Snyder1961] and text below). Qualitatively, a behavior similar to that of the bulk state is observed for confined C$_{16}$H$_{33}$OH (see Fig. \[fig:CHspectrum\]b). In the high-temperature liquid phase two overlapping peaks are visible. The stronger one, i. e. that at higher wavenumbers, undergoes an increase in intensity at about 304 K (see Fig. \[fig:CHInt\]b), indicating the transition from the liquid phase to the $R_{II}$ phase, while the secondary peak at lower wavenumbers gets weaker and finally disappears. At the second transition temperature of $T=291$ K the remaining strong peak splits (see also Fig. \[fig:CHTemp\]b). The separation is not as distinct as for bulk material. These transition temperatures, $T=304$ K and $T=291$ K (see Figs. \[fig:CHTemp\]b and \[fig:CHInt\]b), agree well with those obtained via x-ray measurements (compare with Table \[tab:MeltingPoints\]).\ ![(a) IR spectrum showing the $\text{CH}_2$ scissor-vibration for bulk C$_{16}$H$_{33}$OH at various temperatures. (b) IR spectrum showing the $\text{CH}_2$ scissor-vibration of C$_{16}$H$_{33}$OH confined in mesoporous Si at various temperatures. \[fig:CHspectrum\]](scherspektrum.eps) ![Wavenumber of the CH - scissor peak vs temperature for (a) bulk C$_{16}$H$_{33}$OH and (b) confined C$_{16}$H$_{33}$OH (the peak labels refer to Fig. \[fig:CHspectrum\] ). \[fig:CHTemp\] ](scherpeakpos.eps) ![Integrated intensity of the CH - scissor peak vs temperature for (a) bulk C$_{16}$H$_{33}$OH and (b) confined C$_{16}$H$_{33}$OH (the peak labels refer to Fig. \[fig:CHspectrum\] ). \[fig:CHInt\] ](scherint.eps) ------------------------------ ------------ ------------ ------------ ------------ liquid $R_{II}$ liquid $R_{II}$ scissor \[cm$^{-1}$\] 1467 1467 1467 1467 sym. stretch \[cm$^{-1}$\] 2854 2851 2854 2851 assym. stretch \[cm$^{-1}$\] 2927 2921 2924 2918 $f_d$ \[N/m\] 455 453 454 452 $f_\alpha$ \[N/m\] 56 $\pm$ 1 56 $\pm$ 1 57 $\pm$ 1 57 $\pm$ 1 ------------------------------ ------------ ------------ ------------ ------------ : \[tab:ergebnis1\] Wavenumbers $\overline{\nu}=\omega/(2\pi c)$ (with $\omega$ being the angular frequency and $c$ the speed of light) and resulting stretching and bending force constants in the liquid and $R_{II}$ phase of bulk and confined C$_{16}$H$_{33}$OH. $f_\alpha$ has been evaluated using both the Eq. (\[eq:alpha1\]) and the Eq. (\[eq:alpha2\]). The difference yields the specified uncertainty. $f_\alpha$ are in units of $N/m$ (see Eq. (\[eq:pot2\]) in Appendix A and Ref. [@Meister1946]). To get $f_\alpha$ in units $Nm/rad^{2}$ one has to multiply $f_\alpha$ with d$^2$, where $d=1,09 \cdot 10^{-10}$ m is the CH bond length. In the following we want to analyze the dynamics of the CH$_2$-groups [**in order to check whether it is affected by the geometric confinement, e. g. by an interaction with the pore surfaces, by the limited number of neighboring molecules (finite-size-effects) or by structural changes**]{}. In a first approximation we can assume that it is not affected by the stretching of the OH - groups. On the one hand, there is the scissor vibration, where the angle $\alpha$ between the two CH-bonds oscillates around its equilibrium value $\alpha=109.47$° (see Fig. \[fig:moleculeCH2\]). In addition, symmetric and asymmetric stretching vibrations of the CH-bonds are observable (for the values see Table \[tab:ergebnis1\]). Let $f_\alpha$ and $f_d$ denote the respective force constants. These can be calculated from the measured vibration frequencies using Eqs. (\[eq:fd\])-(\[eq:alpha2\]) (see Appendix A; the difference in calculating $f_\alpha$ via Eq. (\[eq:alpha1\]) or Eq. (\[eq:alpha2\]) is below 3.5% confirming that the inner-molecular coupling terms can be neglected). Table \[tab:ergebnis1\] shows the results for the liquid and the $R_{II}$ phase. Neither the phase transition liquid $\rightarrow$ R$_{II}$ nor geometrical confinement does markely affect the innermolecular constants. ![$\text{CH}_2$ molecules with C-H bondlength d, H-C-H angle $\alpha$ and the resulting inner force constants $f_d$ and $f_{\alpha}$\[fig:moleculeCH2\] ](ch2.eps)  In the $R_{II}$ phase the molecules rotate about their long axis, so that the primitive cell consists of only one molecule per layer (see Fig. \[fig:bulkstructure\_rot\]). Therefore, no splitting is observed. But in the C phase (below 310 K for bulk and below 291 K for confined C$_{16}$H$_{33}$OH), where the molecules are arranged in a herringbone structure, there are two molecules per layer in the primitive cell. So the symmetry of the arrangement allows a splitting of the scissoring band and obviously the molecular interactions are sufficiently strong that we are able to observe a double peak (see above, Figs. \[fig:CHspectrum\] and \[fig:CHTemp\]). The strength of interaction depends on the distances between neighboring H-atoms of adjacent chains and can be analyzed using a formalism developed by Snyder (see Ref. [@Snyder1961] and Appendix B). In Fig. \[fig:lattice\] we have sketched the orthorhombic lattice of the crystalline C$_{16}$H$_{33}$OH subcell (a view on the a-b-plane perpendicular to the molecules axis). In what follows we restrict ourselves to this $\beta$-form, that is characteristic for confined C$_{16}$H$_{33}$OH (a quantitative analysis of bulk C$_{16}$H$_{33}$OH is difficult due to the superposition of $\beta$- and $\gamma$-form). Assuming that the inner force constant $f_\alpha$ does not change at the phase transition, the intermolecular force constants $f_{3,j}$ can be evaluated from the observed splitting of the scissor band as described in Appendix B (see Eq. (\[eq:fij\])). The values needed are the lattice parameters (see Table \[tab:ab\]) and the herringbone angle $\zeta$ between the projection of the backbone and the $a$-axis (see Fig. \[fig:lattice\]). The latter one is determined via Eq. (\[eq:Winkel\]) and the measured intensities of the two CH$_2$-scissoring-peaks. For confined C$_{16}$H$_{33}$OH we have $I_a = 13.29$ and $I_b = 22.23$ yielding an angle of $\zeta = 37.7$° (see Fig. \[fig:CHInt\]b for $T=245$ K). We display the intermolecular force constants in Table \[tab:ergebnis4\]. For comparison, we also list literature values for an alkane, C$_{23}$H$_{48}$ at 90 K, which have been evaluated in the same way [@Snyder1961]. This alkane and C$_{16}$H$_{33}$OH exhibit a similar structure: The backbones of the molecules consist of the same CH$_2$-units and both take the $\beta$-form at low-temperatures. In addition, also the values of the lattice constants for C$_{23}$H$_{48}$, $a=7.45$ A and $b=4.96$ A, are close to those of C$_{16}$H$_{33}$OH (see Tab. \[tab:ab\]). Due to this structural similarity the intermolecular distances listed in Table \[tab:ergebnis4\] are similar, however, the respective force constants differ slightly by 10 to 20%. This is mainly due to the orientation of CH$_2$-groups (the projection of the backbones on the a-b-plane) characterized by the herringbone angle $\zeta$. For C$_{16}$H$_{33}$OH $\zeta=37.7$° holds, for the alkane $\zeta=42$°. This difference is probably due to the presence of polar OH-groups in C$_{16}$H$_{33}$OH that are strongly interacting and thus have an impact on the molecular orientation. The above comparison confirms once again that confined C$_{16}$H$_{33}$OH takes the $\beta$-form in contrast to the bulk material ($\gamma$- and $\beta$-form). In order to assess the validity of our analysis, we also calculate the theoretical band splitting of the CH$_2$-scissoring vibration and compare it with the measured values. Using the values from Table \[tab:ergebnis4\] as well as Eqs. (\[eq:splitting\]) and (\[eq:Gab\]), we get a theoretical value of $\Delta\overline{\nu}_{calc}=8.1$ cm$^{-1}$ for confined C$_{16}$H$_{33}$OH at $T = 245$ K. The measured band splitting is $\Delta\overline{\nu}_{meas}=7.8$ cm$^{-1}$. Therefore, the experimental data is in good agreement with the theory. Summary ======= We have studied the structure and molecular dynamics of n-hexadecanol confined in nanochannels of mesoporous silicon and of bulk n-hexadecanol in their respective phases (in the order of decreasing temperature: liquid, rotator R$_{II}$ and C). For this purpose we have performed x-ray and infrared-measurements. The transition-temperatures for confined C$_{16}$H$_{33}$OH are lower than for bulk C$_{16}$H$_{33}$OH ($\Delta T \simeq 20$ K, see Table \[tab:MeltingPoints\]). In addition, under confinement the phase transitions are smeared, probably due to a distribution of pore diameters. Geometrical confinement does not affect the innermolecular force constants of the CH$_2$-scissoring vibration (see Table \[tab:ergebnis1\]) but has an impact on the molecular arrangement. The R$_{II}$ phase of both bulk and confined hexadecanol is characterized by an orthorhombic subcell, where the chain axis are parallel to the layer normal (see Fig. \[fig:bulkstructure\_rot\]). However, in the low-temperature C phase there is a fundamental structural difference. While bulk C$_{16}$H$_{33}$OH exhibits a polycrystalline mixture of $\beta$- and $\gamma$-forms (see Figs. \[fig:bulkstructure\_cryst\] and \[fig:confstructure\_cryst\]), geometrical confinement favors a phase closely related to the $\beta$-form: only crystallites with an orthorhombic subcell are formed, where the chain axes are parallel to the bi-layer normal. However, the $\gamma$-form having a monoclinic subcell, in which the chain axis are tilted with respect to the layer normal, is suppressed. A reason for this might be the irregular shape of the nanochannels, into which the crystallites have to fit, favoring the formation of the geometrically more simple and less bulky form [@Christenson2001; @Morishige2000] (see Fig. \[Porenschnitt\]). Since only the pure $\beta$-form is present under confinement, we were able to evaluate the inter-molecular force constants of the CH$_2$-scissor vibration. 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Meister and F. F. Cleveland, Am. J. Physics **14**, 13 (1946). E. B. Wilson Jr., J. Chem. Physics **7**, 1047 (1939). H. Siebert, Anwendungen der Schwingungsspektroskopie in der anorganischen Chemie, Springer-Verlag, Berlin (1966). M. Tasumi, T. Shimanouchi, J. Chem. Phys. **43**, 1245 (1965). R. S. Stein, J. Chem. Phys. **23**, 734 (1955). D. A. Dows, J. Chem. Phys. **32**, 1342 (1960). J. deBoer, Physica **9**, 363 (1942). Appendix A {#appendix-a .unnumbered} ========== In this section we show how the innermolecular force constants of the CH$_2$-groups can be evaluated using three characteristic vibration frequencies, that are easily measured: the scissor vibration as well as the symmetric and asymmetric CH-bond stretching. For this purpose we apply the Wilson FG - matrix method [@Wilson1939]. We use the notation of Meister and Cleveland for the similar $\text{H}_2\text{O}$ molecule [@Meister1946] and perform the calculations in the same way.\ Fig. \[fig:moleculeCH2\] shows a single $\text{CH}_2$ - molecule. In the following we will neglect the influence of the neighboring molecules on this one. $d=1.09$ A is the length of the C - H bond and $\alpha = 109.47$° the angle between the two C - H bonds [@Abrahamsson1960]. This kind of molecule belongs to the $C_{2_{\nu}}$ point group. This means, there are two vibrations of type $A_1$ (symmetric stretching and bending vibration) and one vibration of type $B_2$ (asymmetric stretching vibration). The internal coordinates of this molecule are $\Delta d_1$, $\Delta d_2$ and $\Delta\alpha$. $\Delta d_1$ and $\Delta d_2$ mean changes in the bond length of the two C - H - bonds and $\Delta\alpha$ changes in the angle between the two bonds. Therefore we get three symmetry coordinates, two for $A_1$ and one for $B_2$. If we assume d being the equilibrium C - H bond length, then we obtain for the three symmetry coordinates: $$R_1=\sqrt{\frac{1}{2}}\Delta d_1+\sqrt{\frac{1}{2}}\Delta d_2\\$$ $$R_2=\Delta\alpha \cdot d$$ $$R_3=\sqrt{\frac{1}{2}}\Delta d_1-\sqrt{\frac{1}{2}}\Delta d_2$$\ Now, we have to calculate the **F** matrix, related to the potential energy, and the **G** matrix related to the kinetic energy. The potential energy can be written as $$2V=\sum f_{ik}r_ir_k \label{eq:pot1}$$ and with the internal coordinates $$\begin{split} 2V=&f_d\left[\left(\Delta d_1\right)^2+\left(\Delta d_2\right)^2\right]\\ &+f_{\alpha}(d\Delta\alpha)^2+2f_{d\alpha}\left(\Delta d_1+\Delta d_2\right)\left(d\Delta\alpha\right)\\ &+2f_{dd}\left(\Delta d_1\right)\left(\Delta d_2\right) \label{eq:pot2} \end{split}$$ Now we set $d_1=d_2=d$ and write Eq. (\[eq:pot2\]) as $$2V=\sum F_{jl}R_jR_l$$ with $F_{jl}=F_{lj}$. In matrix form, Eqs. (\[eq:pot1\]) and (\[eq:pot2\]) become $$2V = \textbf{r}'\textbf{fr} \label{eq:vmatrix1}$$ and $$2V=\textbf{R}'\textbf{FR} \label{eq:vmatrix2}$$ **r**$'$ and **R**$'$ are the transposes of **r** and **R**. With Eqs. (\[eq:vmatrix1\]) + (\[eq:vmatrix2\]) $$\textbf{r}'\textbf{fr}=\textbf{R}'\textbf{FR}$$ The $R_i$’s are linear combinations of the $r_i$’s $$\begin{split} R_i&=\sum_{k}U_{ik}r_k\\ \textbf{R}&=\textbf{Ur} \end{split}$$ Since the $R_i$’s are orthogonal and normalized, then $\textbf{U}^{-1}=\textbf{U}'$ and $$\begin{aligned} \textbf{r}=\textbf{U}'\textbf{R}\\ \textbf{r}'=(\textbf{U}'\textbf{R})'=\textbf{R}'\textbf{U}\end{aligned}$$ This means with Eqn. (10) $$\begin{aligned} \textbf{R}'(\textbf{UfU}')\textbf{R}=\textbf{R}'\textbf{FR}\\ \textbf{F}=\textbf{UfU}'\end{aligned}$$ The **F** matrix is [c|ccc]{} & d\_1 & d\_2 &\ d\_1 & f\_d & f\_[dd]{} & df\_[d]{}\ d\_2 & f\_[dd]{} & f\_d & df\_[d]{}\ & df\_[d]{} & df\_[d]{} & d\^2f\_ The **U** matrix for type $A_1$ is\ [c|ccc]{} A\_1 & d\_1 & d\_2 &\ R\_1 & & & 0\ R\_2 & 0 & 0 & 1 and for $B_2$ [c|ccc]{} B\_2 & d\_1 & d\_2 &\ R\_3 & & - & 0 So, for the type $A_1$ the **F** matrix is\ $$\textbf{F}_{A_1} = \textbf{UfU}'= \left(\begin{array}{cc} F_{11} & F_{12}\\ F_{21} & F_{22} \end{array}\right) = \left( \begin{array}{cc} f_d + f_{dd} & \sqrt{2}df_{\alpha}\\ \sqrt{2}df_{\alpha} & d^2f_{\alpha} \end{array}\right)$$ and for the $B_2$ type $$\textbf{F}_{B_2}=(F_{33})=(f_d-f_{dd})$$ The exact derivation of the $\textbf{G}$ matrix shouldn’t be shown here. It can be gleaned by Meister and Cleveland [@Meister1946]. Only the most important steps shall be explained here.\ If only non-degenerate vibrations are present, the elements of the kinetic energy matrix can be written as $$G_{jl}=\sum_{p}\mu_pg_p\textbf{S}_j^{(t)}\textbf{S}_l^{(t)}$$ where j and l refer to symmetry coordinates used in determining the **S** vector, $p$ refer to a set of equivalent atoms, a typical one of the set being t. $\mu_p$ is the reciprocal of the mass of the typical atom $t_p$ and $g_p$ is the number of equivalent atoms in the $p$th set. The **S** vector is given by $$\textbf{S}_j^{(t)}=\sum_{k}U_{jk}s_{kt}$$ where $j$, $U_jk$ and $\sum_k$ have the same meaning as above. $s_{kt}$ can be expressed in terms of unit vectors along the chemical bonds and depends on the changes in the bond length or the angle between the bonds. So, the **G** matrix for the $A_1$ vibration type has the form $$\begin{split} \textbf{G}_{A_1}&= \left(\begin{array}{cc} G_{11} & G_{12}\\ G_{21} & G_{22} \end{array}\right)\\ &=\left(\begin{array}{cc} \mu_H+\mu_C(1+\cos\alpha) & -\frac{\mu_C\sqrt(2)\sin\alpha}{d}\\ -\frac{\mu_C\sqrt(2)\sin\alpha}{d} & \frac{2\mu_H+\mu_C(1-\cos\alpha)}{d} \end{array}\right) \end{split}$$ and for the $B_2$ vibration type $$\textbf{G}_{B_2} (G_{33})=(\mu_H+\mu_C(1-\cos\alpha))$$ To determine the frequencies, one has to solve the equation $$\left|\textbf{GF}-\lambda\textbf{E}\right|=0 \qquad$$ where $\lambda = \omega^2= (\overline{\nu} 2 \pi c)^2$ denotes the square of the angular frequency. For the $A_1$ type one gets the equation $$\begin{split} \lambda^2&-\lambda(F_{11}G_{11}+2F_{12}G_{12}+F_{22}G_{22})\\ &+\begin{array}{|cc|} F_{11}&F_{12}\\ F_{21}&F_{22} \end{array} \cdot \begin{array}{|cc|} G_{11}&G_{12}\\ G_{21}&G_{22} \end{array}=0 \label{eq:loesung1} \end{split}$$ and for the $B_2$ type $$\lambda_3-F_{33}G_{33}=0 \label{eq:loesung2}$$ Eq. (\[eq:loesung1\]) can be separated with the Vieta expression [@Siebert1966]. Inserting the terms for the $F_{ij}$ and $G_{ij}$, we obtain $$\begin{aligned} \begin{aligned} \lambda_1+\lambda_2 & = (f_d+f_{dd})[\mu_C(1+\cos\alpha)+\mu_H]\\ & +2f_{\alpha}[\mu_C(1-\cos\alpha)+\mu_H]-4f_{d\alpha}\mu_C\sin\alpha \label{l1+l2} \\ \lambda_1\cdot\lambda_2 &= [(f_d+f_{dd})f_{\alpha}-2f_{d\alpha}^2]2\mu_H(2\mu_C+\mu_H) \end{aligned} \label{lamb1lamb2}\end{aligned}$$ For Eq. (\[eq:loesung2\]) one obtains $$\lambda_3=(f_d-f_{dd})[\mu_C(1-\cos\alpha)+\mu_H] \label{lamb3}$$ Neglecting the coupling constants $f_{dd}$ and $f_{d\alpha}$ allows to evaluate the innermolecular force constants using the measured wave numbers, $\overline{\nu}_{d,sym}=\sqrt{\lambda_1}/(2\pi c)$, $\overline{\nu}_\alpha=\sqrt{\lambda_2}/(2\pi c)$ and $\overline{\nu}_{d,asym}=\sqrt{\lambda_3}/(2\pi c)$. Then Eq. (\[lamb3\]) yields $$f_d= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d, asym}^2}{\mu_C(1-\cos\alpha)+\mu_H}\label{eq:fd}$$ Inserting this result into Eq. (\[lamb1lamb2\]) yields $$\label{eq:alpha1} f_{\alpha}= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d,sym}^2\cdot \overline{\nu}_\alpha^2} {\overline{\nu}_{d,asym}^2} \cdot \frac{\mu_C(1-\cos\alpha)+\mu_H}{2\mu_H(2\mu_C+\mu_H)}$$ There is a second possibility to evaluate $f_\alpha$, i. e. by inserting Eq. (\[eq:fd\]) into Eq. (\[l1+l2\]). This yields $$f_{\alpha}= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d,sym}^2 + \overline{\nu}_\alpha^2- \overline{\nu}_{d,asym}^2 \cdot \frac{\mu_C(1+\cos\alpha)+\mu_H}{\mu_C(1-\cos\alpha)+\mu_H}}{2(\mu_C(1-\cos\alpha)+\mu_H)} \label{eq:alpha2}$$ Taking the measured wavenumbers listed in Table \[tab:ergebnis1\] and the average angle between the CH-bonds, $\alpha=109.4$°, as well as the masses of the atoms, $1/\mu_C=12u$ and $1/\mu_H=1u$ ($u=1.6606 \cdot 10^{-27}$ kg), Eqs. (\[eq:fd\])-(\[eq:alpha2\]) yield the force constants listed in Table \[tab:ergebnis1\]. The difference in calculating $f_\alpha$ via Eq. (\[eq:alpha1\]) or Eq. (\[eq:alpha2\]) is below 3.1% confirming that the inner-molecular coupling terms can be neglected. Appendix B {#appendix-b .unnumbered} ========== What follows is a summary of Snyder’s derivation of the intermolecular force constants between the CH$_2$ groups of neighboring molecules that gives rise to a splitting of the scissor band at low temperatures [@Snyder1961]. We show how this formalism can be applied to C$_{16}$H$_{33}$OH. An alternative description can be found in Ref. [@Tasumi1965]. In Fig. \[fig:lattice\] we display a hexagonal subcell of C$_{16}$H$_{33}$OH. While Stein [@Stein1955] has taken only one pair of neighboring $\text{CH}_2$ into account to calculate the splitting of rocking and scissoring bands, Snyder has shown that more pairs have to be included. When we consider the distances of H-atoms from the H-atom no. 3 (see Fig. \[fig:lattice\]), then all atoms except no. $2'$, $6'$, 5 and 6 have distances larger then 3.7 Å. The internal coordinates $\alpha_i$ are always half of the angle between the C - H bonds of a $\text{CH}_2$ molecule. Solid circles are H-atoms in the same plane, dashed circles H-atoms in a plane above or below. ![Lateral view at the long axis of the C$_{16}$H$_{33}$OH chain, x is the projection of the C - C distance on the c-axis of the crystal lattice \[fig:Kette\]](kette.eps) \ Now, we want to write the positions of these five H-atoms as a vector. Fig. \[fig:Kette\] shows the lateral view of a part of the C$_{16}$H$_{33}$OH chain. With values from Abrahamsson [@Abrahamsson1960] for $l_{CC}=1.545$ A and $\alpha_{CCC}=110.4$°, we can calculate the distance of the a-b-plane to the corresponding plane above or below with $$\nonumber x=l_{CC}\sin(\frac{\alpha_{CCC}}{2})=1.2687\,\text{\AA}$$ Assuming that the hydrogen in the central plane has the $c$ component 0, the hydrogen in the plane above has the component $c=1.2687$ Å. ![Lateral view at the long axis of the C$_{16}$H$_{33}$OH chain, d is the projection of half of the C - C distance on the a-b-plane of the subcell. \[fig:seitlich\]](seitenansicht.eps) \ The projection of the C - C bond in the a-b-plane is according to Fig. \[fig:seitlich\] $$\nonumber d=\frac{l_{CC}}{2}\cos(\frac{\alpha_{CCC}}{2})=0.4409\,\text{\AA}$$ Taking the point **0** for the lower left edge of the ab - plane the five atoms have the coordinates: $$\nonumber \begin{split} H_3&= \left(\begin{array}{c} \frac{a}{2}-d\cos(\zeta)-l\cos(\alpha_3-\zeta)\\ \frac{b}{2}+d\sin(\zeta)-l\sin(\alpha_3-\zeta)\\ 0 \end{array}\right)\\ \nonumber H_{2'}&= \left(\begin{array}{c} -d\cos(\zeta)+l\sin(\alpha_{2'}-\frac{\pi}{2}+\zeta)\\ b-d\sin(\zeta)-l\cos(\alpha_{2'}-\frac{\pi}{2}+\zeta)\\ 0 \end{array}\right)\\ \nonumber H_{5}&= \left(\begin{array}{c} d\cos(\zeta)-l\sin(\alpha_{5}-\frac{\pi}{2}+\zeta)\\ d\sin(\zeta)+l\cos(\alpha_{5}-\frac{\pi}{2}+\zeta)\\ 1.2674 \end{array}\right)\\ \nonumber H_{6}&= \left(\begin{array}{c} d\cos(\zeta)+l\cos(\alpha_6-\zeta)\\ d\sin(\zeta)-l\sin(\alpha_6-\zeta)\\ 1.2674 \end{array}\right)\\ \nonumber H_{6'}&= \left(\begin{array}{c} d\cos(\zeta)+l\cos(\alpha_{6'}-\zeta)\\ b+d\sin(\zeta)-l\cos(\alpha_{6'}-\zeta)\\ 1.2674 \end{array}\right) \end{split}$$ with $a$ and $b$ being the lattice constants of the crystalline phase. We get the distances between the atom 3 and the other ones (see Fig. \[fig:lattice\]) with $$r_{3j}=\left|H_3-H_j\right| \label{eq:distances}$$ where $j=2', 6', 5, 6$ is. The herringbone angle $\zeta$ (see the upper left corner of Fig. \[fig:lattice\]) can be determined with the relation [@Snyder1961] $$\frac{I_a}{I_b}=\tan^2\zeta \label{eq:Winkel}$$ where $I_a$ is the integrated intensity of the scissoring mode, which is polarized in the a direction (higher mode at 1473 cm$^{-1}$) and $I_b$ the one of the mode, which is polarized in the $b$ direction (lower mode at 1462 cm$^{-1}$).\ With the distances of two hydrogen atoms $H_3$ and $H_j$ (=$r_{3,j}$) \[in our case 3 denotes the central H-atom (see Fig. \[fig:lattice\]) and $j=2',5,6,6'$ the neighboring H-atoms that interact\] we obtain the intermolecular force constants $f_{3,j}$: $$\begin{aligned} f_{3,j} & = & \frac{\partial^2 V_{HH}}{\partial\alpha_3\partial\alpha_j} \nonumber \\ & = & \left(\frac{\partial^2 V_{HH}}{\partial r^2}\right)_{r_{3j}}\left(\frac{\partial r}{\partial \alpha_3}\right)\left(\frac{\partial r}{\partial \alpha_j}\right) \label{eq:fij}\end{aligned}$$ where $V_{HH}$ is the hydrogen repulsion potential introduced by Dows [@Dows1960]: $$V_{HH}=1.2\cdot 10^{-10}e^{-3.52r} \label{eq:potential}$$ with r in Å. The values of $(\partial^2 V_{HH}/\partial r^2)_{r_{ij}}$ are obtained from Eq. (\[eq:beta\]). $$\beta=\frac{\partial^2 V_{HH}}{\partial r^2}=1.486848\cdot 10^{-9}e^{-3.52r} \label{eq:beta}$$ with $\beta$ in $\frac{ergs}{\text{\AA}^2}=10^{16}\frac{dyne}{cm}=10^{13}\frac{N}{m}$. The measured intensity ratio (Eq. (\[eq:Winkel\])) allows us to calculate the distances $r_{ij}$ as well as the partial derivatives $\partial r/\partial \alpha_i$ (see Eq. (\[eq:distances\]) and above). Finally, by knowing the intermolecular force constants $f_{3,j}$ from Eq. (\[eq:beta\]) we can evaluate the band splitting of the scissoring vibration [@Snyder1961]. For the angular frequencies $$\overline{\nu}_1^2 - \overline{\nu}_2^2 = \left( \frac{1}{2 \pi c}\right)^2 \cdot \underbrace{G_a^B\cdot \left\{ 2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5} \right\}}_{\Delta\lambda^B} \label{eq:splitting}$$ holds [^3] with $$G_a^B =\frac{4}{3}Q_R^2\mu_C +Q_r^2\mu_H \qquad \label{eq:Gab}$$ Here $1/\mu_C=12u$ and $1/\mu_H=1u$ denote the masses of the atoms ($u=1.6606 \cdot 10^{-27}$ kg), $1/Q_R=1.545 \cdot 10^{-10}$ m the C-C distance and $1/Q_r=1.09 \cdot 10^{-10}$ m the C-H distance, so that $G_a^B=(0.88825/u)$ Å$^{-2}= 5.349 \cdot 10^{46}$ (N m)$^{-1}$s$^{-2}$. For the band splitting of the wavenumbers we thus obtain $$\begin{split} \Delta \overline{\nu}&=\overline{\nu}_1 - \overline{\nu}_2\\ &= \left( \frac{1}{2 \pi c}\right)^2 \cdot \frac{ G_a^B \cdot \left\{ 2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5} \right\} }{ \overline{\nu}_1 + \overline{\nu}_2} \end{split}$$ Inserting the values of Table \[tab:ergebnis4\], i. e.$(2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5})=15.784 \cdot 10^{-21}$ Nm for confined C$_{16}$H$_{33}$OH, as well as the respective wave numbers, that we take from Fig. \[fig:CHTemp\] at low temperatures (labels 1a and 1b: $\overline{\nu}_1 + \overline{\nu}_2 \simeq 2 \cdot 146700$ 1/m) we get $\Delta\overline{\nu}=810$ m $^{-1}\equiv 8.1$ cm$^{-1}$ for confined C$_{16}$H$_{33}$OH. [^1]: producer: SiMat, Landsberg, Germany; specific conductivity: $\rho=0.01-0.025$ $\Omega$cm. [^2]: For several alkanes, there also exists a Rotator-(I)-Phase $R_I$, where the molecules switch between two equal positions. [^3]: In Snyder’s general theory the force constants for the scissoring vibration are denominated as $f_a^3$ (= $f_{3,2'}$), $f_b^2$ (= $f_{3,6}+f_{3,6'}$) and $f_b^3$ (= $f_{3,5}$).

ArXiv

--- abstract: 'Many real world problems can now be effectively solved using supervised machine learning. A major roadblock is often the lack of an adequate quantity of labeled data for training. A possible solution is to assign the task of labeling data to a crowd, and then infer the true label using aggregation methods. A well-known approach for aggregation is the Dawid-Skene (DS) algorithm, which is based on the principle of Expectation-Maximization (EM). We propose a new simple, yet effective, EM-based algorithm, which can be interpreted as a ‘hard’ version of DS, that allows much faster convergence while maintaining similar accuracy in aggregation. We show the use of this algorithm as a quick and effective technique for online, real-time sentiment annotation. We also prove that our algorithm converges to the estimated labels at a linear rate. Our experiments on standard datasets show a significant speedup in time taken for aggregation - upto $\sim$8x over Dawid-Skene and $\sim$6x over other fast EM methods, at competitive accuracy performance. The code for the implementation of the algorithms can be found at <https://github.com/GoodDeeds/Fast-Dawid-Skene>.' author: - 'Vaibhav B Sinha, Sukrut Rao, Vineeth N Balasubramanian' bibliography: - 'fastdawidskene.bib' title: 'Fast Dawid-Skene: A Fast Vote Aggregation Scheme for Sentiment Classification' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10003120.10003130&lt;/concept\_id&gt; &lt;concept\_desc&gt;Human-centered computing Collaborative and social computing&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010257&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Machine learning&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010257.10010258.10010259&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Supervised learning&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010257.10010282.10010284&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Online learning settings&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10002951.10003227&lt;/concept\_id&gt; &lt;concept\_desc&gt;Information systems Information systems applications&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; Introduction ============ Supervised learning has been highly effective in solving challenging tasks in sentiment analysis over the last few years. However, the success of supervised learning for the domain in recent years has been premised on the availability of large amounts of data to effectively train models. Obtaining a large labeled dataset is time-consuming, expensive, and sometimes infeasible; and this has often been the bottleneck in translating the success of machine learning models to newer problems in the domain. An approach that has been used to solve this problem is to crowdsource the annotation of data, and then aggregate the crowdsourced labels to obtain ground truths. Online platforms such as Amazon Mechanical Turk and CrowdFlower provide a friendly interface where data can be uploaded, and workers can annotate labels in return for a small payment. With the ever-growing need for large labeled datasets and the prohibitive costs of seeking experts to label large datasets, crowdsourcing has been used as a viable option for a variety of tasks, including sentiment scoring [@CSsentimentscoring], opinion mining [@CScommodityreview], general text processing [@Snow:2008:CFG:1613715.1613751], taxonomy creation [@Bragg2013CrowdsourcingMC], or domain-specific problems, such as in the biomedical field [@DBLP:journals/corr/GuanGDH17; @Albarqouni2016AggNetDL], among many others. In recent times, there is a growing need for a fast and real-time solution for judging the sentiment of various kinds of data, such as speech, text articles, and social media posts. Given the ubiquitous use of the internet and social media today, and the wide reach of any information disseminated on these platforms, it is critical to have a efficient vetting process to ensure prevention of the usage of these platforms for anti-social and malicious activities. Sentiment data is one such parameter that could be used to identify potentially harmful content. A very useful source for identifying harmful content is other users of these internet services, that report such content to the service administrators. Often, these services are set up such that on receiving such a flag, they ask other users interacting with the same content to classify whether the content is harmful or not. Then, based on these votes, a final decision can be made, without the need for any human intervention. Some such works include: crowdsourcing the sentiment associated with words [@CSsentimenttoword], crowdsourcing sentiment scoring for online media [@CSsentimentscoring], crowdsourcing the classification of words to be used as a part of lexicon for sentiment analysis [@CSlexicon], crowdsourcing sentiment judgment for video review [@CSvideoreview], crowdsourcing for commodity review [@CScommodityreview], and crowdsourcing for the production of word level annotation for opinion mining tasks [@CSsyntacticrelatedness]. However, with millions of users creating and adding new content every second, it is necessary that this decision be quick, so as to keep up with and effectively address all flags being raised. This indicates a need for fast vote aggregation schemes that can provide results for a stream of data in real time. The use of crowdsourced annotations requires a check on the reliability of the workers and the accuracy of the annotations. While the platforms provide basic quality checks, it is still possible for workers to provide incorrect labels due to misunderstanding, ambiguity in the data, carelessness, lack of domain knowledge, or malicious intent. This can be countered by obtaining labels for the same question from a large number of annotators, and then aggregating their responses using an appropriate scheme. A simple approach is to use majority voting, where the answer which the majority of annotators choose is taken to be the true label, and is often effective. However, many other methods have been proposed that perform significantly better than majority voting, and these methods are summarized further in Section \[related\]. Despite the various recent methods proposed, one of the most popular, robust and oft-used method to date for aggregating annotations is the Dawid-Skene algorithm, proposed by [@dawid1979maximum], based on the Expectation Maximization (EM) algorithm. This method uses the M-step to compute error rates, which are the probabilities of a worker providing an incorrect class label to a question with a given true label, and the class marginals, which are the probabilities of a randomly selected question to have a particular true label. These are then used to update the proposed set of true labels in the E-step, and the process continues till the algorithm converges on a proposed set of true labels (further described in Section \[dawidskenealgo\]). In this work, we propose a new simple, yet effective, EM-based algorithm for aggregation of crowdsourced responses. Although formulated differently, the proposed algorithm can be interpreted as a ‘hard’ version of Dawid-Skene (DS) [@dawid1979maximum], similar to Classification EM [@celeux1992classification] being a hard version of the original EM. The proposed method converges upto 7.84x faster than DS, while maintaining similar accuracy. We also propose a hybrid approach, a combination of our algorithm with the Dawid-Skene algorithm, that combines the high rate of convergence of our algorithm and the better likelihood estimation of the Dawid-Skene algorithm as part of this work. Related Work {#related} ============ The Expectation-Maximization algorithm for maximizing likelihood was first formalized by [@10.2307/2984875]. Soon after, Dawid and Skene [@dawid1979maximum] proposed an EM-based algorithm for estimating maximum likelihood of observer error rates, which became very popular for crowdsourced aggregation and is still considered by many as a baseline for performance. Many researchers, to this day, have worked on analyzing and extending the Dawid-Skene methodology (henceforth, called DS), of which we summarize the more recent efforts below. The work on crowdsourced data aggregation have not been confined only for sentiment analysis or opinion mining tasks, instead most of the methods are generic and can easily used for sentiment analysis and opinion mining tasks. A new model, GLAD, was proposed in [@NIPS2009_3644], that could simultaneously infer the true label, the expertise of the worker, and the difficulty of the problem, and use this to improve on the labeling scheme. [@Raykar:2010:LC:1756006.1859894] improved upon DS by jointly learning the classifier while aggregating the crowdsourced labels. However, the efforts of [@NIPS2009_3644] were restricted to binary choice settings; and in the case of [@Raykar:2010:LC:1756006.1859894], they focused on classification performance, which is however not the focus of this work. [@ipeirotis2010quality] presented improvements over DS to recover from biases in labels provided by the crowd, such as cases where a worker always provides a higher label than the true label when labels are ordinal. More recently, [@NIPS2016_6124] analyzed and characterized the tradeoff between the cost of obtaining labels from a large group of people per data point, and the improved accuracy on doing so, as well as the differences in adaptive vs non-adaptive DS schemes. In addition to these efforts, there has also been a renewed interest in recent years to understand the rates of convergence of the Dawid-Skene method. [@minimax-optimal-convergence-rates-for-estimating-ground-truth-from-crowdsourced-labels] obtained the convergence rates of a projected EM algorithm under the homogeneous DS model, which however is a constrained version of the general DS model. [@NIPS2014_5431] proposed a two-stage algorithm which uses spectral methods to offset the limitations of DS to achieve near-optimal rate convergence. [@article] recently proposed a permutation-based generalization of the DS model, and derived optimal rates of convergence for these models. However, none of these efforts have explicitly focused on increasing the speed of convergence, or making Dawid-Skene more efficient in practice. The work in [@IWMV] is the closest in this regard, where they proposed an EM-based Iterative Weighted Majority Voting (IWMV) algorithm which experimentally leads to fast convergence. We use this method for comparison in our experiments. In addition to methods based on Dawid-Skene, other methods for vote aggregation have been developed, such as using Gaussian processes [@Rodrigues:2014:GPC:3044805.3044941] and online learning methods [@Welinder2010OnlineCR]. The scope of the problem addressed by Dawid-Skene has also been broadened, to allow cases such as when a data point may have multiple true labels [@DUAN20145723]. (In this work, we show how our method can be extended to this setting too.) For ensuring reliability of the aggregated label, a common approach is to use a large number of annotators, which may however increase the cost. To mitigate this, work has also been done to intelligently assign questions to particular annotators [@0768fc60fef84637864e13671a981243], reduce the number of labels needed for the same accuracy [@Welinder2010OnlineCR], consider the biases in annotators [@NIPS2011_4311] and so on. Recent work on vote aggregation also includes deep learning-based approaches, such as [@Albarqouni2016AggNetDL; @training-deep-neural-nets-aggregate-crowdsourced-responses; @DBLP:journals/corr/abs-1709-01779]. A survey of many earlier methods related to vote aggregation can be found in the work of [@10.1007/978-3-642-41154-0_1] and [@sheshadri2013square]. Moreover, a benchmark collection of methods and datasets for vote aggregation is defined in [@sheshadri2013square], which we use for evaluating the performance of our method. While many new methods have been developed, the DS algorithm still remains relevant as being one of the most robust techniques, and is used as a baseline for nearly every new method. Inspired by [@celeux1992classification], our work proposes a simple EM-based algorithm for vote aggregation, that provides a similar performance as Dawid-Skene but with a much faster convergence rate. We now describe our method. Proposed Algorithm {#algos} ================== We propose an Expectation-Maximization (EM) based algorithm for efficient vote aggregation. The E-step estimates the dataset annotation based on the current parameters, and the M-step estimates the parameters which maximize the likelihood of the dataset. Starting from a set of initial estimates, the algorithm alternates between the M-step and the E-step till the estimates converge. Although formulated using a different approach to the aggregation problem, we call our algorithm Fast Dawid-Skene (FDS), because of its similarity to the DS algorithm (described in Section \[dawidskenealgo\]). Preliminaries {#subsec_preliminaries} ------------- For convenience, we use the analogy of a question-answer setting to model the crowdsourcing of labels. The data shown to the crowd is viewed as a question, and the possible labels as choices of answers from the crowd worker/participant. Let the questions (data points, problems) that need to be answered be $q = \{1,2,3,\dots,Q\}$ and the annotators (participants, workers) labeling them be $a = \{1,2,3,\dots,A\}$. The task requires the participants to label each question by selecting one of the predefined set of choices (options), $c = \{1,2,3,\dots, C\}$, which has the same length across all questions. A participant is said to answer a given question when s/he chooses an option as the answer for that question. A participant need not answer all the questions, and in fact, for a large pool of questions, it is reasonable to assume that a participant might be invited to answer only a small subset of all the questions. Each question is assumed to be answered by at least one participant (ideally, more). We also assume that the choice selected by a participant for a question is independent of the choice selected by any other participant. This assumption holds for real-world applications that use contemporary crowdsourcing methods, where participants generally do not know each other, and are often physically and geographically separated, and thus do not influence each other. Besides, while answering a question, the participants have no knowledge of the choices chosen by previous participants in these settings. The Fast Dawid-Skene Algorithm {#ouralgo} ------------------------------ We now derive the proposed Fast Dawid-Skene (FDS) algorithm under the assumption that each question has only one correct choice, and that a participant can select only one choice for each question. (In Section \[discussions\], we show how our method can be extended to relax this assumption.) Our goal is to aggregate the choices of the crowd for a question and to approximate the correct choice. Consider the question $q$. Let the $K$ participants that answered this question be $\{q_1, q_2, \dots, q_K\}$. The value of $K$ may vary for different questions. Let the choices chosen by these $K$ participants for question $q$ be $\{c_{q_1}, c_{q_2}, \dots, c_{q_K}\}$, and the correct (or aggregated) answer to be estimated for the question $q$ be $Y_q$. We define the answer to the question $q$ to be the choice $c \in \{1,2,\dots,C\}$ for which $P\left(Y_{q} = c | c_{q_1}, c_{q_2}, \dots, c_{q_K}\right)$ is maximum. Using Bayes’ theorem and the independence assumption among participants’ answers, we obtain: $$\begin{aligned} \label{e1} P&(Y_{q} = c | c_{q_1},c_{q_2},\dots, c_{q_K})\nonumber \\ &= \frac{P(c_{q_1}, c_{q_2}, \dots, c_{q_K} | Y_{q} = c)P(Y_{q} = c)}{\sum\limits_{c=1}^{C} P(c_{q_1}, c_{q_2}, \dots, c_{q_K} | Y_{q} = c)P(Y_{q} = c)}\nonumber\\ &= \frac{\left(\prod\limits_{k = 1}^{K} P(c_{q_k} | Y_{q} = c)\right)P(Y_{q} = c)}{\sum\limits_{c = 1}^{C} \left(\prod\limits_{k = 1}^{K} P(c_{q_k} | Y_{q} = c)\right)P(Y_{q} = c) } \end{aligned}$$ Let $T_{qc}$ be the indicator that the answer to question $q$ is choice $c$. Using our formulation: $$\label{e2} T_{qc} = \begin{cases} 1 &c = \underset{j \in \{1,2,\dots,C\}}{\arg\max} P(Y_{q} = c | c_{q_1}, c_{q_2}, \dots, c_{q_K}) \\ 0 & \text{otherwise} \end{cases}$$ These $T_{qc}$s serve as the proposed answer sheet. To determine the correct (or aggregated) choice for a question $q$, we need the values of $P(c_{q_k} | Y_{q} = c)$ for all $k$ and $c$, which however is not known given only the choices from the crowd annotators. However, if the correct choices are known for all the questions, we can compute these parameters. Let $q_k$ be the annotator $a$. To compute the parameters, we first define the following sets: $$S_{a}^{(c)} = \left\{ i\, |\, Y_i = c \wedge a \text{ has answered question } i \right\}$$ and $$T_{c_a}^{(c)} = \left\{ i \,|\, Y_i = c \wedge a \text{ has answered } c_a \text{ on question } i \right\}$$ Then, we have: $$\label{e3} P(c_a | Y_{q} = c) = \frac{ \left| T_{c_a}^{(c)} \right|}{ \left| S_a^{(c)} \right|}$$ where $\left| \cdot \right| $ denotes the cardinality of the set. Also, $P(Y_{q} = c)$ can be defined as: $$\label{e4} P(Y_{q} = c) = \frac{\text{Number of questions having answer as }c}{\text{Total number of questions}}$$ The above quantities can be estimated if we have the correct choices, and conversely, the correct choices can be obtained using the above quantities. We hence use an Expectation-Maximization (EM) strategy, where the E-step calculates the correct answer for each question, while the M-step determines the maximum likelihood parameters using equations \[e3\] and \[e4\]. There are no pre-calculated values of parameters to begin with, and so in the first E-step, we estimate the correct choices using majority voting. We continue applying the EM steps until convergence. We use the total difference between two consecutive class marginals being under a fixed threshold as the convergence criterion. We discuss the convergence criterion in more detail in Section \[experiments\]. The proposed algorithm is summarized below in Algorithm \[fdsalgorithm\]. Crowdsourced choices of $Q$ questions by $A$ participants (annotators) from $C$ choices Proposed true choices - $T_{qc}$ Estimate $T$s using majority voting. *M-step:* Obtain the parameters, $P(c_a | Y_{q} = c)$ and $P(Y_{q} = c)$ using Equations \[e3\] and \[e4\] *E-step:* Estimate $T$s using the parameters, $P(c_a | Y_{q} = c)$ and $P(Y_{q} = c)$, and with the help of Equations \[e2\] and \[e1\]. convergence Connection to Dawid-Skene Algorithm {#dawidskenealgo} ----------------------------------- The Dawid-Skene algorithm [@dawid1979maximum] was one of the earliest EM-based methods for aggregation, and still remains popular and competitive to newer approaches. In this subsection, we briefly describe the Dawid-Skene methodology, and show the connection of our approach to this method. As defined in [@dawid1979maximum], the maximum likelihood estimators for the DS method are given by: $$\begin{aligned} \hat{\pi}_{cl}^{(a)} &= \frac{\text{number of times participant $a$ chooses $l$ when $c$ is correct}}{\text{number of questions seen by participant $a$ when $c$ is correct}} \end{aligned}$$ and $\hat{p_c}$, which is the probability that a question drawn at random has a correct label of $c$. Let $n_{ql}^{(a)}$ be the number of times participant $a$ chooses $l$ for question $q$. Let $\{T_{qc} : q = 1,2,\dots, Q\}$ be the indicator variables for question $q$. If choice $m$ is true, for question $q$, $T_{qm} = 1$ and $\forall j \ne m,\,T_{qj} = 0$. Given the assumptions made in Section \[subsec\_preliminaries\], when the true responses of all questions are available, the likelihood is given by: $$\label{e8} \prod_{q=1}^{Q} \prod_{c=1}^{C} \left\{ p_c \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}}\right\}^{T_{qc}}$$ where $n_{ql}^{(a)}$ and $T_{qc}$ are known. Using equation \[e8\], we obtain the maximum likelihood estimators as: $$\label{e9} \hat{\pi}_{cl}^{(a)} = \frac{\sum_q T_{qc} n_{ql}^{(a)}}{\sum_l \sum_q T_{qc} n_{ql}^{(a)}}$$ $$\label{e10} \hat{p}_c = \frac{\sum_q T_{qc}}{Q}$$ We then obtain using Bayes’ theorem: $$\label{e11} p(T_{qc} = 1 | \text{data}) = \frac{\prod_{a=1}^{A} \prod_{l=1}^{C} (\pi_{cl}^{(a)})^{n_{ql}^{(a)}} p_c }{ \sum_{r=1}^{C} \prod_{a=1}^{A} \prod_{l=1}^{C} (\pi_{rl}^{(a)})^{n_{ql}^{(a)}} p_r}$$ The DS algorithm is then defined by using equations \[e9\] and \[e10\] to obtain the estimates of $p$s and $\pi$s in the M-step, followed by using equation \[e11\] and the estimates of $p$s and $\pi$s to calculate the new estimates of $T$s in the E-step. These two steps are repeated until convergence (when the values don’t change over an iteration). A close examination of the DS and proposed FDS algorithms shows that our algorithm can be perceived as a ‘hard’ version of DS. The DS algorithm derives the likelihood assuming that the correct answers (which are ideally binary-valued) are known, but uses the values for $T_{qc}$ (which form a probability distribution over the choices) directly as obtained from equation \[e11\]. Instead, in our formulation, we always have $T_{qc}$ as either $0$ or $1$ after each E-step. Our method is similar to the well-known Classification EM proposed in [@celeux1992classification], which shows that a ‘hard’ version of EM significantly helps fast convergence and helps scale to large datasets [@jollois2007speed]. We show empirically in Section \[experiments\] that this subtle difference between DS and FDS ensures that changes in the answer sheet dampens down quickly, and allows our method to converge much faster than DS with comparable performance. A careful implementation for both FDS and DS provides a solution in $O(QACn)$ time under the assumption that there is only one correct choice for each question, where $n$ is the number of iterations required by the algorithm to converge. As the cost per iteration of FDS would be similar to DS by the nature of its formulation, this implies that the speedup of our algorithm is proportional to the ratio of the number of iterations required to converge by the two algorithms, which we also confirm experimentally. Theoretical Guarantees for Convergence -------------------------------------- In this subsection, we establish guarantees for convergence. We prove that if we start from an area close to a local maximum of the likelihood, we are guaranteed to converge to the maximum at a linear rate. For the analysis of our algorithm’s convergence, we first frame it in a way similar to the Classification EM algorithm as proposed by [@celeux1992classification]. Classification EM introduces an extra C-step (Classification step) after the E-step. This is the step that assigns each question a single answer, thus doing a ‘hard’ clustering of questions based on options instead of the ‘soft’ clustering by DS. To continue with the proof we will use the notation used for DS. The term $ P(c_{q_k} | Y_{q} = c)$ for FDS is replaced by $\pi_{cc_{qk}}^{q_k}$ and the term $ P(Y_q = c) $ for FDS is replaced by $p_c$. $n_{ql}^{(a)}$ used by DS would be either $1$ or $0$ for the setting considered. Having established the analogy, we restate the algorithm in CEM form (Algorithm \[cemalgorithm\]). Crowdsourced choices of $Q$ questions by $A$ participants (annotators) from $C$ choices Proposed true choices - $T_{qc}$ Estimate $T$s using majority voting. This essentially does the first E and C step. *M-step:* Obtain the parameters, $\pi$s and $p$s using Equations \[e3\] and \[e4\] *E-step:* Estimate $T$s using the parameters, $\pi$ and $p$, and with the help of Equation \[e1\]. *C-step:* Assign $T$s using the values obtained in the E-step and Equation \[e2\]. convergence We prove the convergence of the CEM algorithm similar to [@celeux1992classification]. For the proof, let us first form partitions. We form $C$ partitions out of all the questions based on their correct answer in a step. $$P_c = \{q | Y_q = c\}$$ In the CEM approach, each question can belong to only one partition. Now, we define the CML (Classification Maximum Likelihood) criterion: $$C_2(P,p,\pi) = \sum_{c=1}^{C} \sum_{q \in P_c} \log \left({ p_c f(q, \pi_c)}\right)$$ In the above equation, $\pi_c = \{\pi_{cj}^{(a)} | \forall j \in \{1\dots C\} \text{ and a } \in \{1\dots A\} \}$ and $$f(q,\pi_c) = \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}}$$ To prove convergence, we define a few more notations. Note that we begin the algorithm by first doing a majority vote. This assigns each question to a class and forms the first partition. We denote this partition as $P^0$. We then proceed to the M-step and estimate $\pi$ and $p$. Let us denote this first set of parameters by $\pi^1$ and $p^1$. The next EC step gives the next partition, $P^1$. Thus, the algorithm continues to calculate $(P^{m}, p^{m+1}, \pi^{m+1})$ from $(P^{m}, p^{m}, \pi^{m})$ in the M step. Then, in the EC step, it calculates $(P^{m+1}, p^{m+1}, \pi^{m+1})$ from $(P^{m}, p^{m+1}, \pi^{m+1})$. For the sequence $(P^{m}, p^{m}, \pi^{m})$ obtained by FDS, the value of $C_2(P^{m}, p^{m}, \pi^{m})$ increases and converges to a stationary value. Under the assumption that $p$s and $\pi$s are well defined, the sequence $(P^{m}, p^{m}, \pi^{m})$ converges to a stationary point. To prove the above theorem we prove that\ $C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m}) \, \forall m > 1$.\ Note that equations \[e3\] and \[e4\] maximize the likelihood given the values of $T$ and $n$ (as shown by [@dawid1979maximum]), i. e. $T$ is known, and so $\pi$s and $p$s obtained by the M-step maximize the likelihood. We need to show that maximizing the likelihood is the same as maximizing the CML criterion, $C_2$. In the case of hard clustering, for each $q$, only one class, $c$, can have $T_{qc}$ as $1$; all other classes will have $T_{qc}$ as 0. With this observation, we can rewrite the CML criterion as: $$\begin{aligned} C_2(P,p,\pi) &= \sum_{c=1}^{C} \sum_{q \in P_c} \log (p_c f(q, \pi_c))\\ &= \log \left\{\prod_{q=1}^{Q} \prod_{c=1}^{C} \left( p_c f(q, \pi_c) \right)^{T_{qc}} \right\}\\ &= \log \left\{ \prod_{q=1}^{Q} \prod_{c=1}^{C} \left( p_c \prod_{a=1}^{A} \prod_{l=1}^{C} \left(\pi_{cl}^{(a)}\right)^{n_{ql}^{(a)}} \right)^{T_{qc}} \right\} \end{aligned}$$ Thus, maximizing maximum likelihood is equivalent to maximizing $C_2$. So, we have that after the M step, $C_2(P^{m}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m})$.\ Now, we consider the EC step. Observe that for each question $q$, we choose the answer as the option $c'$ for which $p_c' f(q,\pi_c') \ge p_c f(q,\pi_c)$ for all $c$ (By definition of the criterion for the C-step). Thus, $\log { p_c f(q, \pi_c)}$ increases individually for each question, and so cumulatively, $C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m+1}, \pi^{m+1})$.\ Combining the two inequalities, we obtain, $$C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) \ge C_2(P^{m}, p^{m}, \pi^{m})$$ This proves that $C_2$ increases at each step. Since the number of questions are finite and so the number of partitions as well are finite; the value of $C_2$ must converge after a finite number of iterations.\ On convergence, we obtain $ C_2(P^{m+1}, p^{m+1}, \pi^{m+1}) = \\C_2(P^{m}, p^{m+1}, \pi^{m+1}) = C_2(P^{m}, p^{m}, \pi^{m})$ for some $m$. By definition of the C-step, the first equality implies that $P^{m+1} = P^{m}$. Also under the assumption that $p$s and $\pi$s are well defined, we have that $p^m = p^{m+1}$ and $\pi^{m+1} = \pi^m$. This proves the convergence to a stationary point. To prove the rate of convergence, we define $M$ to be the set of matrices $U \in \mathbb{R}^{C \times Q}$ of nonnegative values. The matrices are defined such that the summation of values in each column is 1 and the summation along each row is nonzero.\ Consider the criterion to be maximized as: $$C_2'(U,p,\pi) = \sum_{c=1}^{C} \sum_{q=1}^{Q} u_{qc} \log (p_c f(q, \pi_c))$$ With the above definitions, proposition 3 of [@celeux1992classification] guarantees a linear rate of convergence for FDS to a local maximum from a neighborhood around the maximum. Hybrid Algorithm {#hybridalgo} ---------------- While the proposed FDS method is quick and effective, by using the softer marginals, DS can obtain better likelihood values (which we found in some of our experiments too). A comparison of the likelihood values over multiple datasets (described in Section 4) is provided in Table 2. To bring the best of both DS and FDS, we propose a hybrid version, where we begin with DS, and at each step, we keep track of sum of the absolute values of the difference in class marginals ($p_c$s). When this sum falls below a certain threshold, we switch to the FDS algorithm and continue (Algorithm \[hybalgorithm\]). Our empirical studies showed that this hybrid algorithm can maintain high levels of accuracy along with faster convergence (Section \[experiments\]). We however observe that a similar likelihood to DS does not necessarily translate to better accuracy, and in fact FDS outperforms Hybrid on some datasets. Crowdsourced choices for $Q$ questions by $A$ participants given $C$ choices per question, threshold $\gamma$ Aggregated choices: $T_{qc}$ Estimate $T$s using majority voting. *M-step:* Obtain parameters, $\hat{\pi}_{cl}^{(a)}$ and $\hat{p}_c$ using equations \[e9\] and \[e10\] *E-step:* Estimate $T$s using parameters, $\hat{\pi}_{cl}^{(a)}$ and $\hat{p}_c$ using equation \[e11\]. $\sum_c | p_c^t - p_c^{t-1} | < \gamma$ EM steps of Algorithm \[fdsalgorithm\] (FDS) convergence Experimental Results {#experiments} ==================== We validated the proposed method on several publicly available datasets for vote aggregation, and the results are presented in this section. We first describe the datasets, competing methods used for comparison and the performance metrics used before presenting the results. #### Datasets: We used seven real-world datasets to compare the performance of the proposed method against other methods. These include *LabelMe* [@Russell2008; @R7807338], *SentimentPolarity (SP)* [@Pang:2005:SSE:1219840.1219855; @Rodrigues:2014:GPC:3044805.3044941], *DAiSEE* [@d2016daisee; @kamath2016crowdsourced], and four datasets from the SQUARE benchmark [@sheshadri2013square]: *Adult2* [@ipeirotis2010quality], *BM* [@DBLP:journals/corr/abs-1209-3686], *TREC2010* [@Buckley10-notebook], and *RTE* [@Snow:2008:CFG:1613715.1613751]. Many of the datasets had varying number of annotators per data point. For uniformity, we set a threshold for each dataset, and all data points with fewer annotators than the threshold were removed. In our experiments, we studied the performance of all the methods by varying the number of annotators from one till the threshold, by taking a random subset of all annotators for a data point at each step (We maintained the same random seed across the methods, and conducted multiple trials to verify the results presented herewith). Also, the *TREC2010* dataset has an ‘unknown’ class, which we removed for our experiments. Table 1 lists the size, the number of classes, and the number of annotators in each dataset. [|P[1.1cm]{}||P[0.5cm]{}|P[0.8cm]{}|P[1cm]{}||P[1.1cm]{}|P[1.2cm]{}|P[1.1cm]{}|]{} & \# qns &\# options (per qn)& Maximum \# of annotators (per qn) & Speedup of FDS over DS in Time (Iterations) & Speedup of FDS over IWMV in Time (Iterations) & Speedup of Hybrid over DS in Time (Iterations)\ Adult2 & 305 & 4 & 9 & 6.61(7.87) & 1.32(1.15) & 2.30(2.43)\ BM & 1000 & 2 & 5 & 2.69(4.51) & 1.70(1.02) & 1.49(2.03)\ TREC2010 & 3670 & 4 & 5 & 7.84(8.64) & 6.09(2.93) & 4.39(4.59)\ DAiSEE & 4628 & 4 & 10 & 6.57(7.37) & 4.40(2.04) & 4.11(4.37)\ LabelMe & 589 & 8 & 3 & 7.55(8.59) & 0.54(1.14) & 5.15(5.47)\ RTE & 800 & 2 & 10 & 3.14(4.95) & 2.63(1.24) & 1.88(2.24)\ SP & 4968 & 2 & 5 & 3.00(3.95) & 2.78(0.94) & 2.40(2.54)\ \[datasettable\] #### Baseline Methods: A total of six aggregation algorithms were used in our experiments for evaluation - Majority Voting (MV), Dawid-Skene (DS) [@dawid1979maximum], IWMV [@IWMV], GLAD [@NIPS2009_3644], proposed Fast Dawid-Skene (FDS), and the proposed hybrid algorithm. IWMV is among the fastest methods using EM for aggregation under general settings. [@IWMV] compared IWMV against other well-known aggregation methods, including [@Raykar:2010:LC:1756006.1859894], [@Karger] and [@LPI], and showed that IWMV gives an accuracy comparable to these algorithms but does so in a much lesser time. We hence compare our performance to IWMV in this work. GLAD [@NIPS2009_3644], another popular method, was proposed only for questions with two choices, and we hence use this method for comparison only on the binary label datasets in our experiments. #### Performance Metrics: For each experiment, the following metrics were observed: the accuracy of the aggregated results (against provided ground truth), time taken and number of iterations needed for empirical convergence. For DS, FDS, and Hybrid, the negative log likelihood after each iteration was also observed. For MV, only the accuracy was observed. The experiments were conducted on a 4-core system with Intel Core i5-5200U 2.20GHz processors with 8GB RAM. \[fig\_result\_graphs\] [@ccccccc@]{} Adult2 & BM & TREC2010 & DAiSEE & LabelMe & RTE & SP \ & & & & & &\ & & & & & &\ & & & & & &\ & & & & & #### Results: The results of our experiments are presented in Figure 1 and Table \[logltable\]. Table \[datasettable\] shows the speedup in time and number of iterations needed to converge of FDS over DS and IWMV and of Hybrid over DS, averaged over all observations with varying number of annotators. ---------- ---------- ---------- ---------- FDS DS Hybrid Adult2 1283.75 1153.09 1154.97 BM 2110.16 2094.76 2100.32 TREC2010 13109.26 12180.84 12346.91 DAiSEE 39968.08 36178.16 36350.61 LabelMe 1714.50 1655.94 1660.06 RTE 3741.61 3679.63 3680.32 SP 12472.00 12433.70 12440.70 ---------- ---------- ---------- ---------- \[logltable\] #### Performance Analysis of Fast Dawid-Skene: The results show that FDS gives similar accuracies when compared to DS, Hybrid, GLAD, and IWMV, and a significant improvement over MV, on most datasets except for the BM and LabelMe datasets. In LabelMe, the aggregation accuracy is not at par with DS or Hybrid but is still significantly higher than MV and comparable to IWMV. In the BM dataset, the accuracies of FDS and IWMV are slightly lower than MV but both are comparable to each other. In terms of time taken, we notice that apart from the LabelMe dataset, FDS performs much better than DS, Hybrid, IWMV and GLAD all through. In the case of LabelMe, IWMV outperforms in terms of speed but the margin is very small (around 0.1 sec). This leads us to infer that in general, FDS gives comparable accuracies to other methods while taking significantly lesser time. #### Performance Analysis of the Hybrid Method: The goal of the Hybrid algorithm is to converge to a similar likelihood as DS in much lesser time. From the experiments (especially Table 2), we see that this is indeed the case - the log likelihood of the Hybrid algorithm is close to that of DS and consistently better than FDS. This naturally leads to accuracies almost similar to those obtained by DS, as is confirmed in the results. The total time taken for convergence is much lower for Hybrid as compared to DS. Moreover, the time taken for convergence by Hybrid is consistently low and does not deviate as much as IWMV. While IWMV outperforms Hybrid with respect to time in a few datasets, the proposed Hybrid outperforms IWMV on accuracy on those datasets. These observations support Hybrid to be an algorithm which performs with accuracies similar to DS in a much lesser time consistently over datasets. #### Implementation Details: We discuss two important implementation details of the proposed methods in this section: *initialization* and *stopping conditions*. As argued in [@dawid1979maximum], a symmetric initialization of the parameters (all $P(Y_q = c)$s to be $1 / C$) corresponds to a start from a saddle point, from where the EM algorithm faces difficulty in converging. Instead, a good initialization is to start with the majority voting estimate. While performing majority voting, it could often happen that there is a tie between two or more options with the highest number of votes. In such situations, we randomly choose an option among those which received the highest votes[^1]. We maintained the same random seed for all methods which required this decision. The ideal convergence criterion would be when the answer sheet proposed by an algorithm stops changing. This condition is met within a few iterations for FDS and Hybrid, but DS does not converge using this criterion in a reasonable number of steps. For example, in case of the *DAiSEE* dataset, DS did not converge even after 100 iterations (as compared to $\le 10$ for FDS). To address this issue, we set the convergence criterion as the point when the difference in class marginals is less than $10^{-4}$. We do not include the changes in participant error rates in the final convergence criterion because we observed that its fluctuations could lead to stopping prematurely. Similarly, the criterion for switching from DS to FDS in the Hybrid algorithm is the point when the change in class marginals is less than 0.005 (which happened approximately between 45-75% of total iterations across the datasets). Online Vote Aggregation ======================= Online aggregation of crowdsourced responses is an important setting in today’s applications, where data points may be streaming in large data applications. We consider a setting in which we have access to an initial set of questions and have obtained the proposed answer key using FDS. We also have $P(Y = c)$ and $P(c_a| Y = a) \,\forall\, c, a$ at this time. When we receive a new question and the answers from multiple participants for this new question, we first estimate the answer for this question directly using majority voting. We then update the parameters using the M-step in Algorithm \[fdsalgorithm\]. After the M-step, we run the E-step only for this question to re-obtain the aggregated choice. To update the new knowledge which we have regarding the new participants, we run the M-step for one last time. We conducted experiments on the *SP* dataset[^2], and observed almost the same accuracy for online FDS as offline FDS (Table 4) for different number of annotators. Table 3 shows the results for the max number of annotators (= 5). \[onlinetable\] ---------------------------- -------- -------- -------- DS FDS Hybrid Accuracy 90.94% 90.60% 90.64% Time taken to converge (s) 4.40 3.76 4.09 \# Iterations to converge 26 4 5 ---------------------------- -------- -------- -------- \[onvsofftable\] ------------ -------- -------- -------- -------- Accuracy 2 3 4 5 FDS 85.59% 88.41% 90.02% 90.74% Online FDS 83.57% 88.06% 89.90% 90.60% ------------ -------- -------- -------- -------- Extension to Multiple Correct Options {#discussions} ===================================== The proposed FDS method can be extended to solve the aggregation problem under different settings. We describe an extension below, using the same notations as in Section \[subsec\_preliminaries\]. In real-world machine learning settings such as multi-label learning, a data point might belong to multiple classes, which would result in more than one true choice per question. For such cases, we now assume that participants are allowed to choose more than one choice for each question. Our Algorithm \[fdsalgorithm\] originally assumes that every question has exactly one correct choice. To overcome this limitation, we can make a simple modification in how we interpret questions when multiple options are correct. We assume that every (question, option) pair is a separate binary classification problem, where the label is true if the option is chosen for that question, and false otherwise. This transforms a task with $Q$ questions and $C$ options each to a task with $QC$ questions and two options each. This is valid because the correctness of an option is independent of the correctness of all other options for that question in this setting. We ran experiments using this model on the Affect Annotation Love dataset *(AffectAnnotation)* used in [@DUAN20145723] (which was specifically developed for this setting) on FDS, and compared our performance with DS and Hybrid. Our results are summarized in Table 5 (annotators=5, averaged over five subsets), showing the significantly improved results of FDS over DS. Hybrid attempts to follow DS in the likelihood estimation, and thus does not perform as well as FDS in this case. Besides, our results for FDS also performed better than the methods proposed in [@DUAN20145723], which showed a best accuracy of $\approx92\%$ on this dataset. \[multtable\] ---------------------------- -------- -------- -------- DS FDS Hybrid Accuracy 88.66% 94.14% 89.26% Time taken to converge (s) 0.44 0.057 0.14 \# Iterations to converge 29.6 2 5.8 ---------------------------- -------- -------- -------- Conclusion ========== In this paper we introduced a new EM-based method for vote aggregation in crowdsourced data settings. Our method, Fast Dawid-Skene (FDS), turns out to be a ‘hard’ version of the popular Dawid-Skene (DS) algorithm, and shows up to 7.84x speedup over DS and up to 6.09x speedup over IWMV in time taken for convergence. We also propose a hybrid variant that can switch between DS and FDS to provide the best in terms of accuracy and speed. We compared the performance of the proposed methods against other state-of-the-art EM algorithms including DS, IWMV and GLAD, and our results showed that FDS and the Hybrid approach indeed provide very fast convergence at comparable accuracies to DS, IWMV and GLAD. We proved that our algorithm converges to the estimated labels at a linear rate. We also showed how the proposed methods can be used for online vote aggregation, and extended to the setting where there are multiple correct answers, showing the generalizability of the methods. [^1]: We also tried a variant, in which the option with the highest running class marginal was used to break ties. But this variant did not perform as well as the randomized majority voting across all methods. We also ran many trials with different random seeds, and found the results to almost the same as those presented. [^2]: More results, including on other datasets, on <https://sites.google.com/view/fast-dawid-skene/>

ArXiv

--- abstract: 'We consider operators arising from regular Dirichlet forms with vanishing killing term. We give bounds for the bottom of the (essential) spectrum in terms of exponential volume growth with respect to an intrinsic metric. As special cases we discuss operators on graphs. When the volume growth is measured in the natural graph distance (which is not an intrinsic metric) we discuss the threshold for positivity of the bottom of the spectrum and finiteness of the bottom of the essential spectrum of the (unbounded) graph Laplacian. This threshold is shown to lie at cubic polynomial growth.' address: - | Mathematisches Institut\ Friedrich Schiller Universit[ä]{}t Jena\ 07743 Jena, Germany - | Mathematisches Institut\ Friedrich Schiller Universit[ä]{}t Jena\ 07743 Jena, Germany - | York College of the City University of New York\ Jamaica, NY 11451\ USA author: - Sebastian Haeseler - Matthias Keller - 'Rados[ł]{}aw K. Wojciechowski' title: Volume growth and bounds for the essential spectrum for Dirichlet forms --- Introduction and Main Results ============================= In 1981 Brooks proved that the bottom of the essential spectrum of the Laplace Beltrami operator on a complete non compact Riemannian manifold with infinite measure can be bounded by the exponential volume growth rate of the manifold [@Br]. Following this, similar results were proven in various contexts, see [@DK; @Fuj; @Hi; @Hi2; @OU; @Stu]. Very recently it was shown in [@KLW] that such a result fails to be true in the case of the (non-normalized) graph Laplacian when the volume is measured with respect to the natural graph distance. Indeed, there are graphs of cubic polynomial volume growth that have positive bottom of the spectrum and slightly more than cubic growth already allows for purely discrete spectrum. This suggests that one should look for other candidates for a metric on a graph. In this work we use the context of regular Dirichlet forms (without killing term) and the corresponding concept of intrinsic metrics, see [@Stu] and [@FLW], to prove a Brooks-type theorem. The purpose of this approach is threefold. First, we provide a set up which includes all known examples (and various others, e.g., quantum graphs) and give a unified treatment. Additionally, our estimates are slightly better than most of the previous results. Secondly, our method of proof seems to be much clearer and simpler than most of the previous works. Finally, graph Laplacians are now included and the disparity discussed above is resolved by considering suitable metrics. As an application, we can now prove that the examples found in [@KLW] for Laplacians on graphs do indeed give the borderline for positive bottom of the spectrum. In particular, for the natural graph distance the threshold for zero bottom of the essential spectrum and the discreteness of the spectrum lies at cubic growth. Let $X$ be a locally compact separable metric space and $m$ a positive Radon measure of full support. Let ${{\mathcal E}}$ be a closed, symmetric, non-negative form on the Hilbert space $L^{2}(X,m)$ of real-valued square integrable functions with domain $D$. We assume that ${{\mathcal E}}$ is a regular Dirichlet form without killing term (for background on Dirichlet forms see [@Fuk], more details are given in Section \[s:DF\]). Let $L$ be the positive self adjoint operator arising from ${{\mathcal E}}$. Define $$\begin{aligned} {{\lambda}}_{0}(L):=\inf{{\sigma}}(L)\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L):=\inf{\sigma_{\!\mathrm{ess}}}(L)\end{aligned}$$ where ${\sigma_{\!\mathrm{ess}}}(L)$ denotes the essential spectrum of $L$. We let $\rho$ be an intrinsic pseudo metric in the sense of [@FLW]. For $x_{0}\in X$ and $r\geq 0$, we define the distance ball $B_{r}=B_{r}(x_{0})=\{x\in X\mid \rho(x,x_{0})\leq r\}$. Let the *exponential volume growth* be defined as $$\begin{aligned} \mu=\liminf_{r\to\infty}\frac{1}{r}\log m(B_{r}(x_{0})).\end{aligned}$$ Note that, in contrast to previous works on manifolds [@Br], graphs [@Fuj] and strongly local forms [@Stu], we consider a $\liminf$ here, rather than a $\limsup$. If $\rho$ takes values in $[0,\infty)$, then $X=\bigcup_{r} B_{r}(x_{0})$. In this case $\mu$ does not depend on the particular choice of $x_{0}$. There is another constant first introduced in [@Stu] which we call the *minimal exponential volume growth* and which is defined as $$\begin{aligned} {\widetilde{ \mu}}=\liminf_{r\to\infty}\frac{1}{r}\inf_{x\in X}\log \frac{m(B_{r}(x))}{m(B_{1}(x))}.\end{aligned}$$ In this paper we prove the following theorem. \[t:main\] Let $L$ be the positive self adjoint operator arising from a regular Dirichlet form ${{\mathcal E}}$ without killing term and let $\rho$ be an intrinsic metric such that all distance balls are compact. Then, $$\begin{aligned} {{\lambda}}_{0}(L)\leq \frac{{\widetilde{ \mu}}^{2}}{4}.\end{aligned}$$ If additionally $m(\bigcup_{r}B_{r}(x_{0}))=\infty$ for some $x_{0}$, then $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{\mu^{2}}{4}.\end{aligned}$$ This has the following immediate corollary. The corollary has various consequences, for example, the exponential instability of the semigroup $(e^{-tL})_{t\geq0}$ on $L^{p}(X,m)$, $p\in [1,\infty]$, see [@Stu Corollary 2]. Suppose that $(X,d)$ is of subexponential growth, i.e., ${\widetilde{ \mu}}=0$ (respectively, $\mu=0$). Then, ${{\lambda}}_{0}(L)=0$ (respectively, ${{\lambda}}_{0}^{\mathrm{ess}}(L)=0$). \(a) Let us discuss Theorem \[t:main\] in the perspective of the present literature: For the Laplace Beltrami operator on a Riemannian manifolds an estimate for ${{\lambda}}_{0}^{\mathrm{ess}}$ can be found in [@Br], see also [@Hi2]. In [@Stu] the statement for ${{\lambda}}_{0}$ is proven for strongly local Dirichlet forms. For non-local operators such results were known only for normalized Laplacians on graphs, see [@DK; @Fuj; @Hi; @OU]. These operators are of a very special form, in particular, they are always bounded. For unbounded Laplacians on graphs the conclusions of the theorem do not hold if one considers volume with respect to the natural graph metric, see [@KLW]. However, by [@FLW] (see also [@GHM]), there is now a suitable notion of intrinsic metric for non-local forms. Let us stress that our result covers the results in [@Br; @DK; @Fuj; @OU; @Stu]. The results of type [@Hi; @Hi2] could certainly also be obtained with slightly more technical effort which we avoid here for clarity of presentation. \(b) Despite the fact that our result is much more general, we have a unified method of proof for the bounds on the spectrum and the essential spectrum. Moreover, for the essential spectrum, the proof is significantly simpler than the one of [@Br; @Fuj] as we use test functions that converge weakly to zero and, therefore, avoid a cut-off procedure. \(c) Indeed, we prove a slightly more general result than above for non-local forms in Section \[s:nonlocal\]. In particular, for some special cases we prove much better estimates and recover the results of [@DK; @Fuj; @OU] in Corollary \[c:normalized\] in Section \[s:graph\]. \(d) If we assume that $\rho$ takes values in $[0,\infty)$, then we can clearly replace the assumption that $m(\bigcup_{r}B_{r}(x_{0}))=\infty$ with $m(X)=\infty$. The case when $m(X) < \infty$ is notably different, see [@HKLW2] for more details. \(e) If $\inf_{x\in X}m(B_{1}(x))>0$, then one can also show that $ {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq {{\widetilde{ \mu}}^{2}}/{4}$. \(f) Our result deals exclusively with Dirichlet forms with vanishing killing term. The major challenge in the case of non vanishing killing term is to give a proper definition of volume which incorporates the killing term. We shortly discuss a strategy of how one could approach this case: We need an positive generalized harmonic function $u$, i.e., ${{\mathcal E}}(u,{{\varphi}})=0$ for all ${{\varphi}}\in D$, where $u$ is assumed to be locally in the domain of ${{\mathcal E}}$ (this space is introduced in [@FLW] as $\mathcal{D}_{\mathrm{loc}}^{*}$). Such a function exists in many settings, see e.g. [@DK; @HK; @LSV], and the result which guarantees the existence of such a function is often referred to as a Allegretto-Piepenbrink type theorem. Then, by a ground state representation, see Theorem 10.1 [@FLW], one obtains a form ${{\mathcal E}}_{u}$ with vanishing killing term such that ${{\mathcal E}}={{\mathcal E}}_{u}$ on the intersection of their domains. Now, we can apply the methods above for ${{\mathcal E}}_{u}$ to derive the result for ${{\mathcal E}}$. However, as shown in [@HK], there are examples of non-locally finite weighted graphs that do not have such a generalized harmonic function. Therefore, it would be interesting to find sufficient conditions under which the approach above can be carried out. Let us highlight one of the applications of our results for graphs. Let $\Delta $ be the graph Laplacian on $\ell^{2}(X)$ acting as $$\begin{aligned} \Delta{{\varphi}}(x)= \sum_{y \sim x}({{\varphi}}(x)-{{\varphi}}(y))\end{aligned}$$ (for more details, see Sections \[s:graph\] and \[s:graph2\]). Moreover, let $B_{r}^{d}$, for $r\geq0$, be balls with respect to the natural graph distance $d$ defined as the length of the shortest path of edges between two vertices. It has to be stressed that this metric is not an intrinsic metric for $\Delta$. However, we will show in Theorem \[t:graph2\] that, if the growth of the balls $B_{r}^{d}$ is $r^{3-{{\varepsilon}}}$ for any ${{\varepsilon}}>0$, then ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$ and if it is less than $r^{3}$, then ${{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$. We demonstrate by examples that this result is sharp, see Section \[s:graph2\]. The paper is structured as follows. In Section \[s:Preliminaries\] we recall some basic facts about Dirichlet forms and intrinsic metrics. Moreover, we give a bound on the bottom of the essential spectrum via weak null sequences and introduce the test functions. In Section \[s:proof\] we prove the crucial estimate for the strongly local and the non-local parts of the Dirichlet form and prove the main theorem. In Section \[s:applications\] we discuss the result for weighted graphs and prove the polynomial growth bound discussed above. Note added: After this work was completed we learned about the very recent preprint of Matthew Folz “Volume growth and spectrum for general graph Laplacians" which contains related material in the special case of graphs. Preliminaries {#s:Preliminaries} ============= In this section we introduce the basic notions and concepts. The first subsection is devoted to recalling the setting of Dirichlet forms. In the second subsection we prove an estimate for the bottom of the essential spectrum and in the third subsection we discuss the basic properties of the test functions that are used to prove our result. Dirichlet forms {#s:DF} --------------- In this section we recall some elementary facts about Dirichlet forms, see e.g. [@Fuk] and, for recent work on non-local forms, [@FLW]. As above let $X$ be a locally compact separable metric space and let $m$ be a positive Radon measure of full support. We consider all functions on $X$ to be real-valued, but, by complexifying the corresponding Hilbert spaces and forms, we could also consider complex-valued functions. A closed non-negative form on $L^{2}(X,m)$ consists of a dense subspace $D\subseteq L^{2}(X,m)$ and a sesqui-linear non-negative map ${{\mathcal E}}:D\times D\to{{\mathbb R}}$ such that $D$ is complete with respect to the form norm $\|\cdot\|_{{{\mathcal E}}}=\sqrt{{{\mathcal E}}(\cdot,\cdot)+\|\cdot\|^{2}}$ where $\|\cdot\|$ always denotes the $L^{2}$ norm. We write ${{\mathcal E}}(u):={{\mathcal E}}(u,u)$ for $u\in D$. A closed non-negative form $({{\mathcal E}},D)$ is called a *Dirichlet form* if for any $u\in D$ and any normal contraction $c:{{\mathbb R}}\to{{\mathbb R}}$ we have $c\circ u\in D$ and ${{\mathcal E}}(c\circ u)\leq {{\mathcal E}}(u)$. Here, $c$ is a normal contraction if $c(0)=0$ and $|c(x)-c(y)|\leq|x-y|$ for $x,y\in{{\mathbb R}}$. A Dirichlet form is called *regular* if $D\cap C_{c}(X)$ is dense both in $(D,\|\cdot\|_{{{\mathcal E}}})$ and $(C_{c}(X),\|\cdot\|_{\infty})$ where $C_{c}(X)$ is the space of continuous compactly supported functions. A function $f:X\to{{\mathbb R}}$ is said to be *quasi continuous* if for every ${{\varepsilon}}>0$ there is an open set $U\subseteq X$ with $$\begin{aligned} \mathrm{cap}(U):=\inf\{\|v\|_{{{\mathcal E}}}\mid v\in D,\, 1_{U}\leq v\}\leq {{\varepsilon}},\end{aligned}$$ such that $f\vert _{X\setminus U}$ is continuous (where $\inf\emptyset=\infty$ and $1_{U}$ is the characteristic function of $U$). For a regular Dirichlet form $({{\mathcal E}},D)$ every $u\in D$ admits a quasi continuous representative, see [@Fuk Theorem 2.1.3]. In the following we assume that when considering $u$ as a function we always choose a quasi continuous representative. There is a fundamental representation theorem for regular Dirichlet forms called the Beurling-Deny formula, see [@Fuk Theorem 3.2.1.]. It states that there is a non-negative Radon measure $k$ on $X$, a non-negative Radon measure $J$ on $X\times X\setminus d$ which is $X\times X$ without the diagonal $d:=\{(x,x)\mid x\in X\}$ and a positive semi-definite bilinear form ${{\Gamma}}^{(c)}$ on $D\times D$ with values in the signed Radon measures on $X$ which is *strongly local*, i.e., satisfies ${{\Gamma}}^{(c)}(u,v)=0$ if $u$ is constant on the support of $v$, such that $$\begin{aligned} {{\mathcal E}}(u)=\int_{X}d{{\Gamma}}^{(c)}(u)+\int_{X\times X\setminus d} (u(x)-u(y))^{2}dJ(x,y)+\int_{X}u(x)^{2}dk(x),\end{aligned}$$ where we choose a quasi continuous representative of $u$ in the second and third integral. The first term on the right hand side is called the *strongly local part* of ${{\mathcal E}}$, the second term is called the *jump part* and the third term is called the *killing term*. The measure $J$ gives rise to a Radon measure ${{\Gamma}}^{(j)}$ (where the $j$ refers to ‘jump’) which is characterized by $$\begin{aligned} \int_{K}d{{\Gamma}}^{(j)}(u)=\int_{K\times X\setminus d}(u(x)-u(y))^{2}dJ(x,y)\end{aligned}$$ for $K\subseteq X$ compact and $u\in D$. The focus of this paper is on regular Dirichlet forms ${{\mathcal E}}$ without killing term, i.e., $k\equiv0$. Thus, we denote $$\begin{aligned} {{\Gamma}}={{\Gamma}}^{(c)}+{{\Gamma}}^{(j)}.\end{aligned}$$ The space $D_{\mathrm{loc}}^{*}$ of *functions locally in the domain* of ${{\mathcal E}}$ was introduced in [@FLW] and is important for the definition of intrinsic metrics. It is defined as the set of functions $u\in L^{2}_{\mathrm{loc}}(X,m)$ such that for all open and relatively compact sets $G$ there is a function $v\in D$ such that $u$ and $v$ agree on $G$ and for all compact $K\subseteq X$ $$\begin{aligned} \int_{K\times X\setminus d}(u(x)-u(y))^{2}dJ(x,y)<\infty.\end{aligned}$$ We can extend ${{\Gamma}}^{(c)}$ and ${{\Gamma}}^{(j)}$ to $D_{\mathrm{loc}}^{*}$, see [@Fuk Remarks after the proof of Theorem 3.2.1.] and [@FLW Proposition 3.3]. For the strongly local part we have a *chain rule* (see [@Fuk Theorem 3.2.2.]) as follows: for ${{\varphi}}:{{\mathbb R}}\to{{\mathbb R}}$ continuously differentiable with bounded derivative ${{\varphi}}'$, $$\begin{aligned} {{\Gamma}}^{(c)}({{\varphi}}(u),v) = {{\varphi}}'(u){{\Gamma}}^{(c)}(u,v),\quad u,v\in D_{\mathrm{loc}}^{*}\cap L^{\infty}(X,m).\end{aligned}$$ A *pseudo metric* is a map $\rho:X\times X\to[0,\infty]$ which is symmetric, satisfies the triangle inequality and $\rho(x,x)=0$ for all $x\in X$. For $A\subseteq X$ we define the map $\rho_{A}:X\to[0,\infty]$ by $$\begin{aligned} \rho_{A}(x)=\inf_{y\in A}\rho(x,y).\end{aligned}$$ If $\rho$ is a pseudo metric and $T>0$, then $\rho\wedge T$ is a pseudo metric and we have that $(\rho\wedge T)_{A}=\rho_{A} \wedge T$ and $|\rho_{A}(x)\wedge T-\rho_{A}(y)\wedge T|\leq\rho(x,y)$. By [@FLW Definition 4.1.] a pseudo metric $\rho$ is called an *intrinsic metric* for the Dirichlet form ${{\mathcal E}}$ if there are Radon measures $m^{(c)}$ and $m^{(j)}$ with $m^{(c)}+m^{(j)}\leq m$ such that for all $A\subseteq X$ and all $T>0$ the functions $\rho_{A}\wedge T$ are in $D_{\mathrm{loc}}^{*}\cap C(X)$ and satisfy $$\begin{aligned} {{\Gamma}}^{(c)}(\rho_{A}\wedge T)\leq m^{(c)}\quad\mbox{and}\quad{{\Gamma}}^{(j)}(\rho_{A}\wedge T)\leq m^{(j)}.\end{aligned}$$ This implies that if $A\subseteq X$ is such that $\rho_{A}(x)<\infty$ for all $x\in X$, then $\rho_{A}\in D_{\mathrm{loc}}^{*}\cap C(X)$ and ${{\Gamma}}(\rho_{A})\leq m$. We assume that $\rho$ is continuous with respect to the original topology. An estimate for the bottom of the essential spectrum ---------------------------------------------------- The following Persson-type theorem seems to be standard in some settings, see [@Per; @Gri]. However, since we are not able to find a proper reference in the literature which covers our case, we include a short proof. \[p:h\] Let $h$ be a closed quadratic form on $L^{2}(X,m)$ that is bounded from below and let $H$ be the corresponding self adjoint operator. Assume that there is a normalized sequence $(f_{n})$ in $D(h)$ that converges weakly to zero. Then, $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(H)\leq\liminf_{n\to\infty} h(f_{n}).\end{aligned}$$ Without loss of generality assume that $h\geq0$ and that ${{\lambda}}_0^{\mathrm{ess}}(H)>0$. Let $0< {{\lambda}}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$. We will show that there is an $N\geq0$ such that $h(f_{n})>{{\lambda}}$ for all $n \geq N$. Let ${{\lambda}}_{1}$ be such that ${{\lambda}}<{{\lambda}}_{1}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$ and let ${{\varepsilon}}>0$ be arbitrary. Since $D(H)$ is a core for $D(h)$ there exist $g_{n}\in D(H)$ for all $n\geq0$ such that ${\left\Vert f_{n}-g_{n}\right\Vert}_{h}^{2}=h(f_{n}-g_{n})+{\left\Vert f_{n}-g_{n}\right\Vert}^{2}\leq {{\varepsilon}}$ and $(g_{n})$ converges weakly to zero as well. As ${{\lambda}}_{1}<{{\lambda}}_{0}^{\mathrm{ess}}(H)$, the spectral projection $E_{(-\infty,{{\lambda}}_{1}]}$ of $H$ and the interval $(-\infty,{{\lambda}}_{1}]$ is a finite rank operator. Therefore, as $(g_{n})$ converges weakly to zero, there is an $N\geq0$ such that ${\left\Vert E_{(-\infty,{{\lambda}}_{1}]}g_{n}\right\Vert}^{2}< {{\varepsilon}}$ for $n\geq N$. Letting $\nu_{n}$ be the spectral measure of $H$ with respect to $g_{n}$, we estimate for $n\ge N$ $$\begin{aligned} h(g_{n})\geq\int_{{{\lambda}}_{1}}^{\infty}td\nu_{n}(t)\geq {{\lambda}}_{1} \int_{{{\lambda}}_{1}}^{\infty}d\nu_{n}(t)={{\lambda}}_{1}(\|g_{n}\|^{2} -\|E_{(-\infty,{{\lambda}}_{1}]}g_{n}\|^{2})> {{\lambda}}_{1}(1-{{\varepsilon}}),\end{aligned}$$ where we used ${{\lambda}}_{1}\ge0$ as $h\geq0$. Since $h(f_{n})\geq h(g_{n})-{{\varepsilon}}$ by the choice of $g_{n}$, we conclude the asserted inequality by choosing ${{\varepsilon}}=({{\lambda}}_1-{{\lambda}})/(1+{{\lambda}}_{1})>0$. The test functions ------------------ In this section we introduce the sequence of test functions which we will use to estimate the bottom of the (essential) spectrum. If $\mu = \infty$ or ${\widetilde{ \mu}} =\infty$ the statements of our theorem become obvious, therefore, from now on, we assume that $\mu, {\widetilde{ \mu}} < \infty$. For $r\in{{\mathbb N}},x_{0}\in X,{{\alpha}}>0$, define $$\begin{aligned} f_{r,x_{0},{{\alpha}}}&:X\to[0,\infty),\quad x\mapsto \big( (e^{{{\alpha}}r}\wedge e^{{{\alpha}}(2r-\rho(x_{0},x))}) - 1 \big)\vee 0.\end{aligned}$$ Then, for fixed $r$, ${{\alpha}}$, $x_{0}$, we have $f\vert_{B_{r}}\equiv e^{{{\alpha}}r}-1$, $f\vert_{B_{2r}\setminus B_{r}}= e^{ {{\alpha}}(2r- \rho(x_{0},\cdot))}-1$ and $f\vert_{X\setminus B_{2r}}\equiv0$. Clearly, $f$ is spherically homogeneous, i.e., there exists $h:[0,\infty)\to[0,\infty)$ such that $f(x)=h(\rho(x_{0},x))$. The definition of $f$ combines ideas from [@Br], [@Fuj] and [@Stu]. Moreover, for $r\in{{\mathbb N}},x_{0}\in X,{{\alpha}}>0$, let $g_{r,x_{0},{{\alpha}}}:X\to[0,\infty)$, be given by $$\begin{aligned} g_{r,x_{0},{{\alpha}}}=(f_{r,x_{0},{{\alpha}}}+2)1_{B_{2r}}\end{aligned}$$ \[l:f\] Let ${{\alpha}}>\mu/2$, $x_{0}\in X $ and $f_{r}=f_{r,x_{0},{{\alpha}}}$ and $g_{r}=g_{r,x_{0},{{\alpha}}}$ for $r\geq0$. Then, - $f_{r},g_{r}\in L^{2}(X,m)$ for all $r\ge0$. - If $m(\bigcup_{r}B_{r})=\infty$, then $f_{r}/\|f_{r}\|$ converges weakly to $0$ as $r\to\infty$. - There is a sequence $(r_{k})$ such that $\|g_{r_{k}}\|/\|f_{r_{k}}\|\to 1$ as $k\to\infty$. If ${{\alpha}}>{\widetilde{ \mu}}/2$, then - There are a sequences $(x_{k})$ in $X$ and $(r_{k})$ such that $f_{k}=f_{r_{k},x_{k},{{\alpha}}},g_{k}=g_{r_{k},x_{k},{{\alpha}}}\in L^{2}(X,m)$ and we have that $\|g_{k}\|/\|f_{k}\|\to 1$ as $k\to\infty$. \(a) As $\mu<\infty$ it follows that $m(B_{r}(x_{0}))<\infty$ for all $r\geq0$. Therefore, $f_{r},g_{r}\in L^{2}(X,m)$ for all $r\ge0$ since $f_{r},g_{r}$ are supported in $B_{2r}$ and bounded.\ (b) Let $\psi\in L^{2}(X,m)$ with ${\left\Vert \psi\right\Vert}=1$, ${{\varepsilon}}>0$ and set ${{\varphi}}=\psi1_{\bigcup B_{r}}$. There exists $R>0$ such that ${\left\Vert {{\varphi}}1_{X\setminus B_{R}}\right\Vert}\leq {{\varepsilon}}/2$. Moreover, let $r\geq R$ be such that $m(B_{R})\leq{{\varepsilon}}^{2} m(B_{r})/4$ (this choice is possible since $m(\bigcup B_{r})=\infty$). We conclude by the Cauchy-Schwarz inequality and $\|f_{r}1_{B_{R}}\|\leq\frac{{{\varepsilon}}}{2}\|f_{r}\|$ that $$\begin{aligned} {\left\langle {{\varphi}},f_{r}\right\rangle} ={\left\langle {{\varphi}}1_{B_{R}},f_{r}\right\rangle} + {\left\langle {{\varphi}}1_{X\setminus B_{R}},f_{r}\right\rangle}\leq {\left\Vert {{\varphi}}\right\Vert}{\left\Vert f_{r}1_{B_{R}}\right\Vert}+{\left\Vert {{\varphi}}1_{X\setminus B_{R}}\right\Vert} {\left\Vert f_{r}\right\Vert}\leq {{\varepsilon}}{\left\Vert f_{r}\right\Vert}.\end{aligned}$$ As ${{\mathrm {supp}\,}}f_{r}\subseteq \bigcup_{s} B_{s}$, it follows that ${\left\langle \psi,f_{r}\right\rangle}={\left\langle {{\varphi}},f_{r}\right\rangle}$ for $r\geq0$ which proves (b).\ Before we prove (c) we show (d) and indicate how to adapt the proof to (c) afterwards. Let $0<{{\varepsilon}}<{{\alpha}}-{\widetilde{ \mu}}/2$. By the definition of ${\widetilde{ \mu}}$ there are sequences $(r_{k})$ of increasing positive numbers and $(x_{k})$ of elements in $X$ such that $$\begin{aligned} \frac{ m(B_{2r_{k}}(x_{k}))}{m(B_{1}(x_{k}))}&\leq e^{(2{\widetilde{ \mu}}+{{\varepsilon}})r_{k}},\quad k\geq0.\end{aligned}$$ We set $f_{k}=f_{r_{k},x_{k},{{\alpha}}}$, $g_{k}=g_{r_{k},x_{k},{{\alpha}}}$. As $m(B_{2r_{k}}(x_{k}))<\infty$ and the functions $f_{k},g_{k}$ are supported in $B_{2r_{k}}(x_{k})$ and bounded, they are in $L^{2}(X,m)$. By definition we have $g_{k}=g_{k}1_{B_{2r_{k}}}=(f_{k}+2)1_{B_{2r_{k}}}$, $k\geq0$. Using the inequalities $(a+b)^{2}\leq \frac{1}{(1-{{\varepsilon}})}a^{2}+\frac{1}{{{\varepsilon}}}b^{2}$ and $\|f_{k}\|^{2}\geq m({B_{r_{k}}}(x_{k}))(e^{{{\alpha}}r_{k}}-1)^{2}\geq m({B_{r_{k}}}(x_{k}))e^{2{{\alpha}}r_{k}}/c$ for some $c>0$ we get $$\begin{aligned} \frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}} &\leq\frac{(\|f_{k}\|+2\sqrt{m(B_{2r_{k}}(x_{k}))})^{2}}{\|f_{k}\|^{2}} \leq\frac{\frac{1}{(1-{{\varepsilon}})}\|f_{k}\|^{2}+\frac{4}{{{\varepsilon}}}{m({B_{2r_{k}}(x_{k})})}} {\|f_{k}\|^{2}}\\ &\leq\frac{1}{(1-{{\varepsilon}})}+\frac{4c}{{{\varepsilon}}} {\frac{{m({B_{2r_{k}}(x_{k})})}}{m({B_{r_{k}}}(x_{k}))}} e^{-2{{\alpha}}r_{k}}.\end{aligned}$$ For $r_{k}$ large enough we have $$\begin{aligned} \frac{ m(B_{r_{k}}(x_{k}))}{m(B_{1}(x_{k}))}\geq\inf_{x\in X}\frac{ m(B_{r_{k}}(x))}{m(B_{1}(x))}\geq e^{({\widetilde{ \mu}}-{{\varepsilon}})r_{k}}.\end{aligned}$$ Thus, by the choice of $(r_{k})$ and $(x_{k})$, we have ${\frac{{m({B_{2r_{k}}})}}{m({B_{r_{k}}})}} \leq e^{({\widetilde{ \mu}}+2{{\varepsilon}})r_{k}}$. As $0<{{\varepsilon}}<{{\alpha}}-{\widetilde{ \mu}}/2$ $$\begin{aligned} \frac{\|g_{{k}}\|^{2}}{\|f_{{k}}\|^{2}}\leq\frac{1}{(1-{{\varepsilon}})}+\frac{4c}{{{\varepsilon}}} e^{({\widetilde{ \mu}}+2{{\varepsilon}}-2{{\alpha}}) r_{k}}\to \frac{1}{(1-{{\varepsilon}})}\quad\mbox{as $k\to\infty$}.\end{aligned}$$ Since ${{\varepsilon}}$ can be chosen to be arbitrarily small and ${\|g_{{k}}\|}\geq{\|f_{{k}}\|}$ we deduce the statement.\ For (c) we choose $(x_{k})$ to be $x_{0}$ and follow the lines of the proof replacing ${\widetilde{ \mu}}$ by $\mu$. If $\inf_{x\in X}m(B_{1}(x))>0$, then $f_{k}/\|f_{k}\|$ of (d) also converges weakly to zero as $k\to\infty$. The following auxiliary estimates will later give us bounds for the Lipshitz constants of $f_{r,x,{{\alpha}}}$. \[l:e\] Let ${{\alpha}}>0$. For all $R\geq 0$ one has $$\begin{aligned} \frac{{\left({e}^{\alpha R} -1\right)}^{2}}{\left({e}^{2\alpha R} +1\right)} \leq\frac{{\alpha }^{2}{R}^{2}}{2}.\end{aligned}$$ Moreover, for $ R\in [0, 1] $ one has $$\begin{aligned} \frac{{\left({e}^{\alpha R} -1\right)}^{2}}{\left({e}^{2\alpha R} +1\right)} \leq\frac{{R}^{2} {\left({e}^{\alpha}-1\right)}^{2}}{\left( R^2{e}^{2\alpha}+1\right)}.\end{aligned}$$ For the first statement let $s={{\alpha}}R$ and check via a series expansion that $s\mapsto{s}^{2}\left({e}^{2s} +1\right)- 2{\left({e}^{s}-1\right)}^{2}$ is non-negative. The second statement follows by direct calculation since we have $e^{{{\alpha}}R}-1\leq R(e^{{{\alpha}}}-1)$ for $R\in[0,1]$ and ${{\alpha}}> 0$. \[l:f\_Lip\] Let $r\in{{\mathbb N}}$, $x_{0}\in X$, ${{\alpha}}>0$ and set $f:=f_{r,x_{0},{{\alpha}}}$, $g:=g_{r,x_{0},{{\alpha}}}$. Then, for all $ x,y\in X$ $$\begin{aligned} (f(x)-f(y))^{2}&\leq c({{\alpha}}) (g(x)^{2}+g(y)^2) \rho(x,y)^{2}\end{aligned}$$ where $c({{\alpha}})= \frac{{{{\alpha}}^{2}}}{2} $. If additionally $\rho(x,y)\leq 1$, then $c({{\alpha}})$ can be chosen to be $c({{\alpha}},\rho(x,y))=\frac{(e^{{{\alpha}}}-1)^{2}}{\rho(x,y)^{2}e^{2{{\alpha}}}+1}$. In particular, $f$ is Lipshitz continuous with Lipshitz constant ${{\alpha}}( e^{{{\alpha}}r}+1)$. We fix $r$, ${{\alpha}}$ and $x_{0}$ for the proof. Let $x,y\in X$ be given and let $s=\rho(x_{0},x)$ and $t=\rho(x_{0},y)$. We define $ D_{s,t}:=(f(x)- f(y))^{2}$. Moreover, we use the estimate on $F(R):=\frac{(e^{{{\alpha}}R}-1)^{2}}{e^{2{{\alpha}}R}+1}$, $R\geq0,$ by $c({{\alpha}})R^{2}$ (and by $c({{\alpha}},R)R^{2}$ for $R\leq1$) from Lemma \[l:e\]. By symmetry we may assume, without loss of generality, that $s\leq t$ so that we have six cases to check.\ Case 1: If $s\leq t\leq r$, then $D_{s,t}=0$.\ Case 2: If $s\leq r\leq t\leq 2r$, then since $t-r\leq t-s=\rho(x_{0},y)-\rho(x_{0},x)\leq \rho(x,y)$ and $g(x)=e^{{{\alpha}}r}+1$, $g(y)=e^{{{\alpha}}(2r-t)}+1$, $$\begin{aligned} D_{s,t}&=(e^{{{\alpha}}r} -e^{{{\alpha}}(2r-t)})^{2} =( e^{2{{\alpha}}r} +e^{2{{\alpha}}(2r-t)})F(t-r) \leq ( e^{2{{\alpha}}r} +e^{2{{\alpha}}(2r-t)})c({{\alpha}})(t-r)^{2}\\ &\leq c({{\alpha}})(g(x)^{2}+ g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 3: If $s\leq r\leq 2r\leq t$, then since $r\leq t-s\leq \rho(x,y)$, $g(x)=e^{{{\alpha}}r}+1$ and $g({y})=0$, $$\begin{aligned} D_{s,t}=(e^{{{\alpha}}r} -1)^{2} = (e^{2{{\alpha}}r}+1)F(r) \leq (e^{2{{\alpha}}r}+1)c({{\alpha}}) r^{2}\leq2c({{\alpha}}) (g(x)^{2}+g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 4: If $ r\leq s\leq t\leq 2r$, then since $t-s\leq \rho(x,y)$ and $g(x)=e^{{{\alpha}}(2r- s)}+1$, $g(y)=e^{{{\alpha}}(2r-t)}+1$, $$\begin{aligned} D_{s,t}&=(e^{{{\alpha}}(2r-s)} -e^{{{\alpha}}(2r-t)})^{2}= ( e^{2{{\alpha}}(2r- s)}+ e^{2{{\alpha}}(2r -t)}) F(t-s)\\ &\leq c({{\alpha}})(g(x)^{2}+ g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 5: If $ r\leq s\leq 2r\leq t$, then since $2r-s\leq t-s\leq \rho(x,y)$, $g(x)=e^{{{\alpha}}(2r- s)}+1$ and $g(y)=0$, $$\begin{aligned} D_{s,t}=(e^{{{\alpha}}(2r-s)}-1)^{2} = ( e^{2{{\alpha}}(2r-s)}+1)F(2r-s)\leq c({{\alpha}})(g(x)^{2}+g(y)^{2})\rho(x,y)^{2}.\end{aligned}$$ Case 6: If $2r\leq s\leq t$, then $D_{s,t}=0$.\ The Lipshitz bound follows since $g$ is bounded by $e^{{{\alpha}}r}+1$. \[l:f\_in\_D\] Let $({{\mathcal E}},D)$ be a regular Dirichlet form and $\rho$ an intrinsic metric. For all $r>0$, $x_{0}\in X$ and ${{\alpha}}>0$ we have $f:=f_{r,x_{0},{{\alpha}}}\in D_{\mathrm{loc}}^{*}$. Moreover, if $B_{2r}(x_{0})$ is compact, then $f\in D$. By Lemma \[l:f\_Lip\] the functions $f:=f_{r,x_{0},{{\alpha}}}$ are Lipshitz continuous for all $r>0$, $x_{0}$ and ${{\alpha}}>0$. Thus, by a Rademacher type theorem, see e.g. [@Sto Theorem 5.1] for strongly local forms or [@FLW Theorem 4.8] for general Dirichlet forms, we have $f\in D_{\mathrm{loc}}^{*}$ and ${{\Gamma}}(f)\leq m$. If $B_{2r}(x_{0})$ is compact, then the function $f$ is compactly supported which implies that $f\in D$. Proof of the main theorem {#s:proof} ========================= The strongly local estimate --------------------------- In this subsection we give an estimate which will be used to prove the theorem for the strongly local part of the Dirichlet form. For given $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ we denote $f:=f_{r,x_{0},{{\alpha}}}$ and $g:=g_{r,x_{0},{{\alpha}}}$. \[l:SL\] Let $\rho$ be an intrinsic metric for a regular strongly local Dirichlet form $\mathcal{E}$. Then, for all $r>0$, $x_{0}\in X$ and ${{\alpha}}>0$ such that $f\in D$ we have $$\begin{aligned} {{\mathcal E}}(f)\leq{{\alpha}}^{2}\int_{X}g^{2}dm^{(c)}.\end{aligned}$$ As ${{\mathcal E}}$ is strongly local, we get by the chain rule and the fact that $\rho $ is an intrinsic metric that $$\begin{aligned} {{\mathcal E}}(f)&=\int_{B_{2r}\setminus B_{r}}d{{\Gamma}}^{(c)}(f)= \int_{B_{2r}\setminus B_{r}}d{{\Gamma}}^{(c)}(e^{{{\alpha}}(2r-\rho(x_{0},\cdot))} -1)\\ &={{\alpha}}^{2}\int_{B_{2r}\setminus B_{r}} e^{2{{\alpha}}(2r-\rho(x_{0},\cdot))}d{{\Gamma}}^{(c)}(\rho(x_{0},\cdot))\\ &\leq{{\alpha}}^{2}\int_{B_{2r}\setminus B_{r}} e^{2{{\alpha}}(2r-\rho(x_{0},\cdot))}dm^{(c)}\leq {{\alpha}}^{2}\int_{X}g_{r,x_{0},{{\alpha}}}^{2}dm^{(c)}.\end{aligned}$$ The non-local estimate {#s:nonlocal} ---------------------- Next, we treat the non-local case. With applications to graphs in the next section in mind, we do not assume that the jump part is a regular Dirichlet form for now. For this subsection, let $m$ be a Radon measure on $X$ and let $J$ be a symmetric Radon measure on $X\times X\setminus d$ such that for every $m$-measurable $A\subseteq X$ the set $A\times X\setminus d$ is $J$ measurable and vice versa. Let $\rho$ be a pseudo metric on $X$ which is $J$ measurable and assume that for all measurable $A\subseteq X$ $$\begin{aligned} \label{e:adapted}\tag{$\clubsuit$} \int_{A\times X\setminus d}\rho(x,y)^{2}dJ(x,y)\leq m(A)\end{aligned}$$ which immediately implies that for all measurable functions ${{\varphi}}$ $$\begin{aligned} \int_{X\times X\setminus d}{{\varphi}}(x)^{2}\rho(x,y)^{2}dJ(x,y)\leq \int_{X} {{\varphi}}^{2} dm.\end{aligned}$$ We say that the pseudo metric $\rho$ has *jump size in* $[a,b]$, $0\leq a\leq b$, if for the set $A_{a,b}:=\{(x,y)\in X \times X \mid\rho(x,y)\in[a,b]\}\setminus d$ $$\begin{aligned} \int_{X\times X\setminus d}\rho(x,y)^{2}dJ(x,y)=\int_{A_{a,b}} \rho(x,y)^{2}dJ(x,y).\end{aligned}$$ For given $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ we denote $f:=f_{r,x_{0},{{\alpha}}}$ and $g:=g_{r,x_{0},{{\alpha}}}$. \[l:NL\]Assume that $\rho$ satisfies . For all $r\in{{\mathbb N}}$, $x_{0}\in X$ and ${{\alpha}}>0$ $$\begin{aligned} \int_{X\times X\setminus d}(f(x)-f(y))^{2}dJ(x,y)\leq\ 2c({{{\alpha}}}) \int_{X} g^2dm,\end{aligned}$$ where $c({{\alpha}})= \frac{{{\alpha}}^{2}}{2}$. If $\rho$ has jump size in $[{{\delta}},1]$ for some $0\leq{{\delta}}\leq1$, then $c({{\alpha}})$ can be chosen to be $c({{\alpha}},{{\delta}})=\frac{(e^{{{\alpha}}}-1)^{2}}{1+{{\delta}}^{2}e^{2{{\alpha}}}}$. By Lemma \[l:f\_Lip\] and since $\rho$ satisfies $$\begin{aligned} \int_{X\times X\setminus d} (f(x)-f(y))^{2}dJ(x,y)\leq {{{\alpha}}^{2}}\int_{X\times X\setminus d} g(x)^{2}\rho(x,y)^{2}dJ(x,y)\leq{{{\alpha}}^{2}} \int_{X} g^2dm.\end{aligned}$$ Let ${{\delta}}>0$. If the jump size is in $[{{\delta}},1]$, then $$\begin{aligned} \int_{X\times X\setminus d} &(f(x)-f(y))^{2}dJ(x,y)= \int_{{A_{{{\delta}},1}}} (f(x)-f(y))^{2} dJ(x,y)\\ \leq& \int_{{A_{{{\delta}},1}}} |g(x)|^{2}\frac{2(e^{{{\alpha}}}-1)^{2}}{(1+\rho(x,y)^{2}e^{2{{\alpha}}})} \rho(x,y)^{2}dJ(x,y) \\ \leq& \frac{2(e^{{{\alpha}}}-1)^{2}}{(1+{{\delta}}^{2}e^{2{{\alpha}}})} \int_{X\times X\setminus d} g(x)^{2}\rho(x,y)^{2}dJ(x,y)\\ \leq& \frac{2(e^{{{\alpha}}}-1)^{2}}{(1+{{\delta}}^{2}e^{2{{\alpha}}})} \int_{X}g^{2}dm.\end{aligned}$$ Proof of Theorem \[t:main\] --------------------------- We now have all of the ingredients to prove our main result. By [@FLW Lemma 4.7] an intrinsic metric satisfies . Moreover, under the assumption that the distance balls are compact we have that $f_{r,x,{{\alpha}}}\in D$ for all $r>0$, $x\in X$, ${{\alpha}}>0$ by Lemma \[l:f\_in\_D\]. By Lemma \[l:f\] (d) there are a sequences $(x_{k})$ and $r_{k}$ such that for $f_{k}=f_{r_{k},x_{k},{{\alpha}}}$, $g_{k}=g_{r_{k},x_{k},{{\alpha}}}$ with ${{\alpha}}>{\widetilde{ \mu}}/2$ $$\begin{aligned} {{\lambda}}_{0}(L)\leq \lim_{k\to\infty}\frac{{{\mathcal E}}(f_{k})}{\|f_{k}\|^{2}}\leq {{\alpha}}^{2}\lim_{k\to\infty}\frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}}={{\alpha}}^{2}, \end{aligned}$$ where the second inequality follows from Lemmas \[l:SL\] and \[l:NL\] and the equality follows from Lemma \[l:f\] (d). Hence, ${{\lambda}}_{0}(L)\leq {\widetilde{ \mu}}^{2}/4$. Let now $(r_{k})$ be the sequence given by Lemma \[l:f\] (c) for some fixed $x_{0}\in X$ and let $x_{k}=x_{0}$ for all $k\geq0$. By Lemma \[l:f\] (b) the sequence $(f_{k}/\|f_{k}\|)$ converges weakly to zero and, therefore, we get by Proposition \[p:h\] and Lemma \[l:f\] (c), that $$\begin{aligned} {{\lambda}}_{0}^{\mathrm{ess}}(L) \leq\lim_{k\to\infty}\frac{{{\mathcal E}}(f_{k})}{\|f_{k}\|^{2}} \leq {{\alpha}}^{2}\lim_{k\to\infty}\frac{\|g_{k}\|^{2}}{\|f_{k}\|^{2}}={{\alpha}}^{2}. \end{aligned}$$ Therefore, ${{\lambda}}_{0}^{\mathrm{ess}}(L) \leq\mu^{2}/4$. A more general non-local estimate --------------------------------- Let $L$ be the positive selfadjoint operator associated to ${{\mathcal E}}$. \[t:jump\] Assume that $\rho$ satisfies and $f_{r,x,{{\alpha}}}\in D$ for all $r\geq0$, $x\in X$ and ${{\alpha}}>{\widetilde{ \mu}}/2$. Then, $$\begin{aligned} {{\lambda}}_{0}(L)\leq\frac{{\widetilde{ \mu}}^{2}}{4}\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{\mu^{2}}{4}\end{aligned}$$ if $m(\bigcup B_{r}(x_{0}))=\infty$ for $x_{0}$ used to define $\mu$.\ If the jump size is bounded in $[{{\delta}},1]$ for some $0\leq{{\delta}}\leq1$, then $$\begin{aligned} {{\lambda}}_{0}(L)\leq\frac{2{(e^{{\widetilde{ \mu}}/2}-1)^{2}}}{{{{\delta}}^{2}e^{{\widetilde{ \mu}}}}+1} \quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L)\leq \frac{2{(e^{\mu/2}-1)^{2}}}{{{{\delta}}^{2}e^{\mu}+1}}\end{aligned}$$ if $m(\bigcup B_{r}(x_{0}))=\infty$ for $x_{0}$ used to define $\mu$. The proof follows analogously to the proof of the main theorem from Proposition \[p:h\], Lemma \[l:f\] and Lemma \[l:NL\]. Applications {#s:applications} ============ Weighted graphs {#s:graph} --------------- In this section we derive consequences of Theorem \[t:main\] and Theorem \[t:jump\] for graphs. We briefly introduce the setting and refer for more background to [@KL1]. Let $X$ be a countable discrete set. Every Radon measure of full support on $X$ is given by a function $m:X\to(0,\infty)$. Then, $L^{2}(X,m)$ is the space $\ell^{2}(X,m)$ of $m$-square summable functions with norm ${\left\Vert u\right\Vert}=(\sum_{x}u(x)^{2}m(x))^{\frac{1}{2}}$, $u\in \ell^{2}(X,m)$. From [@KL1 Theorem 7] it can be seen that all regular Dirichlet forms without killing term are determined by a symmetric map $b:X\times X\to[0,\infty)$ with vanishing diagonal that satisfies $$\begin{aligned} \sum_{y\in X}b(x,y)<\infty,\qquad\mbox{for all } x\in X,\end{aligned}$$ which gives rise to a measure $J$ on $X\times X\setminus d$ by $J=\frac{1}{2}b$. The one half stems from the convention that in the form we consider each edge only once. The map $b$ can then be interpreted as a weighted graph with vertex set $X$. Namely, the vertices $x,y\in X$ are connected by an edge with weight $b(x,y)$ if $b(x,y)>0$. In this case, we write $x\sim y$. A graph is called *connected* if for all $x,y\in X$ there are vertices $x_i \in X$ such that $x=x_{0}\sim x_{1}\sim\ldots \sim x_{n}=y$. Let a map ${\widetilde{ {{\mathcal E}}}}:\ell^{2}(X,m)\to[0,\infty]$ be given by $$\begin{aligned} {\widetilde{ {{\mathcal E}}}}(u)=\frac{1}{2}\sum_{x,y\in X}b(x,y)(u(x)-u(y))^{2}.\end{aligned}$$ The regular Dirichlet form ${{\mathcal E}}$ associated to $J$ is the restriction of ${\widetilde{ {{\mathcal E}}}}$ to $\overline{C_{c}(X)}^{\|\cdot\|_{\mathcal{E}}}$. Moreover, let $$\begin{aligned} \mbox{ ${{\mathcal E}}^{\max}={\widetilde{ {{\mathcal E}}}}\vert_{D^{\max}},\quad D^{\max}=\{u\in \ell^{2}(X,m)\mid {\widetilde{ {{\mathcal E}}}}(u)<\infty\}$}\end{aligned}$$ which is also a Dirichlet form. We denote the operator arising from ${{\mathcal E}}$ by $L$ and the operator arising from ${{\mathcal E}}^{\max}$ by $L^{\max}$. Let $\rho$ be an intrinsic pseudo metric on $X$. In this context this is equivalent to (see [@FLW Lemma 4.7, Theorem 7.3]) which reads as $$\begin{aligned} \frac{1}{2}\sum_{y\in X}b(x,y)\rho(x,y)^{2}\leq m(x),\quad x\in X.\end{aligned}$$ For simplicity we restrict ourselves to the case when $\rho$ takes values in $[0,\infty)$. (Otherwise, we can easily consider the graph componentwise.) \[r:rho\] Very often it is convenient to consider intrinsic metrics which satisfy $\sum_{y\in X}b(x,y)\rho(x,y)^{2}\leq m(x)$ for all $ x\in X$ (i.e., we drop the $\frac{1}{2}$ on the left hand side). For example, in [@Hu] an explicit example of such a metric $\rho$ is given, for $x,y \in X$, by $$\begin{aligned} \rho(x,y):=\inf\{l(x_{0},\ldots,x_{n})\mid n\geq 1,x_{0}=x, x_{n}=y, x_{i}\sim x_{i-1}, i=1,\ldots,n\}\end{aligned}$$ where the length $l$ is given by $l(x_{0},\ldots,x_{n})= \sum_{i=1}^{n}\min\{{{\mathrm{Deg}}}(x_{i})^{-\frac{1}{2}}, {{\mathrm{Deg}}}(x_{i-1})^{-\frac{1}{2}}\}$ and ${{\mathrm{Deg}}}(z)=\sum_{w}b(z,w)/m(z)$ is a generalized vertex degree. In this case all estimates in the theorem above can be divided by $2$. In general, it is hard to determine whether distance balls with respect to a certain metric are compact, which means finite in the original topology, in the situation of graphs. However, we always have a statement for the operator $L^{\max}$ related to ${{\mathcal E}}^{\max}$. \[t:graph\]Assume that $b$ is connected and $m(X) = \infty$. Then, $$\begin{aligned} {{\lambda}}_{0}(L^{\max})\leq\frac{{\widetilde{ \mu}}^{2}}{4}\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L^{\max})\leq \frac{\mu^{2}}{4}.\end{aligned}$$ If $\rho(x,y)\in[{{\delta}},1]$ for all $x\sim y$, then $$\begin{aligned} {{\lambda}}_{0}(L^{\max})\leq\frac{{2(e^{{\widetilde{ \mu}}/2}-1)^{2}}}{{{{\delta}}^{2}e^{{\widetilde{ \mu}}}}+1} \quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}(L^{\max})\leq \frac{{2(e^{\mu/2}-1)^{2}}}{{{{\delta}}^{2}e^{\mu}+1}}.\end{aligned}$$ In this case where the assumption on the adapted metric above is posed without the $1/2$ on the left hand side, all estimates in the theorem above can be divided by $2$. Let $B_{r_{k}}(x_{0})$, (respectively $B_{{\widetilde{ r}}_{k}}(x_{k})$) be a sequence of distance balls that realizes $\mu$ (respectively ${\widetilde{ \mu}}$), i.e., $\mu=\lim_{k\to\infty} r_{k}^{-1}\log m(B_{r_{k}}(x_{0}))$ (respectively ${\widetilde{ \mu}}=\lim_{k\to\infty} r_{k}^{-1}\log m(B_{{\widetilde{ r}}_{k}}(x_{k}))$). If the measure of $B_{r_{k}}(x_{0})$, (respectively $B_{{\widetilde{ r}}_{k}}(x_{k})$) is infinite for some $k$, then $\mu=\infty$ (respectively ${\widetilde{ \mu}}=\infty$) and we are done. Otherwise, $f_{r_{k},x_{0},{{\alpha}}},g_{r_{k},x_{0},{{\alpha}}} \in \ell^{2}(X,m)$ (respectively $f_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}$, $g_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}\in \ell^{2}(X,m)$) and $f_{r_{k},x_{0},{{\alpha}}}\in D^{\max}$ (respectively $f_{{\widetilde{ r}}_{k},x_{k},{{\alpha}}}\in D^{\max}$) by Lemma \[l:NL\]. Thus, the statement follows directly from Theorem \[t:jump\]. In the case when we know more about the measure or the metric structure we can say something about the operator $L$. This is the case under either of the following additional assumptions: - Every infinite path of vertices has infinite measure. - $\rho$ is any adapted path metric on a locally finite graph such that $(X, \rho)$ is metrically complete. In particular, (A) is satisfied if $\inf_{x\in X}m(x)>0$ and (B) is satisfied if all infinite geodesics have infinite length. \[c:graph\]Assume that either (A) or (B) is satisfied. Then, the statement of Theorem \[t:graph\] holds for $L=L^{\max}$. By [@KL1 Theorem 6], respectively [@HKMW Theorem 2], (A), respectively (B), imply that ${{\mathcal E}}={{\mathcal E}}^{\max}$ and $L=L^{\max}$. Under the slightly stronger assumption that connected infinite sets have infinite measure we can prove the corollary directly. Namely, if one of the relevant distance balls is infinite, then it has infinite measure and the exponential volume growth is infinite. In the other case the corollary follows from Theorem \[t:jump\]. We also recover the result of [@Fuj] which already covers [@DK; @OU]. In their very particular situation, $m$ is the vertex degree and $b$ takes values in $\{0,1\}$. The natural graph distance $d$ is given as the minimum length of a path of edges connecting two vertices where the length is the number of edges contained in the path. (Normalized Laplacians)\[c:normalized\] Let $b$ be a connected weighted graph over $(X,n)$, with $n(x)=\sum_{y\in X}b(x,y)$, $x\in X$ and let $d$ be the natural graph metric. Then, ${{\lambda}}_{0}^{\mathrm{ess}}(L)\leq 1- 2e^{{\widetilde{ \mu}}/2}/({1+e^{{\widetilde{ \mu}}}})$ and ${{\lambda}}_{0}(L)\leq 1-2e^{\mu/2}/({1+e^{\mu}})$. Clearly, $L$ is a bounded operator and thus $L=L^{\max}$. Moreover, the natural graph metric is an intrinsic metric for $2L$ and its jump size in exactly $1$. Thus, the statement follows from the previous theorem. Unweighted graphs and the natural graph distance {#s:graph2} ------------------------------------------------ Let $b:X\times X\to\{0,1\}$ and $m\equiv 1$. Then, the operator $L$ becomes the graph Laplacian $\Delta $ acting on $D(\Delta)=\{{{\varphi}}\in \ell^{2}(X)\mid (x\mapsto\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y)))\in\ell^{2}(X)\}$, see [@KL1; @Woj1], as $$\begin{aligned} \Delta{{\varphi}}(x)=\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y)),\end{aligned}$$ where $x\sim y$ means that $b(x,y)=1$. By $m\equiv 1$ we have that $m(A)=|A|$ for all $A\subseteq X$. For simplicity we assume that the graph is connected. \[t:graph2\] Let the $d$ be the natural graph distance on an infinite graph and $B_{r}^{d}=\{x\in X\mid d(x,x_{0})\leq r\}$ for some $x_{0}\in X$ and $r\ge0$. If $$\begin{aligned} \liminf_{r\to\infty}\frac{\log |B_{r}^{d}(x_{0})|}{\log r}<3,\end{aligned}$$ then, ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$. Moreover, if $$\begin{aligned} \limsup_{r\to\infty}\frac{ |B_{r}^{d}(x_{0})|}{ r^{3}}<\infty,\end{aligned}$$ then ${{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$ and, in particular, ${{\sigma}}_{\mathrm{ess}}(\Delta)\neq \emptyset$. \(a) The result above is sharp. This can be seen by the examples of antitrees discussed below the proof. \(b) In [@GHM Theorem 1.4] it is shown that less than cubic growth implies stochastic completeness. \(c) In the case where the vertex degree is bounded by some $K$, the situation is very different: the $n$ in Corollary \[c:normalized\] becomes $\deg$ in our situation, where $\deg:X\to{{\mathbb N}}$ is the function assigning to a vertex the number of adjacent vertices, and the corresponding normalized operator is ${\widetilde{ \Delta}}$ acting on $\ell^{2}(X,\deg)$ as ${\widetilde{ \Delta}}{{\varphi}}(x)=\frac{1}{\deg(x)}\sum_{y\sim x}({{\varphi}}(x)-{{\varphi}}(y))$. Then, $$\begin{aligned} {{\lambda}}_{0}({\widetilde{ \Delta}})\leq{{\lambda}}_{0}(\Delta)\leq K{{\lambda}}({\widetilde{ \Delta}})\quad\mbox{and}\quad {{\lambda}}_{0}^{\mathrm{ess}}({\widetilde{ \Delta}})\leq{{\lambda}}_{0}^{\mathrm{ess}}(\Delta)\leq K{{\lambda}}^{\mathrm{ess}}({\widetilde{ \Delta}}),\end{aligned}$$ see, e.g., [@K]. Thus, in the bounded situation, the threshold lies again at subexponential growth by Corollary \[c:normalized\] (as the measures $m\equiv 1$ and $n=\deg$ also give the same exponential volume growth.) Explicit estimates for the exponential volume growth of planar tessellations in terms of curvature can be found in [@KP]. \(d) In the case of bounded vertex degree we also have a threshold for recurrence of the corresponding random walk at quadratic volume growth, see [@Woe Lemma 3.12]. Let $\rho$ be the intrinsic metric from [@Hu] introduced above in Remark \[r:rho\] which, in the case of unweighted graphs, is given by $$\rho(x,y)=\inf\{ \sum_{i=0}^{n-1}\min\{\deg(x_{i})^{-\frac{1}{2}}, \deg(x_{i+1})^{-\frac{1}{2}}\}\mid (x_{0},\ldots, x_{n})\mbox{ is a path from $x$ to $y$}\}.$$ Let $ B_{r}^{\rho}=\{x\in X\mid \rho(x,x_{0})\leq r\}$, while $B_{r}^{d}$ are the balls with respect to the natural graph distance $d$. The proof of the theorem is based on the following lemma which is inspired by the proof of [@GHM Theorem 1.4]. Indeed, the second statement is taken directly from there. If $\liminf\limits_{r\to\infty}{\log |B_{r}^{d}|}/{\log r}=\beta\in[1,3)$, then $\liminf\limits_{r\to\infty}{\log|B_{r}^{\rho}|}/{\log r}\leq {\frac{2\beta}{3-\beta}}$. Moreover, if $\limsup\limits_{r\to\infty}{ |B_{r}^{d}|}/{ r^{3}}<\infty$, then $\limsup\limits_{r\to\infty}\frac{1}{r}\log|B_{r}^{\rho}|<\infty$. Let $S_{r}^{d}=B_{r}^{d}\setminus B_{r-1}^{d}$, $r\geq0$, and for convenience set $S_{-r}^{d}=B_{-r}^{d}=\emptyset$ for $r>0$. Let $1\leq{{\alpha}}<3$ and $(r_{k})$ be an increasing sequence such that ${\log |B_{r_{k}}^{d}(x_{0})|}/{\log r_{k}}<{{\alpha}}$ for all $k\ge0$. Then, $$\begin{aligned} |B_{r_{k}}^{d}|=\sum_{r=0}^{r_{k}}|S_{r}^{d}|< r^{{{\alpha}}}_{k}\end{aligned}$$ for large $k\ge0$. For ${{\varepsilon}}>0$ and $k\geq0$ set $$\begin{aligned} A_{k}:=\{r\in[0,r_{k}]\cap {{\mathbb N}}_{0}\mid |S_{r}^{d}|> \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}.\end{aligned}$$ We can estimate $|A_{k}|\leq{{\varepsilon}}r_{k}$ via $$\begin{aligned} r_{k}^{{{\alpha}}}>|B_{r_{k}}^{d}| \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\sum_{r\in A_{k}}r^{{{\alpha}}-1} \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\sum_{r=0}^{|A_{k}|}r^{{{\alpha}}-1} \geq\frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}}\int_{0}^{|A_{k}|}r^{{{\alpha}}-1}dr =\frac{|A_{k}|^{{{\alpha}}}}{{{\varepsilon}}^{{{\alpha}}}}.\end{aligned}$$ Thus, $$\begin{aligned} |\{r\in[1,r_{k}]\cap {{\mathbb N}}_{0}\mid \max_{i=0,1,2,3}|S_{r-i}^{d}|> \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}|\leq 4{{\varepsilon}}r_{k}\end{aligned}$$ and $$\begin{aligned} |\{r\in[1,r_{k}]\cap {{\mathbb N}}_{0}\mid \max_{i=0,1,2,3}|S_{r-i}^{d}|\leq \frac{{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\}|\geq (1-4{{\varepsilon}})r_{k}.\end{aligned}$$ As we have $\deg \leq |S_{r-1}^{d}\cup S_{r}^{d}\cup S_{r+1}^{d}|$ on $S_{r}^{d}$, we get $|D_{k}|\geq (1-4{{\varepsilon}})r_{k}$, where $$\begin{aligned} D_{k}:=\{(r+1)\in[0,r_{k}-1]\cap {{\mathbb N}}_{0}\mid \deg\leq \frac{3{{\alpha}}}{{{\varepsilon}}^{{{\alpha}}}} r^{{{\alpha}}-1}\mbox{ on } S_{r-1}^{d}\cup S_{r}^{d}\}.\end{aligned}$$ Hence, for $(r+1)\in D_{k}$ we have for $x\in S_{r-1}^{d}$, $y\in S_{r}^{d}$ $$\rho(x,y)\geq cr^{-\frac{{{\alpha}}-1}{2}},\quad\mbox{ with } c=\sqrt{{{{\varepsilon}}^{{{\alpha}}}}/{3{{\alpha}}}}.$$ Since any path from $x_{0}$ to $S_{r_{k}}^{d}$ contains such edges we have for any $x\in S_{r_{k}}^{d}$ $$\begin{aligned} \rho(x_{0},x)\geq c\sum_{(r+1)\in D_{k}}r^{-\frac{{{\alpha}}-1}{2}} \geq c \sum_{r=4{{\varepsilon}}r_{k}}^{r_{k}-1}r^{-\frac{{{\alpha}}-1}{2}} \geq c\int_{4{{\varepsilon}}r_{k}} ^{r_{k}-1}r^{-\frac{{{\alpha}}-1}{2}} dr\geq C_{0} r_{k}^{{\frac{3-{{\alpha}}}{2}}}\end{aligned}$$ with $C_{0}>0$ for ${{\varepsilon}}>0$ chosen sufficiently small and $r_{k}$ large. Let $R_{k}:=C_{0} r_{k}^{{\frac{3-{{\alpha}}}{2}}}$ and $C:=C_{0}^{-\frac{2{{\alpha}}}{3-{{\alpha}}}}$. Then, $B_{R_{k}}^{\rho}\subseteq B_{r_{k}}^{d}$ and since $|B_{r_{k}}^{d}|=\sum_{r=0}^{r_{k}}|S_{r}^{d}|< r^{{{\alpha}}}_{k}$, we conclude $$\begin{aligned} |B_{R_{k}}^{\rho}|\leq |B_{r_{k}}^{d}|<r^{{{\alpha}}}_{k}\leq CR_{k}^{\frac{2{{\alpha}}}{3-{{\alpha}}}}.\end{aligned}$$ Thus, the first statement follows. The second statement is shown in the proof of [@GHM Theorem 1.4]. In the case where the polynomial growth is strictly less than cubic we get by the lemma above that $\mu=0$ with respect to the intrinsic metric $\rho$ and in the case where it is less than cubic we still have $\mu<\infty$. Thus, the statement follows from Corollary \[c:graph\], where (A) is clearly satisfied as $m\equiv 1$. Let us discuss the example of antitrees which show the sharpness of the result. They were first introduced in [@Woj3] and further studied in [@BK; @KLW]. An antitree is a spherically symmetric graph, where a vertex in the $r$-th sphere is connected to all vertices in the $(r+1)$-th sphere for $r\ge0$, and there are no horizontal edges. Thus, an antitree is characterized by a sequence $(s_{r})$ taking values in ${{\mathbb N}}$ which encodes the number of vertices in the sphere $S_{r}^{d}=B_{r}^{d}\setminus B_{r-1}^{d}$. *Stronger growth than cubic:* In [@KLW Corollary 6.6] it is shown that if the polynomial volume growth of an antitree is more than cubic, i.e., as $r^{3+{{\varepsilon}}}$ for ${{\varepsilon}}>0$, then ${{\lambda}}_{0}(\Delta)>0$ and ${{\sigma}}_{\mathrm{ess}}(\Delta)=\emptyset$. Indeed, in the intrinsic metric $\rho$, these antitrees have finite diameter and thus $\mu=\infty$, see [@Hu]. *Cubic growth:* If the distance spheres of an antitree satisfy $|S_{r}^{d}|=(r+1)^{2}$, then $|B_{r}^{d}|\sim (r+1)^{3}$. Moreover, the function which takes the value $r^{-2}$ on vertices of the $(r-1)$-th sphere, $r\geq1$, is a positive generalized super-solution for $\Delta$ to the value $2$, that is, $\Delta {{\varphi}}\geq 2 {{\varphi}}$. Thus, by a discrete Allegretto-Piepenbrink theorem (see [@Woj2 Theorem 4.1] or [@HK Theorem 3.1]) it follows that ${{\lambda}}_{0}(\Delta)\geq2$. By Theorem \[t:graph2\] we thus have $2\leq {{\lambda}}_{0}^{\mathrm{ess}}(\Delta)<\infty$. *Weaker growth than cubic:* In this case Theorem \[t:graph2\] shows that ${{\lambda}}_{0}(\Delta)={{\lambda}}_{0}^{\mathrm{ess}}(\Delta)=0$. **Acknowledgements.** The authors are grateful to J[ó]{}zef Dodziuk and Daniel Lenz for their continued support and for generously sharing their knowledge. The research of RKW was partially sponsored by the Fundação para a Ciência e a Tecnologia through project PTDC/MAT/101007/2008 and by the Research Foundation of CUNY through the PSC-CUNY Research Award 42. [10]{} J. Breuer, M. 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Math. J. [**58**]{} (2009), no. 3, 1419–1441. R. K. Wojciechowski, *Stochastically incomplete manifolds and graphs*, Random walks, boundaries and spectra, Progr. Prob., vol. 64, Birkhäuser Verlag, Basel, 2011, pp. 163–179.

ArXiv

--- abstract: 'We first show that the pions produced at high $p_T$ in heavy-ion collisions over a wide range of high energies exhibit a scaling behavior when the distributions are plotted in terms of a scaling variable. We then use the recombination model to calculate the scaling quark distribution just before hadronization. From the quark distribution it is then possible to calculate the proton distribution at high $p_T$, also in the framework of the recombination model. The resultant $p/\pi$ ratio exceeds one in the intermediate $p_T$ region where data exist, but the scaling result for the proton distribution is not reliable unless $p_T$ is high enough to be insensitive to the scale-breaking mass effects.' author: - 'Rudolph C. Hwa$^1$ and C. B. Yang$^{1,2}$' date: January 2003 title: 'Scaling Behavior at High $p_T$ and the $p/\pi$ Ratio' --- Introduction ============ There are three separate and independent aspects about the hadrons produced at large transverse momentum $(p_T)$ in heavy-ion collisions at high energies that collectively contribute to a coherent picture to be addressed in this paper. One is the existence of a scaling behavior at large $p_T$ that we have found by presenting the data in terms of a new variable. Another is the issue about the surprisingly large proton-to-pion ratio at moderate $p_T$ ($\sim$ 2 - 3 GeV/c) discovered by PHENIX [@ts] in central $AuAu$ reactions at $\sqrt{s} =$ 130 and 200 GeV. The third issue concerns the hadronization process relevant for the formation of hadrons at large $p_T$ and the applicability of the recombination model [@dh]. It is our goal to show that, in light of the scaling behavior of the $\pi^0$ produced, the recombination mechanism naturally gives rise to a $p/\pi$ ratio that exceeds 1 in the $2 < p_T < 3$ GeV/c range. Particle production in heavy-ion collisions at very high energies is usually described in terms of hydrodynamical flow [@hyd], jet production at high $p_T$ [@jet], thermal statistical model [@the], or a combination of various hadronization mechanisms [@all]. In none of the conventional approaches does one expect protons to be produced at nearly the same rate as the pions. If all hadrons with $p_T > 2$ GeV/c are regarded as products of jet fragmentation, then the known fragmentation functions of quark or gluon jets would suppress proton relative to pion by the sheer weight of the proton mass. Such a discrepancy from the observed data led some to regard the situation as an anomaly and proposed the gluonic baryon junction as a mechanism to enhance the proton production rate [@vg]. Their predictions remain to be checked by experiments. The parton fragmentation functions have been used even at low $p_T$ in string models where the production of particles in hadronic collisions is treated as the fragmentation of diquarks, as done in the dual parton model [@dpm]. There has been a long-standing dichotomy on whether particle production in the fragmentation region can better be described by fragmentation [@dpm; @lpy] or recombination [@dh; @hy]. It is possible that the two pictures might be unified in a more comprehensive treatment of hadronization in the future. Here we extend the recombination model to the central region at large $p_T$. It should be recognized that an essential part of the recombination model is the determination of the distribution functions of the quarks and antiquarks that are to recombine. In the case of large-$p_T$ hadrons the underlying physics is undoubtedly hard collisions of partons and the associated radiation of gluons. If the parton distributions can be calculated just before hadronization, then the final step of recombination can readily be completed. If those distributions cannot be determined in pQCD, then the step between the initiating large-$p_T$ parton and the resultant hadrons may efficiently be described by a fragmentation function, determined phenomenologically from experiments. Thus in that sense the two approaches, recombination and fragmentation, are not contradictory, but complementary. We state from the outset that no attempt will be made here to perform a first-principle calculation of the parton distributions at large $p_T$ before recombination. However, from the observed data on pion production in central $AuAu$ collisions at the Relativistic Heavy-Ion Collider (RHIC), it is possible to work backwards in the recombination model to determine the quark (and antiquark) distribution at large $p_T$. On the basis of the quark distributions inferred, it is then possible to calculate the proton distribution in the recombination model. The basic idea is that if there is a dense system of quarks and antiquarks produced in a heavy-ion collision whatever the dynamics responsible for them may have been (gluons having been converted to $q\bar{q}$ pairs before hadronization), then the formation of pions and protons (and whatever else) is prescribed by the recombination model without any arbitrariness in normalization and momentum dependence. One limitation of the recombination model as it stands at present is that it is formulated in a frame-independent way in terms of momentum fractions and is therefore inapplicable to a system where the particle momenta are low and the mass effects are large. The physics of recombination is still valid at low momentum, but the details of the wave functions of the constituent quarks become important; they have not been built into the recombination function that takes the simplest form in the infinite momentum frame. Thus our calculation of particles produced at midrapidity is not reliable when $p_T$ is of the order of the masses of the hadrons under consideration. For protons we can trust the results only for $p_T > 3$ GeV/c. For pions the lower limit of validity can be pushed much lower. Since our approach makes crucial use of the experimental data on the pion spectrum as the input, it is essential to relate the spectra determined at different energies to an invariant distribution so that the scale-invariant recombination model can be applied. To discover the existence of an invariant distribution with no theoretical prejudices is a problem worthy in its own right. Fortunately, that turns out to be possible. The analysis for that part of the study will be presented below first to emphasize its independence from the theoretical modeling of hadronization. It should be mentioned that the scaling of transverse mass spectra has been investigated recently [@sb]. The emphasis there has been on the dependences on the particle species and centrality for $m_T<3.8$ GeV, while our focus is on the dependence on energy ($17<\sqrt s<200 $ GeV) for $p_T<8$ GeV/c. Thus the two studies are complementary to each other. A Universal Scaling Distribution ================================ The preliminary data of the $p_T$ distributions of $\pi^0$ produced at RHIC at $\sqrt{s} = 130$ and 200 GeV were shown by the PHENIX Collaboration at Quark Matter 2002 [@ddl] for central $AuAu$ collisions together with the WA98 data for $PbPb$ collisions at $\sqrt{s} = 17$ GeV [@rey]. They show that the level of the tail at large $p_T$ rises , as $\sqrt{s}$ is increased. We want to consider the possibility that the three sets of data points can be combined to form a universal curve. The $\pi^0$ inclusive distributions at midrapidity are integrated over $\eta$ for a range of $\Delta \eta = 1$ so that the data points are given for the following quantity [@ddl]: $$\begin{aligned} f(p_T, s) = {1 \over 2 \pi p_T}{dN \over dp_T}= \int_{\Delta\eta} d\eta \left(2 \pi p_TN_{evt} \right)^{-1} {d^2N_{\pi^0} \over d p_T d\eta} . \label{1}\end{aligned}$$ In comparing the PHENIX data with those of WA98 one should recognize that in addition to the difference in the colliding nuclei there is a slight mismatch in centrality (top 10% for PHENIX and top 12.7% for WA98) [@we]. To unify the three data sets it is natural to first consider a momentum fraction variable similar to $x_F$ in longitudinal momentum. However, so much momenta are taken by the other particles outside the $\Delta \eta = 1$ range, it is unwise to also use $\sqrt{s}/2$ as the scale to calculate the transverse momentum fraction. We assume that for every $\sqrt{s}$ there is a relevant scale $K$ to describe the $p_T$ behavior relative to that scale. Let us define $$\begin{aligned} z = p_T/K, \label{2}\end{aligned}$$ and transform $f(p_T, s)$ to a new function $\Phi (z,K)$, where $$\begin{aligned} \Phi (z,K) = K^2 f(p_T, s) = {1 \over 2 \pi z} {dN \over dz} . \label{3}\end{aligned}$$ We adjust $K$ for each $s$ and check whether all three data sets coalesce into one universal dependence on $z$, which we would simply label as $\Phi(z)$, if it is possible. In Fig. 1 we show $\Phi(z)$, where the three symbols represent the three data sets for the three energies. Evidently, the universality exists and is striking. While this behavior needs to be confirmed by more data, and the theoretical implication remains to be explored, the existence of this scaling behavior is a significant phenomenological property of the $p_T$ distributions that suggests some underlying simplicity. It is like the KNO scaling of the multiplicity distributions $P(n, s)$ in $pp$ collisions, where for $\sqrt{s} < 200$ GeV they can be expressed by one universal scaling function $\psi (z)$, with $z = n/\left<n\right>$ [@kno; @lat]. The values of $K$ that are used for the plot in Fig. 1 are in units of GeV: $K = 1 \, (200)$, $0.9 \, (130)$ and $0.717 \, (17)$, the quantities in the parentheses being the values of $\sqrt{s}$. The $\sqrt{s}$ dependence of $K$ forms nearly a straight line, as shown in Fig.2. Since the high and low energy data differ both in colliding nuclei and in centrality, one does not expect strict regularity in how $K$ depends on $\sqrt{s}$. Nevertheless, an approximate linear dependence is a simple behavior expected on dimensional grounds. The straight line in Fig. 2 corresponds to the best fit $$\begin{aligned} K(s) = 0.69 + 1.55 \times 10^{-3} \sqrt{s} , \label{4}\end{aligned}$$ where $\sqrt{s} $ is in units of GeV. It should be recognized that the normalization of $K(s)$ is arbitrary; it is chosen to be $1$ at $\sqrt{s} = 200$ GeV for simplicity. If it is normalized to some other value at that point, the linear behavior in Fig. 2 is unchanged, only the scale of the vertical axis is shifted accordingly. The scaling property in Fig. 1 is also unchanged, the only modifications being the scales of the horizontal and vertical axes. Thus the absolute magnitude of the dimensionless variable $z$ has no significance. If the $z$ dependence of $\Phi(z)$ in Fig. 1 were strictly linear, so that it is a power-law dependence $$\begin{aligned} \Phi(z) \propto z^{\alpha} , \label{5}\end{aligned}$$ then there would be no relevant scale in the problem. The fact that it is not a straight line implies that there is an intrinsic scale in the $p_T$ problem, which is hardly surprising. What is significant is that while there is no strict scaling in $z$, there is no explicit dependence on $s$. That is, at any energy we have the same universal function $\Phi (z)$, which will be referred to as the scaling behavior in $s$. That function can be parametrized by $$\begin{aligned} \Phi (z) = 1500 \left(z^2 + 2 \right)^{-4.9} , \label{6}\end{aligned}$$ which is represented by the smooth curve in Fig. 1. For large enough $z$ Eq. (\[6\]) does have the form of the power law given in Eq. (\[5\]) with $\alpha = 9.8$. It is a succinct statement of the universal properties at high $p_T$. The departure from Eq. (\[5\]) at small $z$ reflects the physics at low $p_T$. Since there is no data on $\pi^0$ for $p_T<1$ GeV/c, the extrapolation of $\Phi(z)$ to $z<1$ is not reliable. However, there is a more accurate determination of $\Phi(z)$ that includes the low $z$ region when the charge $\pi^+$ data are considered; it is given in [@hy4], and is not needed here. ![Scaled transverse momentum distribution of produced $\pi^0$. Data are from Ref. [@ddl; @rey]. The solid line is a fit of the data by Eq. (\[6\]).](fig1.eps){width="55.00000%"} Note that there is no fixed scale in $p_T$ that separates the high- and low-$p_T$ physics. Equation (\[6\]) gives a smooth transition from one to the other in the variable $z$, thus implying different ranges of values of the transition $p_T$ at different $s$. While Eq. (\[6\]) gives a good parametrization of the scaling function $\Phi (z)$ throughout the whole range of $z$, one notices, however, that the WA98 data at 17 GeV shows a slight departure from $\Phi (z)$ at the high $z$ end of that data set. It should be recognized that those data points have $p_T >3$ GeV/c, which represents a huge fraction of the available energy at $\sqrt{s} = 17$ GeV. In fact, one expects the violation of universality to be more severe at higher $z$ at that $\sqrt{s}$, since energy conservation would suppress the inclusive cross section at higher $p_T$. What is amazing is that most of the WA98 data points are well described by $\Phi (z)$, even though the corresponding $p_T$ values take up a much larger fraction of the available energy than the other data points from RHIC. It demonstrates the significance of the variable $z$ in revealing the scaling property. {width="55.00000%"} Pion and Quark Distributions in the Recombination Model ======================================================= Having found a scaling distribution for the produced $\pi^0$ independent of $s$, we now consider the hadronization process in the recombination model in search for an origin of such a scaling behavior. In previous investigations the recombination model has been applied only to the fragmentation region where the longitudinal momenta are large and the transverse momenta are either held fixed at low $p_T$ or integrated over [@dh; @hy; @hy2]. We now consider the creation of pions in the central region of $AA$ collisions and study the $p_T$ dependence. Unlike the former case where the longitudinal momentum fractions of the partons are essentially known (from the structure functions), the $p_T$ distributions of the partons in the latter case are essentially unknown. Indeed, it is the aim of this section to determine the parton $p_T$ distributions from the $\pi^0$ distribution found in the previous section. Let us start by writing down the basic equation for recombination in the 3-space $$\begin{aligned} E{d^3N_{\pi}\over d^3p} = \int {d^3p_1 \over E_1}{d^3p_2 \over E_2}\ {\cal F}(\vec{p}_1, \vec{p}_2)\,{\cal R}_{\pi}(\vec{p}_1, \vec{p}_2, \vec{p}) \label{7}\end{aligned}$$ where the left-hand side (LHS) is the inclusive distribution of pion with energy-momentum $(E, p)$. ${\cal F}(\vec{p}_1, \vec{p}_2)$ is the probability of having a quark at $p^{\mu}_1$ and an antiquark at $p^{\mu}_2$ just before hadronization. ${\cal R}_{\pi}(\vec{p}_1, \vec{p}_2, \vec{p})$ is the invariant distribution, $E{d^3N_{\pi}^{q\bar{q}} /d^3p}$, of producing a pion at $p^{\mu}$ given a $q$ at $p^{\mu}_1$ and a $\bar{q}$ at $p^{\mu}_2$. Note that ${\cal R}_{\pi}$ has the dimension (momentum)$^{-2}$, same as the LHS. Writing the phase-space density in the form $$\begin{aligned} {d^3p\over E} = dy \, d\phi \, p_T \, dp_T , \label{8}\end{aligned}$$ we define the inclusive distribution in $p_T$, averaged over $y$ and $\phi$, $$\begin{aligned} {d^3N_{\pi} \over p_T \,dp_T} = {1 \over \Delta y} \int_{\Delta y } dy \ {1 \over 2 \pi} \int^{2\pi}_0 \, d\phi \ E{d^3N_{\pi}\over d^3p} , \label{9}\end{aligned}$$ where $\Delta y$ is limited to one unit of rapidity in the central region. Our focus will be on the $p_T$ distribution at high $p_T$. For the recombination distribution ${\cal R}_{\pi}(\vec{p}_1, \vec{p}_2, \vec{p})$ we need only consider the partons in the same transverse plane that contains $\vec{p}$, since at high $p_T$ the partons with different $y_i$ are not likely to recombine. Indeed, we assume not only $y_1 = y_2 = y$, but also $\phi _1 = \phi _2 = \phi$ so that the partons and the pion are all colinear, and the kinematics can be reduced to that of a 1-dimensional problem. As in the usual parton model, the parton momentum fractions in the hadron can vary between 0 and 1, but the deviation in the momentum components of the partons transverse to the hadron $\vec{p}$ must be severely limited because of the limited transverse size of the hadron. Thus we write $$\begin{aligned} {\cal R}_{\pi}(\vec{p}_1, \vec{p}_2,\vec{p}) = {\cal R}_{\pi} ^0 \ \delta \left(y_1 - y_2\right) \delta \left(\phi _1 - \phi _2\right)\nonumber\\ \delta \left({y_1 + y_2 \over 2} -y \right) \delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} - \vec{p}_T ) , \label{10}\end{aligned}$$ where ${\cal R}_{\pi}^0 $ is dimensionless, since $\delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} - \vec{p}_T)$ has the dimension of ${\cal R}_{\pi} (\vec{p}_1, \vec{p}_2, \vec{p})$. If this $\delta$-function is further written in the colinear form due to the $\delta(\phi _1 - \phi _2)$ in Eq. (\[10\]) $$\begin{aligned} \delta^2 (\vec{p}_{1_T} + \vec{p}_{2_T} - \vec{p}_T) &=& \delta \left({\phi _1 + \phi _2 \over 2} - \phi\right) {1 \over p_T} \, \nonumber\\ &&\delta\left(p_{1_T} + p_{2_T} - p_T\right) , \label{11}\end{aligned}$$ then Eq. (\[7\]) can be reduced to the 1D form $$\begin{aligned} {dN_{\pi}\over p_T dp_T} &=& \int dp_{1_T} dp_{2_T}p_{1_T}p_{2_T}\ {\cal F}(p_{1_T}, p_{2_T})\,{\cal R}_{\pi} ^0 \, p_T^{\ -2}\nonumber\\ &&\delta\left( { p_{1_T} + p_{2_T} \over p_T} -1 \right), \label{12}\end{aligned}$$ where ${\cal F}(p_{1_T}, p_{2_T})$ is the $q\bar{q}$ distribution in $p_{i_T}$ averaged over $y$ and $\phi$. We can reexpress this equation in terms of the scaling variable $z = p_T/K$, introduced in Eq. (\[2\]), and obtain $$\begin{aligned} {dN_{\pi}\over zdz} = \int dz_1 dz_2\,z_1\,z_2\ F(z_1, z_2)\ R_{\pi} (z_1, z_2, z) \label{13}\end{aligned}$$ where $$\begin{aligned} F(z_1, z_2) = K^4\ {\cal F}( p_{1_T} , p_{2_T}) \label{14}\end{aligned}$$ $$\begin{aligned} R_{\pi} (z_1, z_2, z) = {\cal R}_{\pi}^0 \ z^{-2}\ \delta\left( { z_1 + z_2 \over z}-1 \right) . \label{15}\end{aligned}$$ Since ${\cal F}( p_{1_T} , p_{2_T} )$ is the parton density in $p_{1_T} dp_{1_T} p_{2_T} dp_{2_T}$, $F(z_1, z_2)$ is the corresponding dimensionless density in $z_1 dz_1 z_2 dz_2$. Equation (\[13\]) is now our basic formula for recombination in the scaled transverse-momentum variable. The total number of $q$ and $\bar{q}$ is $\int dz_1 dz_2\,z_1z_2 F(z_1, z_2)$, which should be invariant under a change of scale $$\begin{aligned} z = \lambda x \label{16}\end{aligned}$$ so that $$\begin{aligned} x = p_T/K^{\prime}, \qquad \qquad K^{\prime} = \lambda K . \label{17}\end{aligned}$$ The corresponding change on $F(z_1, z_2)$ is that it becomes $$\begin{aligned} F^{\prime} (x_1, x_2 ) = \lambda^4 F(z_1, z_2). \label{18}\end{aligned}$$ Thus the normalization of $F(z_1, z_2)$ is scale dependent, as it should in view of Eq. (\[14\]). So far the recombination function $R_{\pi}(z_1, z_2, z)$ is not fully specified because ${\cal R}_{\pi}^0$ has not been. In Eq.(\[15\]) the factor $z^{-2}$ is associated with the dimension of the pion density, and the $\delta$-function with momentum conservation. To introduce the pion wave function in terms of the constituent quarks, we rewrite Eq. (\[15\]) as $$\begin{aligned} R_{\pi}(z_1, z_2, z )=R_{\pi}^0\ z^{-2}\ G_{\pi}(\xi_1,\xi _2), \label{19}\end{aligned}$$ where $R_{\pi}^0$ is a normalization constant to be determined and $G_{\pi}(\xi _1,\xi _2)$ is the valon distribution of the pion [@dh; @hy]. Since the recombination of a $q$ and $\bar{q}$ into a pion is the time-reversed process of displaying the pion structure, the dependence of $R_{\pi}(z_1, z_2, z)$ on the pion structure is expected. During hadronization the initiating $q$ and $\bar{q}$ dress themselves and become the valons of the produced hadron without significant change in their momenta. The variable $\xi _i$ in Eq. (\[19\]) denotes the momentum fraction of the $i$th valon, i.e., $$\begin{aligned} \xi _i = z_i/z , \label{20}\end{aligned}$$ which is denoted by $y_i$ in the valon model [@dh; @hy], a notation that cannot be repeated here on account of the rapidity variables already used in Eq. (\[10\]). In general, the valon distribution of a hadron $h$ has a part specifying the wave-function squared, $\tilde{G}_h$, and a part specifying momentum conservation $$\begin{aligned} G_h(\xi _1,\cdots ) = \tilde{G}_h(\xi _1,\cdots)\ \delta\left(\sum_i \xi _i -1 \right) , \label{21}\end{aligned}$$ where the functional form of $\tilde{G}_h$ is determined phenomenologically. Although for proton $\tilde{G}_p$ is found to be highly nontrivial [@hy3], for pion $\tilde{G}_{\pi}$ turns out to be very simple [@hy] $$\begin{aligned} \tilde{G}_{\pi} (\xi _1, \xi _2) = 1 , \label{22}\end{aligned}$$ which is a reflection of the fact that the pion mass is much lower than the constituent quark masses, so tight binding results in large uncertainty in the momentum fractions of the valons. Equation (\[22\]) implies that the valon momenta of the pion is uniformly distributed in the range $0 < \xi _i < 1$. What remains in Eq. (\[19\]) for us to determine is $R_{\pi}^0$. At this point we need to be more specific about the quark and antiquark that recombine. If the colors of $q$ and $\bar{q}$ are considered, then the probability of forming a color singlet pion is $1/9$ in $3 \times \bar{3}$. Similarly, for three quarks forming a proton the probability is $1/27$ in $3 \times 3 \times 3$. In the parton distributions, $F_{q\bar{q}}$ for pion production involves two color triplets and $F_{qqq}$ for proton production involves three color triplets so the color factors work out just right in that the factors of $9$ for $q\bar{q}$ and $27$ for $qqq$ are cancelled by the corresponding inverse factors in the recombination probabilities. In other words, for the $p/\pi$ ratio to be considered later, we can ignore the factors associated with the color degrees of freedom and proceed with the determination of $F_{q\bar{q}}$ without specifying the quark colors and summing over them. The situation with flavor is not the same. For a $u \bar{u}$ pair and a $d\bar{d}$ pair, they can form $\pi^0$ and $\eta$ in the flavor octet. The branching ratio of $\eta$ to $3 \pi^0$ is 32.5% and to $\pi ^+ \pi ^- \pi ^0$ is 22.6%. Thus for every $\eta$ produced there is on average $1.2 \pi ^0$. Due to the higher mass of $\eta$ we make the approximation that the rate of indirect production of $\pi ^0$ via $\eta$ is roughly the same as the direct production from $u \bar{u}$ and $d \bar{d}$. If we now use $q\bar{q}$ to denote either $u \bar{u}$ or $d \bar{d}$, but not both $u \bar{u}$ and $d \bar{d}$, then each pair of $q\bar{q}$ leads to one $\pi ^0$. Since in a heavy-ion collision there are many quarks and antiquarks produced in the central region, it is reasonable to assume that the $q$ distribution is independent of the $\bar{q}$ distribution so that we can write $F_{q\bar{q}}$ in the factorizable form $$\begin{aligned} F_{q\bar{q}} \left(z _1, z _2\right) = F_q (z _1)\ F_{\bar{q}} ( z _2) , \label{23}\end{aligned}$$ where $F_q$ stands for either $u$ or $d$ distributions, and similarly for $F_{\bar{q}} $, but for $\pi ^0$ production $\bar{q}$ should be the antiquark partner of $q$. The fact that we consider $\eta$ production above, but not the vector meson $\rho$ requires an explanation. We defer that discussion until the next section, after we have presented the formalism for the production of protons. Returning now to the normalization of $R_{\pi} \left(z _1, z _2, z\right)$, we note that, using Eqs. (\[19\]), (\[21\]) and (\[22\]), $$\begin{aligned} \int dzzR_{\pi} (z _1, z_2, z) &=& \int {dz\over z} R_{\pi}^0 \delta \left({z _1 + z_2\over z}-1\right)\nonumber\\&& = R_{\pi}^0 \label{24}\end{aligned}$$ is the probability that a $q$ at $z_1$ and a $\bar{q}$ at $z_2$ recombine to form a pion at any $z$. According to our counting in the second paragraph above, the total probability for $q\bar{q} \rightarrow \pi ^0$ integrated over all momenta is $$\begin{aligned} \int^Z_0 {d z _1 \over Z} \int^Z_0 {d z _2 \over Z} \int dz\ z\, R_{\pi}(z _1, z_2, z) = 1 , \label{25}\end{aligned}$$ where $Z$ is the maximum $z_i$, whatever it is. This normalization condition is scale invariant, and we find, using Eq. (\[24\]), that $$\begin{aligned} R_{\pi}^0 = 1 . \label{26}\end{aligned}$$ Putting Eqs. (\[19\]) - (\[23\]) and (\[25\]) in (\[15\]) we obtain $$\begin{aligned} {dN_\pi \over zdz} = \int dz_1 dz_2{z_1z_2\over z}F_q(z_1)\, F_{\bar{q}}(z_2)\,\delta(z_1 + z_2 - z). \label{27}\end{aligned}$$ This is obtained from Eq. (\[9\]) where $y$ and $\phi$ are both explicitly averaged over. The LHS is to be identified with $\Phi(z)$. Note that the $1/2\pi$ factors in Eqs. (\[1\]) and (\[3\]), where $\Phi(z)$ is defined, are there to render $f(p_T, s)$ an average distribution in $\phi$; that is the notation for the experimental distribution, defined in [@ddl]. The distribution defined by us in Eq. (\[9\]) already includes the $1/2\pi$ factor, so our $dN_\pi/z\,dz$ is just the experimental $\Phi(z)$. As we have mentioned earlier, the normalization of $z$ has no significance. By means of a scale change in Eq. (\[16\]) we can move from $z$ to $x$, or vice-versa, without changing the scale invariant form of Eq.(\[27\]). In Eq. (\[6\]) we found $\Phi (z)$ to have the form $$\begin{aligned} \Phi (z) = A \left(z^2 + c \right)^{-n} . \label{28}\end{aligned}$$ If we change $z$ to $x$ according to Eq. (\[16\]), then by keeping the total number of pions invariant, i.e., $$\begin{aligned} \int dz\, z \,\Phi (z, K) = \int dx\, x \,\Phi^{\prime} (x, K^{\prime}) , \label{29}\end{aligned}$$ we have $$\begin{aligned} \Phi^{\prime} (x, \lambda K) = \lambda^2\Phi (\lambda x, K). \label{30}\end{aligned}$$ It thus follows that $$\begin{aligned} \Phi^{\prime} (x) = \lambda^{2(1-n)} A \left(x^2 + c/\lambda^2\right)^{-n}. \label{31}\end{aligned}$$ Similarly, in the $x$ variable the transformed quark distributions is $$\begin{aligned} F^{\prime}_q (x_1, K^{\prime}) = \lambda^2 F_q (z_1, K). \label{32}\end{aligned}$$ Without having to specify the arbitrary scale factor $\lambda$, let us work with the $z$ variable and rewrite Eq. (\[27\]) as $$\begin{aligned} \Phi(z) = \int^z_0 dz_1\ z_1 \left(1 - { z_1 \over z} \right)\ F_q (z_1)\ F_{\bar{q}}(z - z_1) . \label{33}\end{aligned}$$ We must now consider how the $q$ and $\bar{q}$ distributions differ. Unlike the structure functions of the nucleon, where $q$ and $\bar{q}$ have widely different distributions, we are here dealing with the partons at high $p_T$ in heavy-ion collisions just before recombination. The dynamics underlying their $p_T$ dependences is complicated. Many subprocesses are involved, which include hard scattering, gluon radiation, jet quenching, gluon conversion to quark pairs, thermalization, hydrodynamical expansion, to name a few familiar ones. At very large $p_T$ there are far more quark jets than antiquark jets, since the valence quarks have larger longitudinal momentum fractions than the sea quarks. By hard scattering the quarks therefore can acquire larger $p_T$ than the antiquarks. Thus in that way one would expect the $p_T$ distribution of the quarks to be very different from that of the antiquarks. However, that view does not apply to our problem. Those are the $q$ and $\bar{q}$ that initiate jets, along with jets initiated by gluons. The conventional approach is to follow the jet production by jet fragmentation, which can be modified by the dense matter that the initiating partons traverse. As discussed earlier, our approach is not to delve into the dynamical origins of the $q$ and $\bar{q}$ distributions, but to consider the recombination of $q$ and $\bar{q}$ just at the point of hadronization. Such $q$ and $\bar{q}$ are not the partons that initiate jets, but are the parton remnants after the hard partons radiate gluons which subsequently convert to $q\bar{q}$ pairs. Those parton remnants have similar momentum distribution for $q$ and $\bar{q}$, since gluon conversion creates $q$ and $\bar{q}$ on equal basis; those partons are the ones that recombine to form hadrons. They are not to be confused with the jet-initiating hard partons that fragment into hadrons in the fragmentation model. In the recombination picture those hard partons that acquire large $p_T$ immediately after hard scattering are not ready for recombination; they lose momenta and virtuality through gluon radiation until a large body of low-virtuality quarks and antiquarks are assembled for recombination — a view that is complementary to the fragmentation picture. Of course, there are more quarks than antiquarks, since the number of valence quarks of the participating nucleons cannot diminish. For that reason we allow $F_q(z)$ to differ in normalization from $F_{\bar{q}}(z)$. However, as a first approximation we assume that their $z$ dependences are the same. There is some indirect experimental evidence in support of our assumption. In Ref. [@ts] the $\bar{p}/p$ ratio for central collisions is reported to be essentially constant within errors; more precisely, it ranges between $0.6$ and $0.8$ for $p_T$ in the range $0.5 < p_T < 3.8$ GeV/c. Since $\bar{p}$ is formed by the recombination of three $\bar{q}$, while $p$ is formed from three $q$, a quick estimate of the $\bar{q}/q$ ratio is that it varies between $0.6^{1/3}$ and $0.8^{1/3}$, i.e., from $0.843$ to $0.928$. Such a narrow range of variation is sufficient for us to assume that $F_{\bar{q}}(z)$ has the same $z$ dependence as $F_q(z)$. For their relative normalization we take the mean $\bar{p}/p$ ratio to be $0.7$. Thus we adopt the $\bar{q}/q$ ratio to be $$\begin{aligned} F_{\bar{q}}(z)/F_q(z) = F^{\prime}_{\bar{q}}(x)/F^{\prime}_q(x) =0.7^{1/3} . \label{34}\end{aligned}$$ With this input we are finally ready to infer the quark distribution from the pion distribution. {width="55.00000%"} We parameterize $F_q(z)$ by $$\begin{aligned} F_q(z) = a \left(z^2 + z + z_0 \right)^{-m} \label{35}\end{aligned}$$ and adjust the three parameters $a$, $z_0$ and $m$ to fit $\Phi (z)$ by using Eq. (\[33\]). We obtain an excellent fit with the values $$\begin{aligned} a = 90 , \qquad z_0 = 1, \qquad m= 4.65 . \label{36}\end{aligned}$$ In Fig. 3 we show in solid line the data represented by the formula in Eq. (\[6\]) and in dashed line the result of the theoretical calculation using Eqs. (\[33\])-(\[36\]). They coalesce nearly completely in the interval $1 < z < 8$. The quark distribution $F_q(z)$ is shown in Fig. 4. To appreciate the $p_T$ range corresponding to $z$ in Fig. 4, recall Eq. (\[2\]), $p_T = zK$, and Fig. 2 for $K$. Thus at $\sqrt{s} = 200$ GeV, $p_T$ is $z$ in GeV. Equations (\[35\]) and (\[36\]) represent a main result of this study. What is important is that we have found a scaling quark distribution that is independent of $s$ from SPS to RHIC, and perhaps to LHC. It is a succinct summary of the effects of all the dynamical subprocesses in heavy-ion collisions. The non-trivial $z$ dependence in Eq. (\[35\]) indicates that there are intrinsic scales in the low-$p_T$ problem. {width="55.00000%"} The $p/\pi$ Ratio ================= The quark distribution obtained in the preceding section cannot be checked directly. Since it is the distribution at the end of its evolution, massive dileptons would not be sensitive to it due to their production at the early stages. Proton production provides the most appropriate test, since hadronization occurs near the end. We shall therefore calculate the proton distribution at high $p_T$ and compare with the data on the $p/\pi$ ratio. This is not a completely satisfactory venture, since the proton mass is large, so only at very high $p_T$ can our scale invariant calculation be valid without explicit consideration of the mass effect. Present data on the $p/\pi$ ratio do not extend beyond $p_T \sim 3.8$ GeV/c [@ts]. Nevertheless, our calculation should provide some sense on the magnitude of the rate of proton production at the high $p_T$ end. The inclusive distributions in the scaled $p_T$ variable can be obtained in the recombination model by generalizing Eq.(\[13\]) to the recombination of three quarks $$\begin{aligned} {dN_p \over zdz} &=& \int dz_1 dz_2 dz_3\ z_1\,z_2\,z_3\nonumber\\ && F(z_1, z_2, z_3)\ R_p(z_1, z_2, z_3, z) \label{37}\end{aligned}$$ where $F(z_1, z_2, z_3)$ is given the factorizable form $$\begin{aligned} F(z_1, z_2, z_3) = F_u (z_1) F_u (z_2) F_d (z_3). \label{38}\end{aligned}$$ As in Eq. (\[19\]) we relate the recombination function $R_p$ to the valon distribution, $G_p$, of the proton $$\begin{aligned} R_p(z_1, z_2, z_3, z) = R_p^0 \ z^{-2}\,G_p(\xi_1,\xi_2, \xi_3) , \label{39}\end{aligned}$$ where $G_p$ has the general form given in Eq. (\[21\]), and $R_p^0$ remains to be determined. In Ref. [@hy3] a detailed study of the proton structure functions has been carried out in deriving the valon distribution from the parton distributions that fit the deep inelastic scattering data. It is $$\begin{aligned} \tilde{G}_p (\xi_1, \xi_2, \xi_3) = g \ (\xi_1\, \xi_2 ) ^\alpha\ \xi_3^{\beta} , \label{40}\end{aligned}$$ where $$\begin{aligned} \alpha = 1.755, \qquad \beta = 1.05 , \label{41}\end{aligned}$$ $$\begin{aligned} g = \left[B \left(\alpha + 1, \beta +1 \right) B \left( \alpha + 1, \alpha + \beta +2\right)\right]^{-1} . \label{42}\end{aligned}$$ Single-valon distributions $G_p(\xi_i)$ can be obtained from the three-valon distribution by integration and are peaked around $\xi = 1/3$, indicating that each of the three valons carries on average roughly $1/3$ the momentum of the proton, their sum being strictly 1. Details of the valon model, described in [@hy3], are not needed for the following. It is only necessary to recognize that the recombination of two $u$ quarks with a $d$ quark to form a proton has a probability proportional to the proton’s valon distribution that accounts for the proton structure. The other point to bear in mind is that the valon distribution in the proton is obtained in the frame where the proton momentum is infinitely large so the finite masses of the proton and valons are unimportant. However, the validity of that result when the proton momentum is only two or three times larger than its mass is questionable. With that caveat we proceed with our scale invariant calculation and see what can emerge. As discussed in the preceding section, there is no need to consider the color factors for either pion or proton formation since hadrons are color singlets, but the flavor octets for these hadrons do introduce some factors. The $\left.|uud\right>$ state appears in $10 + 8 + 8 ^{\prime}$ of $3 \times 3 \times 3$; among them the first two contain $\Delta ^+$ and $p$. Thus the flavor parts of $|\left<\Delta ^+\left|uud\right>|^2\right.$ and $|\left<p \left| uud\right>|^2\right.$ are $1/3$ for each. Since $\Delta ^+$ decays to $p + \pi^0$ and $n + \pi^+$, $|\left<p\left| \Delta ^+\right>|^2\right.$ gives another factor $1/2$. The spin decomposition of $2 \times 2 \times 2$ is $4 + 2 + 2$, among which the $\Delta ^+$ component is $4/8$ and $p$ is $2/8$. Putting the flavor and spin factors together, we have $$\begin{aligned} &&\left| \left<p \left| uud \right. \right>\right|^2 + \left| \left<p\, | \Delta ^+\right>\left< \Delta ^+ \left| uud \right. \right>\right|^2 \nonumber\\ &&\, \, \, \, \,= {1 \over 3}\times {1 \over 4} + {1 \over 3}\times {1 \over 2} \times {1 \over 2} = {1 \over 6} . \label{43}\end{aligned}$$ We thus normalize $R_p$, as we have done in Eq. (\[25\]), by $$\begin{aligned} \int^Z_0 \prod^3_{i = 1} {dz_i \over Z} \int dz\, z \,R_p (z_1, z_2, z_3, z) = {1 \over 6} . \label{44}\end{aligned}$$ In view of Eqs. (\[21\]) and (\[39\]) we have $$\begin{aligned} R_p^0\int^Z_0 \prod^3_{i = 1} {dz_i \over Z}\ \tilde{G}_p \left({z_1 \over z_t}, {z_2\over z_t}, {z_3\over z_t}\right) = {1 \over 6}, \label{45}\end{aligned}$$ where $z_t = \sum_i z_i$. Using Eq. (\[40\]), the above integral can be transformed to $$\begin{aligned} g \int^1_0 \prod^3_{i = 1} d \zeta _i \left({ \zeta_1 \zeta_2 \over \zeta ^2_t} \right)^{\alpha} \left({ \zeta_3 \over \zeta _t} \right)^{\beta} = 2.924 \label{46}\end{aligned}$$ with $\zeta _i = z_i/Z$ and $\zeta _t = \sum _i \zeta _i$. There is no explicit dependence on $Z$, and Eqs. (\[41\]) and (\[42\]) have been used in getting the numerical value in Eq.(\[46\]). It thus follows that $$\begin{aligned} R_p^0 = 0.057 . \label{47}\end{aligned}$$ At this point we should address the question why we consider $\Delta^+$ production above, but not $\rho$ production in the preceding section. For the production of $\pi^0$, if we are to consider the contribution from $\rho^\pm$ (since $\rho^0$ does not decay strongly into $\pi^0$), we would be extending our scope to other flavored states besides $u\bar u$ and $d\bar d$. Then other vector mesons and higher resonances, such as $K^*$, that can decay into $\pi^0$ must also be included. Similarly, for $p$ production the consideration of other states beside $uud$ would involve many resonances that can decay into $p$. The system is not closed without more phenomenological input beside $\pi^0$. Thus for a closed system in which a prediction can be made, we limit ourselves to only the $u\bar u$ and $d\bar d$ in the meson states and $uud$ in the baryon states; hence, only $\pi^0, \eta, p$ and $\Delta^+$ are considered. To include $u\bar d$ and $d\bar u$, we must also include $uuu$ and $udd$, and so on. We surmise that if more resonances are included in both the meson and baryon sectors, the $p/\pi$ ratio to be determined below would change somewhat; however, the result is not likely to differ by a factor greater than 2. With the recombination function $R_p$ completely determined, and the quark distribution $F_q \left(z_i\right)$ given by Eqs.(\[35\]) and (\[36\]), we can now use Eq. (\[37\]) to calculate the proton distribution in $z$. The result is shown by the solid line in Fig. 5, where only the portion $z > 2$ is exhibited. We have stated at the outset that the scale invariant form of $dN_p/zdz$ cannot be expected to be valid when the mass effect is important. The relevant value of $z$ corresponding to the proton mass (let alone the $\Delta ^+$ mass) is $$\begin{aligned} z_m = m_p/K , \label{48}\end{aligned}$$ which ranges from 1.3 at $\sqrt{s} = 17$ GeV to 0.94 at 200 GeV. As expected, the scaling violating effects are energy dependent. Thus we should not regard the calculated result to be reliable for $z < 3$. At very low $p_T$ the distributions of all hadrons can be given exponential fits in the transverse mass. The STAR data for most central collisions at $\sqrt{s}= 130$ GeV [@pj] give for $\bar{p}$ production for $p_T < 0.6$ GeV $$\begin{aligned} {1 \over 2 \pi m_T}{d^2N_{\bar{p}} \over dm_T dy} = 4 \exp \left[-\left(m_T-m_p\right)/T_p\right] \label{49}\end{aligned}$$ where $m_T = \left(m^2_p + p^2_T\right)^{1/2}$ and $T_p = 565$ MeV. To convert this distribution to that for $p$ we assume that only the normalization at $p_T = 0$ needs to be adjusted. The $\bar{p}/p$ ratio at low $p_T$ is 0.6 [@ts]. Since $m_Tdm_T = p_T dp_T$ and the distribution in $p_T$ changes by a scale factor $K^2$ given in Eq. (\[2\]), where $K = 0.9$ for $\sqrt{s} = 130$ GeV, the factor 4 in Eq. (\[49\]) should therefore be changed to $4 \times 0.81/0.6 = 5.4$. Expressing $m_T$ in terms of $z$ by use of Eq. (\[2\]) with $K = 0.9$, we show the $z$ dependence of the distribution for $p$ in Fig. 5 by the short dashed line. The region $0.5 < z <2$ is left blank because our scaling result cannot be reliably extended into that region. Nevertheless, it is gratifying to observe that the theoretical calculation without any free parameters produces a proton distribution at large $z$ that is reasonable in normalization and shape and can smoothly be connected with the low-$p_T$ distribution by interpolation. ![Proton distribution in $z$. Solid line is the theoretical result; the dashed line is the fit of data at low-$p_T$ [@pj].](fig5.eps){width="55.00000%"} With the proton distribution now at hand, we can calculate the $p/\pi$ ratio. For the pion distribution we use $\Phi (z)$ given in Eq. (\[6\]). For proton we use the calculated result based on Eq. (\[37\]). Their ratio, defined by $$\begin{aligned} R_{p/\pi}(z) = {dN_p \over zdz} / \Phi (z) \label{50}\end{aligned}$$ is shown by the solid line in Fig. 6. The preliminary data on the $p/\pi$ ratio were reported in Ref. [@ts], which we show also in Fig. 6 for both $\sqrt{s} = 130$ and $200$ GeV. Note that because it is a ratio there is no change in the normalizations of $R_{p/\pi}$ for the two energies, but in transforming from $p_T$ to $z$ the factor $K$ in Eq. (\[2\]) must be taken into account. Unlike the pion case the effects of the proton mass are not negligible for $p_T \stackrel{<}{\sim} 3$ GeV/c, and one sees no scaling in $s$ or $z$ in Fig. 6. Our scale invariant calculation is unreliable for $z < 3$ and shows a result that is obviously too high at $z \stackrel{<}{\sim} 2$. There seems to be a good chance that the theory and experiment can agree well for $z > 4$. In Fig. 6 we show two curves that can connect our scaling result with the data. The dotted curve is an eyeball fit of the 130 GeV data with a connection at $z= 3.5$, while the dashed curve fits the 200 GeV data with a connection at $z = 4$. In the absence of a theoretical study that takes the mass-dependent effects into account in the intermediate $p_T$ region, the only point we can make here is that it is not hard to produce a $p/\pi$ ratio that exceeds 1 in the scale invariant calculation in the recombination model, but it does so in a region where both theory and experiment need refinement. Judging by what is self-evident in Fig. 6, we see no strong need for any exotic mechanism for proton production (as proposed in [@vg]) beyond the conventional subprocess where three quarks recombine to form a proton. ![Proton-to-pion ratio: solid line is the scaling distribution from calculation; data (preliminary) are from Ref.[@ts]. The dotted and dashed lines are eyeball fits of the data as extrapolations from the scaling result.](fig6.eps){width="55.00000%"} Conclusion ========== The discovery of a scale invariant distribution $\Phi (z)$ for pion production at intermediate and high values of $p_T$ in heavy-ion collisions ranging over energies in excess of an order of magnitude of variation is an important phenomenology observation that should be checked experimentally in great detail. Additional energy points should be added not only to strengthen the validity of the scaling behavior, but also to find the onset of scaling violation, if it exists. The phenomenological properties of hadron production provide useful insights into the hadronization process and into the nature of the quark system just before they turn into hadrons. The usual approach to the study of heavy-ion collisions is from inside out, following the evolution of the dense matter, either in terms of hydrodynamical flow or of hard parton scattering and subsequent hadronization by fragmentation [@gvw]. Our approach pursued here is from outside in, by starting from the observed scaling behavior of the pions produced and deriving the momentum distribution of the quarks that can give rise to such a behavior. That is accomplished by use of the recombination model. There is no direct way to check the validity of the quark distribution thus obtained. However, we have used it to determine the proton distribution at high $p_T$ where the mass effects are unimportant. The data on proton production have not yet reached that regime where the predicted scaling distribution can be checked. In the region where data exist on the $p/\pi$ ratio we find that our calculated result, though not reliable, is in rough agreement with the imprecise data to the extent that the ratio exceeds 1, a feature that is notable. While the recombination model needs further work to take the proton mass into account at intermediate $p_T$, its formulation in the invariant form has been developed here to treat the very high $p_T$ region. We have made the assumption that the quark and antiquark distributions are the same, apart from normalization, just before recombination. That assumption is supported by the constancy of the $\bar{p}/p$ ratio in the PHENIX data in the central region. That experimental fact can also be used to lend credence to our general approach to hadronization that is treated as a recombination process, for which we have given arguments why the distributions of quarks and antiquarks should be similar before they recombine. In contrast, the fragmentation model would suggest a decreasing function of $\bar{p}/p$ in $p_T$ because of the dominance of quark jets over antiquark jets at large $p_T$ [@jet]. In this paper we have only considered the energy dependence of the $p_T$ spectrum at fixed maximum centrality. It is natural to ask what the dependence is on centrality. We have investigated that problem by making a phenomenological analysis of the data on centrality dependence without using any hadronization model, and found a scaling behavior very similar to what is reported here. The scaling distribution found there [@hy4] includes the very small $p_T$ region in the fit, and is therefore more accurate. But the fits in the intermediate and large $p_T$ region are the same. The implication on the centrality dependence of the $p/\pi$ ratio in the recombination model is still under study. To have an invariant quark distribution independent of $s$ just before hadronization provides an unexpected picture of the quark system. It suggests that the evolution of the system proceeds toward a universal form whatever the collision energy may be. We expect that universal form to depend on rapidity. The origin of such a scaling distribution in $z$ is not known at this point and can form the focus of a program of future theoretical investigations. Acknowledgment {#acknowledgment .unnumbered} ============== We wish to thank D. d’Enterria for a helpful communication. This work was supported, in part, by the U. S. Department of Energy under Grant No. DE-FG03-96ER40972. [99]{} T. Sakaguchi (PHENIX Collaboration), nucl-ex/0209030, talk given at Quark Matter 2002, Nantes, France (2002). K. P. Das and R. C. Hwa, Phys. Lett. [**68B**]{}, 459 (1977); R. C. Hwa, Phys. Rev. D[**22**]{}, 1593 (1980). For a recent review see P. Houvinen, Proc. of Quark Matter 2002, Nantes, France (2002). 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D. d’Enterria (PHENIX Collaboration), hep-ex/0209051, talk given at Quark Matter 2002, Nantes, France (2002). K. Reygers (WA98 and PHENIX Collaborations), nucl-ex/0202018. We are grateful to D. d’Enterria for a helpful communication. Z. Koba, H. B. Nielsen, and P. Olesen, Nucl Phys. [**B40**]{}, 317 (1972). UA5 Collaboration, G. J. Alner [*et. al.*]{}, Phys. Lett. [**138B**]{}, 304 (1984); E735 Collaboration, T. Alexopoulous [*et. al.*]{}, Phys. Lett. [**B435**]{}, 453 (1998). R. C. Hwa and C. B. Yang, nucl-th/0301004 v2. R. C. Hwa, and C. B. Yang, Phys. Rev. C [**65**]{}, 034905 (2002). R. C. Hwa, and C. B. Yang, Phys. Rev. C [**66**]{}, 025204 (2002). P. G. Jones (STAR Collaboration), Proc. of [*The Physics of the Quark-Gluon Plasma*]{}, Palaiseau, France, 2001, p.9. M. Gyulassy, I. Vitev, and X.-N. Wang, Phys.Rev. Lett. [**86**]{}, 2537 (2001).

ArXiv

--- abstract: 'An operator $T$ from vector lattice $E$ into vector topology $(F,\tau)$ is said to be order-to-topology continuous whenever $x_\alpha\xrightarrow{o}0$ implies $Tx_\alpha\xrightarrow{\tau}0$ for each $(x_\alpha)_\alpha\subset E$. The collection of all order-to-topology continuous operators will be denoted by $L_{o\tau}(E,F)$. In this paper, we will study some properties of this new classification of operators. We will investigate the relationships between order-to-topology continuous operators and others classes of operators such as order continuous, order weakly compact and $b$-weakly compact operators.' author: - Kazem Haghnejad Azar date: 'Received: date / Accepted: date' title: 'Order-to-topology continuous operators' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction ============ In locally solid vector lattice, topologies for which order convergence implies topological convergence are very useful. They are known as order continuous topologies. A linear topology $\tau$ on a vector lattice is said to be order continuous whenever $x_\alpha\xrightarrow{o}0$ implies $x_\alpha\xrightarrow{\tau}0$. In normed vector lattice, it is also favourite to us, when order convergence is norm convergent. A normed lattice $E$ has order continuous norm if $\| x_\alpha\|\rightarrow 0$ for every decreasing net $(x_\alpha)_\alpha$ with $\inf_\alpha x_\alpha=0$. Let $E$ be a vector lattice and $(F,\tau)$ be a vector topology. In this manuscript, we will investigate on operators $T:E\rightarrow F$ which carrier every order convergence net into topological convergence. To state our results, we need to fix some notation and recall some definitions. A net $(x_{\alpha})_{\alpha \in A}$ in a vector lattice $ E $ is said to be strongly order convergent to $x\in E$ if there is a net $(z_{\beta})_{\beta \in B} $ in $ E $ such that $ z_{\beta} \downarrow 0 $ and for every $ \beta \in B$, there exists $\alpha_{0} \in A$ such that $ | x_{\alpha} - x |\leq z_{\beta}$ whenever $ \alpha \geq \alpha_{0}$. For short, we will denote this convergence by $ x_{\alpha} \xrightarrow{so} x $ and write that $ x_{\alpha} $ is $so$-convergent to $x$. Obviusely every order convergence net in a vector lattice is strongly order convergent, but converse not holds and for Dedekind complete vector lattice both definitions are the same, for detile see [@1b]. A net $ (x_{\alpha})_{\alpha}$ in vector lattice $ E $ is unbounded order convergent to $ x \in E $ if $ | x_{\alpha} - x | \wedge u \xrightarrow{so} 0$ for all $ u \in E^{+} $. We denote this convergence by $ x_{\alpha} \xrightarrow{uo}x $ and write that $ x_{\alpha} $ $uo$-convergent to $ x $. It is clear that for order bounded nets, $uo$-convergence is equivalent to $so$-convergence. In [@7], Wickstead characterized the spaces in which $w$-convergence of nets implies $uo$-convergence and vice versa and in [@5g1], characterized the spaces $E$ such that in its dual space $ E^{\prime} $, $uo$-convergence implies $w^{*}$-convergence and vice versa. A Banach lattice $E$ is said to be an $AM$-space if for each $x,y\in E$ such that $|x|\wedge |y|=0$, we have $\|x+y\|= max \{\|x\|, \|y\|\}$. A Banach lattice $E$ is said to be $KB$-space whenever each increasing norm bounded sequence of $E^+$ is norm convergent. An operator $T: E\rightarrow F$ between two vector lattices is positive if $T(x)\geq 0$ in $F$ whenever $x\geq 0$ in $E$. Note that each positive linear mapping on a Banach lattice is continuous. In this manuscript $L_b(E,F)$ is the all of bounded operators and the collection of all order continuous operators of $L_b(E,F)$ will be denoted by $L_n(E,F)$; the subscript $n$ is justified by the fact that the order continuous operators are also known as normal operators. That is, $$L_n(E,F) :=\{T \in L_b(E,F): T~ \text{is~ order~ continuous}\}.$$ Similarly, $L_c(E,F)$ will denote the collection of all order bounded operators from $E$ to $F$ that are $\sigma-$order continuous. An operator $T$ from a Banach space $X$ into a Banach space $Y$ is compact (resp. weakly compact) if $\overline{{T(B _ X)}}$ is compact (resp. weakly compact) where $B _ X$ is the closed unit ball of $X$. A continuous operator from Banach lattice $E$ into Banach space $X$ is called $M$-weakly compact if $\lim \Vert Tx_n\Vert=0$ holds for every norm bounded disjoint sequence $(x_n)_n$ of $E$. A subset $A$ of a vector lattice $E$ is called $b$-order bounded in $E$ if it is order bounded in $E^{\sim\sim}$. An operator $T:E\rightarrow X$, mapping each $b$-order bounded subset of $E$ into a relatively weakly compact subset of $X$ is called a $b$-weakly compact operator, see [@3]. An operator $T:E\rightarrow X$ from vector lattice into normed space is called interval-bounded if the image of every order interval is norm bounded. For every interval-bounded linear operator $T:E\rightarrow X$, set $$q_T(x)=\sup\{\Vert Ty\Vert:~\vert y\vert\leq \vert x\vert\},$$ be the absolute monotone seminorm induced by $T$ where $x\in E$. For terminology concerning Banach lattice theory and positive operators, we refer the reader to the excellent book of [@1].\ Main results ============ Let $E$ be a vector lattice and $F$ be a vector topology with topology $\tau$. An operator $T$ from $E$ into $F$ is said to be order-to-topology continuous whenever $x_\alpha\xrightarrow{o}0$ implies $Tx_\alpha\xrightarrow{\tau}0$ for each $(x_\alpha)_\alpha\subset E$. For each sequence $(x_n)\subset E$, if $x_n\xrightarrow{o}0$ implies $Tx_n\xrightarrow{\tau}0$, then $T$ is called $\sigma$-order-to-topology continuous operator. The collection of all order-to-topology continuous operators will be denoted by $L_{o\tau}(E,F)$; the subscript $o\tau$ is justified by the fact that the order-to-topology continuous operators; that is, $$L_{o\tau}(E,F)=\{T\in L(E,F):~T~\text{is order-to-topology continuous }\}.$$ Similarly, $L^\sigma_{o\tau}(E,F)$ will be denote the collection of all $\sigma$-order-to-topology continuous operators, that is, $$L^\sigma_{o\tau}(E,F)=\{T\in L(E,F):~T~\text{is} ~\sigma-\text{order-to-topology continuous }\}.$$ For a normed space $F$, we write $L_{on}(E,F)$ and $L_{ow}(E,F)$ for collection of order-to-norm topology continuous operators and order-to-weak topology continuous operators, respectively. $L^\sigma_{on}(E,F)$ and $L^\sigma_{ow}(E,F)$ have similar definitions. Clearly $L^\sigma_{on}(E,F)$ is a subspace of $L^\sigma_{ow}(E,F)$ and if $F$ has the Schur property, then $L^\sigma_{on}(E,F)=L^\sigma_{ow}(E,F)$. Let $T$ be an order-to-norm topology continuous operators from a vector lattice $E$ into a normed vector lattice $F$ and $0\leq S\leq T$ where $S\in L(E,F)$. Then observe that $S$ is an order-to-norm topology continuous operator. It is clear that for a locally solid vector lattice $E$, if $E$ has order continuous topology, then every continuous operator from $E$ into a vector topology $F$ is order-to-topology continuous. If an operator $T:E\rightarrow F$ from a Banach lattice into a normed vector lattice is positive (and so continuous), in general, $T$ is not order-to-norm continuous as shown in the following examples. 1. Consider the operator $T:\ell^\infty\rightarrow\ell^\infty$ defined by $T((a_n)_n)=(ca_n)_n$ where $c>0$ and $\ell^\infty$ equipped with norm $\|.\|_\infty$. We observe that $T$ is positive (and so continuous). Now let $x_n=(0,0,...0,1,1,..)$, which have first $n$ zero terms and all the other terms equal one. It is clear that $x_n \downarrow 0$ and $\Vert x_n\Vert=1$. It follows that $T$ is not order-to-norm continuous. 2. If $E=C([0,1])$ and $f_n:~f_n(t)=t^n$ for all $t\in [0,1]$, then $f_n \downarrow 0$ as $n\rightarrow \infty$ and $\Vert f_n\Vert=1$. Now we define an operator $T:C([0,1])\rightarrow C([0,1])$ with $T(f)=f$ which is positive (and so continuous), but $T$ is not order-to-norm continuous. The following example that is inspired from example 3.3 in [@alha2018] shows that an order-to-norm continuous operator need not to be continuous. Let $E=\left\{ (a_n)\in c_0 \mid (na_n)\in c_0\right\}$. It is easy to see that $E$ is a normed vector sublattice of Banach lattice $c_0$. We define operator $T$ from $E$ into $c_0$ as follows $$T(a_1, a_2, a_3,\cdots)=(a_1, 2a_2,3a_3,\cdots).$$ Clearly, $T$ is not continuous. Now, let $(x_\alpha)$ be a net in $E$ such that $x_\alpha\xrightarrow{o}0$. So there exists some $(y_\alpha)\subset E$ such that $|x_\alpha|\leq y_\alpha\downarrow 0$ for all $\alpha$. We have $|Tx_\alpha |\leq T|x_\alpha|\leq Ty_\alpha$. On the other hand, if we denote $n$-th term of $y_\alpha$ by $y_\alpha(n)$, then we have $$\begin{aligned} y_\alpha\downarrow 0 & \iff y_\alpha(n)\downarrow 0 \qquad (\forall n\in\mathbb{N}) \\ & \iff ny_\alpha(n)\downarrow 0 \qquad (\forall n\in\mathbb{N})\\ & \iff Ty_\alpha \downarrow 0.\end{aligned}$$ Therefore, $|Tx_\alpha |\leq Ty_\alpha \downarrow 0$. Since $c_0$ has order continuous norm, $\|Ty_\alpha\|\to 0$. Now, it follows from $\|Tx_\alpha\|\leq\|Ty_\alpha\|\to 0$ that $T$ is order-to-norm continuous. As next example, we observe that a $\sigma$-order-to-topology continuous operators need not be order-to-topology continuous operators, see Example 1.55 from [@1]. Let $E$ be the vector space of all Lebesgue integrable real valued function defined on $[0,1]$. Define the operator $T:E\rightarrow \mathbb{R}$ by $$T(f)=\int_0^\infty f(x)dx.$$ If $f_n\downarrow 0$, clearly $ \vert Tf_n \vert\rightarrow 0$. Thus $T\in L_{on}^\sigma(E,\mathbb{R} )$. Now, let $\Lambda$ denotes the collection of all finite subsets of $[0,1]$. Then $\{\chi_\alpha~:~\alpha\in \Lambda\}\subseteq E$ (where $\chi_\alpha$ is the characteristic function of $\alpha$) satisfies $1-\chi_\alpha\downarrow 0$. On the other hand, observe that $T(1-\chi_\alpha )=1$ which shows that $T\notin L_{on}(E,\mathbb{R})$. \[2.5\] Let $T:E\rightarrow F$ be an order bounded operator between two normed vector lattices. By one of following conditions $T^-, T^+, T \text{and } \vert T\vert\in L_{on}(E,F)$. 1. $F$ has order continuous norm and $T\in L_n(E,F)$ 2. $E$ is a Banach lattice with order continuous norm and $F$ is Dedekind complete. <!-- --> 1. Since $F$ has order continuous norm by Corollary 4.10 from [@1], $F$ is Dedekind complete. It follows that $T^-$ and $T^+$ exist. Now, let $(x_\alpha)\subset E^+$ and $x_\alpha\downarrow 0$. Since $T$ is order continuous, it follows $T^-x_\alpha\downarrow 0$ and $T^+x_\alpha\downarrow 0$. As $F$ has order continuous norm, we have $\|T^-x_\alpha\|\rightarrow 0$ and $\|T^+x_\alpha\|\rightarrow 0$, and so by using inequalities $\|Tx_\alpha\|\leq\|T^+x_\alpha\|+\|T^-x_\alpha\|$ and $\| \vert T \vert x_\alpha\|\leq\|T^+x_\alpha\|+\|T^-x_\alpha\|$ proof holds. 2. Let $(x_\alpha)\subset E^+$ and $x_\alpha\downarrow 0$. By Theorem 4.3, from [@1], $T^+$ is norm continuous. Consequently $\|T^+x_\alpha\|\leq \|T^+\|\|x_\alpha\|\rightarrow 0$, since $E$ has order continuous norm. Similarly, $\|T^-x_\alpha\|\rightarrow 0$, and so proof follows immediately. The next example shows that order continuity of $T$ in above proposition, can not in general be dropped. Let $E$ be the collection of real functions on $[0,1]$ that are continuous except on a finite subset of $[0,1]$. Let $T:E\rightarrow L^1([0,1])$ be an operator with $T(f)=f$. We know that $L^1([0,1])$ has order continuous norm and $T$ is order bounded. Let $Q\cap [0,1]=\{r_1,~r_2,~r_3,\ldots\}$ and $F_n=\{r_1,~r_2,\ldots,r_n\}$. We define the sequence of functions in $E$ as follows: $$\begin{aligned} f_n(x)=\left\lbrace \begin{array}{lc} 0\quad \text{if}~~~x\in F_n\\ 1\quad\text{if}~~~x\in [0,1]\setminus F_n \end{array}\right. \end{aligned}$$ Clearly $f_n\downarrow 0$, but $\|Tf_n\|=1$. Thus $T$ is not $\sigma$-order-to-topology continuous operator. Recall that an operator $T:E\rightarrow F$ between two vector lattices is said to preserve disjointness whenever $x\bot y$ in $E$ implies $Tx\bot Ty$ in $F$. By using Theorem 3.1.4, from [@6], we have the following result. \[2.7\] Let $E$ and $F$ be normed vector lattice and $T$ be an order bounded disjointness operator from $E$ into $F$. Then $T\in L_{on}(E,F)$ if and only if $T^-, T^+ \text{and } \vert T\vert\in L_{on}(E,F)$. By notice to preceding theorem, for $1\leq p, q<\infty$ with $\frac{1}{p}+ \frac{1}{q}=1$, we have the following assertions. 1. $L_{on}(\ell_p, \mathbb{R})=\ell_q$. 2. $L_{on}(L_p([0,1]), \mathbb{R})=L_q([0,1])$. <!-- --> 1. Dose the modulus of an operator $T$ from vector lattice $E$ into normed vector lattice $F$ exists and $\vert T\vert\in L_{on}(E, F)$ or $\vert T\vert\in L_{ow}(E, F)$ whenever $T\in L_{on}(E, F)$ or $T\in L_{ow}(E, F)$, respectively? 2. Is $ L_{on}(E, F)$ a band in $ L_{ow}(E, F)$? \[2.7\] Let $E$ be a vector lattice and $F$ be a Dedekind complete normed vector lattice. If $T$ is positive then $T\in L_{on}(E,F)$ if and only if $x_\alpha\downarrow 0$ implies $\|Tx_\alpha\|\rightarrow 0$. We just prove one side. The other side is clear. Let $(x_\alpha)$ be a net in $E$ such that $x_\alpha\xrightarrow{o}0$. So there exists some $(y_\alpha)\subset E$ such that $|x_\alpha|\leq y_\alpha\downarrow 0$ for all $\alpha$. So $|Tx_\alpha |\leq T|x_\alpha|\leq Ty_\alpha$ for all $\alpha$. Thus, $\|Tx_\alpha \|\leq \|Ty_\alpha \|$ for all $\alpha$. By our hypothesis $\|Ty_\alpha \| \to 0$. Hence $\|Tx_\alpha \|\to 0$; that is, $T\in L_{on}(E,F)$. \[t:2.8\] Let $E$ be a $\sigma$-Dedekind complete vector lattice and $F$ be a normed vector lattice. An order bounded operator $T:E\rightarrow F$ is $\sigma-$order-to-norm continuous if and only if $0<x_n\uparrow \leq x$ in $E$ implies $(Tx_n)$ is norm convergent to $T(\sup_nx_n)$. Let $T$ be a $\sigma$-order-to-norm continuous operator and let $(x_n)\subset E^+$ and $x\in E$ where $0<x_n\uparrow \leq x$. We set $\sup x_n=y$. It follows $(y-x_n)\downarrow 0$, and so $\|T(y-x_n)\|\rightarrow 0$. Thus $(T x_n)$ is norm convergent to $Ty$.\ Conversely, by preceding Lemma, without lose generality assume that $T$ is a positive operator. Let $(x_n)\subset E^+$ with $x_n\downarrow 0$. We set $y_n=x_1-x_n$. Observe that $0\leq y_n\uparrow \leq x_1$. Thus $(Ty_n)$ is norm convergent to $Tx_1$, since $\sup_ny_n=x_1$, which follows that $(Tx_n)$ is norm convergent. Therefore, $T\in L^\sigma_{on}(E,F)$. \[2.8\] Let $E$ be a $\sigma$-Dedekind complete vector lattice and $F$ be a normed vector lattice. If $T$ is interval-bounded, then $T\in L^\sigma_{on}(E,F)$ if and only if $T$ is order weakly compact. By Theorem 3.4.4 from [@6], $T$ is order weakly compact operator if and only if $(Tx_n)$ is convergent for every order bounded increasing sequence $(x_n)$ in $E^+$. So by the Theorem \[t:2.8\] proof holds. \[2.12\] Let $E$ and $F$ be normed vector lattice with $F$ Dedekind complete. If $E$ or $F$ has order continuous norm, then $L_n(E,F)$ is a band in $L_{on}(E,F)$. Let $T\in L_n(E,F)$. It follws $T^-,~T^+\in L_n(E,F)$, and so by using Theorem 4.3, [@1], $T^-$ and $T^+$ are norm continuous. As $E$ has order continuous norm, $T^-,~T^+\in L_{on}(E,F)$, and so $T\in L_{on}(E,F)$. Thus $L_n(E,F)$ is a subspace of $L_{on}(E,F)$. Now let $\vert S\vert\leq\vert T\vert$ where $S\in L_{n}(E,F)$ and $T\in L_{on}(E,F)$. Then for each $x\in E$, we have $\vert Sx\vert\leq \vert S \vert(\vert x\vert )\leq \vert T \vert (\vert x\vert )$. It follows that $$\|Sx\|\leq \| \vert Sx\vert \|\leq \| \vert S \vert(\vert x\vert ) \|\leq \| \vert T \vert (\vert x\vert )\|\leq \| T \| \|x\|$$ The above inequalities shows that $S$ is norm-to-norm continuous. Let $x_\alpha \downarrow 0$. As $E$ has order continuous norm, it follows that $\| x_\alpha\|\rightarrow 0$, and so $\| Sx_\alpha\|\rightarrow 0$. Than $S\in L_{on}(E,F)$. Thus $L_n(E,F)$ is an ideal in $L_{on}(E,F)$. To see that the ideal $L_n(E,F)$ is a band, let $0\leq T_\lambda\uparrow T$ in $L_{on}(E,F)$. As $T$ is positive, $T$ is norm continuous. Since $E$ has order continuous norm, proof follows immediately.\ Now if $F$ has order continuous norm, we have similar argument. \[2.13\] Let $E$ be a vector lattice and $F$ a normed vector lattice which every norm null net in $F$ is order bounded. Then 1. $L_{on}(E,F)$ is an ideal in $L_{b}(E,F)$. 2. If $L_{on}(E,F)$ is order dense in $L_b(E,F)$, then $L_b(E,F)=L_n(E,F)$, and if $E$ has also norm continuous and $F$ is Dedekind complete, then $L_{on}(E,F)=L_n(E,F)=L_b(E,F)$. <!-- --> 1. Let $x\in E^+$ and consider a net $(x_\alpha)_\alpha$, where $x_\alpha =x-\alpha$ for each $\alpha\in [0,x]$. It follows that $x_\alpha\downarrow 0$. If $T\in L_{on}(E,F)$, then $(Tx_\alpha)$ is null-norm convergent in $F$. It follows that $(Tx_\alpha)_\alpha$ is norm bounded in $F$. By assumption, $(Tx_\alpha)_\alpha$ is order bounded, which shows that $L_{on}(E,F)$ is a subspace of $L_{b}(E,F)$. Now let $\vert S\vert\leq\vert T\vert$ where $T\in L_{b}(E,F)$ and $S\in L_{on}(E,F)$. Then for each $x\in E$, we have $\vert Sx\vert\leq \vert S \vert(\vert x\vert )\leq \vert T \vert (\vert x\vert )$. It follows that $S$ is order bounded, and so $L_{on}(E,F)$ is an ideal in $L_{b}(E,F)$. 2. Assume that $T\in L_b(E,F)$. Since $L_{on}(E,F)$ is order dense in $L_b(E,F)$, there is a $(T_\lambda)_\lambda\subset L_{on}(E,F)$ such that $0\leq T_\lambda\uparrow T$ in $L_{b}(E,F)$. Let $x_\alpha \downarrow 0$ in $E$. Then for each fixed index $\lambda$ we have $\Vert T_\lambda (x_\alpha)\Vert\rightarrow 0$ and $T_\lambda(x_\alpha)\downarrow$, and so by Theorem 5.6, from [@2], we have $T_\lambda(x_\alpha)\downarrow 0$. It follows that $(T_\lambda)_{\lambda}\subset L_n(E,F)$. Then by using Theorem 1.57 in [@1], we have $T\in L_n(E,F)$. It follows that $L_b(E,F)=L_n(E,F)$. Now if $E$ has order continuous, then proof follows immediately from Theorem 5.6 from [@2]. As notice to Theorem \[2.12\] and Theorem \[2.13\], we have the following consequence \[2.8\] Let $E$ be a normed vector lattice. Then 1. $L_n(c_0,\ell^\infty)$ is a band in $L_{on}(c_0,\ell^\infty)$. 2. $L_{on}(c_0,\ell^\infty)$ is an ideal in $L_b(c_0,\ell^\infty)$. 3. $L_{on}(E,\mathbb{R})$ and $L_{on}(E,\ell^\infty)$ are ideal in $ E^\prime$ and $ L_b(E,\ell^\infty)$, respectively where $E^\prime$ is norm dual of $E$. 4. if $E$ has order continuous norm, then $E_n^\thicksim =E^\thicksim=L_{on}(E,\mathbb{R}) $. \[2.14\] Let $E$ and $F$ be a vector normed lattices and $F$ Dedekind complete. Let $T\in L_b(E,F)$. Then the following assertions are equivalent. 1. $E$ is Dedekind $\sigma$-complete and $x_n\downarrow 0$ in $E$ implies $\Vert Tx_n\Vert\rightarrow 0$. 2. If $0<x_n\uparrow \leq x$ holds in $E$, then $(Tx_n)$ is norm convergent. 3. $T\in L_{on}(E,F)$ In the following, without lose generality, we assume that $T$ is positive operator.\ $(1)\Rightarrow (2)$ Let $(x_n)\subset E$ and $0<x_n\uparrow \leq x$ holds in $E$. Set $\sup x_n=y$. It follows $y-x_n\downarrow 0$. By hypothesis, we have $\Vert T(y-x_n)\Vert \to 0$. It follows that $(Tx_n)$ is norm convergent.\ $(1)\Rightarrow (3)$ By lemma \[2.7\], it is enough to prove that $Tx_\alpha$ is norm convergent to zero in $F$ whenever $x_\alpha\downarrow 0$ in $E$. Let $(x_\alpha)_\alpha\subset E^+$ with $x_\alpha \downarrow 0$. If $(Tx_\alpha)_\alpha$ is not norm convergent, then it is not a norm Cauchy net. Thus there exists some $\epsilon>0$ and a sequence $(\alpha_n)$ of indices with $ \alpha_n\uparrow$, such that $\| Tx_{\alpha_n}-Tx_{\alpha_{n+1}}\|>\epsilon$ for all $n$. On the other hand, since $E$ is Dedekind $\sigma$-complete there is $x\in E$ such that $x_{\alpha_n}\uparrow x$. It follows $x-x_{\alpha_n}\downarrow 0$. By hypothesis, we see that $Tx_{\alpha_n}$ is norm Cauchy sequence, which contradicts with $\|Tx_{\alpha_n}-Tx_{\alpha_{n+1}}\|>\epsilon$. Thus $(Tx_\alpha)_\alpha$ is norm convergent to some point $y\in F$. By Theorem 5.6, [@2], we see that $y=0$, and so $T\in L_{on}(E,F)$.\ $(3)\Rightarrow (1)$ Obviously.\ $(2)\Rightarrow (1)$ Let $x_n\downarrow 0$. Then $0\leq (x_1-x_n)\uparrow x_1$ holds in $E$. Then we have $0\leq T(x_1-x_n)\uparrow Tx_1$. By assumption the sequence $\{T(x_1-x_n))$ is norm convergent, and so Theorem 4.9 from [@2], follows that $\Vert Tx_n\Vert\rightarrow 0$. \[2.9\] Assume that $T:E\rightarrow F$ is an order bounded operator from $\sigma$-Dedekind complete vector lattice into normed vector lattice. 1. $T\in L^\sigma_{on}(E,F)$ if and only if $T$ is order weakly compact. 2. If $T\in L^\sigma_{on}(E,F)$, then $T$ is $M$-weakly compact. <!-- --> 1. By using Theorem 3.4.4 from [@6] and Theorem \[2.14\] proof follows. 2. By Theorem 5.57, [@1], proof follows. \[2.y\] Let $T: E\rightarrow X$ be an order bounded bounded operator from $\sigma$-Dedekind complete vector lattice into Banach space. Then the following assertions are equivalent. 1. $q_T(x_n)\rightarrow 0$ as $n\rightarrow 0$ for every order bounded disjoint sequence. 2. $\vert Tx_n\vert\rightarrow 0$ as $n\rightarrow 0$ for every order bounded disjoint sequence. 3. If $0<x_n\uparrow \leq x$ holds in $E$, then $(Tx_n)$ is norm convergent. 4. $T$ is order-weakly compact. 5. $T\in L^\sigma_{on}(E,F)$. By using Theorem 3.4.4 from [@6] and Corollary \[2.8\] proof holds. Alpay-Altin-Tonyali introduced the class of $b$-weakly compact operators for vector lattices having separating order duals [@3]. In [@4], Alpay and Altin proved that a continuous operator $T$ from a Banach lattice $E$ into a Banach space $X$ is $b$-weakly compact if and only if $(Tx_n)_n$ is norm convergent for each $b$-order bounded increasing sequence $(x_n)_n$ in $E^+$ if and only if $(Tx_n)_n$ is norm convergent to zero for each $b$-order bounded disjoint sequence $(x_n)_n$ in $E^+$. In [@5] authors proved that an operator $T$ from a Banach lattice $E$ into a Banach space $X$ is $b$-weakly compact if and only if $(Tx_n)_n$ is norm convergent for every positive increasing sequence $(x_n)_n$ of the closed unit ball $B_E$ of $E$. Now in the following we study the relationships between two classifications of operators, $b-$weakly compact and $\sigma-$order-to-norm continuous operators. \[2\] Let $E$ and $F$ be normed vector lattice, and let $T$ be an operator from $E$ into $F$. Then the following assertions hold 1. If $F$ is Dedekind $\sigma$-complete, then each positive $b$-weakly compact operator is $\sigma$-order-to-norm continuous operator. 2. If $E$ has a order unit with Dedekind $\sigma$-complete, then each $\sigma$-order-to-norm continuous operator is $b$-weakly compact operator <!-- --> 1. Let $T$ be a $b$-weakly compact operator and $(x_n)_n\subset E$ with $x_n \downarrow 0$. Set $y_n=x_1-x_n$. Then $y_n\uparrow$ and $\sup_n\|y_n\|\leq \Vert x_1\Vert$. It follows that $(T(y_n))_n$ is norm convergent and $Ty_n\uparrow$. By using Theorem 3.46 from [@1], $(T(y_n))_n$ norm convergence to $Tx_1$. Thus $(T(x_n))_n$ is norm convergent to $0$. 2. Let $(x_n)_n$ be an increasing positive sequence in $E$ with $\sup_n\Vert x_n\Vert<\infty$. Since $E$ has order unit, $(x_n)_n$ is order bounded, and so there is a positive element $x\in E$ such that $x_n \uparrow \leq x$. Then there exists $y\in E$ such that $\sup_n x_n=y$. It follows that $0\leq y-x_n\downarrow 0$. As $T\in L_{on}^\sigma (E,F)$, we have $\Vert T(y-x_n)\Vert\rightarrow 0$ which implies $(T(x_n))_n$ is norm convergence, and so $T$ is $b$-weakly compact. It is clear that the identity operator $ I:l^1 \rightarrow l^1 $ is an order-to-norm continuous operator, but its adjoint $ I:l^\infty \rightarrow l^\infty $ is not order-to-norm continuous. Note that the identity operator $ I : l^\infty \rightarrow l^\infty $ is not order-to-norm continuous, while its adjoint is order-to-norm continuous. The following results, give a sufficient and necessary condition for which the order-to-norm continuity of an operator implies the order-to-norm continuity of its adjoint and reverse. Let $E$ and $ F$ be two Banach lattices with $E$ Dedekind complete. Then the following conditions are equivalent. 1. Each continuous operator from $ F^\prime $ into $ E^\prime$ is order-to-norm continuous. 2. If a continuous operator $ T:E\rightarrow F $ is an order-to-norm continuous operator, then its adjoint $ T^\prime $ is order-to-norm continuous. 3. $ F^\prime $ is a $KB$-space. $1 \Rightarrow 2$ Obvious. $2 \Rightarrow 3$ Assume by the way of contradiction, that $ F^\prime $ is not a $KB$-space. By Lemma 2.1 of [@dualbweak] there is a positive order bounded disjoint sequence $(g_n)$ of $F^\prime$ satisfying $\|g_n\| =1 $. Since $\|g_n\|=\sup\{g_n(y): 0\leq y \in F \text{ and } \|y\|\leq 1 \}$ holds for all $n$, we choose $y_n \in F^+$ with $\|y_n \|=1$ and $g_n(y_n) \geq \frac{1}{2}$. Now, we consider a positive operator $ T:l^1 \rightarrow F $ defined by $$T((a_n))= \sum_{n=1}^{\infty} a_ny_n \text{ for all } (a_n)\in l^1.$$ Clearly, $ T $ is well defined. Since $ T $ is positive, therefore $T$ is continuous and hence $ T $ is order-to-norm continuous. But its adjoint $ T^\prime : F^\prime \rightarrow l^\infty $ defined by $$T^\prime(h)= (h(y_n)) \text{ for all } h\in F^\prime .$$ As follows, we shows that $T^\prime$ is not order-to-norm continuous. Since $(g_n)$ is disjoint, by using Corollary 3.6 from [@5g] obvious $g_n \xrightarrow{uo} 0$ in $F^\prime$. Now since $ (g_n)$ is order bounded, it is clear that $g_n \xrightarrow{o} 0 $ in $ F^\prime$. On the other hands, we have $$\|T^\prime (g_n)\|= \|(g_n(y_m))_m\| \geq\| g_n(y_n)\|\geq \frac{1}{2} \ holds \ for \ all \ n.$$ Therefore $T^\prime$ is not order-to-norm continuous. $3\Rightarrow 1$ Obvious. Let $ E $ be a vector lattice with Dedekind complete and property (b), and $F$ be a Banach lattice. Then the following statements are equivalent. 1. Each continuous operator from $ E $ into $ F $ is order-to-norm continuous. 2. Each continuous operator $ T : E\rightarrow F $ is order-to-norm continuous whenever its adjoint $ T^\prime $ is order-to-norm continuous. 3. $ E $ is a $KB$-space $ 1 \Rightarrow 2 $ Obvious. $ 2 \Rightarrow 3 $ Let $ E $ is not $KB$-space. To finish the proof, we have to construct an operator $ T:E \rightarrow F $ such that $ T $ is not order-to-norm continuous but its adjoint $ T^\prime $ is order-to-norm continuous. Since $ E $ is not $KB$-space then, it follows from Lemma 2.1 of [@dualbweak] that $ E^+$ contains a b-order bounded disjoint sequence $ (x_n) $ satisfying $\| x_n \| =1$ for all $n$. So by Lemma 3.4 of [@dualbweak], there exists a positive disjoint sequence $ (g_n)$ of $ E^\prime $ with $\|g_n\| \leq 1 $ such that $$g_n (x_n)=1 \ for\ all\ n\ and\ g_n(x_m)=0\ for\ all\ n\neq m.$$ Now we defined the positive operator $ T : E \rightarrow l^\infty $ by $$T(x) = (g_n(x))_{n=1} ^ \infty\ for\ each\ x\in E.$$ Note that $ (g_n(x))_{n=1}^\infty \subset R $ and $ \sup_n | g_n (x)| \leq \sup_n \| g_n \| \|x\| \leq \|x\| < \infty$. So, $ T $ is well defined. It is clear that $(x_n) $ is order bounded. By using Corollary 3.6 from [@5g], obvious $ x_n \xrightarrow{uo} 0$, and thus $ x_n \xrightarrow{o} 0 $ in $ E$. Since $ T(x_n) = e_n $, therefore $T(x_n) \nrightarrow 0 $ in norm. It follows that $ T $ is not order-to-norm continuous. Since the norm of $(l^\infty)^\prime$ is order continuous, it follows that $ T^\prime : (l^\infty)^\prime \rightarrow E^\prime $ is order-to-norm continuous. $3 \Rightarrow 1$ Obvious. [HD]{} Y. Abramovich and G. Sirotkin, *On order convergence of nets*, Positivity [**9**]{}, 287-292 (2005). R. Alavizadeh and K. Haghnejad Azar, *On b-order Dunford-Pettis operators and the b-AM-compactness property*, Acta Math. Univ. Comen., in press C. D. Aliprantis and O. Burkinshaw, *Positive Operators*, Springer, Berlin (2006) C. D. Aliprantis and O. Burkinshaw, *Locally Solid Riesz spaces*, Springer, Berlin (1978) S. Alpay and B. Altin, C. Tonyali, *On property (b) of vector lattices*, Positivity [**7**]{}, 135-139 (2003) S. Alpay and B. Altin, *A note on b-weakly compact operators*, Positivity [**11**]{}, 4, 575-582 (2007) B. Aqzzouz, A. Elbour and J. Hmichane, *The duality problem for the Class of b-weakly compact operators*, Positivity, [**13**]{}, 683-692 (2009) B. Aqzzouz, M. Moussa and J. Hmichane, *Some Characterizations of b-weakly compact operators on Banach lattices*, Math. Reports, [**62**]{}, 315-324 (2010) N. Gao, *Unbounded order convergence in dual spaces*, J.Math. Anal. 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ArXiv

--- abstract: 'We present ALMA observations of two moderate luminosity quasars at redshift 6. These quasars from the Canada-France High-z Quasar Survey (CFHQS) have black hole masses of $\sim 10^{8} M_\odot$. Both quasars are detected in the [\[C[ii]{}\]]{} line and dust continuum. Combining these data with our previous study of two similar CFHQS quasars we investigate the population properties. We show that $z>6$ quasars have a significantly lower far-infrared luminosity than bolometric-luminosity-matched samples at lower redshift, inferring a lower star formation rate, possibly correlated with the lower black hole masses at $z=6$. The ratios of [\[C[ii]{}\]]{} to far-infrared luminosities in the CFHQS quasars are comparable with those of starbursts of similar star formation rate in the local universe. We determine values of velocity dispersion and dynamical mass for the quasar host galaxies based on the [\[C[ii]{}\]]{} data. We find that there is no significant offset from the relations defined by nearby galaxies with similar black hole masses. There is however a marked increase in the scatter at $z=6$, beyond the large observational uncertainties.' author: - 'Chris J. Willott' - Jacqueline Bergeron and Alain Omont bibliography: - 'willott.bib' title: 'Star formation rate and dynamical mass of $10^{8}$ solar mass black hole host galaxies at redshift 6' --- Introduction ============ Improved astronomical observational facilities have enabled the discovery and study of many galaxies at an early phase of the Universe’s history. It is now possible to witness the majority of the stellar and black hole mass growth over cosmic time and identify how physical conditions at early times differ from now. One of the major relations to be determined as a function of time is the tight correlation between black hole mass and galaxy properties observed for nearby galaxies (see @Kormendy:2013 2013 for a review). Observations of this relation at high-redshift are critical to understanding the cause because most of the growth occurred at early times. Attempts to measure black hole and galaxy masses at high-redshift face a number of problems. Black hole mass measurements cannot be made directly by resolved kinematics of gas or stars within the black hole’s sphere of influence, nor by reverberation mapping. Instead black hole masses, $M_{\rm BH}$, of quasars can be measured at any redshift using the single-epoch virial mass estimator that involves measuring a low-ionization broad emission line, such as [Mg[ii]{}]{} or [H$\beta$]{}, and calibrating the location of the emitting gas with low-$z$ reverberation-mapped quasars [@Wandel:1999a]. For AGN with obscured broad lines $M_{\rm BH}$ can only be estimated from the luminosity making an assumption about the accretion rate relative to the Eddington limit. Measuring galaxy properties, such as luminosity or velocity dispersion, $\sigma$, of distant quasars is hampered by surface brightness dimming, the bright glare of the quasar and AGN (active galactic nuclei) emission line-contamination of spectral features. Up to $z\approx 1$ there has been considerable success in measuring AGN host galaxy luminosities, morphologies and in some cases velocity dispersions [@Cisternas:2011; @Park:2014]. At higher redshifts ($1<z<4$) the galaxy light is more difficult to separate from the quasar, which, combined with greater mass-to-light corrections, lead to larger uncertainties [@Merloni:2010; @Targett:2012]. The results of these studies are mixed with some evidence in favour of higher $M_{\rm BH}$ at a given galaxy mass. At yet higher redshifts it has proved impossible to measure the galaxy light of quasars [@Mechtley:2012] before launch of the  and instead the main method of determining galaxy mass is kinematics of cool gas in star-forming regions [@Carilli:2013]. Facilities such as the IRAM Plateau de Bure Interferometer, the Jansky Very Large Array and the Atacama Large Millimeter Array (ALMA) have sufficient sensitivity and resolution to resolve the gas in distant quasar hosts and provide dynamical masses [@Walter:2004; @Walter:2009; @Wang:2010; @Wang:2013]. In particular, ALMA has the sensitivity to probe $z=6$ quasar hosts with star formation rates, SFR, in the tens of solar masses per year, rather than only in the extreme starbursts previously observable [@Willott:2013]. The studies above focussed on $z\approx 6$ Sloan Digital Sky Survey (SDSS) and UKIRT Infrared Deep Sky Survey (UKIDSS) quasars with high UV and far-IR luminosities and found that their black holes are on average 10 times greater than the corresponding $\sigma$ for local galaxies, roughly consistent with a continuation of the evolution seen in lower redshift studies. Although observationally there appears to be an increase in $M_{\rm BH}$ with redshift at a given galaxy mass or $\sigma$, it has long been understood that there are selection biases that affect how closely the observations trace the underlying distribution. In particular, the steepness of the galaxy and dark matter mass functions combined with large scatter in their correlations with black hole mass mean that a high black-hole-mass-selected sample of quasars will have a systematic offset in $\sigma$ towards lower values. This effect, first identified by @Willott:2005b and @Fine:2006 was studied in detail in @Lauer:2007 and numerous studies thereafter. The magnitude of the effect depends upon the scatter in the correlation, which has not been conclusively measured at high-redshift, but appears to increase with redshift [@Schulze:2014]. @Willott:2005b and @Lauer:2007 showed that the bias is particularly strong for $M_{\rm BH}>10^9 M_\odot$ quasars such as those in the SDSS at $z \approx 6$ and therefore that the factor of 10 increase in $M_{\rm BH}$ at a given $\sigma$ first seen in the quasar SDSSJ1148+5251 [@Walter:2004] could be accounted for by the bias (see also @Schulze:2014 2014). In comparison, there would be little bias for a sample of high-$z$ quasars with black hole masses of $M_{\rm BH}\sim10^8 M_\odot$ [@Lauer:2007]. An alternative to measuring the evolution of the assembled galaxy and black hole masses is to determine the rate at which mass growth is occurring. For quasars the bolometric luminosity is a measure of the black hole mass growth rate. For galaxies, the star formation rate is proportional to the stellar mass growth. The star formation rate can be determined by the rest-frame far-infrared dust continuum luminosity. Additionally, the interstellar [\[C[ii]{}\]]{} far-infrared emission line is well-correlated with star-formation [@De-Looze:2014; @Sargsyan:2014] so can also be used as a star formation proxy. In @Willott:2013 (2013, hereafter Wi13) we presented Cycle 0 ALMA observations in the [\[C[ii]{}\]]{} line and 1.2mm continuum for two $z=6.4$ quasars from the Canada-France-High-z Quasar Survey (CFHQS, @Willott:2010a 2010b). These quasars have $M_{\rm BH}\sim10^8 M_\odot$, a factor of 10–30 lower than most SDSS quasars known at these redshifts. One quasar was detected in line and continuum and the other remained undetected in these sensitive observations placing an upper limit on its star formation rate of SFR$<40\,M_\odot\,{\rm yr}^{-1}$. In this paper we present ALMA observations of two further CFHQS quasars with similar redshift and black hole mass with the aim of providing a sample large enough to address the issue of how host galaxy properties such as SFR, $\sigma$ and dynamical mass depend upon black hole accretion rate and mass at a time just 1 billion years after the Big Bang. In particular, these quasars are not subject to the bias in the $M_{\rm BH} - \sigma$ relation discussed previously because of their moderate black hole masses. Cosmological parameters of $H_0=67.8~ {\rm km~s^{-1}~Mpc^{-1}}$, $\Omega_{\mathrm M}=0.307$ and $\Omega_\Lambda=0.693$ [@Planck-Collaboration:2014] are assumed throughout. Observations ============ CFHQSJ005502+014618 (hereafter J0055+0146) and CFHQSJ222901+145709 (hereafter J2229+1457) were observed with ALMA on the 28, 29 and 30 November 2013 for Cycle 1 project 2012.1.00676.S. Between 22 and 26 12m diameter antennae were used. The typical long baselines were $\sim 400$m providing similar spatial resolution to our Cycle 0 observations. Observations of the science targets were interleaved with nearby phase calibrators, J0108+0135 and J2232+1143. The amplitude calibrator was Neptune and the bandpass calibrators J2258-2758 and J2148+0657. Total on-source integration times were 4610s for J0055+0146 and 5490s for J2229+1457. The band 6 (1.3mm) receivers were set to cover the frequency range of the redshifted [\[C[ii]{}\]]{} transition ($\nu_{\rm rest}$=1900.5369 GHz) and sample the dust continuum. There are four $\approx 2$GHz basebands, two pairs of adjacent bands with a $11$GHz gap in between. The channel width is 15.625MHz (17kms$^{-1}$). The data were initially processed by North American ALMA Regional Center staff with the [CASA]{} software package. On inspection of these data it became clear that the [\[C[ii]{}\]]{} line of J0055+0146 was located right at the edge of the baseband, 1000kms$^{-1}$ from the targeted frequency defined by the broad, low-ionization [Mg[ii]{}]{} emission line ($z_{\rm MgII} = 5.983$; @Willott:2010 2010a). The [Mg[ii]{}]{} line redshift is usually close to the systemic redshift as measured by narrow optical lines with a dispersion of 270kms$^{-1}$ [@Richards:2002]. A large offset for this quasar was not particularly surprising for two reasons: firstly the signal-to-noise (SNR) of the [Mg[ii]{}]{} detection is not very high and the line appears double-peaked due to noise and/or associated absorption; secondly the [Ly$\alpha$]{} redshift ($z_{\rm Ly \alpha} = 6.02$) is offset from [Mg[ii]{}]{} by 1600kms$^{-1}$ (in the same direction as the [\[C[ii]{}\]]{} offset) and this would make the size of the [Ly$\alpha$]{} ionized near-zone negative, which is not physically sensible for a quasar with such a high ionizing flux and has not been observed in a sample of 27 $z\approx 6$ quasars [@Carilli:2010]. Due to this redshift uncertainty the receiver basebands were set up so that the adjacent band covered the [Ly$\alpha$]{} redshift with zero gap between the two bands. The default ALMA Regional Center reduction excluded 11 channels at each end of the 128 channel band. However, only the first 4 channels need to be excluded, so we re-reduced the data with [CASA]{} to include more spectral channels at the baseband edges. We checked that the noise does not increase in these extra channels, except for the very first and last channels to contain data so we excluded those. In summary our reduced product contains 118 of the original 128 channels per baseband, compared with 106 channels in the default reduction. ![ALMA spectrum of J0055+0146 covering two adjacent basebands. The gap between the bands with no data is shaded in gray. The higher frequency band is centred on the redshift determined from the [Mg[ii]{}]{} emission line whereas the lower frequency band covers the [Ly$\alpha$]{} redshift. The [\[C[ii]{}\]]{} line is found at the edge of the higher frequency band. The blue curve is a Gaussian plus continuum fit as described in the text. The red circle marks the continuum level independently measured in the three line-free basebands. The upper axis is the velocity offset from the best-fit [\[C[ii]{}\]]{} Gaussian peak.[]{data-label="fig:linespecj0055"}](fig1.pdf) Results ======= Figure \[fig:linespecj0055\] shows the reduced spectrum of J0055+0146 from the two adjacent basebands. The final gap between the bands is only $\approx$ 150 kms$^{-1}$ and crucially the peak of the [\[C[ii]{}\]]{} line is contained within the higher frequency band. The lower frequency band contains only a small amount of the line flux but provides an important constraint on the wings and hence the peak and width for a symmetric line. A single Gaussian plus flat continuum model was fit to the available data using a Markov-Chain Monte Carlo (MCMC). This process shows a good fit for a single Gaussian with FWHM=$359 \pm 27$ kms$^{-1}$. The formal uncertainty in the FWHM is very small considering that there is some missing data. This is because a symmetric line model is used and with the peak and wings covered by data there is little margin for deviation in the missing channels. We add in quadrature an extra 10% uncertainty in both the line flux and FWHM due to the missing channels. ![ALMA spectrum of J2229+1457 covering the single baseband containing the [\[C[ii]{}\]]{} line. The blue curve is a Gaussian plus continuum fit and the red circle the independent continuum level. The upper axis is the velocity offset from the best-fit [\[C[ii]{}\]]{} Gaussian peak.[]{data-label="fig:linespecj2229"}](fig2.pdf) The spectrum of J2229+1457 is plotted in Figure \[fig:linespecj2229\]. For this quasar the [\[C[ii]{}\]]{} line is centred in the band with no significant offset from the [Mg[ii]{}]{} redshift. The line is consistent with a single Gaussian with a best fit FWHM=$351 \pm 39$ kms$^{-1}$, similar to the value for J0055+0146. The continuum level of the fit is again consistent with the independent continuum level determined from the three line-free basebands. Measurements from the spectra are given in Table \[tab:data\]. ![The color scale shows the integrated [\[C[ii]{}\]]{} line maps for the two quasars. White contours of 1.2mm continuum emission from the three line-free basebands are over-plotted at levels of 2,4,6$\sigma$beam$^{-1}$. The quasar optical positions are shown with a black plus symbol. The positional offsets between the optical and millimeter are most likely due to astrometric mismatch, rather than a physical offset. J0055+0146 is well-detected in both continuum and line emission. J2229+1457 has only a $2\sigma$ continuum detection that is spatially co-incident with the line emission. The restoring beam is shown in yellow in the lower-left corner.[]{data-label="fig:contlinemaps"}](fig3.pdf) Figure \[fig:contlinemaps\] shows maps of the 1.2mm continuum (white contours) plus the [\[C[ii]{}\]]{} line (color scale) for the quasars. For both quasars there is no significant offset between any of the continuum or line centroids or the optical quasar position. The $< 1{^{\prime\prime}}$ mm-optical offset is within the relative uncertainty of the optical astrometry. The more accurate [\[C[ii]{}\]]{} positions for the two quasars are 00:55:02.92 +01.46.17.80 and 22:29:01.66 +14.57.08.30. For J2229+1457 there is only a marginal $2\sigma$ detection of the continuum located coincident with the peak of the line emission. As seen in Figure \[fig:contlinemaps\] there are several other continuum peaks of this magnitude or greater in the vicinity, so it is not considered a secure detection, however the measured flux and uncertainty are included in Table \[tab:data\]. The quasar 1.2mm continuum flux-densities were converted to far-infrared luminosity, $L_{\rm FIR}$, assuming a typical SED for high-redshift star-forming galaxies. As in we adopt a greybody spectrum with dust temperature, $T_{\rm d} =47$K and emissivity index, $\beta=1.6$. To convert from far-IR luminosity to star formation rate we use the relation SFR $(M_\odot\,{\rm yr}^{-1})=1.5\times10^{-10}L_{\rm FIR}\, (L_\odot)$ appropriate for a Chabrier IMF [@Carilli:2013]. We note that this assumes that all the dust contributing to the 1.2mm continuum is heated by hot stars and not by the quasar. An alternative estimate of the star formation rate comes from the [\[C[ii]{}\]]{} luminosity. We adopt the relation in @Sargsyan:2014 of SFR $(M_\odot\,{\rm yr}^{-1}) = 1.0\times10^{-7}L_{\rm [CII]} \, (L_\odot)$. For the remainder of this paper, uncertainties on $L_{\rm FIR}$ (and inferred SFR) only include the flux measurement uncertainties, not that of the dust temperature and luminosity to SFR conversion. [lll]{} & CFHQSJ0055+0146 & CFHQSJ2229+1457\ $z_{\rm MgII}\,^{\rm a}$ & $5.983 \pm 0.004 $ & $6.152 \pm 0.003 $\ $z_{\rm [CII]}$ & $6.0060 \pm 0.0008$ & $6.1517 \pm 0.0005$\ FWHM$_{\rm [CII]}$ & $359 \pm 45$ kms$^{-1} $& $351 \pm 39$ kms$^{-1} $\ $I _{\rm [CII]} ~($Jykms$^{-1}) $ & $0.839 \pm 0.132$ & $0.582 \pm 0.075$\ $L_{\rm [CII]} ~(L_\odot)$ & $(8.27 \pm 1.30) \times 10^8$ & $(5.96 \pm 0.77) \times 10^8$\ $f_{\rm 1.2mm}\ (\mu$Jy) & $211 \pm 34$ & $54 \pm 29$\ $L_{\rm FIR} ~ (L_\odot)$ & $(4.85 \pm 0.78) \times 10^{11}$ & $(1.24 \pm 0.67) \times 10^{11} $\ SFR$_{\rm [CII]}\,(M_\odot\,{\rm yr}^{-1})$ & $83 \pm 13$ & $60 \pm 8$\ SFR$_{\rm FIR}\,(M_\odot\,{\rm yr}^{-1})$ & $73 \pm 12$ & $19 \pm 10$\ $L_{\rm [CII]} / L_{\rm FIR}$ & $(1.70 \pm 0.38) \times 10^{-3}$ & $(4.80 \pm 2.67) \times 10^{-3}$\ [Notes.]{}—\ $^{\rm a}$ Derived from [Mg[ii]{}]{} $\lambda2799$ observations [@Willott:2010].\ Uncertainties in $L_{\rm FIR}$, SFR$_{\rm [CII]}$ and SFR$_{\rm FIR}$ only include measurement uncertainties, not the uncertainties in extrapolating from a monochromatic to integrated luminosity or that of the luminosity-SFR calibrations. The synthesized beam sizes are $0\farcs63$ by $0\farcs45$ for J0055+0146 and $0\farcs76$ by $0\farcs64$ for J2229+1457. The better resolution for J0055+0146 is mostly due to higher elevation of observation. We used the [CASA IMFIT]{} task to fit 2D gaussian models to these maps. For J0055 both the continuum and line are resolved with deconvolved source sizes of $0\farcs51 \pm 0\farcs13$ by $0\farcs35 \pm 0\farcs26$ at position angle 87 degrees and $0\farcs50 \pm 0\farcs14$ by $0\farcs18 \pm 0\farcs27$ at position angle 62 degrees, respectively. At the distance to this quasar the spatial extent of $0.5{^{\prime\prime}}$ is equal to a linear size of 2.9kpc. We note that the missing data in the red wing of the [\[C[ii]{}\]]{} line may cause a bias in the size and inclination if the emission comes from a rotating disk, but there is no evidence for this based on the similarity of the line and continuum sizes. For J2229+1457 the continuum is too poorly detected to attempt a size measurement and the line emission is only marginally more extended than the beam size. In several other $z\approx 6$ quasars velocity gradients across the sources are observed (; @Wang:2013 2013). Velocity gradients are not seen for either of these two quasars, although for J0055+0146 the missing data for 150kms$^{-1}$ of the red wing hampers our ability to detect such a gradient. Discussion ========== Evolution of far-IR luminosity ------------------------------ In we reported on the low far-IR luminosities of the two previously observed CFHQS quasars and implications for the relatively low SFR of these quasar host galaxies relative to the black hole accretion rate. We now revisit this issue with the sample of four $z\approx 6$ CFHQS quasars with ALMA observations. We note that this sample includes four of the six CFHQS quasars with measured black hole masses within the absolute magnitude range $-25.5<M_{1450}<-24$ at a declination low enough for ALMA observation. The two unobserved quasars have $7\times10^8<M_{\rm BH}<10^9 M_\odot$, and were not observed due to limited time available and the desire to study the lowest mass black holes from CFHQS. Therefore there is a slight bias to low black hole mass in this sample compared to pure UV-luminosity-selection. Two of the four quasars are well detected in the continuum with fluxes of $211 \pm 34$ (J0055+0146) and $120 \pm 35$ (J0210-0456) $\mu$Jy. J2229 has a marginal $2\sigma$ detection of $54 \pm 29\,\mu$Jy (Figure \[fig:contlinemaps\] and Table \[tab:data\]) and J2329-0301 is undetected with a $1\sigma$ rms of 30$\mu$Jy. We combine the four values of far-infrared luminosity derived from these measurements assuming that J2329-0301 has a flux equal to its $2\sigma$ upper limit of $60\,\mu$Jy. The mean and standard deviation of the sample is $L_{\rm FIR} = (2.6 \pm 1.4) \times 10^{11}\,L_\odot$. We note this is much lower than the values of $10^{12} - 10^{13} \,L_\odot$ typically discussed for $z\approx 6$ quasars due to two factors, firstly that the CFHQS sample here have lower AGN luminosity than most known $z\approx 6$ quasars and a correlation between AGN and far-IR luminosities is present [@Wang:2011; @Omont:2013], but also that our small sample is selected on quasar rest-frame UV luminosity and black hole mass, whereas previous studies have focussed on quasars with pre-ALMA millimeter continuum detections. The implication is that these quasars have very high black hole accretion rates as inferred from the AGN bolometric luminosity, $L_{\rm Bol}$, yet relatively low SFR. Such a scenario is consistent with the well-known evolutionary model whereby the optical quasar phase comes after the main star forming phase [@Khandai:2012; @Lapi:2014], possibly due to quasar feedback inhibiting gas cooling and star formation. The measured ratio of $L_{\rm FIR} / L_{\rm Bol}=0.035$ for the four CFHQS quasars is only found in the optical quasar phase of co-evolution at a time $\sim 1$Gyr after the onset of activity for the $z=2$ model of @Lapi:2014. Given that this is the age of the universe at $z=6$ the evolution must occur more rapidly at higher redshift. However, the effect of AGN variability may also be important leading to a selection effect whereby AGN luminosity-selected objects are observed to have lower ratios of $L_{\rm FIR} / L_{\rm Bol}$ than the time-averaged values [@Hickox:2014; @Veale:2014]. We have previously shown that the two $z=6.4$ CFHQS quasars observed with ALMA in Cycle 0 have $L_{\rm FIR}$ lower than quasars of similar AGN luminosity at lower redshift. On the other hand, at fixed AGN luminosity $L_{\rm FIR}$ is observed to rise from $z=0$ to $z=3$ [@Serjeant:2010; @Bonfield:2011; @Rosario:2012; @Rosario:2013]. We next analyze the evolution of $L_{\rm FIR}$ using our expanded ALMA sample at $z=6$ and comparable low-redshift data. At all redshifts we determine the mean $L_{\rm FIR}$ for optically-selected quasars and X-ray AGN in a narrow range of $L_{\rm Bol}$ corresponding to the mean $L_{\rm Bol}$ of the four CFHQS $z \approx 6$ quasars in this paper ($L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$). ![Stacked mean far-infrared luminosity for samples of quasars at different redshifts. Details of the samples are described in the text, but all samples are selected to include roughly the same range in bolometric luminosity centred on $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$, the mean $L_{\rm Bol}$ of the four $z\approx 6$ CFHQS quasars plotted with the blue square. There is a clear rise in $L_{\rm FIR}$ up to a peak at $2<z<3$ followed by a decline to $z=6$. The magenta curve shows the mean $L_{\rm FIR}$ due to star formation for the model of @Veale:2014 including scaling by a factor of 2 to account for stellar mass loss.[]{data-label="fig:lfirevol"}](fig4.pdf) At $z<3$ we use three datasets based on imaging of AGN. @Serjeant:2010 stacked SPIRE data of optically-selected quasars and quoted their results as rest-frame 100$\mu$m luminosity. We adopt $L_{\rm FIR}= 1.43\,\nu L_{\nu} (100\,\mu{\rm m})$ [@Chary:2001] to convert to far-infrared luminosity. The absolute magnitude bin $-26<I_{\rm AB}<-24$ corresponds well to $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$ and we have trimmed the size of the highest redshift bin from $2<z<4$ to $2<z<2.7$ because inspection of the luminosity-redshift plane figure in @Serjeant:2010 shows all but one of the 52 quasars in this bin are at $z<2.7$. @Rosario:2013 analyzed the PACS data for optically-selected quasars in COSMOS. Due to the shorter wavelength of PACS than SPIRE they presented results in rest-frame 60$\mu$m luminosity. We adopt $L_{\rm FIR}= 1.5\,\nu L_{\nu} (60\,\mu{\rm m})$ [@Chary:2001] to convert to far-infrared luminosity. From this study we use only the highest redshift, highest luminosity bin as this compares well with $L_{\rm Bol} \sim 7 \times 10^{12}\,L_\odot$. @Rosario:2012 determined the mean infrared luminosity with PACS for X-ray-selected AGN from the COSMOS survey. We note that the X-ray selected AGN sample contains a mixture of broad-line, narrow-line and lineless AGN and these may have different evolutionary properties, but @Rosario:2013 showed that the mean $L_{\rm FIR}$ of quasars and X-ray-selected are similar at a given AGN luminosity and redshift. We use the @Rosario:2012 data from the AGN luminosity bin $8 \times 10^{11}< L_{\rm Bol}< 2 \times 10^{13}\,L_\odot$. Whilst most sources in this bin have $L_{\rm Bol}< 3 \times 10^{12}\,L_\odot$, we consider the results appropriate to compare to the $z\approx 6$ quasars as the correlation between $L_{\rm FIR}$ and $L_{\rm Bol}$ is very shallow at this luminosity in @Rosario:2012. Figure \[fig:lfirevol\] plots data from these three low-redshift studies with the CFHQS ALMA bin at $6<z<6.5$. The three low-redshift studies show a rise in $L_{\rm FIR}$ of a factor of 4 from $z=0.3$ to $z=2.4$. This rise is attributed to the general increase in massive galaxy specific star formation rate over this redshift range [@Hickox:2014]. The $6<z<6.5$ bin has a large dispersion due to the range in 1.2mm continuum flux measured for the 4 quasars. The mean $L_{\rm FIR}$ at $z \approx 6$ is a factor of about 2 lower than the $z=0.3$ bin and 6 lower than the $z=2.4$ peak at the so-called [*quasar epoch*]{}. There is clear evidence here for a turnaround that mimics the evolution of the quasar luminosity function [@McGreer:2013] and star formation rate density [@Bouwens:2014], albeit with a much less steep high-redshift decline due to the fact we are measuring star formation in special locations within the universe where dark matter halos must have collapsed much earlier than typical in order to build up the observed black hole masses of $\sim 10^8\,M_\odot$. What is the physical reason for this turnaround at $z>3$? In the evolutionary picture where the optical quasar phase follows the starburst phase one would expect the star formation and black hole accretion to be more tightly coupled at high-redshift where there is barely enough time for star formation to have decreased substantially. A clue may come from one of the few differences between quasars at these two epochs. @Willott:2010 showed that the Eddington ratios of matched quasar luminosity samples at $z=2$ and $z=6$ are significantly different with the $z=6$ quasars having a factor of 3$\times$ higher Eddington ratios and therefore 3$\times$ lower black hole masses than at $z=2$. Such a difference exists between the typical black hole mass of our CFHQS ALMA sample and that of the highest luminosity bin of @Rosario:2013. This Eddington ratio evolution is observed in other studies [@De-Rosa:2011; @Trakhtenbrot:2011; @Shen:2012] and predicted by many theoretical works due to the increase in gas supply to black holes at high-redshift [@Sijacki:2014]. In Figure \[fig:lfirevol\] we also plot a theoretical curve of mean $L_{\rm FIR}$ versus redshift for a simulated sample of rest-frame UV-selected quasars in the same $L_{\rm Bol}$ range as the observed quasar samples for the model of @Veale:2014. This model assumes an evolving linear relationship between star formation and black hole growth. The variant of the model plotted here is the “accretion” model where the quasar luminosity is proportional to the black hole growth rate and the Eddington ratio distribution is a truncated power-law with slope $\beta=0.6$ (dashed curve in Figure 8 of @Veale:2014 2014). The model is constrained by the observed evolving quasar luminosity function and the local ratio of black hole to galaxy mass. We have scaled this model with a factor of $2\times$ increase in $L_{\rm FIR}$ to account for stellar mass loss. As seen in Figure \[fig:lfirevol\] this curve increases from low redshift to the peak quasar epoch at $2<z<3$ by about the same factor as the data, although the total normalization of the curve is lower by a factor of 3 to 4. @Veale:2014 discuss some of the reasons why the normalization may be lower than the observations. The decrease in $L_{\rm FIR}$ with increasing cosmic time from $z=2$ to $z=0$ for fixed luminosity quasars is due to the fact that such quasars are rarer at lower redshift and on the steep end of the luminosity function where scatter is more important. This behaviour also follows from the general decrease in specific star formation rate with cosmic time. The high-redshift behaviour of a decline from $z=3$ to $z=6$ matches our observations, so it is instructive to understand why this occurs in the model. It is due to the assumed $(1+z)^2$ evolution of the ratio of accretion growth to stellar mass growth, but this assumed evolution is also degenerate with evolution in the Eddington ratio. As discussed previously there is observational evidence for positive evolution in the Eddington ratio from $z=2$ to $z=6$, meaning that the ratio of accretion growth to stellar mass growth may change more gradually than $(1+z)^2$ . A possible alternative explanation for the low $L_{\rm FIR}$ at $z=6$ is that at these early epochs insufficient dust has been generated so that star formation occurs more often within lower dust environments [@Ouchi:2013; @Tan:2013; @Fisher:2014; @Ota:2014]. In this case there could be a much smaller decline in the typical SFR of a luminous quasar hosting galaxy. However, two lines of evidence point towards this not being the main factor for our quasar sample. First, quasars at $z=6$ are known to have emission line ratios similar to lower redshift quasars inferring high metallicity at least close to the accreting black hole [@Freudling:2003]. Second, the [\[C[ii]{}\]]{}luminosities in three of the four quasars are high (see below), suggesting high carbon abundances throughout the host galaxies. The [\[C[ii]{}\]]{} – far-IR luminosity relation ------------------------------------------------ Three of the four CFHQS ALMA quasars are detected in both [\[C[ii]{}\]]{} line and 1.2mm continuum emission (two new detections in this paper and J0210$-$0456 in ). With the low $L_{\rm FIR}$ discussed in the previous section, these quasar host galaxies probe a new regime in $L_{\rm FIR}$ at high-redshift. In Figure \[fig:lciilfir\] we plot the ratio of [\[C[ii]{}\]]{} to far-IR luminosity as a function of $L_{\rm FIR}$. Also plotted are several samples from the literature which, due to ALMA at high-redshift and at low-redshift, are rapidly increasing in size and data quality. The low-redshift $z<0.4$ sample of galaxies is from @Gracia-Carpio:2011 (2011 and in prep.) and contains a mix of normal galaxies, starbursts and ultra-luminous infrared galaxies (ULIRGs), some of which contain AGN. The ULIRGs show a [*[\[C[ii]{}\]]{} deficit*]{} that has been widely discussed in the literature as due to possible factors including AGN contamination of $L_{\rm FIR}$ [@Sargsyan:2012], high gas fractions [@Gracia-Carpio:2011] or the dustiness, temperature and/or density of star forming regions [@Farrah:2013; @Magdis:2014]. Previous observations of high $L_{\rm FIR}$ $z>5$ SDSS quasars [@Maiolino:2005; @Wang:2013] showed a similar deficit. However many $0.5<z<5$ ULIRGs do not show this deficit and have $L_{\rm [CII]} / L_{\rm FIR}$ ratios comparable to low-redshift star-forming galaxies [@Stacey:2010]. This is visible in Figure \[fig:lciilfir\] for the $0.5<z<5$ compilation of @De-Looze:2014. ![The ratio of [\[C[ii]{}\]]{} to far-IR luminosity as a function of far-IR luminosity. High-redshift ($z>5$) sources, mostly quasar host galaxies, are identified with large symbols. Error bars are only plotted for the CFHQS ALMA sources to enhance the clarity of the figure. The solid and dotted blue lines show the best fit power-law and $1\sigma$ uncertainty for the $z>5$ sources. The $z>5$ relation is largely consistent with the distribution of data at lower redshift. []{data-label="fig:lciilfir"}](fig5.pdf) By adding three $z>6$ quasars with $10^{11}<L_{\rm FIR}<10^{12}\,{\rm L}_\odot$ to Figure \[fig:lciilfir\] we have greatly expanded the range of luminosities at the highest redshift. Large symbols on Figure \[fig:lciilfir\] identify $z>5$ sources. The three $z>5$ @De-Looze:2014 sources are HLSJ091828.6+514223 at $z=5.24$ [@Rawle:2014] , HFLS3 at $z=6.34$ [@Riechers:2013] and SDSSJ1148+5251 at $z=6.42$ [@Maiolino:2005]. We note that these sources were mostly selected for followup based on high $L_{\rm FIR} $. The quasar ULASJ1120+0641 at $z=7.1$ has a more moderate $L_{\rm FIR}$ and $L_{\rm [CII]} / L_{\rm FIR}$ ratio and lies in between the CFHQS and high $L_{\rm FIR}$ objects on the plot. Although we are wary of the selection effects in Figure \[fig:lciilfir\] and the as yet unknown cause for the change in $L_{\rm [CII]} / L_{\rm FIR}$ with $L_{\rm FIR}$ at high-redshift, the new data provide the opportunity to make the first measurement of the slope of this relation at $z>5$. We fit the 12 $z>5$ sources with a single power-law model of the dependence of $L_{\rm FIR}$ on $L_{\rm [CII]}$ incorporating the observational (but not systematic) uncertainties using a MCMC procedure. The best-fit relation is $$\log_{10} L_{\rm FIR}=0.59+1.27 \log_{10} L_{\rm [CII]}.$$ Therefore the $L_{\rm [CII]} / L_{\rm FIR}$ ratio line plotted in Figure \[fig:lciilfir\] has a logarithmic slope of $(1/1.27)-1=-0.21$. The MCMC $1\sigma$ uncertainties, based solely on the observational data, are plotted as dotted lines. These also favor a shallow negative slope, not nearly as steep as the slope that would be fit to the previous $z>0.5$ data that covers only a narrow range of $L_{\rm FIR}$. The $L_{\rm [CII]} / L_{\rm FIR}$ ratios of the CFHQS and ULAS $z>6$ quasars are not greatly different to those of similar $L_{\rm FIR}$ galaxies at low-redshift. @Wang:2013 note that the low $L_{\rm [CII]} / L_{\rm FIR}$ ratios for SDSS $z>5$ quasars may be at least in part due to AGN contamination of the far-IR emission. Future observations at higher spatial resolution will be critical to examine differences in the spatial distribution of the line and continuum emission (e.g. @Cicone:2014 2014). We expect to observe that the dust continuum is more compact than the [\[C[ii]{}\]]{} line in the high $L_{\rm FIR}$ $z>5$ quasars, due to either more centrally-concentrated starbursts with higher dust temperatures, like local ULIRGs, or AGN dust-heating. In contrast we expect the low $L_{\rm FIR}$ $z>5$ quasars have star formation spread more evenly throughout their host galaxies, with similar spatial distribution of line and continuum emission. The $z\approx 6$ $M_{\rm BH}-\sigma$ and $M_{\rm BH}-M_{\rm dyn}$ relationships ------------------------------------------------------------------------------- The combination of black hole mass estimates and [\[C[ii]{}\]]{} line host galaxy dynamics for these $z\approx 6$ quasars allows us to investigate the black hole - galaxy mass correlation at an early epoch in the universe. The evolution of this relationship is a critical constraint on the co-evolution (or not) of galaxies and their nuclear black holes. As discussed in the Introduction there are reasons to believe that past studies using only the most massive black holes from SDSS quasars (e.g. @Wang:2010 2010) were prone to a bias where one would expect the black holes to be relatively more massive than the galaxies [@Willott:2005b; @Lauer:2007], as observed. With new data on $M_{\rm BH} \sim 10^8\,M_\odot$ quasars we are able to test this hypothesis and determine any real offset from the local relationship. Additionally, most previous work in this area has used the molecular CO line to trace the gas dynamics. CO is usually more centrally concentrated than [\[C[ii]{}\]]{}, so [\[C[ii]{}\]]{} potentially probes a larger fraction of the total mass (although we note that @Wang:2013 2013 found similar dynamical masses using CO and [\[C[ii]{}\]]{} for their $z\approx 6$ quasars). In addition to the CFHQS and [@Wang:2013] quasars we add to our study two other $z>6$ quasars observed in the [\[C[ii]{}\]]{} line: SDSSJ1148+5251 at $z=6.42$ and the most distant known quasar, ULASJ1120+0641 at $z=7.08$. Both these quasars have [Mg[ii]{}]{}-derived black hole masses [@De-Rosa:2011; @De-Rosa:2014] with very low measurement uncertainties. The black hole masses for all three CFHQS quasars also come from [Mg[ii]{}]{} measurements [@Willott:2010]. Some of these spectra are of moderate SNR and have substantial measurement uncertainties on the black hole masses. To all the quasars with [Mg[ii]{}]{}-derived black hole masses we add a 0.3 dex uncertainty to the measurement uncertainties to account for the dispersion in the reverberation-mapped quasar calibration [@Shen:2008]. None of the [@Wang:2013] quasars have [Mg[ii]{}]{} measurements so black hole masses are estimated assuming that the quasars radiate at the Eddington limit, as observed for most $z\approx 6$ quasars [@Jiang:2007; @Kurk:2007; @Willott:2010; @De-Rosa:2011]. The dispersion in the lognormal Eddington ratio distribution at $z \approx 6$ is 0.3 dex [@Willott:2010]. We add 0.3 dex uncertainty from the observed dispersion in the Eddington ratio distribution in quadrature to the 0.3 dex due to the dispersion in the reverberation-mapped quasar calibration for a total uncertainty on the [@Wang:2013] quasar black hole masses of 0.45 dex. First we consider the $M_{\rm BH}-\sigma$ relationship. For nearby galaxies $\sigma$ is the velocity dispersion of the galaxy bulge. At high-redshift bulges are less common [@Cassata:2011] and we do not expect the $z\approx 6$ kinematics to match that of a pressure supported bulge. With the limited spatial resolution of current data we cannot be sure the [\[C[ii]{}\]]{} gas is distributed in a rotating disk, although there is evidence of this for some sources [@Wang:2013]. [@Ho:2007a] discusses the relationship and calibration of bulge velocity dispersion and disk circular velocity and concludes that although there is additional scatter one can relate molecular or atomic gas in a disk to stellar bulges. The major complication is the inclination of the disk. For a random sample of inclinations this can be modelled, however there is a possibility that quasars have disks oriented more often face-on, reducing the line-of-sight velocity dispersion [@Ho:2007]. [lccccl]{} Name & $z_{\rm [CII]}$ & $M_{\rm BH} ~(M_\odot)\,^{\rm a}$ & $\sigma ^{\rm b}$ & $M_{\rm dyn}~(M_\odot)$ & Refs.$^{\rm c}$\ CFHQSJ0055+0146 & $6.0060 \pm 0.0008$ & $(2.4^{+2.6}_{-1.4})\times 10^{8} $ & $207 \pm 45$ & $4.2\times 10^{10} $ & 1,2\ CFHQSJ0210$-$0456 & $6.4323 \pm 0.0005$ & $(0.8^{+1.0}_{-0.6})\times 10^{8} $ & $98 \pm 20$ & $1.3\times 10^{10} $ & 1,2,3\ CFHQSJ2229+1457 & $6.1517 \pm 0.0005$ & $(1.2^{+1.4}_{-0.8})\times 10^{8}$ & $241 \pm 51$ & $4.4\times 10^{10} $ & 1,2\ SDSSJ0129$-$0035 & $5.7787 \pm 0.0001$ & $(1.7^{+3.1}_{-1.1})\times 10^{8} $ & $112 \pm 21$ & $1.3\times 10^{10} $ & 4\ SDSSJ1044$-$0125 & $5.7847 \pm 0.0007$ & $(1.1^{+1.9}_{-0.7})\times 10^{10} $ & $291 \pm 76$ & — $^{\rm d}$ & 4\ SDSSJ1148+5251 & $6.4189 \pm 0.0006$ & $(4.9^{+4.9}_{-2.5})\times 10^{9} $ & $186 \pm 38$ & $1.8\times 10^{10} $ & 5,6,7\ SDSSJ2054$-$0005 & $6.0391 \pm 0.0001$ & $(0.9^{+1.6}_{-0.6})\times 10^{9} $ & $364 \pm 67$ & $7.2\times 10^{10} $ & 4\ SDSSJ2310+1855 & $6.0031 \pm 0.0002$ &$(2.8^{+5.1}_{-1.8})\times 10^{9} $ & $325 \pm 61$ & $9.6 \times 10^{10} $ & 4\ ULASJ1120+0641 & $7.0842 \pm 0.0004$ & $(2.4^{+2.4}_{-1.2})\times 10^{9} $ & $ 144 \pm 34$ & $2.4\times 10^{10} $ & 8,9\ ULASJ1319+0950 & $6.1330 \pm 0.0007$ &$(2.1^{+3.8}_{-1.4})\times 10^{9} $ & $381 \pm 91$ & $12.5\times 10^{10} $ & 4\ [Notes.]{}—\ $^{\rm a}$ Derived from [Mg[ii]{}]{} $\lambda2799$ observations if possible, else from Eddington luminosity assumption. Uncertainties include observational errors plus systematics based on calibrations.\ $^{\rm b}$ Derived from Gaussian FWHM fit to [\[C[ii]{}\]]{} spectrum using method of [@Ho:2007] including an inclination correction (see text for individual inclinations assumed). Uncertainties include observational errors plus systematics based on calibrations.\ $^{\rm c}$ References: (1) This paper, (2) @Willott:2010, (3) @Willott:2013, (4) @Wang:2013, (5) @Maiolino:2005, (6) @Walter:2009, (7) @De-Rosa:2011, (8) @Venemans:2012, (9) @De-Rosa:2014.\ $^{\rm d}$ This quasar does not have a dynamical mass calculation in @Wang:2013 due to the difference in the [\[C[ii]{}\]]{} and CO line profiles. We determine $\sigma$ using the method of [@Ho:2007], specifically setting the [\[C[ii]{}\]]{} line full-width at 20% equal to 1.5$\times$ the FWHM as expected for a Gaussian since most of the lines are approximately Gaussian. The [\[C[ii]{}\]]{} emitting gas is assumed to be in an inclined disk where the inclination angle, $i$, is determined by the ratio of minor ($a_{\rm min}$) and major ($a_{\rm maj}$) axes, $i=\cos^{-1}( a_{\rm min}/a_{\rm maj})$. The circular velocity is therefore $v_{\rm cir}=0.75 \,$FWHM$_{\rm [CII]} / \sin i$. For all of the quasars in this study we determine an inclination from the [\[C[ii]{}\]]{}data or assume an inclination if one or both the major and minor axes are unresolved. All the quasars in [@Wang:2013] were spatially resolved, although some had quite large uncertainties on $a_{\rm min}$ and $a_{\rm maj}$. We adopt the inclination angles from their paper. For the [\[C[ii]{}\]]{} emission of J0210$-$0456, $a_{\rm maj} =0\farcs52 \pm 0\farcs25$ (2.9kpc) with $i=64^{\circ}$ . For J0055+0146, $a_{\rm maj} =0\farcs50 \pm 0\farcs14$ (2.9kpc) with $i=69^{\circ}$ (Section 3, assuming no bias from missing red wing data). The [\[C[ii]{}\]]{} emission of J2229+1457 is only marginally spatially resolved ($0\farcs85$ versus beam size of $0\farcs76$) and we estimate an intrinsic FWHM of $\approx 0\farcs4$ (2.4kpc). An inclination of $i=55^{\circ}$ is assumed as this is the median inclination angle for the resolved sources in this paper and [@Wang:2013]. Neither SDSSJ1148+5251 nor ULASJ1120+0641 have published inclination angles, so we also assume $i=55^{\circ}$ for both of them. We adopt FWHM$_{\rm [CII]} =287 \pm 28$kms$^{-1}$ for SDSSJ1148+5251 [@Walter:2009] and FWHM$_{\rm [CII]} =235 \pm 35$kms$^{-1}$ for ULASJ1120+0641 [@Venemans:2012] . Values of black hole masses and $\sigma$ for this sample are provided in Table \[tab:masses\]. Figure \[fig:mbhsig\] shows the $M_{\rm BH} - \sigma$ relationship for the $z\approx 6$ quasar sample. Uncertainties on black hole masses include the scatter in the calibration as described previously. Uncertainties in $\sigma$ include FWHM measurement uncertainty plus a 10% uncertainty for the conversion from FWHM to $v_{\rm cir}$ and 15% for the conversion from $v_{\rm cir}$ to $\sigma$ as seen in the sample of [@Ho:2007a]. The black line is the local correlation of [@Kormendy:2013] with the gray band the $\pm 1\sigma$ scatter. The first thing to note is that the quasars are distributed around the local relationship rather than all being offset to low $\sigma$ as is commonly believed to be the case. As noted by [@Wang:2010], using the method of [@Ho:2007], rather than calculating $\sigma$ as FWHM$/2.35$, leads to much higher $\sigma$. Note that this is without adopting extreme face-on inclinations for most quasars. There are still several quasars, such as SDSSJ1148+5251 and ULASJ1120+0641, that have values of $\sigma$ considerably lower than the local relation. The main result of Figure \[fig:mbhsig\] is that whilst there is little mean shift between the $z=0$ and $z\approx 6$ data, there is a much larger scatter in the data at $z\approx 6$, well beyond the size of the error bars. This larger scatter at an early epoch is expected based on dynamical evolution, incoherence in AGN/starburst activity and the tightening of the relation over time from merging [@Peng:2007]. We note that our hypothesis that the bias described in the Introduction would lead to the lower $M_{\rm BH}$ quasars being located on the local relation with a lower scatter than the high $M_{\rm BH}$ quasars is not supported by these observations. The scatter in $\log_{10} \sigma$ at $M_{\rm BH}\approx 10^8\,M_\odot$ is about the same as that at $M_{\rm BH}> 10^9\,M_\odot$ We go one step further from $\sigma$ to determine dynamical masses using the deconvolved [\[C[ii]{}\]]{} sizes. For consistency, we follow the method of [@Wang:2013]. The dynamical mass within the disk radius is given by $M_{\rm dyn} \approx 1.16\times 10^5\, v_{\rm cir}^2 \,D \,M_\odot$ where $D$ is the disk diameter in kpc and calculated as $1.5 \times$ the deconvolved Gaussian spatial FWHM. The resulting dynamical masses are given in Table \[tab:masses\]. We note that there is considerable uncertainty on these values due to the unknown spatial and velocity structure of the gas, the marginal spatial resolution and limited sensitivity that means we may be missing more extended gas. Due to the these uncertainties we do not place formal error bars on the dynamical masses, following [@Wang:2013]. Higher resolution data in the future are required to confirm the derived masses. ![Black hole mass versus velocity dispersion calculated from the [\[C[ii]{}\]]{} line using the method of [@Ho:2007] for $z\approx 6$ quasars. Quasars from the CFHQS are shown as blue squares and the other symbols show quasars from the SDSS and ULAS surveys. The black line with gray shading is the local correlation $\pm 1\sigma$ scatter of black hole mass and bulge velocity dispersion [@Kormendy:2013]. The $z\approx 6$ quasars are distributed around the the local relationship, but with a much larger scatter and some quasars with significantly lower $\sigma$ for their $M_{\rm BH}$.[]{data-label="fig:mbhsig"}](fig6.pdf) ![Black hole mass versus host galaxy dynamical mass for $z\approx 6$ quasars. Symbols as for Figure \[fig:mbhsig\]. The black line with gray shading is the local correlation $\pm 1\sigma$ scatter from the work of [@Kormendy:2013] equating $M_{\rm dyn}$ to $M_{\rm bulge}$. The CFHQS quasars lie on the local relationship and do not show the large offset displayed by the most massive black holes. Uncertainties in $M_{\rm dyn}$ have not been calculated due to the reasons given in the text.[]{data-label="fig:mbhmdyn"}](fig7.pdf) SDSSJ1148+5251 has been extensively studied in [\[C[ii]{}\]]{} [@Maiolino:2005; @Walter:2009; @Maiolino:2012; @Cicone:2014]. The highest resolution observations by [@Walter:2009] revealed a very compact circumnuclear starburst with radius 0.75kpc and FWHM$_{\rm [CII]} =287 \pm 28$kms$^{-1}$. For an assumed inclination of $i=55^{\circ}$ this gives $M_{\rm dyn}=1.8 \times 10^{10}\,M_\odot$. For comparison [@Walter:2004] determined a dynamical mass from CO emission in this quasar of $5.5\times 10^{10}\,M_\odot$ within a larger radius of 2.5kpc, the larger radius being the main difference between the results. Recent observations have shown more complex [\[C[ii]{}\]]{} emission including evidence for gas extended over tens of kpc and at high velocities indicative of outflow [@Maiolino:2012; @Cicone:2014]. We adopt $M_{\rm dyn}=1.8 \times 10^{10}\,M_\odot$ for SDSSJ1148+5251 noting that the true value could be several times larger. The most distant known quasar, ULASJ1120+0641 at $z=7.08$, has been well detected in [\[C[ii]{}\]]{}, although not yet spatially resolved [@Venemans:2012]. Based on the published FWHM$_{\rm [CII]} =235 \pm 35$kms$^{-1}$ and assuming a spatial FWHM of 3kpc (similar to the other quasars resolved by ALMA) and $i=55^{\circ}$ we determine $M_{\rm dyn}=2.4 \times 10^{10}\,M_\odot$. In Figure \[fig:mbhmdyn\] we plot black hole mass versus galaxy dynamical mass for the most distant known quasars. The black line and gray shading represent the local correlation of $M_{\rm BH}$ with bulge mass $M_{\rm bulge}$ [@Kormendy:2013]. In the absence of gas accretion and mergers the present stellar bulge mass represents the sum of the gas and stellar mass at high-redshift, so it is a good comparison for the dynamical mass within the central few kpc. [@Kormendy:2013] note that their correlation (their equation 10) gives a black hole to bulge mass ratio of 0.5% at $M_{\rm bulge}=10^{11}\,M_\odot$ that is 2 to 4 times higher than previous estimates due to the omission of pseudobulges, galaxies with uncertain $M_{\rm BH}$ and ongoing mergers. The position of the high-$z$ data with respect to low redshift is fairly similar to Figure \[fig:mbhsig\], not surprising because $v_{\rm cir}$ derived from the [\[C[ii]{}\]]{} velocity FWHM is a major factor in both $\sigma$ and $M_{\rm dyn}$. The points are shifted somewhat further from the local bulge mass than for the local velocity dispersion. This shift is due to the smaller size of galaxies at high-redshift, as the size is the only term in the derivation of dynamical mass not in $\sigma$. We note the much greater dynamic range in black hole mass (2 dex) than in dynamical mass (1 dex) in our sample. This is likely due more to our selection over a wide range of quasar luminosity than to a non-linear relationship between these quantities at $z=6$. All three of the CFHQS quasars lie within the local $1\sigma$ scatter and the one $M_{\rm BH}\sim 10^8\,M_\odot$ quasar in [@Wang:2013] is only a factor of 4 greater than the local relationship. In contrast the $M_{\rm BH}> 10^9\,M_\odot$ quasars tend to show a larger scatter and larger offset above the local relationship as previously found [@Walter:2004; @Wang:2010; @Venemans:2012; @Wang:2013]. We caution that there are considerable uncertainties in some of these measurements as already discussed, but in dynamical mass the results look more like we would expect based on the quasar selection bias effect. Conclusions =========== During ALMA Early Science cycles 0 and 1 we have observed a complete sample of four $z>6$ moderate luminosity CFHQS quasars with black hole masses $\sim 10^8\,M_\odot$. Three of the four are detected in both far-IR continuum and the [\[C[ii]{}\]]{} emission line. The far-IR luminosity is found to be substantially lower than that of similar luminosity quasars at $1<z<3$. Assuming that far-IR luminosity traces star formation equally effectively at these redshifts this implies that at $z\approx 6$ quasars are growing their black holes more rapidly than their stellar mass compared to at the peak of the [*quasar epoch*]{} ($1<z<3$). The ratios of \[CII\] to far-IR luminosities for the CFHQS quasars lie in the range 0.001 to 0.01, similar to that of low-redshift galaxies at the same far-IR luminosity. This suggests a similar mode of star-formation spread throughout the host galaxy (rather than in dense circumnuclear starburst regions that have lower values for this ratio in local ULIRGs). Combining with previous $z>5.7$ quasar data at higher $L_{\rm FIR}$ we find that the far-IR luminosity dependence of the [\[C[ii]{}\]]{}/FIR ratio has a shallow negative slope, possibly due in part to an increase in $L_{\rm FIR}$ due to quasar-heated dust in some optically-luminous high-$z$ quasars. The three CFHQS quasars well-detected in the [\[C[ii]{}\]]{} emission line allow this atomic gas to be used as a tracer of the host galaxy dynamics. Combining with published data on higher black hole mass quasars we have investigated the $M_{\rm BH}-\sigma$ and $M_{\rm BH}-M_{\rm dyn}$ relations at $z\approx 6$. We show that the $z=6$ quasars display a $M_{\rm BH}-\sigma$ relation with similar slope and normalization to locally, but with much greater scatter. Similar results are obtained for the $M_{\rm BH}-M_{\rm dyn}$ relation with a somewhat higher normalization at $z=6$ and a higher scatter at high $M_{\rm BH}$. As discussed in Combining our results on the relatively low $L_{\rm FIR}$ for $ M_{\rm BH} \sim 10^8\,M_\odot$ $z\approx 6$ quasars with their location on the $M_{\rm BH}-\sigma$ relation leads to something of a paradox. The fact these quasars lie on the local $M_{\rm BH}-\sigma$ relation suggests that their host galaxies have undergone considerable evolution to acquire such a high dynamical mass. So why is it that this mass accumulation is not leading to a high star formation rate? As discussed in , simulations such as those of @Khandai:2012 and @Lapi:2014 predict that such low ratios of SFR to black hole accretion occur after episodes of strong feedback that inhibits star formation throughout quasar host galaxies. Another possibility mentioned in Section 4.1 is that $L_{\rm FIR}$ fails to trace star formation so effectively in these high-redshift galaxies, due to lower dust content (e.g. @Ouchi:2013 2013). Note that using $L_{\rm [CII]}$ as a star formation rate tracer instead of $L_{\rm FIR}$, would give higher SFR by a factor of three for one of the CFHQS quasars. Higher resolution follow-up [\[C[ii]{}\]]{} observations of these quasars are critical to measure more accurately the distribution and kinematics of the gas used as a dynamical tracer in order to reliably determine the location and scatter of the correlations between black holes and their host galaxies at high-redshift. Thanks to staff at the North America ALMA Regional Center for processing the ALMA data. Thanks to Melanie Veale for useful discussion and providing her models in electronic form. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2012.1.00676.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. [*Facility:*]{} .

ArXiv

--- abstract: 'We derive a universal Hamiltonian for a quantum dot in the presence of spin-orbit interaction in the symmetry limits of a strong spin-dependent Aharonov-Bohm-like term. We also derive a closed expression for the conductance through such a dot, and use it to study the effects of spin-orbit on the conductance peak statistics in the presence of an exchange interaction. For a realistic strength of the exchange interaction, we find that the width of the peak-spacing distribution is sensitive to spin-orbit coupling only in the absence of an orbital magnetic field. We also find that spin-orbit coupling modifies the shape of the peak-spacing distribution and suppresses the peak-height fluctuations.' author: - 'Y. Alhassid and T. Rupp' title: 'A universal Hamiltonian for a quantum dot in the presence of spin-orbit interaction' --- The statistical fluctuations of the single-particle spectrum and wavefunctions in a chaotic or diffusive quantum dot with a large Thouless conductance $g_T$ can be described by random matrix theory (RMT) [@alhassid00; @guhr98]. In open dots, which are strongly coupled to leads, RMT can successfully describe the mesoscopic fluctuations of the conductance assuming non-interacting electrons. However, in almost-isolated dots, which are weakly coupled to leads, electron-electron interactions cannot be ignored. In such dots, most of the residual interaction terms are suppressed in the limit of a large $g_T$, except for a few terms. These terms constitute the interacting part of the universal Hamiltonian [@kurland00; @aleiner02], and include, in addition to the charging energy, a constant exchange interaction. Using RMT to describe the single-particle Hamiltonian, a significantly better agreement with the data of Refs.  is found once this exchange term is included [@exchange; @usaj03]. In general, the exchange interaction suppresses fluctuations of both the conductance peak spacings and peak heights. Spin-orbit (SO) scattering is also expected to affect the conductance fluctuations, although its effects are suppressed in small dots. Enhanced SO coupling in a parallel magnetic field explained the suppression of conductance fluctuations in chaotic open 2D GaAs dots [@folk01; @halperin01]. The possible symmetries of the single-particle Hamiltonian in such dots with SO scattering were recently classified [@aleiner01], and their signatures were observed in open dots [@zumbuhl02]. Realistic studies of SO effects in almost-isolated dots require the inclusion of interactions. However, the universal Hamiltonian for a 2D dot has been derived only in the absence of SO coupling. Here we study how the universal Hamiltonian is modified in the presence of SO scattering, both in the presence and absence of an orbital magnetic field. In particular, we derive the universal Hamiltonian \[Eq. (\[universal-H\])\] in the new symmetry limits introduced by the leading order SO term (a spin-dependent Aharonov-Bohm-like term). We also describe the dot’s Hamiltonian \[Eqs. (\[crossover-H\]),(\[spin\])\] in the crossover induced by this SO term. Electron correlations in the presence of SO scattering were recently studied in metal nanoparticles [@gorokhov03], but such 3D nanoparticles do not possess the new symmetries considered here. In addition, we derive a closed formula for the conductance through a dot described by the new universal Hamiltonian \[Eqs. (\[conductance\]) – (\[spin-prob\])\], and study the corresponding conductance peak statistics. For a realistic strength of the exchange interaction, we find that the standard deviation of the peak-spacing fluctuations is sensitive to SO scattering only in the absence of an orbital magnetic field. SO coupling also modifies the shape of the peak-spacing distribution and suppresses the peak height fluctuations. The symmetries of the one-body Hamiltonian in the presence of SO in GaAs dots were classified by applying a suitable unitary transformation to its original form [@aleiner01]. The transformed Hamiltonian is expanded in the parameter $L/\lambda$ ($L$ is the linear size of the dot and $\lambda$ is a mean SO scattering length), which is assumed to be small. The two leading contributions are described by effective vector potentials that modify the vector potential of the orbital magnetic field $B$. The first is a spin-dependent Aharonov-Bohm-like vector potential ${\bf a}_\perp$ of order $(L/\lambda)^2$, and the second is a spin-flip term ${\bf a}_\parallel$ of order $(L/\lambda)^3$. In small dots, only the first term is relevant and the total effective orbital field is given by $$\label{effective-B} B_{\rm eff} = B + B_{\rm so} s_z\;; \;\;\; B_{\rm so}={c\hbar \over e \lambda^2}\;,$$ where $s_z$ is the electron spin component perpendicular to the plane of the dot, and $B_{\rm so}$ is an effective SO field. The effect of the SO interaction is described by a dimensionless parameter $x^2_\perp = \kappa g_T (\Phi_{\rm so}/\Phi_0)^2=\kappa g_T ({\cal A}/\lambda^2)^2$, where ${\cal A}$ is the area of the dot, $\kappa$ is a geometrical coefficient, $\Phi_{\rm so}$ is the flux associated with the SO field, and $\Phi_0=c\hbar/e$ is the unit flux. In the following we derive the universal Hamiltonian in the symmetry limits $x_\perp \gg 1$ for both $B=0$ and $B \neq 0$. (in practice, the crossover is often achieved for $x_\perp \sim 1$). The single-particle eigenstates in the presence of an effective field (\[effective-B\]) are given by $|\alpha \sigma\rangle$, where $\sigma=\pm$ describes spin up/down electrons with orbital wavefunctions $\psi_{\alpha \pm}({\bf r})$ and energies $\epsilon_{\alpha \pm}$. In the absence of SO coupling ($x_\perp=0$), $\psi_{\alpha +} =\psi_{\alpha -}$. When SO is present, we have to distinguish between two cases. For $B \neq 0$, the wavefunctions $\psi_{\alpha \pm}({\bf r})$ correspond to two different values $B \pm B_{\rm so}/2$ of the effective field, and become uncorrelated for $x_\perp \gg 1$. Thus, we have a crossover from two degenerate Gaussian unitary ensembles (GUE) at $x_\perp=0$ to two uncorrelated GUE at $x_\perp \gg 1$ [@orbital]. However, for $B=0$, time reversal invariance leads to Kramers degeneracy $\epsilon_{\alpha +} =\epsilon_{\alpha -}$ and $\psi_{\alpha -} (\bf r) = \psi^*_{\alpha +}(\bf r)$, and the crossover is from two degenerate Gaussian orthogonal ensembles (GOE) at $x_\perp=0$ to two degenerate GUE at $x_\perp \gg 1$. Since $|\alpha \sigma \rangle$ is a complete single-particle basis, we can write the dot’s Hamiltonian in this basis. We first discuss the limits $x_\perp=0$ and $x_\perp \gg 1$ (more precisely, $x_\perp^2 \agt g_T$) but not the crossover itself. In these limits, we consider the “diagonal” part of the interaction, i.e., direct terms $v_{\alpha \sigma \gamma \sigma';\alpha \sigma \gamma \sigma'}$, exchange terms $v_{\alpha \sigma \gamma \sigma';\gamma \sigma \alpha \sigma'}$, and Cooper channel terms $v_{\alpha + \alpha -;\gamma + \gamma-}$. For $g_T \gg 1$, we separate this “diagonal” interaction into average and fluctuating parts, and identify terms that remain finite for $g_T \to \infty$. We define the following average matrix elements (for $\alpha\neq \gamma$) $$\begin{aligned} \label{averages} v_1 &= & \bar v_{\alpha \sigma \gamma \sigma';\alpha \sigma \gamma \sigma'}\;;\;\; v_2 = \bar v_{\alpha \sigma \gamma \sigma;\gamma \sigma \alpha \sigma} \;; \nonumber \\ v_3 & = & \bar v_{\alpha \sigma \gamma -\sigma;\gamma \sigma \alpha -\sigma} \;;\;\; v_4 = \bar v_{\alpha + \alpha -;\gamma + \gamma-} \;.\end{aligned}$$ While all the direct matrix elements have the same average $v_1$, we have distinguished two types of exchange matrix elements. For $x_\perp=0$, the orbital wavefunctions are spin-independent and $v_2=v_3=J_s$. For $x_\perp \to \infty$, $v_2=J_s$ remains unchanged (the orbital wavefunctions correspond to the same spin $\sigma$), but $v_3 =0$. The vanishing of $v_3$ can be shown, e.g., in a contact model for the screened interaction \[$v(\bf r - \bf r') \propto \delta(\bf r - \bf r')$\] for which $v_{\alpha \sigma \gamma -\sigma;\gamma \sigma \alpha -\sigma} \propto \int d {\bf r}\psi^*_{\alpha \sigma}({\bf r}) \psi^*_{\gamma -\sigma}({\bf r}) \psi_{\gamma \sigma}({\bf r}) \psi_{\alpha -\sigma}({\bf r})$. For $B \neq 0$, the correlator of the wavefunctions $\psi_{\alpha \sigma}$ and $\psi_{\alpha -\sigma}$ is a GUE parametric correlator which decays as a power law for $x_\perp \gg 1$ [@alhassid95; @wilkinson95]. For $B=0$, the orbital wavefunctions have GUE symmetry with $\psi_{\alpha -\sigma}= \psi^*_{\alpha \sigma}$ and thus $v_3 \propto \int d {\bf r}\overline{\psi^*_{\alpha \sigma}({\bf r}) \psi_{\gamma \sigma}({\bf r}) \psi_{\gamma \sigma}({\bf r}) \psi^*_{\alpha \sigma}({\bf r})}=0$ [@average]. Using Eqs. (\[averages\]), we can write the average part of the diagonal interaction in the form $$\begin{aligned} \label{average-int} \bar V_{\rm diag} = \sum_{\alpha \neq \gamma} ( \frac{1}{2} v_1 \hat n_\alpha \hat n_\gamma - \frac{1}{2} v_2 \sum_{\sigma}\hat n_{\alpha \sigma} \hat n_{\gamma \sigma} + \frac{1}{2} v_3 \sum_{\sigma} a^\dagger_{\alpha \sigma} a^\dagger_{\gamma -\sigma} a_{\alpha -\sigma}a_{\gamma \sigma} + v_4 \hat T^\dagger_\alpha \hat T_\gamma) - (v_1 + v_3 + v_4)\sum_\alpha \hat T^\dagger_\alpha \hat T_\alpha \;,\end{aligned}$$ where $\hat n_{\alpha \sigma}$ is the occupation operator of the state $|\alpha \sigma\rangle$ and $\hat n_{\alpha}= \hat n_{\alpha +} + \hat n_{\alpha -}$. Also $\hat T^\dagger_\alpha= a^\dagger_{\alpha +} a^\dagger_{\alpha -}$ is a pair creation operator, and we have used $\overline{ v_{\alpha + \alpha -;\alpha + \alpha-}} = v_1+ v_3+ v_4$ (as is easily verified for a contact interaction). The total spin operator of the dot can be represented in the basis $|\alpha \sigma\rangle$ as ${\bf\hat S} =\frac{1}{2} \sum_{\alpha \gamma; \sigma \sigma'} \langle \alpha \sigma | \gamma \sigma'\rangle a^\dagger_{\alpha \sigma} {\bm \sigma}_{\sigma \sigma'} a_{\gamma \sigma'}$ (${\bm \sigma}$ are Pauli matrices), and is no longer diagonal in the orbital label $\alpha$. However, since $\langle \alpha \sigma | \gamma \sigma\rangle = \delta_{\alpha \gamma}$, the $z$ component of the total spin is diagonal $\hat S_z=\frac{1}{2}\sum_{\alpha} (\hat n_{\alpha +} - \hat n_{\alpha -}) = \sum_{\alpha} \hat s_{\alpha z}$, and we have $\sum_\sigma \hat n_{\alpha \sigma} \hat n_{\gamma \sigma} = \frac{1}{2} \hat n_{\alpha} \hat n_{\gamma} + 2 \hat s_{\alpha z} \hat s_{\gamma z}$. The $v_3$ term in Eq. (\[average-int\]) is absent in the limit $x_\perp \to \infty$, while for $x_\perp=0$ it can be related to the spin operators through $a^\dagger_{\alpha +} a^\dagger_{\gamma -} a_{\alpha -}a_{\gamma +} + a^\dagger_{\alpha -} a^\dagger_{\gamma +} a_{\alpha +}a_{\gamma -} = 4( \hat s_{\alpha z} \hat{\bf s}_{\gamma z} - \hat s_\alpha \cdot \hat{\bf s}_\gamma)$. Thus in both symmetry limits $x_\perp=0$ and $x_\perp \to \infty$ we can write (\[average-int\]) in the form $$\begin{aligned} \bar V_{\rm diag}= \frac{1}{2}(v_1- v_2/2) \hat n^2 & - & \frac{1}{2}(v_1-v_2) \hat n -v_3 \hat{\bf S}^2 \nonumber \\ & - &(v_2-v_3) \hat S_z^2 +v_4 \hat T^\dagger \hat T \;,\end{aligned}$$ where $\hat T^\dagger=\sum_\alpha \hat T^\dagger_{\alpha}$. In the following we denote $\beta=1$ ($\beta=2$) for $B=0$ ($B \neq 0$). For $x_\perp=0$, $v_2=v_3=J_s$ and $v_4=\delta_{\beta 1} J_c$ ($J_c$ is the strength of the Cooper channel interaction), and we recover the universal Hamiltonian [@kurland00; @aleiner02]. However, for $x_\perp \gg 1$, we have $v_3=0$ and the ${\bf \hat{S}}^2$ interaction is replaced by $\hat S_z^2$. Furthermore, $v_4=\delta_{\beta 1} J_c$ still holds for $x_\perp\gg 1$. For $B \neq 0$, $v_4$ is obviously zero, while for $B=0$ we use $\psi_{\alpha -}=\psi^*_{\alpha +}$ to obtain $v_{\alpha + \alpha -;\gamma + \gamma-}\propto \int d {\bf r} |\psi_{\alpha +} ({\bf r})|^2 |\psi_{\gamma +}({\bf r})|^2$ and thus $v_4\neq 0$. As in the absence of SO, the off-diagonal elements of the residual interaction are suppressed at large $g_T$. Thus, for $x_\perp \gg 1$, we obtain a new universal Hamiltonian $$\label{universal-H} \hat H= \sum_{\alpha \sigma} \epsilon_{\alpha\sigma} \hat n_{\alpha \sigma}+ \frac{1}{2}U_d \hat n^2 -J_s \hat S_z^2 + \delta_{\beta 1} J_c \hat T^\dagger \hat T \;.$$ For $B=0$, the spin up/spin down levels are degenerate GUE levels ($\epsilon_{\alpha +} = \epsilon_{\alpha -}$), while for $B \neq 0$, the spin up/spin down levels are uncorrelated GUE levels. The important new feature of (\[universal-H\]) is that the exchange interaction is now given by $-J_s \hat S_z^2$ instead of $-J_s \hat{\bf S}^2$ [@dipolar]. In the crossover itself (finite $x_\perp$), the fluctuations of the off-diagonal matrix elements are enhanced and cannot be ignored [@brouwer02]. Instead, we use the interaction of the universal Hamiltonian in the absence of SO scattering, and add SO coupling to the single-particle Hamiltonian $$\begin{aligned} \label{crossover-H} \hat H_c = \sum_{\alpha \sigma} \epsilon_{\alpha\sigma} \hat n_{\alpha \sigma}& + & \frac{1}{2}U_d \hat n^2 - J_s \hat S_z^2 + \delta_{\beta 1} J_c \hat T^\dagger \hat T \nonumber \\ & - & {1\over 2}J_s (\hat S_+\hat S_- + \hat S_-\hat S_+) \;,\end{aligned}$$ where we have used $\hat{\bf S}^2 = (\hat S_+\hat S_- + \hat S_-\hat S_+)/2 +\hat S_z^2$. The spin and pairing operators can be rewritten in the SO eigenstates $|\alpha \sigma\rangle$. The spin projection $S_z= \frac{1}{2}\sum_\alpha (\hat n_{\alpha+} - \hat n_{\alpha -})$ and the pair operator $\hat T^\dagger = \sum_\alpha a^\dagger_{\alpha +} a^\dagger_{\alpha-}$ (for $\beta=1$) remain diagonal as in the universal Hamiltonian (\[universal-H\]). However, the spin components in the plane of the dot acquire an off-diagonal form $$\begin{aligned} \label{spin} \hat S_+ = \sum_{\alpha \gamma} \zeta^*_{\gamma \alpha} a^\dagger_{\alpha +} a_{\gamma -}\;;\;\; \hat S_- = \sum_{\alpha \gamma} \zeta_{\alpha \gamma} a^\dagger_{\alpha -} a_{\gamma +} \;.\end{aligned}$$ Here $\zeta_{\alpha \gamma}$ are fluctuating quantities whose statistical properties depend on $x_\perp$. For $B\neq 0$, $\zeta_{\alpha \gamma} \equiv \langle \alpha(0) | \gamma (x_\perp)\rangle$ describe parametric overlaps in the Gaussian unitary process (GUP) between the eigenstates at scaled parameter values $x_\perp=0$ and $x_\perp$ [@alhassid95; @wilkinson95]. For $B=0$, $\zeta_{\alpha \gamma}$ are just the orthogonal invariants $\rho_{\alpha \gamma}$ defined by the “real” scalar product of the eigenstates $\alpha$ and $\gamma$ [@brouwer02]. In the GUP, $\overline{|\zeta_{\alpha \gamma}|^2} = x_\perp^2/[(\epsilon_\alpha -\epsilon_\gamma)^2/\Delta^2 +(\pi x_\perp^2)^2]$, where $\epsilon_\alpha$ are the energy levels at a parameter value $x_\perp \gg 1$ [@wilkinson95] (a similar expression holds in the GOE to GUE crossover [@brouwer02]). Thus the number of levels $\gamma$ coupled to a level $\alpha$ in (\[spin\]) is $\sim x_\perp^2$. Since (\[crossover-H\]) describes an effective Hamiltonian of $\sim g_T$ levels around the Fermi energy, it is valid for $x_\perp^2 \ll g_T$. The matrix elements $\zeta_{\alpha \gamma}\zeta_{\mu \nu}$ of the interaction $\hat S_+\hat S_- + \hat S_-\hat S_+$ have rms values $\lesssim 1/x_\perp^2$, and dominate the corrections to (\[crossover-H\]) which are of the order $1/g_T$ . For $x^2_\perp \agt g_T$, the effective Hamiltonian is the universal Hamiltonian (\[universal-H\]). The spin $S$ is no longer a good quantum number of the Hamiltonians (\[universal-H\]) and (\[crossover-H\]) (for $x_\perp\neq 0$), but $S_z=M$ remains a good quantum number. In the following we focus on the universal Hamiltonian (\[universal-H\]) and ignore the Cooper channel term. Since $[\hat n_{\lambda \sigma},\hat S_z]=0$, the occupations ${\bf n} \equiv \{n_{\lambda \sigma}\}$ form a complete set of good quantum numbers. The corresponding eigenstates $| N {\bf n} M \rangle$ have energies $\varepsilon^{(N)}_{{\bf n} M} = \sum_{\lambda \sigma} \epsilon_{\lambda \sigma} n_{\lambda \sigma} +U_d N^2/2 - J_s M^2$, where $N=\sum_{\lambda \sigma} n_{\lambda \sigma}$ is the total number of electrons and $M=\sum_\lambda(n_{\lambda +}- n_{\lambda -})/2$. We have calculated the conductance $G$ at temperature $T$ using the rate equations approach [@master]. Defining a scaled conductance $g = (\hbar k T/ e^2 \bar \Gamma) G$ ($\bar \Gamma$ is an average tunneling width), we find $$\label{conductance} g = \sum_{\lambda \sigma} w_{\lambda \sigma} g_{\lambda \sigma} \;.$$ Here $g_{\lambda \sigma}\! = (2/\bar\Gamma) \Gamma_{\lambda \sigma}^{\rm l} \Gamma_{\lambda \sigma}^{\rm r}/ (\Gamma_{\lambda \sigma}^{\rm l} + \Gamma_{\lambda \sigma}^{\rm r})$ are the single-particle level conductances, where $\Gamma^{\rm r}_{\lambda \sigma}$ ($\Gamma^{\rm r}_{\lambda \sigma}$) is the partial width of an electron in level $\lambda \sigma$ to decay to the left (right) lead. The thermal weights $w_{\lambda \sigma}$ are given by $$\label{weight} w_{\lambda \sigma} = \sum_{{\bf n} \atop n_{\lambda \sigma}=0} \tilde P^{(N)}_{{\bf n} M} f(\varepsilon^\sigma_{\lambda M}) \;,$$ where the sum over all occupation sequences is restricted to the level $\lambda$ with spin $\sigma$ being empty. In Eq. (\[weight\]), $f(x)=(1+e^{x/kT})^{-1}$ is the Fermi-Dirac function evaluated at an energy $\varepsilon^\pm_{\lambda M}= \epsilon_{\lambda \pm} \mp J_s (M \pm 1/4) -\tilde \epsilon_F$, which corresponds to the addition of an electron with spin $\sigma$ is to a level $\lambda$ (${\tilde \epsilon}_{\rm F}$ is an effective Fermi energy). The quantity $\tilde P^{(N)}_{{\bf n} M}= e^{-[\varepsilon^{(N)}_{{\bf n} M} -\tilde \epsilon_F N]/kT}/Z$ is the equilibrium probability of the state $| N {\bf n} M\rangle$, with a partition function $Z$ defined as a Boltzmann-weighted sum over all possible $N$- and $(N+1)$-body states. The sum over all occupation numbers in Eq. (\[weight\]) can be evaluated in closed form at constant $M$. The constraint $n_{\lambda \sigma}=0$ is taken into account by introducing the factor $1-n_{\lambda \sigma}$, and we have $$\label{w-sigma} w_{\lambda \sigma} = \sum_M (1-\langle n_{\lambda}\rangle^\sigma_{n_{\sigma}}) \tilde P_{N, M} f(\varepsilon^\sigma_{\lambda M}) \;;$$ where $-N/2 \leq M \leq N/2$. Here $n_{\pm }= N/2 \pm M$ is the number of spin-up and spin-down electrons, and $\tilde P_{N,M}$ is the probability to find the dot with $N$ electrons and spin projection $S_z=M$. Since the trace at fixed $N,M$ is equivalent to a trace at fixed $n_+,n_-$, we find $$\label{spin-prob} \tilde P_{N,M} = e^{-( F^{+}_{n_+} + F^{-}_{n_-} + U_{N,M} )/kT}/ Z \;,$$ with $U_{N,M} = U_d N^2/2 - J_s M^2 - \tilde \epsilon_{\rm F} N$. In Eqs. (\[w-sigma\]) and (\[spin-prob\]) $F^{\sigma}_{n_\sigma} = -kT \ln {\rm tr}_{n_\sigma} e^{- \sum_{\lambda} \epsilon_{\lambda \sigma} c^\dag_\lambda c_\lambda/kT}$ and $\langle n_{\lambda}\rangle^\sigma_{n_{\sigma}}$ are free energy and canonical occupations of $n_\sigma$ non-interacting spinless fermions with energies $\epsilon_{\lambda \sigma}$. Effects of the SO interaction on the conductance peak height statistics in the absence of exchange interaction were discussed in Ref. . Here we consider SO effects on both the peak spacing and peak height statistics in the presence of exchange. We calculated these statistics in the limit $x_\perp \gg 1$ using the universal Hamiltonian (\[universal-H\]), and compared them with the corresponding statistics in the absence of SO interaction (i.e., $x_\perp =0$) [@exchange]. The left panel of Fig. \[fig1\] shows the width $\sigma(\Delta_2)$ of the peak spacing distribution versus $k T/\Delta$ for several values $J_s$ of the exchange interaction and for both limits $x_\perp \gg 1$ (dashed lines) and $x_\perp=0$ (solid lines). In the absence of exchange ($J_s=0$), the SO interaction leads to a strong suppression of the spacing fluctuations. However, at $J_s=0.3 \Delta$ the width is no longer sensitive to the SO coupling, except for a small suppression at higher temperatures. The peak-spacing distribution itself is affected by SO coupling and decreases more slowly at large spacings. In particular, the bimodality of the $x_\perp=0$ distribution for $kT \alt 0.3\ \Delta$ disappears for $x_\perp \gg 1$ (see inset in Fig. \[fig1\]) [@bimodal]. The right panel of Fig. \[fig1\] shows similar results as in the left panel but for $B=0$. In contrast to the $B \neq 0$ case, we observe that the width $\sigma(\Delta_2)$ is still sensitive to the SO coupling at $J_s=0.3\ \Delta$. SO enhances the spacing fluctuations at low temperatures but suppresses them at higher temperatures. = 0.98 A quantity that characterizes the peak height statistics is the ratio between the standard deviation $\sigma(g_{\rm max})$ and the average $\overline{g_{\rm max}}$ of the peak heights $g_{\rm max}$. The left panel of Fig. \[fig2\] shows this ratio as a function of $kT/\Delta$ for $J_s=0.3\ \Delta$ and $B \neq 0$ in both limits $x_\perp=0$ and $x_\perp \gg 1$. We observe that SO scattering suppresses $\sigma(g_{\rm max})/\overline{g_{\rm max}}$, and this suppression becomes stronger with temperature. At higher temperatures, inelastic scattering becomes important. The right panel of Fig. \[fig2\] shows similar results as in the left panel but in the rapid-thermalization limit of strong inelastic scattering. We observe that SO interaction can lead to a significant suppression of $\sigma(g_{\rm max})/\overline{g_{\rm max}}$ also in the rapid-thermalization limit. At present, no statistical data are available for observing SO effects in the presence of an exchange interaction. Using the value of $\lambda$ determined from experiments in large open dots  [@zumbuhl02], we estimate $x_\perp \ll 1$ for the small dots of Refs. . It would be interesting to measure the conductance peak statistics in almost-isolated dots with large area ${\cal A}$, in which SO effects are enhanced (for a fixed electron density, $x_\perp \propto {\cal A}^{5/4}$). In such large dots, it is difficult to reach the limit $T \ll \Delta$, but our results are not restricted to low temperatures. =0.98 In conclusion, we have derived the universal Hamiltonian of a quantum dot in the new symmetry limits when the leading order SO interaction term is included, both in the presence and absence of an orbital magnetic field. Using this universal Hamiltonian, we have identified the signatures of SO scattering in the conductance peak statistics in the presence of an exchange interaction. The universal Hamiltonians for other symmetries in the presence of SO scattering and the corresponding crossover Hamiltonians will be discussed elsewhere. We acknowledge useful discussions with B.L. Altshuler, P.W. Brouwer, V. Fal’ko, D. Huertas-Hernando, C.M. Marcus, and A.D. Mirlin. This work was supported in part by the U.S. DOE grant No. DE-FG-0291-ER-40608. [99]{} Y. Alhassid, Rev. Mod. Phys. [**72**]{}, 895 (2000). T. Guhr, A. Müller-Groeling, and H. A. Weidenmüller, Phys. Rep. [**299**]{}, 190 (1998). I. L. Kurland, I. L. Aleiner, and B. L. Altshuler, Phys.  Rev.  B [**62**]{}, 14886 (2000). I.L. Aleiner, P.W. Brouwer, and L.I. Glazman, Phys. Rep. [**358**]{}, 309 (2002). S. R. Patel [*et al.*]{}, Phys. Rev. Lett. [**80**]{}, 4522 (1998). S. R. Patel [*et al.*]{}. Phys. Rev. Lett. [**81**]{}, 5900 (1998). Y. Alhassid and T. Rupp, Phys. Rev. Lett. [**91**]{}, 056801 (2003). G. Usaj and H. U. Baranger, Phys. Rev. B [**67**]{}, 121308 (2003). J. A. Folk [*et al.*]{}, Phys. Rev. Lett. [**86**]{}, 2102 (2001). B. I. Halperin [*et al.*]{}, Phys. Rev. Lett. **86**, 2106 (2001). I.L. Aleiner and V.I. Fal’ko, Phys. Rev. Lett. [**87**]{}, 256801 (2001); [**89**]{}, 079902(E) (2002). D. M. Zumbühl [*et al.*]{}, Phys. Rev. Lett. [**89**]{}, 276803 (2002). D. A. Gorokhov and P.W. Brouwer, Phys. Rev. Lett. [**91**]{}, 186602 (2003); cond-mat/0311086. We assume the orbital field to be sufficiently large so that none of the effective fields in (\[effective-B\]) vanish as $B_{\rm so}$ increases. Y. Alhassid and H. Attias, Phys. Rev. Lett. [**74**]{}, 4635 (1995); Phys. Rev. Lett. [**76**]{}, 1711 (1996). M. Wilkinson and P. Walker, J. Phys. A [**28**]{}, 6143 (1995). The quantity $v_{\alpha \sigma \gamma -\sigma; \gamma -\sigma \alpha \sigma}$ is not gauge invariant and its average is generally not well defined. However in the limit $x_\perp \to\infty$, this quantity self averages to zero. In particular, $\overline{|v_{\alpha \sigma\gamma -\sigma; \gamma -\sigma \alpha \sigma}|^2}=0$ (up to $1/g_T$ corrections). The exchange interaction in (\[universal-H\]) is formally similar to the dipolar interaction of the double dot model by S. Adam, P. W. Brouwer, and P. Sharma, cond-mat/0309074. S. Adam [*et al.*]{}, Phys. Rev. B [**66**]{}, 165310 (2002). Y. Alhassid, T. Rupp, A. Kaminski, and L. I. Glazman, arXiv:cond-mat/0212072. K. Held, E. Eisenberg, and B. L. Altshuler, Phys. Rev. Lett., 106802 (2003). Finite-$g_T$ corrections to the universal Hamiltonian can also suppress the bimodality.

ArXiv

--- abstract: 'Bound- and excited-state electronic nonlinearities in CdS quantum dots have been investigated by Degenerate Four-Wave Mixing (DFWM) and Z-scan techniques in the femtosecond time regime. This QD sample shows Kerr-type nonlinearity for incident beam intensity below 0.18 TW/cm$^2$. However, further increment in intensity results in four-photon absorption (4PA) indicated by open- and closed-aperture Z-scan experiments. Comparing open-aperture Z-scan experimental results with theoretical models, the 4PA coefficient $\alpha_4$ has been deduced. Furthermore, third-order nonlinear index $\gamma$ and refractive-index change coefficient $\sigma_r$ corresponding to excited-state electrons due to 4PA have been calculated from the closed-aperture Z-scan results. UV-visible absorption and photoluminescence experimental results are analyzed towards estimating band gap energy and defect state energy. Time Correlated Single Photon Counting (TCSPC) was employed to determine the decay time corresponding to band-edge and defect states. The linear and nonlinear optical techniques have allowed the direct observation of lower and higher-order electronic states in CdS quantum dots.' author: - 'P. Ghosh [^1]' - 'E. Ramya' - 'P. K. Mohapatra' - 'D. Kushavah' - 'D. N. Rao' - 'P. Vasa' - 'K. C. Rustagi' - 'B. P. Singh' bibliography: - 'REFERENCES\_pin3.bib' title: 'Observation of four-photon absorption and determination of corresponding nonlinearities in CdS quantum dots' --- Introduction ============ The physics of quantum dots (QDs) is of great scientific interest from both fundamental and application point of view. A comprehensive knowledge about nonlinear absorption and refraction processes in quasi-zero dimensional semiconductor structures or QDs is important for further development of nonlinear-optical semiconductor devices [@Marcelo2013; @Dakovski2013; @Lad2007; @Guang2008]. Quest for knowledge about this topic can be adequately addressed by nonlinear optical experimental techniques, such as Z-scan [@YoshinoPhysRevLett.91.063902; @Sheik-Bahae1990; @Said1992; @Wei1992], degenerate four-wave mixing (DFWM) [@Canto-Said1991; @Bindra1999], and pump-probe spectroscopy [@Gaponenko1994]. Over the past years, these nonlinear optical experimental techniques have been extensively used as powerful tools towards investigating the excited electron-hole pair states dynamics of semiconductor QDs, providing complementary information that obtained by linear optical experimental techniques. With the access of ultrafast and ultrahigh intense laser pulses, multiphoton absorption $\it i.e.$ simultaneous absorption of two or more photons has been extensively studied. These multiphoton absorption processes are exceedingly promising in many fields including optical limiting [@He:95; @Prasad2008; @Venkatram2008; @Kiran:02], 3D microfabrication [@Maruo:97], optical data storage [@Nature2002PNPrasad; @PARTHENOPOULOS1989], and biomedical applications [@Yanik2006]. In this regard, CdS QDs are of particular interest because of their high intrinsic nonlinearity [@Kalyaniwalla1990]. So far, various nonlinear processes for comprehensive materials were studied [@Sheik-Bahae1990; @Canto-Said1991; @Said1992]. Furthermore, third-order nonlinear index $\gamma$ and refractive-index change coefficient $\sigma_r$ corresponding to free-carriers due to TPA have been calculated from closed-aperture Z-scan results [@Said1992]. To the best of our knowledge, there are hardly any work included discussion on deriving these nonlinear parameters for three or four-photon absorption in QDs. In this paper, we report the detail investigation of nonlinear optical processes in CdS QDs synthesized by gamma-irradiation technique. Towards understanding these processes, intensity dependent DFWM, open, and closed-aperture Z-scan experiments were performed. Furthermore, we derived $\gamma$ and $\sigma_r$ values corresponding to excited-state electrons generated by four-photon absorption. Results of open-aperture Z-scan with 400 nm femtosecond laser pulses has also been presented. In the first section of results and discussion, we report nonlinear studies on this CdS QD sample. In the later part, we present UV-visible absorption, room temperature photoluminescence and TCSPC experimental results for better understanding of the electronic states in the QDs. Experimental ============ The results of ultrafast nonlinear experiments including DFWM, open-aperture and closed-aperture Z-scan on colloidal solution of CdS QD sample have been reported in this paper. These nonlinear studies are performed using a Ti: Sapphire femtosecond laser (Spectra-Physics, Mai Tai, Spitfire amplifier) having wavelength $\lambda = 800$ nm, and repetition rate 1 KHz. The pulse width was determined to be 110 fs through intensity autocorrelation measurements. The nonlinear properties are investigated for the intensity regime 0.02 TW/cm$^2$ to 0.80 TW/cm$^2$ with the femtosecond laser pulses. The input beam intensity is varied using a polarizer and a $\lambda/2$ plate combination. It can be noted that at this intensity range, the water solution does not show any nonlinear behaviour for DFWM as well as Z-scan experiments. The DFWM experiments are performed using folded boxcar geometry [@Wise1998]. In this technique, a three-dimensional phase-matching is implemented, which enables spatial separation of the signal-beam from the input beams. The fundamental beam is divided into three nearly equal intensity beams (intensity ratio of 1:1:0.9) in such a way that they form three corners of a square and are focused into the nonlinear medium. All three beams are synchronized both spatially and temporally. The resultant DFWM signal is generated due to the phase-matched interaction: $\overrightarrow{k}_4=\overrightarrow{k}_1-\overrightarrow{k}_2+\overrightarrow{k}_3$. In Z-scan experiments, a Gaussian laser beam is tightly focused onto an optically non-linear sample using a finite aperture and the transmittance through the medium is measured in the far field. Finally, the resultant transmittance is recorded as function of the sample position Z measured about the focal plane. Open-aperture Z-scan has also been performed at wavelength 400 nm (second harmonic of the fundamental wavelength from a BBO crystal). The details about synthesis and structural characterization of the CdS QDs are reported in [@Soumyendu2012]. Particle size distribution and chemical composition are obtained from the HRTEM images, XPS and Raman spectra analysis. Results and discussion ====================== DFWM signal versus probe delay plots for colloidal solution of CdS QDs are shown in Fig. \[cds\_dfwm\_800nm\] (a). ![\[cds\_dfwm\_800nm\] ](DFWM_signal_delay_a.pdf "fig:"){height="0.25\textheight"} ![\[cds\_dfwm\_800nm\] ](DFWM_signal_intensity_b.pdf "fig:"){height="0.25\textheight"} The signals are fitted with Gaussian function (solid curve). The signal profiles are nearly symmetric about the maximum ($\it i.e.$ zero time delay) illustrating that the response times of the nonlinearities are shorter than the pulse duration (110 fs). This fast response enhances their potential for photonic switching applications. The intensity dependence of the DFWM signal amplitude is presented in Fig. \[cds\_dfwm\_800nm\] (b). At relatively low input intensities ($< 200$ GW/cm$^2$), the DFWM signal amplitude followes a cubic (with a slope of 2.9$\pm$0.1) dependence. It clearly demonstrates that the nonlinearity behaves in a Kerr-like fashion and the origin of DFWM does not have contribution from any multiphoton absorption process, which leads to higher power dependence [@Sutherland1996]. It can be seen from the intensity dependence of the DFWM signal plot that the DFWM signal intensity goes down at input intensity around 180 GW/cm$^2$. This substantial reduction in the DFWM signal intensity is mainly due to the nonlinear absorption of all interacting beams. However, the DFWM signal does not show any higher power dependence, expected for multiphoton absorption, indicating the dominance of $\chi^{(3)}$ process over multiphoton photon absorption at this input intensity regime. To confirm this, we have performed open-aperture Z-scan experiment, which is discussed in the next section. The measurement of $\chi^{(3)}$ values are performed at zero time delay of all the beams. We estimated the magnitude of $\chi^{(3)}_{1111}$ by maintaining the same polarization for all the three incident beams. The third-order nonlinear optical susceptibility $\chi^{(3)}$ is estimated by comparing the measured DFWM signal of the sample with that of $CS_2$ as reference ($\chi^{(3)} = 5 \times 10^{-13}$ esu [@HBLiao1998; @Minoshima1991]) measured with the same experimental conditions. The equation relating $\chi^{(3)}_{ref}$ and $\chi^{(3)}_{samp}$ is given by [@Sutherland1996] $$\chi^{(3)}_{samp}=\Bigg(\frac{n_{samp}}{n_{ref}}\Bigg)^2\Bigg(\frac{I_{samp}}{I_{ref}}\Bigg)^{1/2} \Bigg(\frac{L_{ref}}{L_{samp}}\Bigg)\alpha L_{samp} \Bigg(\frac{e^{\frac{\alpha L_{samp}}{2}}}{1-e^{-\alpha L_{samp}}}\Bigg)\chi_{ref}^{(3)}, \label{chi_3_DFWM}$$ where $I$ is the DFWM signal intensity, $\alpha$ is the linear absorption coefficient, $L$ is sample path length, and $n$ ($n_{samp}=1.329$ and $n_{CS_2}=1.606$ at $\lambda= 800$ nm) is the refractive-index. The effective refractive-index of the sample is essentially that of water solution. The $\chi^{(3)}$ value for the CdS QD sample comes out to be $(4.15\pm0.42) \times 10^{-13}$ esu for an input intensity of 47.5 GW/cm$^2$. Assuming no QD-QD interaction, the measured $\chi^{(3)}$ can be written as $$\chi^{(3)}=\Xi^{(3)} N,$$ where $N$ is the QD concentration in the solution and $\Xi^{(3)}$ is the average nonlinearity per QD. The QD concentration for CdS QD sample is 3.2 M. The $\Xi^{(3)}$ value for the CdS QDs comes out to be $2.15 \times 10^{-30}$ esu per QD. One of the main sources of error that arises in experiments is through the intensity fluctuations of laser pulses. This problem is tackled by taking the averaged data of 1000 pulses. The second major source of error could be from the determination of solution concentration. Considering all the unforced random experimental errors, we estimate an overall error of 10 $\%$ in our calculations by repeating the experiments few times. Towards performing Z-scan experiments, the incident Gaussian laser beam was passed through an aperture of diameter 3 mm and focused by a lens of focal length 12 cm. The beam waist ($\omega_0$) at the focal point (Z = 0) and the Rayleigh range ($Z_0 = k \omega_0^2/2$) were 23.3 $\mu$m and 2.13 cm, respectively. Whereas, the sample cell thickness was 1 mm. Therefore, the sample was considered as ’thin’ and the slowly varying envelope approximation (SVEA) was applied to obtain theoretical fitting of the experimental data points [@Sheik-Bahae1990]. Fig. \[oazs\_800nm\] (a) shows the measured open-aperture Z-scan plots of colloidal solution of CdS QDs for 800 nm wavelength, 110 fs laser pulses with three different input peak intensities (0.53 TW/cm$^2$, 0.67 TW/cm$^2$, and 0.80 TW/cm$^2$). The scattered points are experimental data points and the continuous curves are the theoretical fitting corresponding to 4PA. All the theoretical simulations were performed following the analytic expression for open-aperture Z-scan transmittance under first-order approximation given by Bing Gu [*et al.*]{} [@BingGuJOSA2010]. ![\[oazs\_800nm\] ](OAZSCAN_ALL_a.pdf "fig:"){height="0.25\textheight"} ![\[oazs\_800nm\] ](OAZSCAN_1mW_b.pdf "fig:"){height="0.25\textheight"} Fig. \[oazs\_800nm\] (b) shows the theoretical fitting of the open-aperture Z-scan data corresponding to input peak intensity 0.53 TW/cm$^2$ with n = 2, 3 , and 4. The theoretical fitting obtained with n = 2 and 3 corresponding to two-photon (TPA) and three-photon absorption (3PA) do not exactly reproduce the experimental data. This is a clear indication that the the TPA and 3PA are not the dominant processes at 800 nm excitation. The curves are therefore fitted with theoretically simulated result corresponding to four-photon absorption (4PA) process. The theoretical fitting with 4PA matches well with the experimental data. Fig. \[alpha234\_OL\_FSL\] (a) shows multi-photon absorption coefficient versus incident beam intensity plots. It can be noted that the $\alpha_4$ value remains almost constant for the intensity range 0.53 TW/cm$^2$ to 0.80 TW/cm$^2$. Whereas, $\alpha_2$ and $\alpha_3$ increase quadratically and linearly with incident beam intensity, respectively. Therefore, it can be concluded that at this incident beam intensity range, four-photon absorption process is dominant. Fig. \[alpha234\_OL\_FSL\] (b) shows nonlinear transmittance plot for CdS QD sample. It shows that the nonlinear absorption starts at peak intensity around 0.18 TW/cm$^2$, which supports the results obtained in DFWM experiments. ![\[alpha234\_OL\_FSL\] ](alpha234_a.pdf "fig:"){height="0.25\textheight"} ![\[alpha234\_OL\_FSL\] ](OL_b.pdf "fig:"){height="0.25\textheight"} Towards understanding the role of 4PA in nonlinear refraction, in case of excitation of CdS QDs with 800 nm femtosecond laser pulses, closed-aperture experiment was performed at different irradiances, ranging from 0.17 TW/cm$^2$ to 0.53 TW/cm$^2$. Fig. \[CAZS\_ALL\_deltan\_vs\_Intensity\] (a) shows the theoretical fitting of closed-aperture Z-scan plots corresponding to different incident beam intensity for CdS QD samples. All the theoretical fittings of closed-aperture Z-scan transmittance results were performed following the analytic expression given by Bing Gu [*et al.*]{} [@BingGuJOSA2010]. ![\[CAZS\_ALL\_deltan\_vs\_Intensity\] ](CAZSCAN_ALL_a.pdf "fig:"){height="0.25\textheight"} ![\[CAZS\_ALL\_deltan\_vs\_Intensity\] ](CAZSCAN_deltan_intensity_b.pdf "fig:"){height="0.25\textheight"} The valley-peak configuration of the closed-aperture Z-scan curve indicates positive (self-focusing) nonlinearity due to the electronic Kerr-effect and excited state electrons reached by 4PA process. The corresponding phase equation can be given by [@Sheik-Bahae1990] $$\frac{d\Delta \phi}{dz}= k\Delta n, \label{delta_n}$$ where $\Delta n = \gamma I+\sigma_r N$ is the change in index of refraction. $\gamma$ is the nonlinear index corresponding to the bound electrons and $\sigma_r$ is the change in the refractive-index per unit photo-generated excited state electron density N. In the context of excited state electron generation due to 4PA, we can neglect excited state relaxation as these processes occur at longer time scale than the femtosecond laser pulses used for performing these experiments. Therefore, neglecting relaxation loss, the excited state electron generation rate due to 4PA can be given by $$\frac{d N}{dt}= \frac{\alpha_4 I^4}{4\hbar \omega}. \label{carrier_generation_rate}$$ Using Eqs. \[delta\_n\] and \[carrier\_generation\_rate\], we obtained the formula relating $\Delta n/I_{0}$ and $I_{0}$ for the presence of third-order nonlinearity and photo-generated excited state electrons by 4PA. The equation is given by $$\Delta n/I_{0}=\gamma + C \sigma_r I_0^3, \label{gamma_sigma_r}$$ where $C=0.23 (\alpha_4 \tau_0/4 \hbar \omega)$. Here $\tau_0$ is pulse width of the excitation laser beam. In absence of nonlinear absorption, the difference between peak and valley ($\Delta T_{p-v}$) in closed-aperture Z-scan transmittance can be given by [@Said1992] $$\Delta T_{p-v}=p^{(3)} <\Delta \phi_0>,$$ where $p^{(3)} = 0.406(1-S)^{0.25}$ and $\Delta \phi_0$ is the on-axis phase change at the focus. A closed and an open-aperture Z-scan are performed at same irradiance, and the closed-aperture data are divided by the open-aperture data. $\Delta T_{p-v}$ is obtained from the resultant curve. This value is then divided by $p^{(3)}k L_{eff} I_{0}/2^{1/2}$ to determine $\Delta n/I_0$. For determining $L_{eff}$, $\alpha$ is calculated using the formula $\alpha = \alpha_0 + \alpha_4 I_0^3$, where $\alpha_0$ is the linear absorption coefficient, and $\alpha_4$ is the 4PA coefficient which is obtained from the open-aperture Z-scan experiment results. The experiments are performed at different irradiances, and $\Delta n/I_{0}$ is plotted as function of $I_0$. $\Delta n/I_{0}$ versus $I_{0}$ plot is shown in Fig. \[CAZS\_ALL\_deltan\_vs\_Intensity\] (b). In absence of any higher-order nonlinearity, this plot is expected to be a horizontal line with vertical intercept $\gamma$. From the theoretical fitting (red continuous curve) using Eq. \[gamma\_sigma\_r\], $\gamma$ and $\sigma_r$ are calculated and the values are $(4.45 \pm 0.1) \times 10^{-4}$ $cm^2/TW$ and $(6.0 \pm 0.3) \times 10^{-21}$ cm$^3$, respectively. Therefore, the closed-aperture Z-scan results further establish the 4PA processes. In this section, the results of the linear studies including UV-visible absorption, room temperature photoluminescence, and TCSPC are reported towards establishing the energetic positions of the electronic states and their decay times. Fig. \[Abs\_PL\_TCSPC\] (a) shows absorption and photoluminescence spectra of colloidal solution of CdS QDs. ![\[Abs\_PL\_TCSPC\] ](PL_Abs_FSL_a.pdf "fig:"){height="0.25\textheight"} ![\[Abs\_PL\_TCSPC\] ](TCSPC_NLO_b.pdf "fig:"){height="0.25\textheight"} The peak positions of the absorption band-edge for these semiconductor QD sample appears at around 380 nm wavelength ($\sim$ 3.26 eV). Whereas, the band gap of bulk CdS is 2.42 eV. This large blue shift of absorption band-edge is due to quantum confinement effect in QDs having diameter less than 5.8 nm (Bohr radius of bulk CdS). The broadness of the absorption band-edge suggests broad particle size distribution and confirmed by HRTEM images [@Soumyendu2012]. The average diameter of the QDs is 4.2 nm. The photoluminescence spectrum of these CdS QDs manifests two broad bands corresponding to Stokes shifted band-edge emission and defect state emission. The band-edge photoluminescence band ranges from 350 nm ($\sim$ 3.5 eV) to 500 nm ($\sim$ 2.5 eV). Whereas, the defect state emission band energy ranges from 2.5 eV to 1.6 eV with peak at around 1.9 eV. Time correlated single photon counting (TCSPC) was performed to determine the decay time of the band-edge and defect-state transitions in these QD sample. The fluorescence decay plots of the colloidal solutions of CdS QDs are shown in Fig. \[Abs\_PL\_TCSPC\] (b). A picosecond laser of wavelength 375 nm is used as excitation source. The PL emission is monitored at wavelengths 400 nm and 650 nm which correspond to the band-to-band and defect state transitions respectively. The FWHM of the instrument respose function (IRF) is 254 ps. The curves can be fitted with three exponential decay functions. The fluorescence decay times corresponding to 400 nm emission wavelength are: $\tau_1 \sim 0.2$ ns, $\tau_1 \sim 1.5$ ns, and $\tau_1 \sim 4.5$ ns. The decay times corresponding to 650 nm emission wavelength are: $\tau_1 \sim 0.3$ ns, $\tau_1 \sim 1.4$ ns, and $\tau_1 \sim 4.5$ ns. Whereas, the decay time obtained in DFWM with 800 nm femtosecond laser pulses is of the order of 110 fs. These results confirm that electrons do not get excited to band edge or defect states for 800 nm fs laser pulse excitation. Left panel of Fig. \[OAZSCAN\_400nm\_electronic\_states\_schematic\] shows open-aperture Z-scan with 400 nm femtosecond laser pulses with peak intensity 1.0 TW/cm$^2$. The theoretically simulated result corresponding to two-photon absorption (TPA) adequately reproduces the experimental data. The TPA coefficient value comes out to be 3.6 cm/TW. The schematic description of two-photon and four-photon transition processes and all the electronic states probed by linear and nonlinear optical techniques are shown in the right panel of Fig. \[OAZSCAN\_400nm\_electronic\_states\_schematic\]. The energy corresponding to 3PA for 800 nm excitation wavelength is 4.65 eV. ![\[OAZSCAN\_400nm\_electronic\_states\_schematic\] ](OAZSCAN_400nm_Electronic_states_FSL.png){height="0.28\textheight"} Whereas, Dhayal [*et al.*]{} [@Dhayal2014] have reported that there are real excited states at this energy level for CdS QDs. Therefore, it can be concluded that the electrons absorb three-photons initially and reach these real excited states. Thereafter the electrons absorb one more photon to reach the terminal state. Therefore, this multiphoton absorption process can be called as 3PA assisted 4PA. Conclusion ========== The ultrafast nonlinear optical properties including the time response of CdS QD sample using degenerate four-wave mixing technique at a wavelength of 800 nm with 110 fs pulses were thoroughly investigated. The nonlinear experiments were performed for the intensity regime 0.02 TW/cm$^2$ to 0.80 TW/cm$^2$. The CdS QD sample shows Kerr-type nonlinearity for intensity below 0.18 TW/cm$^2$. However, the intensity dependent open-aperture and closed-aperture Z-scan studies with 800 nm femtosecond laser pulses indicate 4PA above this input intensity. The closed-aperture Z-scan also manifests positive nonlinearity (self-focusing) for the CdS QDs. Open-aperture Z-scan with 400 nm femtosecond laser pulses shows two-photon absorption (TPA). Band gap energy and the defect state energy of the CdS QDs were estimated from the UV-visible absorption and PL spectrum. Whereas, information about the energy positions of the higher-order electronic states is obtained from the multiphoton absorption processes. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Prof. A. S. Pente, BARC, Mumbai, for providing gamma-irradiation facility for synthesis of CdS QDs. We also thank CRNTS and Central Surface Analytical Facility, IIT Bombay for facilitating HR-TEM characterizations. References {#references .unnumbered} ========== [^1]: Corresponding author

ArXiv

--- abstract: 'This paper focuses on interior penalty discontinuous Galerkin methods for second order elliptic equations on very general polygonal or polyhedral meshes. The mesh can be composed of any polygons or polyhedra which satisfies certain shape regularity conditions characterized in a recent paper by two of the authors in [@WangYe2012]. Such general meshes have important application in computational sciences. The usual $H^1$ conforming finite element methods on such meshes are either very complicated or impossible to implement in practical computation. However, the interior penalty discontinuous Galerkin method provides a simple and effective alternative approach which is efficient and robust. This article provides a mathematical foundation for the use of interior penalty discontinuous Galerkin methods in general meshes.' author: - 'Mu Lin[^1]' - 'Junping Wang[^2]' - 'Yanqiu Wang[^3]' - 'Xiu Ye[^4]' title: Interior penalty discontinuous Galerkin method on very general polygonal and polyhedral meshes --- discontinuous Galerkin, finite element, interior penalty, second-order elliptic equations, hybrid mesh. 65N15, 65N30. Introduction ============ Most finite element methods are constructed on triangular and quadrilateral meshes, or on tetrahedral, hexahedral, prismatic, and pyramidal meshes. To extend the idea of the finite element method into meshes employing general polygonal and polyhedral elements, one immediately faces the problem of choosing suitable discrete spaces on general polygons and polyhedrons. This issue has rarely been addressed in the past, partly because it can usually be circumvented by dividing the polygon or polyhedron into sub-elements using only one or two basic shapes. However, allowing the use of general polygonal and polyhedral elements does provide more flexibility, especially for complex geometries or problems with certain physical constraints. One of such example is the modeling of composite microstructures in material sciences. A well-known solution to this problem is the Voronoi cell finite element method [@Ghosh94; @Ghosh95; @Ghosh04; @Moorthy98], in which the mesh is composed of polygons or polyhedrons representing the grained microstructure of the given material. The main difficulty of constructing conforming finite element methods on Voronoi meshes is that, the finite element space has to be carefully chosen so that it is continuous along interfaces. Although the constructions on triangles, quadrilaterals, or three-dimensional simplexes are straight forward, it is not easy for general polygons and polyhedrons. Probably the only practically used solution is the rational polynomial interpolants proposed by Wachspress [@Wachspress75], in which rational basis functions are defined using distances from several “nodes”. An important constraint in the construction of the Wachspress basis is that, the rational basis functions need to be piecewise linear along the boundary of every element, in order to ensure $H^1$ conformity of the finite element space. This not only limits the approximation order of the entire Wachspress finite element space, but also complicates the construction. The Wachspress element has gained a renewed interest recently [@Dasgupta03; @Dasgupta03b; @Sukumar06]. However, as we have pointed out above, its construction is complicated and usually requires the aid of computational algebraic systems such as Maple. Another practically important issue is to define finite element methods on hybrid meshes. Hybrid meshes are frequently used nowadays. It can handle complicated geometries, and can sometimes reduce the total number of unknowns. Another possible reason for using the hybrid mesh is that, some engineers argue that in three-dimensions, a hexahedral mesh yields more accurate solution than a tetrahedral mesh for the same geometry [@Yamakawa03; @Yamakawa09], as partly verified by numerical experiments. However, pure hexahedral meshes lack the ability of handling complicated geometries. Hence a hybrid mesh becomes a welcomed compromise between accuracy and flexibility. For conforming finite element methods based on hybrid meshes, continuity requirements on interfaces must be satisfied. Such a coupling is straight-forward for the $H^1$-conforming finite elements on a triangular-quadrilateral hybrid mesh. However, for three-dimensional meshes, high order finite elements, or other complicated finite element spaces, it usually requires special treatments. An alternative solution, that can address both issues mentioned above, is to use the weak Galerkin method proposed in [@WangYe2012]. The weak Galerkin method uses discontinuous piecewise polynomials inside each element and on the interfaces to approximate the variational solution. In [@WangYe2012], the authors have proved optimal convergence of the weak Galerkin method for the mixed formulation of second order elliptic equations on very general polygonal and polyhedral meshes. Most of the existing error analysis of finite element methods assume triangular, quadrilateral, or some commonly-seen three-dimensional meshes. To our knowledge, it is the first time that optimal convergence for the finite element solutions has been rigorously proved in [@WangYe2012] for general meshes of arbitrary polygons and polyhedrons. The discontinuous Galerkin method imposes the interface continuity weakly, and is known to be able to handle non-conformal, hybrid meshes as well as a variety of basis functions. There have been many research works in this direction, for example, nodal discontinuous Galerkin methods [@Bergot10; @Cohen00; @Hesthaven00] for hyperbolic conservation laws. However, we would like to point out that so far there has been no theoretical analysis on the convergence rate of discontinuous Galerkin method, on very general polygonal or polyhedral meshes yet. Motivated by the work in [@WangYe2012], here we would like to fill the gap. The objective of this paper is to establish the theoretical analysis of the interior penalty discontinuous Galerkin method [@Arnold02] for elliptic equations on very general meshes and discrete spaces. The paper is organized as follows. In Section 2, we briefly describe the interior penalty discontinuous Galerkin method in an abstract setting. In Section 3, several assumptions on the discrete spaces are listed, which form a minimum requirement for the well-posedness and the approximation property of the discrete formulation. Abstract error estimations are given. In Section 4, we discuss choices of meshes and discrete spaces that satisfy the assumptions given in Section 3. Finally, numerical results are presented in Section 5. The model problem and the interior penalty method ================================================= Consider the model problem $$\label{eq:ellipticeq} \begin{cases} -\Delta u=f\qquad &\mbox{in }\Omega,\\ u=0 &\mbox{on }\partial\Omega, \end{cases}$$ where $\Omega\in\mathbb{R}^d(d=2,3)$ is a closed domain with Lipschitz continuous boundary, and $f\in L^2(\Omega)$. For any subdomain $K\subset \Omega$ with Lipschitz continuous boundary, we use the standard definition of Sobolev spaces $H^s(K)$ with $s\ge 0$ (e.g., see [@adams; @ciarlet] for details). The associated inner product, norm, and seminorms in $H^s(K)$ are denoted by $(\cdot,\cdot)_{s,K}$, $\|\cdot\|_{s,K}$, and $|\cdot|_{s,K}$, respectively. When $s=0$, $H^0(K)$ coincides with the space of square integrable functions $L^2(K)$. In this case, the subscript $s$ is suppressed from the notation of norm, semi-norm, and inner products. Furthermore, the subscript $K$ is also suppressed when $K=\Omega$. Finally, all above notations can easily be extended to any $e\subset \partial K$. For the $L^2$ inner product on $e$, we usually denote it as $\langle\cdot,\cdot\rangle_{e}$ in stead of $(\cdot,\cdot)_{e}$, as it can be replaced by the duality pair when needed. For simplicity, we assume that $\Omega$ satisfy certain conditions such that Equation (\[eq:ellipticeq\]) has at least $H^{r}$ regularity with $r>3/2$, that is, the solution to Equation (\[eq:ellipticeq\]) satisfies $u\in H^{r}(\Omega)$ and $$\label{eq:regularity} \|u\|_r \le C_R \|f\|.$$ This assumption is standard in the practice of interior penalty discontinuous Galerkin methods, as it ensures that the exact solution $u$ also satisfies the discontinuous Galerkin formulation, and thus the a priori error estimation can be easily derived in a Lax-Milgram framework. However, such a regularity assumption is not necessary in the practice of interior penalty methods. A well-known technique, which was first proposed by Gudi [@Gudi10], is to use a posteriori error estimation to derive an a priori error estimation for the interior penalty method, with only minimum regularity requirement $u\in H^1(\Omega)$. We believe that the same technique applies for the general polygonal and polyhedral meshes, as long as a working a posteriori error estimation is available. However, here we choose to completely skip this issue, as it is not the main purpose of this paper. Assume that for all set $K$ discussed in this paper, including $\Omega$ itself, the unit outward normal vector $\bn$ is defined almost everywhere on $\partial K$. Note this is true for all polygonal and polyhedral elements with Lipschitz continuous boundaries. Since the exact solution $u\in H^{r}(\Omega)$ with $r>3/2$, it is clear that for any smooth function $v$ defined on $K$, $$(\nabla u, \, \nabla v)_K - \langle\nabla u\cdot\vn,\, v\rangle_{\partial K} = (f,\, v)_K,$$ where $(\cdot,\cdot)_K$ is the $L^2$-inner product in $L^2(K)$ and $\langle\cdot,\cdot\rangle_\pK$ is the $L^2$-inner product in $L^2(\pK)$ Let $\mathcal{T}_h$ be a partition of the domain $\Omega$ into non-overlapping subdomains/elements, each with Lipschitz continuous boundary. Here $h$ denotes the characteristic size of the partition, which will be defined in details later. The interior interfaces are denoted by $e = \bar{K_1}\cap\bar{K_2}$, where $K_1$, $K_2\in \mathcal{T}_h$. Boundary segments are similarly denoted by $e = \bar{K}\cap\partial\Omega$, where $K\in \mathcal{T}_h$. Denote by $\mathcal{E}_h$ the set of all interior interfaces and boundary segments in $\mathcal{T}_h$, and by $\mathcal{E}_h^0=\mathcal{E}_h\setminus\partial\Omega$ the set of all interior interfaces. For every $K\in\mathcal{T}_h$, let $|K|$ be the area/volume of $K$, and for every $e\in \mathcal{E}_h$, let $|e|$ be its length/area. Denote $h_e$ the diameter of $e\in \mathcal{E}_h$ and $h_K$ the diameter of $K\in \mathcal{T}_h$. Clearly, when $e\subset \partial K$, we have $h_e\le h_K$. Finally, define $h=\max_{K\in \mathcal{T}_h} h_K$ to be the characteristic mesh size. Notice that $\mathcal{T}_h$ defined above is a very general mesh/partition on $\Omega$, as we do not specify the shape and conformal property of $K\in \mathcal{T}_h$. The interior penalty discontinuous Galerkin (IPDG) method can be extended to such a general mesh, without any modification of the formulation. However, to ensure its approximation rate, certain conditions must be imposed on $\mathcal{T}_h$ and the discrete function spaces. In this paper, we are interested in discussing the minimum requirements of such conditions. First, we shall give the formulation of the interior penalty discontinuous Galerkin method. Let $V_K$ be a finite dimensional space of smooth functions defined on $K \in \mathcal{T}_h$. Define $$V_h=\{v\in L^2(\Omega):v|_K\in V_K,\textrm{ for all } K\in\mathcal{T}_h\},$$ and $$V(h)=V_h+\left( H_0^1(\Omega)\cap \prod_{K\in\mathcal{T}_h} H^{r}(K) \right),\qquad \textrm{where }r>\frac{3}{2}.$$ For any internal interface $e = \bar{K_1}\cap\bar{K_2} \in \mathcal{E}_h$, let $\bn_1$ and $\bn_2$ be the unit outward normal vectors on $e$, associated with ${K_1}$ and ${K_2}$, respectively. For $v\in V(h)$, define the average $\{\nabla v\}$ and jump $[v]$ on $e$ by $$\{\nabla v\} = \frac{1}{2}\left(\nabla v|_{K_1} + \nabla v|_{K_2} \right),\qquad [v] = v|_{K_1} \bn_1 + v|_{K_2} \bn_2.$$ On any boundary segment $e= \bar{K}\cap\partial\Omega$, the above definitions of average and jump need to be modified: $$\{\nabla v\} = \nabla v|_{K},\qquad [v] = v|_{K} \bn_K,$$ where $\bn_K$ is the unit outward normal vector on $e$ with respect to $K$. Define a bilinear form on $V(h)\times V(h)$ by $$\begin{aligned} A(u,v) =& \sum_{K\in \mathcal{T}_h}(\nabla u,\nabla v)_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla u\},\, [v]\rangle_e \\ &\quad -\delta \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [u]\rangle_e + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \langle[u],\, [v]\rangle_e, \end{aligned}$$ where $\delta = \pm 1,\, 0$ and $\alpha>0$. when $\delta=1$, the bilinear form $A(\cdot,\cdot)$ is symmetric. The constant $\alpha$ is usually required to be large enough, but still independent of the mesh size $h$, in order to guarantee the well-posedness of the discontinuous Galerkin formulation. Details will be given later. It is clear that the exact solution $u$ to Equation (\[eq:ellipticeq\]) satisfies $$\label{eq:dg-exactsol} A(u,v) = (f,v)\qquad\textrm{for all } v\in V_h,$$ as $[u]$ vanishes on all $e\in\mathcal{E}_h$. Hence the following interior penalty discontinuous Galerkin formulation is consistent with Equation (\[eq:ellipticeq\]): find $u_h \in V_h$ satisfying $$\label{eq:dg} A(u_h,v) = (f,v)\qquad\textrm{for all } v\in V_h.$$ Finally, we would like to point out that the formulation (\[eq:dg\]) is computable, as long as each finite dimensional space $V_K$ has a clearly defined and computable basis. Abstract theory =============== Define a norm $\3bar\cdot\3bar$ on $V(h)$ as following: $$\begin{aligned} \3bar v\3bar^2=\sum_{K\in\mathcal{T}_h}\|\nabla v\|_K^2+\sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2+\alpha\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v]\|_e^2.\end{aligned}$$ By the Poincaré inequality, $\3bar\cdot\3bar$ is obviously a well-posed norm on $V(h)$. Next, we give a set of assumptions, which form the minimum requirements guaranteeing the well-posedness and the approximation properties of the interior penalty discontinuous Galerkin method. - (The trace inequality) There exists a positive constant $C_T$ such that for all $K\in\mathcal{T}_h$ and $\theta\in H^1(K)$, we have $$\label{eq:TraceIn} \|\theta\|_{\partial K}^2\le C_{T}(h_K^{-1}\|\theta\|_K^2+h_K\|\nabla\theta\|_K^2).$$ - (The inverse inequality) There exists a positive constant $C_I$ such that for all $K\in\mathcal{T}_h$, $\phi\in V_K$ and $\phi\in \frac{\partial}{\partial x_i}V_K$ where $i=1,\ldots, d$, we have $$\label{eq:InverseIn} \|\nabla\phi\|_K\le C_I\, h_K^{-1}\|\phi\|_K.$$ - (The approximability) There exist positive constants $s$ and $C_A$ such that for all $v\in H^{s+1}(\Omega)$, we have $$\label{eq:approximability} \inf_{\chi_h\in V_h} \3bar v-\chi_h \3bar \le C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|v\|_{s+1,K}^2\right)^{1/2}.$$ The abstract theory of the interior penalty discontinuous Galerkin method can be entirely based on Assumptions [**[I1]{}**]{}-[**[I3]{}**]{}. \[lem:wellposedness\] Assume [**[I1]{}**]{}-[**[I2]{}**]{} hold. The bilinear form $A(\cdot,\cdot)$ is bounded in $V(h)$, with respect to the norm $\3bar\cdot\3bar$. Indeed, $$A(u,v)\le \frac{1+\alpha}{\alpha} \3bar u\3bar\, \3bar v\3bar\qquad \textrm{for all } u,\, v\in V(h).$$ Furthermore, denote $C_1 = C_T(1+C_I)^2$. Then for any constant $0<C<1$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, the bilinear form $A(\cdot,\cdot)$ is coercive on $V_h$. That is, $$A(v, v) \ge \frac{C}{1+C_1} \3bar v\3bar^2\qquad \textrm{for all } v\in V_h.$$ The boundedness of $A(\cdot,\cdot)$ follows immediately from the Schwarz inequality. Here we only prove the coercivity. First, notice that for all $v\in V_h$, by assumptions [**[I1]{}**]{}-[**[I2]{}**]{} and the fact that $h_e\le h_K$ for all $e\in \partial K\cap \mathcal{E}_h$, $$\begin{aligned} \sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 & \le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} h_e \|\nabla v\|_e^2\right) \\ &\le \sum_{K\in\mathcal{T}_h} h_K \|\nabla v\|_{\partial K}^2 \\ & \le \sum_{K\in\mathcal{T}_h} h_K \bigg( C_T(1+C_I^2) h_K^{-1} \|\nabla v\|_K^2 \bigg) \\ & = C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2. \end{aligned}$$ Then, by the Schwarz inequality, the Young’s inequality and assumptions [**[I1]{}**]{}-[**[I2]{}**]{}, we have $$\begin{aligned} \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [v]\rangle_e &\le \varepsilon \sum_{e\in\mathcal{E}_h}h_e\|\{\nabla v\}\|_e^2 + \frac{1}{4\varepsilon} \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\ &\le \varepsilon C_1 \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \frac{1}{4\varepsilon\alpha} \left(\alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \right), \end{aligned}$$ where $\varepsilon$ is chosen to be $\frac{1-C}{(1+\delta)C_1}$ for any given constant $0<C<1$. Clearly, for such an $\varepsilon$, we have $1-(1+\delta)\varepsilon C_1 = C$ and $$1-\frac{1+\delta}{4\varepsilon\alpha} \ge C \quad \Longleftrightarrow \quad \alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}.$$ Combine the above and let $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, we have $$\begin{aligned} A(v, v) &= \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 - (1+\delta) \sum_{e\in\mathcal{E}_h}\langle\{\nabla v\},\, [v]\rangle_e + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\ &\ge \bigg( 1-(1+\delta)\varepsilon C_1\bigg)\sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \bigg(1-\frac{1+\delta}{4\varepsilon\alpha}\bigg)\alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \\ &\ge C \left( \sum_{K\in\mathcal{T}_h} \|\nabla v\|_K^2 + \alpha \sum_{e\in\mathcal{E}_h}\frac{1}{h_e} \|[v]\|_e^2 \right)\\ &\ge \frac{C}{1+C_1} \3bar v\3bar^2. \end{aligned}$$ Lemma \[lem:wellposedness\] guarantees the existence and uniqueness of the solution to Equation (\[eq:dg\]). In the rest of this paper, we shall always assume $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. By subtracting (\[eq:dg-exactsol\]) from (\[eq:ellipticeq\]), one gets the standard orthogonality property of the error, $$A(u-u_h,\, v_h) = 0\qquad\textrm{for all } v\in V_h.$$ Then clearly, for all $\chi_h\in V_h$, $$\begin{aligned} \3bar \chi_h - u_h\3bar^2 &\le \frac{1+C_1}{C}A(\chi_h - u_h,\, \chi_h - u_h) \\ &= \frac{1+C_1}{C}A(\chi_h - u,\, \chi_h - u_h) \\ &\le \frac{(1+C_1)(1+\alpha)}{C\alpha} \3bar \chi_h - u\3bar \, \3bar \chi_h - u_h\3bar. \end{aligned}$$ Then, using the triangle inequality, $$\begin{aligned} \3bar u-u_h\3bar &\le \inf_{\chi_h\in V_h} \bigg( \3bar u-\chi_h\3bar + \3bar \chi_h-u_h\3bar \bigg) \\ &\le \bigg(1 + \frac{(1+C_1)(1+\alpha)}{C\alpha} \bigg) \inf_{\chi_h\in V_h}\3bar u-\chi_h\3bar. \end{aligned}$$ Combine this with assumption [**[I3]{}**]{}, we get the following abstract error estimation: Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $C$ be a given constant in $(0,1)$ and $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\3bar u-u_h\3bar \lesssim C_A\bigg(1 + \frac{(1+C_1)(1+\alpha)}{C\alpha} \bigg) \left(\sum_{K\in \mathcal{T}_h} h_K^{2s} \|u\|_{s+1,K}^2\right)^{1/2}.$$ Finally, we derive the $L^2$ error estimation by using the standard duality argument. Let $\delta=1$, that is, the bilinear form $A(\cdot,\cdot)$ is symmetric. Consider the following problem $$\begin{cases} -\Delta \phi=u-u_h\qquad &\mbox{in }\Omega,\\ \phi=0 &\mbox{on }\partial\Omega. \end{cases}$$ Here again, we assume that the domain $\Omega$ satisfies certain condition such that $\phi$ has $H^r$ regularity, with $r>3/2$. Let $\phi_h\in V_h$ be an approximation to $\phi$ such that they satisfy Assumption [**[I3]{}**]{}. Clearly $$\begin{aligned} \|u-u_h\|^2 &= (-\Delta \phi,\, u-u_h) = \sum_{K\in \mathcal{T}_h}(\nabla \phi,\nabla (u-u_h))_K-\sum_{e\in\mathcal{E}_h}\langle\{\nabla \phi\},\, [u-u_h]\rangle_e \\ &= A(\phi, \, u-u_h) = A(\phi-\phi_h,\, u-u_h)\\ &\le \frac{1+\alpha}{\alpha} \3bar \phi-\phi_h \3bar\, \3bar u-u_h\3bar \\ &\le \frac{1+\alpha}{\alpha}C_A \left(\sum_{K\in \mathcal{T}_h} h_K^{2\min\{r-1,s\}} \|\phi\|_{\min\{r,s+1\},K}^2\right)^{1/2} \3bar u-u_h\3bar. \end{aligned}$$ This gives the following theorem Assume [**[I1]{}**]{}-[**[I3]{}**]{} hold, $\delta=1$, $C$ be a given constant in $(0,1)$, $\alpha \ge \frac{(1+\delta)^2C_1}{4(1-C)^2}$, and the elliptic equation (\[eq:ellipticeq\]) has $H^r$ regularity with $r>3/2$. Let $u$ and $u_h$ be the solution to equations (\[eq:ellipticeq\]) and (\[eq:dg\]), respectively. Then $$\|u-u_h\| \le \frac{1+\alpha}{\alpha}C_AC_R h^{\min\{r-1,s\}} \3bar u-u_h\3bar.$$ Requirements on meshes and discrete spaces ========================================== On triangular or quadrilateral meshes, the usual tool for proving assumptions [**[I1]{}**]{}-[**[I3]{}**]{} is to use a scaling argument built on affine transformations. However, on general polygons and polyhedrons, it is not clear how to define such affine transformations. The assumptions [**[I1]{}**]{}-[**[I3]{}**]{} were first validated in [@WangYe2012] for general polygonal and polyhedral meshes that satisfy a set of conditions introduced in [@WangYe2012]. Such conditions can be stated as follows. All the elements of $\mathcal{T}_h$ are assumed to be closed and simply connected polygons or polyhedrons. We make the following shape regularity assumptions for the partition $\mathcal{T}_h$. - Assume that there exist two positive constants $\rho_v$ and $\rho_e$ such that for every element $K\in\mathcal{T}_h$ and $e\in \mathcal{E}_h$, we have $$\begin{aligned} \rho_vh_K^d\le |K|,\ \ \rho_eh_e^{d-1}\le |e|.\end{aligned}$$ - Assume that there exists a positive constant $\kappa$ such that for every element $K\in\mathcal{T}_h$ and $e\in \partial K\cap \mathcal{E}_h$, we have $$\begin{aligned} \kappa h_K\le h_e.\end{aligned}$$ - Assume that for every $K\in\mathcal{T}_h,$ and $e\in\partial K \cap \mathcal{E}_h$, there exists a pyramid $P(e,K,A_e)$ contained in $K$ such that its base is identical with $e$, its apex is $A_e\in K$, and its height is proportional to $h_K$ with a proportionality constant $\sigma_e$ bounded away from a fixed positive number $\sigma^*$ from below. In other words, the height of the pyramid is given by $\sigma_eh_K$ such that $\sigma_e\ge \sigma^*>0.$ The pyramid is also assumed to stand up above the base $e$ in the sense that the angle between the vector ${\bf x}_e-A_e,$ for any ${\bf x}_e\in e$, and the outward normal direction of $e$ is strictly acute by falling into an interval $[0,\theta_0]$ with $\theta_0<\pi/2$. - Assume that each $K\in\mathcal{T}_h$ has a circumscribed simplex $S(K)$ that is shape regular and has a diameter $h_{S(K)}$ proportional to the diameter of $K$; i.e., $h_{S(K)}\le\gamma_*h_K$ with a constant $\gamma_*$ independent of $K.$ Furthermore, assume that each circumscribed simplex $S(K)$ intersects with only a fixed and small number of such simplexes for all other elements $K\in\mathcal{T}_h.$ Under the above assumptions, the following results have been proved in [@WangYe2012]: (The trace inequality). Assume [**[A1]{}**]{}-[**[A3]{}**]{} hold on a polygonal or polyhedral mesh. Then [**[I1]{}**]{} is true. (The inverse Inequality). Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Then [**[I2]{}**]{} is true with $C_I$ depending on $n$, but not on $h_K$ or $|K|$. \[lem:L2proj\] Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Let $Q_h$ be the $L^2$ projection onto $V_h$. Then for all $0\le s\le n$ and $v\in H^{s+1}(\Omega)$, $$\begin{aligned} \sum_{K\in\mathcal{T}_h}\|v-Q_hv\|_K^2 &\le C_{Q0} h^{2(s+1)} \|v\|_{s+1}^2. \\ \sum_{K\in\mathcal{T}_h}\|\nabla (v-Q_hv)\|_K^2 &\le C_{Q1} h^{2s} \|v\|_{s+1}^2. \end{aligned}$$ It is not hard to see that [**[I3]{}**]{} follows immediately from Lemma \[lem:L2proj\]. Indeed, notice that as long as [**[I1]{}**]{} and [**[I2]{}**]{} are true and $v\in H^r(\Omega)$ with $r>3/2$, we have $$\3bar v-Q_h v\3bar \le (1+C_1) \sum_{K\in\mathcal{T}_h}\|\nabla (v-Q_hv)\|_K^2 + \alpha\sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v-Q_h v]\|_e^2,$$ where $C_1=C_T(1+C_I)^2$. Next, notice that by [**[A2]{}**]{}, [**[I1]{}**]{} and Lemma \[lem:L2proj\], $$\begin{aligned} \sum_{e\in\mathcal{E}_h}\frac{1}{h_e}\|[v-Q_h v]\|_e^2 &\le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} \frac{1}{h_e} \|(v-Q_h v)|_K\|_e^2\right) \\ & \le \sum_{K\in\mathcal{T}_h} \left(\sum_{e\in \partial K\cap \mathcal{E}_h} \frac{1}{\kappa h_K} \|(v-Q_h v)|_K\|_e^2\right) \\ & = \sum_{K\in\mathcal{T}_h} \frac{1}{\kappa h_K} \|(v-Q_h v)|_K\|_{\partial K}^2 \\ & \le \sum_{K\in\mathcal{T}_h} \frac{C_T}{\kappa h_K} \left(h_K^{-1} \|v-Q_h v\|_K^2 + h_K\|\nabla(v-Q_h v)\|_K^2\right) \\ &\le \frac{C_T}{\kappa} (C_{Q0} + C_{Q1}) h^{2s} \|v\|_{s+1}^2. \end{aligned}$$ Combine the above, we have (The aproximability) Assume [**[A1]{}**]{}-[**[A4]{}**]{} hold on a polygonal or polyhedral mesh and each $V_K$ is the space of polynomials with degree less than or equal to $n$. Then for all $\frac{1}{2}< s\le n$ and $v\in H^{s+1}(\Omega)$, there exists a constant $C_A$ independent of $h$ such that $$\inf_{\chi_h\in V_h} \3bar v-\chi_h \3bar \le C_A h^{s} \|v\|_{s+1}.$$ Here $s>\frac{1}{2}$ is added so that $\3bar v - \chi_h\3bar$ is well-defined. Numerical Examples ================== Finally, we present numerical results that support the theoretical analysis of this paper. We fix the coefficients $\delta = 1$ and $\alpha = 10$, since the purpose of the numerical experiments is to examine the accuracy of the interior penalty discontinuous Galerkin method on arbitrary polygonal meshes, not for different coefficients. Consider the Poisson’s equation on $\Omega = (0,1)\times(0,1)$ with the exact solution $u = sin(2\pi x)\cos(2\pi y)$. Clearly $u=0$ on $\partial\Omega$. For simplicity of the notation, we denote $$|u-u_h|_{1,h} = \left(\sum_{K\in\mathcal{T}_h} |\nabla(u-u_h)|_K^2\right)^{1/2}.$$ The first test is performed on a non-conformal triangular-quadrilateral hybrid mesh. The initial mesh and the mesh after one uniform refinement are given in Figure \[fig:mesh1\]. A sequence of uniform refinements are then applied to generate a set of nested meshes. Notice that the meshes are non-conformal and there are hanging nodes. However, the interior penalty discontinuous Galerkin method can deal with such meshes without special treatments. We solve the Poisson equation using the interior penalty discontinuous Galerkin formulation (\[eq:dg\]) on these meshes, where the local discrete spaces $V_K$ are taken to be $P_1$ polynomials on each $K\in \mathcal{T}_h$, no matter whether $K$ is a triangle or quadrilateral. The $H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test1\] and Figure \[fig:test1\]. These errors are computed using a 5th order Gaussian quadrature on triangles. For quadrilateral elements, the errors can be conveniently computed by dividing the quadrilateral into two triangles and then applying the Gaussian quadrature. Our results show that the $H^1$ semi-norm has an approximate order of $O(h)$, while the $L^2$ norm has an approximate order of $O(h^2)$, as predicted by the theoretical analysis. ![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-1 "fig:"){width="6cm"}![Initial and refined mesh for test 1.[]{data-label="fig:mesh1"}](mesh1-2 "fig:"){width="6cm"} ------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ---------------- $h$ $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $O(h^r)$, $r=$ \[1mm\] $|u-u_h|_{1,h}$ 1.2006 0.5904 0.2917 0.1452 0.0725 1.0124 $\|u-u_h\|$ 0.0551 0.0159 0.0042 0.0011 0.0003 1.9270 ------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ---------------- : Convergence rates for test 1.[]{data-label="tab:test1"} ![Convergence rates for test 1.[]{data-label="fig:test1"}](error1){width="8cm"} In the second test, we consider a hybrid mesh containing mainly hexagons, but with a few quadrilaterals and pentagons. Indeed, it is derived by taking the dual mesh of a simple triangular mesh. In Figure \[fig:mesh2\], the initial triangular mesh and its dual mesh are shown. By refining the triangular mesh and computing its dual mesh, we get a sequence of hexagon hybrid meshes. Again, we solve the interior penalty discontinuous Galerkin formulation (\[eq:dg\]) on these hexagon hybrid meshes, with the local discrete spaces $V_K$ of $P_1$ polynomials. The $H^1$ semi-norm and the $L^2$ norm of the errors are reported in Table \[tab:test2\] and Figure \[fig:test2\]. Optimal convergence rates are achieved. ![The original triangular mesh and its dual mesh used in test 2.[]{data-label="fig:mesh2"}](mesh2-1 "fig:"){width="6cm"}![The original triangular mesh and its dual mesh used in test 2.[]{data-label="fig:mesh2"}](mesh2-2 "fig:"){width="6cm"} ------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ---------------- $h$ $\frac{1}{16}$ $\frac{1}{32}$ $\frac{1}{64}$ $\frac{1}{128}$ $\frac{1}{256}$ $O(h^r)$, $r=$ \[1mm\] $|u-u_h|_{1,h}$ 0.8139 0.3868 0.1894 0.0941 0.0470 1.0270 $\|u-u_h\|$ 0.0461 0.0129 0.0034 0.0009 0.0002 1.9393 ------------------------- ---------------- ---------------- ---------------- ----------------- ----------------- ---------------- : Convergence rates for test 2.[]{data-label="tab:test2"} ![Convergence rates for test 2.[]{data-label="fig:test2"}](error2){width="8cm"} [99]{} , [*Sobolev Spaces*]{}, Academic press, 2003. , [*Unified analysis of discontinuous Galerkin methods for elliptic problems*]{}, SIAM J. Numer. Anal. 39 (2002), pp. 1749–1779. , [*Higher-order finite elements for hybrid meshes using new nodal pyramidal elements*]{}, J. Sci. Comput. 42 (2010), pp. 345–381. , [*The Finite Element Method for Elliptic Problems*]{}, North-Holland, New York, 1978. , [*Higher-Order Numerical Methods for Transient Wave Equations*]{}, Springer, Berlin, 2000. , [*Interpolation within convex polygons: Wachspress’ shape functions*]{}, Hournal of Aerospace Engineering, 16 (2003), pp. 1-8. , [*Integration within polygonal finite elements*]{}, Journal of Aerospace Engineering, 16 (2003), pp. 9-18. , [*Voronoi cell finite elements*]{}, Computers & Structures, 50 (1994), pp. 33–46. , [*Elastic-plastic analysis of arbitrary heterogeneous materials with the Voronoi cell finite-element method*]{}, Comput. Methods Appl. Mech. Engrg., 121 (1995), pp. 373–409. , [*Three dimensional Voronoi cell finite element model for microstructures with ellipsoidal heterogeneties*]{}, Computational Mechanics, 34 (2004), pp. 510–531. , [*A new error analysis for discontinuous finite element methods for linear elliptic problems*]{}, Math. Comp., 79 (2010), pp. 2169–2189. , [*Stable spectral methods on tetrahedral elements*]{}, SIAM J. Numer. Anal., 21 (2000), pp. 2352–2380. , [*A Voronoi cell finite element model for partical cracking in elastic-plastic composite materials*]{}, Comput. Methods Appl. Mech. Engrg., 151 (1998), pp. 377–400. , [*Recent advances in the construction of polygonal finite element interpolants*]{}, Arch. Comput. Meth. Engrg., 13 (2006), pp. 129–163. , [*Comparison of hexahedral and tetrahedral elements in finite element analysis of the foot and footwear*]{}, J Biomech., 44 (2011), pp. 2337–2343. , [*A rational finite element basis*]{}, Academic Press, New York, 1975. , [*A weak Galerkin mixed finite element method for second-order elliptic problems*]{}, [arXiv:1202.3655v1 \[math.NA\] 16 Feb 2012.]{} , [*Fully-automated hex-dominant mesh generation with directionality control via packing rectangular solid cells*]{}, Int. J. Numer. Meth. Engng., 57 (2003), pp. 2099–2129. , [*Converting a tetrahedral mesh to a prism–tetrahedral hybrid mesh for FEM accuracy and efficiency*]{}, Int. J. Numer. Meth. Engng., 80 (2009), pp. 74–102. [^1]: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204 ([email protected]). [^2]: Division of Mathematical Sciences, National Science Foundation, Arlington, VA 22230 ([email protected]). The research of Wang was supported by the NSF IR/D program, while working at the Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. [^3]: Department of Mathematics, Oklahoma State University, Stillwater, OK 74075 ([email protected]). [^4]: Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204 ([email protected]). This research was supported in part by National Science Foundation Grant DMS-1115097.

ArXiv

--- author: - title: Making Use of Affective Features from Media Content Metadata for Better Movie Recommendation Making ---

ArXiv

--- abstract: 'Proton acceleration by using a 620-TW, $18$-J laser pulse of peak intensity of $5\times 10^{21}$ W/cm$^{2}$ irradiating a disk target is examined using three-dimensional particle-in-cell simulations. It is shown that protons are accelerated efficiently to high energy for a “light” material in the first layer of a double-layer target, because a strongly inhomogeneous expansion of the first layer occurs by a Coulomb explosion within such a material. Moreover, a large movement of the first layer for the accelerated protons is produced by radiation-pressure-dominant acceleration. A time-varying electric potential produced by this expanding and moving ion cloud accelerates protons effectively. In addition, using the best material for the target, one can generate a proton beam with an energy of $200$ MeV and an energy spread of 2$\%$.' author: - Toshimasa Morita title: Laser ion acceleration by using the dynamic motion of a target --- INTRODUCTION ============ Recently, there has been great progress in compact laser systems, with dramatic improvements in both laser power and peak intensity. Ion acceleration by laser pulses has proved to be very useful in applications using compact laser systems. Laser-driven fast ions are expected to be useful in many applications such as hadron therapy, [@SBK; @MVL] fast ignition for thermonuclear fusion, [@ROT; @BRM; @ATH] laser-driven heavy ion colliders, [@ESI1; @BEE] and other applications that use the high-energy ions. Although the achieved proton energy at present is not high enough for some applications such as hadron therapy, which requires 200-MeV protons, other methods can be considered for generating higher energy protons. One simple way is by using a higher power laser. However, current power capabilities of compact lasers are insufficient; moreover, laser power enhancement will result in a cost increase of the accelerator. Therefore, it is important to study conditions for generating higher energy protons with lower laser power and energy by using some special techniques. [@BWP; @FVM; @Toncian; @HAC; @YAH; @PRK; @PPM; @HSM] In this paper, I show a way to obtain $200$-MeV protons by using a laser pulse whose intensity is $I_0 \approx 10^{21}$ W/cm$^{2}$, energy is $\mathcal{E}_{las} \leq 20$ J, and power is $P \approx 500$ TW. I use three-dimensional (3D) particle-in-cell (PIC) simulations to investigate how high-energy, high-quality protons can be generated by a several-hundred-terawatt laser. I study the proton acceleration during the interaction of the laser pulse with a double-layer target composed of a high-$Z$ atom layer coated with a hydrogen layer (see Fig. \[fig:fig01\]). As suggested in Refs. and , a quasimonoenergetic ion beam can be obtained using targets of this type. Our aim is to obtain a high-energy ($\mathcal{E} \approx 200$ MeV) and high-quality ($\Delta \mathcal{E}/\mathcal{E}\leq 2\%$) proton beam using a relatively moderate power laser. In the following sections, I show the dependence of the proton energy on the material of the first layer and that the high-energy protons can be generated by optimally combining a couple of ion acceleration schemes. ION ACCELERATION ================= I consider ion acceleration by a charged disk. The charged disk is produced by a laser pulse with sufficiently high intensity irradiating a thin foil. Many electrons are driven from the foil by the laser pulse, although the ions of the foils almost stay at their initial positions because they are much heavier than the electrons. Therefore, the thin foil will have a charge, which induces an electrostatic field. Ions located on the foil surface are accelerated by this electric field. The $x$ component of the electric field of a positively charged thin disk is $$E_x(x)=\frac{\rho l}{2\epsilon_0} \left(1-\frac{x}{\sqrt{x^{2}+R^{2}}} \right), \label{exx}$$ where $\rho$ is the charge density, $l$ is the disk thickness, $\epsilon_0$ is the vacuum permittivity, and $R$ is the charged disk radius. I assume that the $x$ axis is normal to the disk surface placed at the disk center. The solid curve in Fig. \[fig:fig01\] shows this electric field. The ions, i.e. protons, are accelerated in this electric field, although it rapidly decreases as a function of distance from the target surface. The electric field decreases to $10\%$ at $x =2R$, which is the distance equal to the diameter of the target and can be considered to be the spot size of a laser pulse. Therefore, generating higher energy protons requires producing a higher surface charge density, $\rho l$, or increasing $R$. The former requires a higher intensity laser and the latter requires a higher power laser. The rapidly decreasing accelerating field and its narrow width lead to inefficient proton acceleration by the charged disk. In this paper, I present ways to improve these inefficiences and to generate high-energy protons effectively. Here, let us define the some terms. In laser ion acceleration, the ions are accelerated in some electric field, $E$. We assume that for an ion of mass of $m$ and charge $q$, the force on it from the electric field is $qE$. The equation of motion is $qE=\frac{d}{dt}(mv)$, where $v$ is the ion velocity. This equation can be written as $$E=\frac{d}{dt}(\tilde{m}v), \label{emv}$$ where $\tilde{m}=m/q$. $\tilde{m}$ is the resistance to movement of an ion in a certain electric field, $E$; therefore we call $\tilde{m}$ “mass” in this paper. This expression shows that the smaller $\tilde{m}$ ions can experience greater acceleration in a certain electric field, $E$. Therefore, small-“mass” ions will be called “light,” and big-“mass” ions will be called “heavy.” Ions of the same “mass” undergo the same movement in a certain electric field. Note that $\tilde{m}$ is equal to the inverse of the well-known parameter $q/m$, the charge-to-mass ratio. I use $\tilde{m}$ in this paper because it makes it very simple and easy to image the movement of charged particles in an electric field. ![ Configuration of a double-layer target. The $x$ component of the electric field, $\tilde{E}_x(x)$, is normalized by its maximum, $\rho l/2\epsilon_0$, of an electrically charged disk on the $x$ axis (solid curve). Protons are accelerated in this electric field. []{data-label="fig:fig01"}](fig-01.pdf){width="10.0cm"} Figure \[fig:fig01\] shows that the accelerating protons exit the electric field in a short time. This means that the electric field produced is not used enough for proton acceleration. Therefore, we should create a situation in which the protons experience this electric field longer for efficient acceleration. If the electric potential moves in the direction of the moving protons, the protons will experience the electric field longer. In other words, the charged first layer keeps pushing the moving protons. I present two ways to create this situation. ![ The first layer using “light” materials produces a strongly inhomogeneous expansion due to the Coulomb explosion (light pattern). The expanding first layer moves at average velocity $V$ in the direction of laser propagation by RPDA. The electric potential moves in the $x$ direction as a result of these effects. []{data-label="fig:fig02"}](fig-02.pdf){width="10.0cm"} One way this situation can be created by the use of a Coulomb explosion of the first layer. Figure \[fig:fig02\] shows that the first layer disk undergoes a strongly inhomogeneous expansion owing to the Coulomb explosion. This expansion raises the moving electric potential for the accelerating protons. In other words, many ions in the first layer are distributed close to the accelerating protons keeping a comparatively high density and move in the proton direction. The acceleration rate is higher when the expansion velocity of the first layer ions is higher. This means that the strong Coulomb expansion operates effectively for proton acceleration. The Coulomb explosion level is determined by the “mass” of the ions composing the first layer. Equation (\[emv\]) shows that “light” ions have a high expansion velocity. That is, “light” ions undergo a stronger a Coulomb explosion and should be generating higher energy protons. Another way to induce movement of the first layer is by radiation-pressure-dominant acceleration (RPDA). Figure \[fig:fig02\] shows that the first layer, which expands by a Coulomb explosion (with an ellipsoidal light pattern), is moving with velocity $V$ in the laser propagation direction (proton direction) by RPDA. This movement leads the moving electric potential. Higher $V$ values generate higher energy protons, since the protons experience the accelerating electric field over a longer time by following the electric potential. A portion of the energy and momentum transferred from the laser pulse to the electrons is imparted to the ions via a charge separation field. That is, the ions get accelerated by this field, and the “light” ions have higher velocity (Eq. (\[emv\])). Thus the “light” ions experience a higher first layer velocity and should be generating higher energy protons. One can obtain higher energy protons by using a “light” material in the first layer, as is corroborated by the simulations described below. The simulations were performed with a 3D massively parallel electromagnetic code, based on the PIC method. [@CBL] SIMULATION OF FIRST LAYER MATERIALS {#sim-a} =================================== In this section, I study the dependence of the proton energy on the first layer materials of the double-layer target by using simulations. Simulation parameters --------------------- Here, I show the parameters used in the simulations. The spatial coordinates are normalized by the laser wavelength $\lambda=0.8$ $\mu$m and time is measured in terms of the laser period, $2\pi/\omega$. I use an idealized model, in which a Gaussian linearly polarized laser pulse is incident on a double-layer target represented by a collisionless plasma. The laser pulse with dimensionless amplitude $a=q_eE_{0}/m_{e}\omega c=50$, which corresponds to a laser peak intensity of $5\times 10^{21}$ W/cm$^{2}$, is $10\lambda $ long in the propagation direction, $27$ fs in duration, and focused to a spot with size $4\lambda $ (FWHM), which corresponds to a laser peak power of $620$ TW and a laser energy of $18$ J. Here, the laser peak power is calculated by using $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(y,z)dydz$ and the laser energy is calculated by using $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(y,z,t)dydzdt$, where $I$ is the laser intensity. The laser pulse is normally incident on the target. The electric field is oriented in the $y$ direction. I use this laser pulse in all simulations in this paper. Both layers of the double-layer target are shaped as disks. The first layer has a diameter of $8\lambda $ and a thickness of $0.5\lambda $. The second, hydrogen, layer is narrower and thinner; its diameter is $4\lambda $ and its thickness is $0.03\lambda $. The electron density inside the first layer is $n_{e}=3\times 10^{22}$ cm$^{-3}$ and inside the hydrogen layer it is $n_{e}=9\times 10^{20}$ cm$^{-3}$. The total number of quasiparticles is $8\times 10^{7}$. The number of grid cells is equal to $3300\times1024\times 1024$ along the $X$, $Y$, and $Z$ axes, respectively. Correspondingly, the simulation box size is $120\lambda \times 36.5\lambda \times 36.5\lambda$. The boundary conditions for the particles and for the fields are periodic in the transverse ($Y$,$Z$) directions and absorbing at the boundaries of the computation box along the $X$ axis. $xyz$ coordinates are used in the text and figures; the origin of the coordinate system is located at the center of the rear surface of the initial first layer, and the directions of the $x$, $y$, and $z$ axes are the same as those of the $X$, $Y$, and $Z$ axes, respectively. That is, the $x$ axis denotes the direction perpendicular to the target surface and the $y$ and $z$ axes lie in the plane of the target surface. Although the first layer material can be varied, the number of ions is the same in all cases and the ionization state of each ion is assumed to be $Z_{i}=+6$, [@HSB; @HKM; @JHK] since the laser parameters and the target geometry are fixed. Proton energy as a function of first layer materials ---------------------------------------------------- To examine the dependence of the proton energy, $\mathcal{E}$, on the first layer material, I performed simulations using different materials at normal incidence to the laser pulse. First, I show two simulation results, one in which the first layer consists of carbon (a “light” material) and the other is in which it consists of gold (a “heavy” material). The results are shown in Figs. \[fig:fig03\]–\[fig:fig09\]. ![ Particle distribution and electric field magnitude (isosurface for value $a=2$) for carbon (a) and gold (b). Half of the electric field box has been removed to reveal the internal structure. Shown are the initial shape of the target and the laser pulse ($t=0$), the interaction of the target and laser pulse ($t=25,50\times 2\pi/\omega$), and the first layer shape and the accelerated protons (color scale) ($t=75,100\times 2\pi/\omega$). []{data-label="fig:fig03"}](fig-03.pdf){width="10.0cm"} Figure \[fig:fig03\] shows the particle distribution and electric field magnitude of both cases at each time. We see that the carbon ions are distributed over a much wider area by its Coulomb explosion than are the gold ions. However, the deformation of the laser pulse is similar in both cases at all times. The Coulomb explosion process of the first layer is much slower than the laser pulse progress. The first layer is almost undeformed at $t=25\times 2\pi/\omega$ when the laser pulse is just around the target and it has the strong interactions with the target. A big deformation of the first layer appears at $t>25\times 2\pi/\omega$ when the laser pulse passes through or reflects from the target. Therefore, the explosion of the first layer is an almost simple Coulomb explosion without other effects. The proton energy obtained in the carbon case is much higher than that in the gold case. The average proton energy at $t=100\times 2\pi/\omega$, $\mathcal{E}_\mathrm{ave}$, is $63$ MeV for the carbon case and $38$ MeV for the gold case (a factor of 1.7 higher). ![ Distribution of the ions of the first layer and protons (color scale) in the carbon (a) and gold (b) cases at $t=100\times 2\pi/\omega$; a two-dimensional projection is shown looking along the $z$ axis. In the carbon case, the distribution area of the carbon ion cloud is very wide and it moves in the $x$ direction. []{data-label="fig:fig04"}](fig-04.pdf){width="8.0cm"} The reason why higher energy protons are obtained in the carbon case is that the first layer deformation effectively contributes to the proton acceleration. Let us examine the deformation of the first layer. Figure \[fig:fig04\] shows a cross section of the ion density distribution near the $(x,y,z=0)$ plane at $t=100\times 2\pi/\omega$, as seen by looking along the $z$ axis. In the carbon case, the strong Coulomb explosion distributes the carbon ions ellipsoidally, and the distribution area is much wider than in the gold case. The expansion of the cloud of carbon ions is strongly inhomogeneous and it appears to be substantially elongated in the longitudinal direction. The surface of the carbon ion cloud is close to the acceleration protons, which means that the protons keep getting pushed by a comparatively strong force. Moreover, the center point of the ion cloud is moving in the $x$ direction, at a coordinate of $3.1\lambda$, so that the electrostatic potential is moving in the direction of the protons. In contrast, in the gold case, the ion distribution is very compact and the distance between the ion cloud surface and the protons is much greater than in the carbon case. In addition, it almost does not move, with the coordinate of the center point of gold ions being $0.0\lambda$. Moreover, the proton positions in the carbon case have travelled further along the $x$ axis than in the gold case, since the protons in the carbon case have higher energy (higher velocity) than in the gold case. ![ (a) Average proton energy $\mathcal{E}$ versus time, as obtained in the simulation shown in Figs. \[fig:fig03\]. The proton energy of the carbon case is higher than in the gold case at all times. (b) Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$, for the cases of gold and carbon, normalized by the maximum in the former case. []{data-label="fig:fig05"}](fig-05.pdf){width="8.0cm"} Here, let us consider the variation of the proton energy in time for both cases. Figure \[fig:fig05\](a) shows the average proton energy versus time. The proton energy in the carbon case is always higher than in the gold case and difference in energy grows in time. The protons are accelerated relatively quickly until a time of about $t=50\times2\pi/\omega$ in both cases. In the gold case, the acceleration almost saturates at $t>50\times2\pi/\omega$, although in the carbon case it is still increasing compared with the gold case. This is because the electric field seen by protons in the carbon case is greater and continues longer than in the gold case, owing to the expansion and movement of first layer (see Fig. \[fig:fig04\]). Correspondingly, their energy gain is also greater. In the carbon case the proton energy is 1.7 times higher than in the gold case and the energy spread is almost the same in the both cases, as seen in Fig. \[fig:fig05\](b). The energy spread, $\Delta\mathcal{E}/\mathcal{E}_\mathrm{ave}$, is 6%and 8$\%$ for the cases of carbon and gold, respectively. Higher energy protons can be obtained in the carbon first layer, as a result of the stronger electric field and movement of the first layer. I consider the differences in the electric field and the first layer movement in the two cases. ![ The $x$ component of the electric field, $E_x(x)$ (solid lines) and the proton position (circles) at each time. In the gold case, in the inset, the electric field simply decreases with distance from the target surface. In the carbon case, it does not decrease much and has a peak that moves to the right side. []{data-label="fig:fig07"}](fig-06.pdf){width="8.0cm"} First, I show the differences in the electric field between the two cases. Figure \[fig:fig07\] shows the $x$ component of the electric field along the $x$ coordinate (solid lines), $E_x(x)$, and the proton position (circular dots) at each time. In the gold case (see Fig. \[fig:fig07\](b)), the electric field simply decreases with the distance from the surface of the initial target at all times. Therefore, the electric field that protons experience (see the circular dots) simply decreases too. In contrast, in the carbon case (Fig. \[fig:fig07\](a)), the electric field peaks at points near the protons for each time. The electric field moves in the direction of the accelerating protons keeping the same shape (see Fig. \[fig:fig07\](a): $t=75$ and $t=100\times2\pi/\omega$). The electric fields at the proton positions slowly decrease, and those are higher than in the gold case at all times. At the proton position at $t=50\times2\pi/\omega$, the value of $E_x$ is $2.4$ and $1.4$ TV/m for the cases of carbon and gold ions, respectively. ![ Spatial distribution of particles and the electric field magnitude at early simulation times for the carbon case. A two-dimensional projection is shown looking along the $z$ axis. The solid bar is a projection of the first layer ($t=0$). The gas pattern denotes electrons that are pushed out from the target by the laser pulse ($t=10, 20\times 2\pi/\omega$) and distributed to the rear area (propagation direction) of the target. This produces the movement of the first layer ions. []{data-label="fig:fig08"}](fig-07.pdf){width="13.0cm"} Next, I consider the movement of the first layer. Figure \[fig:fig08\] shows the laser pulse and the target at an early time, $t<20\times2\pi/\omega$, for the carbon case, as seen by looking along the $z$ axis. The laser pulse moves from the left-hand size to the right-hand side, and the color shows the electric field magnitude. We can see that many electrons (gas pattern) are pushed out from the target by the ponderomotive force from the laser pulse. Those pushed-out electrons are distributed to the rear area (propagation direction) of the target and they move forward in the laser propagation direction, although the carbon ions stay at their initial position. This charge separation produces the strong electric field. Then, the carbon ions are moved in the laser propagation direction by this electric field (which is RPDA). ![ (a) Velocity of the first layer of the target in the $x$ direction normalized by the speed of light, $V_x/c$, as a function of time. The velocity of the carbon target is much higher than that for the gold case. (b) Movements of the first layer in the $x$ direction normalized by the wavelength, $x/\lambda$, as a function of time. []{data-label="fig:fig09"}](fig-08.pdf){width="8.0cm"} Figure \[fig:fig09\] shows the first layer velocity, $V_x$, normalized by the speed of light, and the first layer position, normalized by the wavelength, for the $x$ direction as a function of time. These are averaged values for all ions of the first layer. The first layer velocity rises rapidly at the initial time, $t \sim 20\times2\pi/\omega,$ when the laser pulse is still around the target (see Fig. \[fig:fig08\]) and the velocity is constant at time $t>25\times2\pi/\omega,$ after the laser pulse passes through or reflects off the target. The increase in the fist layer velocity stops at $t \approx 20\times 2\pi/\omega$, since by this time there is no clearly one-sided distribution of the electrons like that at time $t \approx 10\times2\pi/\omega$ (see Fig. \[fig:fig08\]). The first layer velocity in the carbon case is $14$ times that in the gold case at $t>25\times 2\pi/\omega$. This means strong RPDA occurs in the carbon case. The movement in the carbon case is much greater than in the gold case too, and the difference in distance between the carbon case and the gold case grows with time. This indicates that the moving electric potential operates efficiently in the carbon case. Incidentally, the velocity for the $y$ direction, $V_y$, of the first layer is relatively very small. It is about $1/1000$ of $V_x$ in both the carbon and gold cases. Movement for the $y$ direction is very small too, because they are normal to the incidence of the laser pulse. ![ Average proton energy, normalized by energy in the gold case, shown with respect to the ratio between the charge state, $Z_{\rm i}$, and the atomic number, $A$, of ions making up the first layer. []{data-label="fig:fig10"}](fig-09.pdf){width="7.0cm"} In above considerations, I showed that the higher energy protons can be obtained for “lighter” material in the first layer by comparing carbon and gold. Here, I investigate the effect using additional materials. Figure \[fig:fig10\] shows the average proton energy for different materials comprising the first layer. The proton energy is normalized by the energy in the case of gold ions. We see that the average proton energy almost linearly depends on the ratio of $Z_{\rm i}/A$, where $Z_{\rm i}$ is the charge state and $A$ is the atomic mass number of the ion. Higher energy protons can be obtained by using a larger ratio of $Z_{\rm i}/A$, i.e., “light,” material for the first layer in the double-layer target. In experiments on laser-driven ion acceleration, a CH polymer target exhibited higher energy protons [@SNAV] than a metallic target. [@CLAR] COMPONENTS OF THE PROTON ENERGY {#sim-b} =============================== In the previous section, I showed that higher energy protons can be obtained by using “light” material in the first layer, because a strong Coulomb explosion and a greater first layer movement toward the accelerating protons occur in such materials. These effects augment the acceleration by the electric field of the charged first layer disk. In this section, I show the amount of each effect of the acceleration on the total proton energy in the carbon case. ![ (a) 3D view and (b) the projection onto the $(x,y)$ plane for a laser pulse irradiating onto the other side, the proton layer side, of a double-layer target. Shown are the particle distribution and electric field magnitude (isosurface for value $a=2$); half of the electric field box has been removed. The protons are accelerated in the $+x$ direction, and the C ions move in the $-x$ direction by RPDA. []{data-label="fig:fig11"}](fig-10.pdf){width="10.0cm"} To examine the effect of the first layer velocity, $V$, I performed simulations with a laser pulse by reversing the irradiation direction of the laser (reverse irradiating) in the previous carbon case. Thus the hydrogen layer was put on the front side of the target (see Fig. \[fig:fig11\]). ![ Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$, for the reverse irradiation case, $C^-$, and the positive irradiation (previous carbon) case, $C^+$, normalized by the maximum in the former case. []{data-label="fig:fig12"}](fig-11.pdf){width="7.0cm"} The results are shown in Figs. \[fig:fig11\] and \[fig:fig12\]. The protons are accelerating in the $+x$ direction even when the laser pulse irradiates the target in the reversed way. The average proton energy in this case, $\mathcal{E}^{-}$, is $32$ MeV. Since the proton layer is very thin and small, it has less effect on the first layer velocity. Therefore, the first layer velocity, $V^-$, has the same absolute value as in positive irradiation, the case of the previous section, but the sign is opposite, $V^-(t)=-V(t)$, where $V(t)$ is the first layer velocity in time of the positive case. Therefore, the first layer movement has a negative effect on the proton energy in the reverse irradiating case. Using this result, we can estimate the amount of proton energy attributable to the first layer velocity, $\mathcal{E}_{V}$. The proton energy in the positive irradiating case, $\mathcal{E}^{+}$, and in the reverse irradiation case, $\mathcal{E}^{-}$, are written as $$\mathcal{E}^{+}=\mathcal{E}_a+\mathcal{E}_V, \label{evp}$$ $$\mathcal{E}^{-}=\mathcal{E}_a-\mathcal{E}_V, \label{evm}$$ where $\mathcal{E}_a$ is the proton energy without the work done by the first layer velocity (i.e., attributable to the acceleration by the electric field of the charged disk of the first layer and the Coulomb explosion of the first layer ions). Hence we obtain the work done by the first layer velocity, $\mathcal{E}_{V}=(\mathcal{E}^{+}-\mathcal{E}^{-})/2$, which is $16$ MeV. Therefore, the ratio of the effect of the first layer velocity, RPDA, is $25\%$ of the total proton energy in the carbon case. In our simulations, the laser intensity and energy of $I_0=5\times 10^{21}$W/cm$^{2}$ and $\mathcal{E}_{las}=18$ J, respectively, are not enough for the RPDA regime in full scale, but this shows that the RPDA regime has a strong effect even at this laser power level. The proton energy without the work done by the first layer velocity, $\mathcal{E}_a$, is $48$ MeV. Next, I examine the work done by the Coulomb explosion of the first layer. To do so, I performed a simulation with a laser pulse reverse irradiating in the gold case. In this case, the average proton energy, $\mathcal{E}^{'-}$, is $32$ MeV. In the positive irradiation case, the gold case of the previous section, the average proton energy, $\mathcal{E}^{'+}$, was $38$ MeV. The difference between these values is very small, because the first layer velocity, $V$, is very small (see Figs. \[fig:fig04\] and \[fig:fig09\]). This means that the work done by the first layer velocity is negligibly small in the gold case. The Coulomb explosion effect is negligible, as shown in previous section, too. Therefore, we estimate that the work without the first layer velocity and Coulomb explosion in the carbon case, $\mathcal{E}_0 \approx (\mathcal{E}^{'+}+\mathcal{E}^{'-})/2$, is $35$ MeV. This is almost the energy from acceleration by only the electric field of the charged disk of the first layer. The ratio of this effect in the total energy of the carbon case is $55\%$. Then we obtain the work done by the Coulomb explosion as $\mathcal{E}_C=\mathcal{E}_a-\mathcal{E}_0=13$ MeV. The ratio of the Coulomb explosion effect, $\mathcal{E}_{C}/\mathcal{E}^{+}$, is $20\%$ in the carbon case (see Fig. \[fig:fig13\]). ![ Proton energy of each acceleration mechanism in the carbon case. The proton energy produced by the electric field of the nonmoving first layer is almost half the total proton energy, and RPDA and the Coulomb explosion each amount to almost a quarter of the total proton energy. []{data-label="fig:fig13"}](fig-12.pdf){width="10.0cm"} Another effect to consider is that of the protons being dragged by the electrons that are pushed out of the target by the laser pulse. I estimate this effect by using the gold case results. The electrons are mainly pushed out in the laser propagation direction. Therefore, in the reverse irradiation case, the dragging effect by electrons acts to reduce the proton energy. The proton energy in the positive irradiating case, $\mathcal{E}^{'+}$, and in the reverse irradiation case, $\mathcal{E}^{'-}$, are written as $\mathcal{E}^{'+}=\mathcal{E}^{'}_a+\mathcal{E}^{'}_V+\mathcal{E}^{'}_d$ and $\mathcal{E}^{'-}=\mathcal{E}^{'}_a-\mathcal{E}^{'}_V-\mathcal{E}^{'}_d$, where $\mathcal{E}^{'}_V$ is the work done by the first layer velocity, $\mathcal{E}^{'}_d$ is the work done by the dragging by the electrons, and $\mathcal{E}^{'}_a$ is other work. The effect of the first layer velocity plus the dragging by electrons, $\mathcal{E}^{'}_V+\mathcal{E}{'}_d=(\mathcal{E}^{'+}-\mathcal{E}^{'-})/2$, is $3$ MeV. Since $\mathcal{E}^{'}_V>0$, the effect of the dragging by the electrons is $\mathcal{E}^{'}_d<3$ MeV, and the ratio is less than $5\%$ of the total energy of the carbon case. Therefore, the effect whereby the protons are dragged by the electrons is very small in our simulations. This is because the electrons move very fast compared with the ions and protons, the distance between electrons and protons quickly becomes very large compared with that between the first layer ions and protons, and, moreover, the electrons are distributed over a very wide area. The electric force decreases with distance by second order. Therefore, the force the electrons exert on the protons is very small compared with the force exerted by the ions. Next, I consider the theory for the work done by the first layer velocity. I assume that the proton velocity $v > V$ and $v^2 \ll c^2$. The theoretical formula [@MBEKK] is written as $\mathcal{E}_V=mV^2(1+\sqrt{2\mathcal{E}_0/mV^2+1})$, where $m$ is the proton mass and $\mathcal{E}_0$ is the proton energy in the case of a nonmoving first layer. Denoting the proton velocity in the case of a nonmoving first layer by $v_0$ and assuming $v_0^2 \ll c^2$ we obtain $$\tilde{\mathcal{E}}_V=2\tilde{V}(\tilde{V}+\sqrt{1+\tilde{V}^2}), \label{eve0}$$ where $\tilde{\mathcal{E}_V}$ is the normalized proton energy of the work done by the first layer velocity, $\mathcal{E}_V/\mathcal{E}_0$, and $\tilde{V}$ is the normalized velocity of the first layer, $V/v_0$. ![ Proton energy produced by RPDA, $\tilde{\mathcal{E}}_V$, as a function of first layer velocity, $\tilde{V}$. The proton energy, $\mathcal{E}_V$, is normalized by the proton energy of the nonmoving first layer case, $\mathcal{E}_V/\mathcal{E}_0$, and the first layer velocity, $V$, is normalized by the proton velocity of the nonmoving first layer case, $V/v_0$. The theoretical result (solid line) is given by equation (\[eve0\]). The simulation results for the gold and carbon cases are plotted with a square and a circle, respectively. []{data-label="fig:fig14"}](fig-13.pdf){width="8.0cm"} In Fig. \[fig:fig14\], I present this theoretical dependence of $\tilde{\mathcal{E}}_V$ on $\tilde{V}$, by using formula (\[eve0\]). The proton energy by RPDA, $\mathcal{E}_V$, grows rapidly with increasing first layer velocity $V$. The simulation results (circle and square dot) are plotted on the figure. The theoretical calculations and the simulation results agree well. Since the first layer velocity in the carbon case is not so high, $V/v_0=0.13$, Fig. \[fig:fig14\] shows that if we produce a high velocity in the first layer, protons with a few times higher energy can be obtained. In addition, assuming $\tilde{V}^2 \ll 1$ we obtain the simpler formula $\tilde{\mathcal{E}}_V=2\tilde{V}$. Generating A 200-MeV proton beam ================================ I show the way to obtain a $200$-MeV proton beam, using the same laser pulse as in the previous section with the same normal incidence. In the previous section, I showed that higher energy protons can be obtained by using “light” material in the first layer. The “lightest” material is hydrogen. Therefore, we could get higher energy protons by using hydrogen for the first layer. I evaluate this contribution by simulation. In this case, even if we put the second layer of thin hydrogen on the hydrogen first layer, the second layer has no meaning. Therefore, I use a simple hydrogen disk, without a second layer. The hydrogen disk target has the same shape and size as that of the first layer of the double-layer target used previously. The electron density inside the target is $n_{e}=9\times 10^{22}$ cm$^{-3}$. Because the hydrogen cloud generated by the target is distributed over a wider area than in the previous case, I define a wider simulation box for $Y$ and $Z$ directions. The number of grid cells is equal to $5000\times 3000\times 3000$ along the $X$, $Y$, and $Z$ axes, respectively. Correspondingly, the simulation box size is $113\lambda \times 67.5\lambda \times 67.5\lambda$. The total number of quasiparticles is $4\times 10^{8}$. The other simulation parameters are the same as those used in previous sections. ![ (a) Particle distribution and electric field magnitude (isosurface for value $a=2$), showing the initial shape of the target and the laser pulse ($t=0$) and the interaction of the target and laser pulse ($t=25,50\times 2\pi/\omega$). Half of the electric field box has been removed to reveal the internal structure. For protons, the color corresponds to energy. (b) Two-dimensional projection of the particle distribution, shown looking along the $z$ axis. Half of the proton cloud has been removed to reveal the internal structure. []{data-label="fig:fig-h1"}](fig-h1.pdf){width="10.0cm"} Figure \[fig:fig-h1\](a) shows the particle distribution and the electric field magnitude in time. At $t=25\times 2\pi/\omega$, the laser pulse is just around the target and it has the strong interactions with a target. The target maintains its initial disk shape at this time. After $t=25\times 2\pi/\omega$, the laser pulse passes through or reflects off of the target, and the proton cloud produced by Coulomb explosion is growing in time. Figure \[fig:fig-h1\](b) shows a cross section of the ion cloud at each time. Hydrogen ions (protons) are classified by color in terms of energy. We see that the exploded hydrogen disk (hydrogen ions) is distributed over a very wide area. The expansion of the cloud of hydrogen ions is inhomogeneous and the cloud is elongated in the longitudinal direction. ![ Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$. []{data-label="fig:fig-h2"}](fig-h2.pdf){width="8.0cm"} Figure \[fig:fig-h2\] shows the proton energy spectrum at $t=100\times 2\pi/\omega$. The maximum energy is $\mathcal{E}_\mathrm{max}=200$ MeV and the average energy is $\mathcal{E}_\mathrm{ave}=25$ MeV. The vertical axis is given in units of number of protons per $1$ MeV. We can estimate the number of obtained protons by using the required energy and energy width from Fig. \[fig:fig-h2\]. We see that we can obtain five times higher proton energy than in the gold case, even by using the same laser pulse, by using an optimum target material. ![ (a) Velocity of the hydrogen target in the $x$ direction, normalized by the speed of light, $V_x/c,$ as a function of time. (b) Movement of the hydrogen target in the $x$ direction, normalized by the wavelength, $x/\lambda$, as a function of time. []{data-label="fig:fig-h3"}](fig-h3.pdf){width="8.0cm"} Figure \[fig:fig-h3\] shows the target average velocity, normalized by the speed of light, and the target average position, normalized by the wavelength, in the $x$ direction as a function of time. The target velocity rises rapidly at the initial time, $t \sim 20\times2\pi/\omega,$ when the laser pulse is still around the target and the velocity is constant at time $t>25\times2\pi/\omega,$ after the laser pulse passes through or reflects off of the target. This is similar to the carbon case (Fig. \[fig:fig09\]), although the target velocity of this case is $2.5$ times that in the carbon case at time $t>25\times 2\pi/\omega$. That means strong RPDA appears in this case, because the target velocity is attributable to RPDA. The movement of the target is much greater than in the carbon case too. In the hydrogen disk target, the protons are accelerated more efficiently than in the carbon case, because hydrogen is much “lighter” than carbon. Because I used the same laser pulse in all simulations, the numbers of electrons pushed out from the target are estimated to be almost the same in carbon and hydrogen disk cases at the initial simulation time. Therefore, the target surface charge is almost the same in the two cases, and the proton energy by the charged disk electric field can be estimated using the same considerations as in Section \[sim-b\], yielding $35$ MeV. Therefore, we can estimate that the proton energy by RPDA and Coulomb explosion is $\approx$165 MeV. We can say that the RPDA and Coulomb explosion effect is much stronger in the hydrogen disk case compared with the carbon case. We see that protons are distributed in different areas based on each energy level (see Fig. \[fig:fig-h1\]). Moreover, high-energy protons are distributed on the $+x$ side edge of the proton cloud and are moving in the $+x$ direction. Therefore, we could select only high-energy protons by using a pinhole with a shutter. The shutter must have high enough accuracy and the ability to shield unwanted particles and radiation. The shutter speed must be very fast near the target. The accuracy becomes coarser if the shutter position moves to a position farther from the target, because the distance between the high-energy protons and the low-energy protons grows in time. It may be difficult to construct a shutter that satisfies both the timing accuracy and shielding ability, however, we could get a similar result by using a magnet and a slit. The path of a proton is changed for each energy level by a magnetic field, and high-energy protons can be taken out by passing through a slit. I show the results using the previous method. ![ Distributions of protons at $t=100\times 2\pi/\omega$ and the selected area of the proton bunch. []{data-label="fig:fig-h4"}](fig-h4.pdf){width="7.0cm"} ![ Proton energy spectrum obtained by cutting off the proton bunch at $t=100\times 2\pi/\omega$, normalized by the maximum. []{data-label="fig:fig-h5"}](fig-h5.pdf){width="8.0cm"} The cutoff position is shown in Fig. \[fig:fig-h4\]. Figure \[fig:fig-h5\] shows energy spectrum of cutoff protons at $t=100\times 2\pi/\omega$. We obtain a proton beam with a maximum energy of $\mathcal{E}_\mathrm{max}=200$ MeV and an average energy of $\mathcal{E}_\mathrm{ave}=193$ MeV with an energy spread of $\Delta\mathcal{E}/\mathcal{E}_\mathrm{ave}=2.3\%$ and a particle number of $2.6\times10^7$. This proton beam has high enough energy and quality for some applications (e.g., in medical applications). CONCLUSIONS =========== Proton acceleration driven by a laser pulse irradiating a disk target is investigated with the help of 3D PIC simulations. I have found higher energy protons are obtained by using “light” materials for the target. As seen in simulations, for these materials a strongly inhomogeneous expansion of the disk target occurs owing to the Coulomb explosion, which plays an important role, and RPDA has a strong effect. The time-varying electric potential of the inhomogeneous expanding ion cloud and the movement of the ion cloud for the protons efficiently accelerate protons. The proton beam energy can be substantially increased by using a “light” material for the target. In our simulations, the laser intensity and energy, $I_0=5\times 10^{21}$ W/cm$^{2}$ and $\mathcal{E}_{las}=18$ J, are not enough to reach the RPDA regime in full scale, but the RPDA regime has a big effect even at this laser power level. Although I show simulation results by using a simple hydrogen disk, it may be difficult to fabricate such a hydrogen disk; a CH$_n$ foil target with a high $n$ value should be a good substitute. The laser parameters used in this paper—intensity, power, energy, and spot size—are ones already existing in current laser systems. Therefore, we should be able to generate $200$-MeV protons now. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== I thank P. Bolton, S. V. Bulanov, T. 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ArXiv

--- abstract: 'We present photometry with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) of stars in the Magellanic starburst galaxy NGC 4449. The galaxy has been imaged in the F435W (B), F555W (V) and F814W (I) broad-band filters, and in the F658N (H$\alpha$) narrow-band filter. Our photometry includes $\approx$ 300,000 objects in the (B, V) color-magnitude diagram (CMD) down to V $\la$ 28, and $\approx$ 400,000 objects in the (V, I) CMD, down to I $\la$ 27 . A subsample of $\approx$ 200,000 stars has been photometrized in all the three bands simultaneously. The features observed in the CMDs imply a variety of stellar ages up to at least 1 Gyr, and possibly as old as a Hubble time. The spatial variation of the CMD morphology and of the red giant branch colors point toward the presence of an age gradient: young and intermediate-age stars tend to be concentrated toward the galactic center, while old stars are present everywhere. The spatial variation in the average luminosity of carbon stars suggests that there is not a strong metallicity gradient ($\lesssim 0.2$ dex). Also, we detect an interesting resolved star cluster on the West side of the galaxy, surrounded by a symmetric tidal or spiral feature consisting of young stars. The positions of the stars in NGC 4449 younger than 10 Myr are strongly correlated with the H$\alpha$ emission. We derive the distance of NGC 4449 from the tip of the red giant branch to be ${\rm D=3.82 \pm 0.27}$ Mpc. This result is in agreement with the distance that we derive from the luminosity of the carbon stars.' author: - 'F. Annibali , A. Aloisi , J. Mack, M. Tosi , R.P. van der Marel, L. Angeretti, C. Leitherer, M. Sirianni' title: 'Starbursts in the Local Universe: new HST/ACS observations of the irregular galaxy NGC 4449[^1]' --- Introduction ============ Starbursts are short and intense episodes of star formation (SF) that usually occur in the central regions of galaxies and dominate their integrated light. The associated star-formation rates (SFR) are so high that the existing gas supply can sustain the stellar production only on timescales much shorter than a cosmic time ($\lesssim 1$ Gyr). The importance of the starburst phenomenon in the context of cosmology and galaxy evolution has been dramatically boosted in recent years by deep imaging and spectroscopic surveys which have discovered star-forming galaxies at high redshift: a population of dusty and massive starbursts, with SFRs as high as $\sim$ 100 – 1000 M$_{\odot}$ yr$^{-1}$, has been unveiled in the submillimeter and millimeter wavelengths at z$>$2 [@blain02; @scott02] and star-forming galaxies at $z >$ 3 have been discovered with the Lyman break selection technique [@steidel96; @pet01] and through Lyman-$\alpha$ emission surveys (@rhoads, see also @lefevre05 for a more recent independent approach). In the local Universe, starbursts are mostly found in dwarf irregular galaxies, and contribute $\sim$ 25% of the whole massive SF [@heck98]. Both observations and theoretical models [@larson78; @genz98; @ni86] show that strong starbursts are usually triggered by processes such as interaction or merging of galaxies, or by accretion of gas, which probably played an important role in the formation and evolution of galaxies at high redshift. Thus, nearby starbursts can serve as local analogs to primeval galaxies to test our ideas about SF, evolution of massive stars, and physics of the interstellar medium (ISM) in “extreme” environments. The high spatial resolution and high sensitivity of Hubble Space Telescope offer the possibility to study the evolution of nearby starbursts in details. This is fundamental in order to address many of the still open questions in cosmological astrophysics: What are the main characteristics of primeval galaxies? What is the nature of star-forming galaxies at high redshift? How important are accretion and merging processes in the formation and evolution of galaxies? The Magellanic irregular galaxy NGC 4449 ($\alpha_{2000} =Ê12^h 28^m 11^{s}.9$, $\delta_{2000} =Ê+ 44^{\circ} 05^{'} 40^{"}$, $l=136.84$ and $b=72.4$), at a distance of $3.82 \pm 0.27$ Mpc (see Section 5), is one of the best studied and spectacular nearby starbursts. It has been observed across the whole electromagnetic spectrum and displays both interesting and uncommon properties. It is one of the most luminous and active irregular galaxies. Its integrated magnitude $M_B = -18.2$ makes it $\approx$ 1.4 times as luminous as the Large Magellanic Cloud (LMC) [@hunter97]. @th87 estimated a current SFR of $\sim 1.5$ M$_{\odot}$ yr$^{-1}$. NGC 4449 is also the only local example of a global starburst, in the sense that the current SF is occurring throughout the galaxy [@hunter97]. This makes NGC 4449 more similar to Lyman break Galaxies (LBGs) at high redshift ($z \simeq3$), where the brightest regions of SF are embedded in a more diffuse nebulosity and dominate the integrated light also at optical wavelengths [@gi02]. Abundance estimates in NGC 4449 were derived in the HII regions by [@talent], [@hgr82] and [@mar97], and for NGC 4449 nucleus by @bok01. The published values are in good agreement with each other, and provide 12 + log(O/H) $\approx 8.31$. Adopting the oxygen solar abundance from @sun98, 12 + log(O/H)$_{\odot} =$ 8.83, we obtain \[O/H\] $=$ -0.52, i.e. NGC 4449 oxygen content is almost one third solar, as in the LMC. New solar abundance estimates, based on 3D hydrodynamic models of the solar atmosphere, accounting for departures from LTE, and on improved atomic and molecular data, provide 12 + log(O/H)$_{\odot} =$ 8.66 [@sun07]. However, the new lower abundances seem to be inconsistent with helioseismology data, unless the majority of the inputs needed to make the solar model are changed [@basu07]. Thus, we will adopt the old abundances from @sun98 throughout the paper. Radio observations of NGC 4449 have shown a very extended HI halo ($\sim 90$ kpc in diameter) which is a factor of $\sim 10$ larger than the optical diameter of the galaxy and appears to rotate in the opposite direction to the gas in the center [@baj94]. Hunter et al. (1998, 1999) have resolved this halo into a central disk-like feature and large gas streamers that wrap around the galaxy. Both the morphology and the dynamics of the HI gas suggest that NGC 4449 has undergone some interaction in the past. A gas-rich companion galaxy, DDO 125, at the projected distance of $\sim 40$ kpc, could have been involved [@theis]. NGC 4449 has numerous ($\sim 60$) star clusters with ages up to 1 Gyr [@gel01] and a young ($\sim$ 6-10 Myr) central cluster [@bok01], a prominent stellar bar which covers a large fraction of the optical body [@hun99], and a spherical distribution of older (3-5 Gyr) stars [@both96]. The galaxy has also been demonstrated to contain molecular clouds from CO observations [@ht96] and to have an infrared (10-150 $\micron$) luminosity of $2 \times 10^{43}$ erg s$^{-1}$ [@th87]. The ionized gas shows a very turbulent morphology with filaments, shells and bubbles which extend for several kpc (Hunter & Gallagher 1990, 1997). The kinematics of the HII regions within the galaxy is chaotic, again suggesting the possibility of a collision or merger [@va02]. Some 40% of the X-ray emission in NGC 4449 comes from hot gas with a complex morphology similar to that observed in H$\alpha$, implying an expanding super-bubble with a velocity of $\sim 220$ kms$^{-1}$ [@sum03]. All these observational data suggest that the late-type galaxy NGC 4449 may be changing as a result of an external perturbation, i.e., interaction or merger with another galaxy, or accretion of a gas cloud. A detailed study of the star-formation history (SFH) of this galaxy is fundamental in order to derive a coherent picture for its evolution, and understand the connection between possible merging/accretion processes and the global starburst. With the aim of inferring its SFH, we have observed NGC 4449 with the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope (HST) in the F435W, F555W, F814W and F658N filters. In this paper we present the new data and the resulting color–magnitude diagrams (CMDs) (Sections 2, 3 and 4). We derive a new estimate of the distance modulus from the magnitude of the tip of the red giant branch (TRGB) and the average magnitude of the carbon stars in Section 5. In Section 6, the empirical CMDs are compared with stellar evolutionary tracks. With the use of the tracks, we are able to derive the spatial distribution of stars of different age in the field of NGC 4449. The Conclusions are presented in Section 7. The detailed SFH of NGC 4449 will be derived through synthetic CMDs in a forthcoming paper. Observations and data reduction =============================== The observations were performed in November 2005 with the ACS Wide Field Camera (WFC) using the F435W (B), F555W (V) and F814W (I) broad-band filters, and the F658N (H$\alpha$) narrow-band filter (GO program 10585, PI Aloisi). We had two different pointings in a rectangular shape along the major axis of the galaxy. Each pointing was organized with a 4 - exposure half $+$ integer pixel dither pattern with the following offsets in arcseconds: (0,0) for exposure 1, (0.12, 0.08) for exposure 2, (0.25, 2.98) for exposure 3, and (0.37, 3.07) for exposure 4. This dither pattern is suitable to remove cosmic rays and hot/bad pixels, fill the gap between the two CCDs of the ACS/WFC, and improve the PSF sampling. The exposure times in the different broad-band filters were chosen to reach at least one magnitude fainter than the TRGB, at I $\approx $24, with a photometric error below 0.1 mag (S/N $>$ 10). Eight exposures of $\sim$ 900 s, 600 s, 500 s and 90 s were acquired for each of the B, V, I and H$\alpha$ filters, respectively. For each filter, the eight dithered frames, calibrated through the most up-to-date version of the ACS calibration pipeline (CALACS), were co-added into a single mosaicked image using the software package MULTIDRIZZLE [@Koe02]. During the image combination, we fine-tuned the image alignment, accounting for shifts, rotations, and scale variations between images. The MULTIDRIZZLE procedure also corrects the ACS images for geometric distortion and provides removal of cosmic rays and bad pixels. The total field of view of the resampled mosaicked image is $\sim$ 380 $\times$ 200 arcsec$^2$, with a pixel size of 0.035 (0.7 times the original ACS/WFC pixel size). To choose the optimal drizzle parameters, we experimented with different combinations of the MULTIDRIZZLE parameters [*pixfrac*]{} (the linear size of the “drop" in the input pixels) and [*pixscale*]{} (the size of output pixels). One must choose a [*pixfrac*]{} value that is small enough to avoid degrading the final image, but large enough that, when all images are dropped on the final frame, the flux coverage of the output image is fairly uniform. Statistics performed on the final drizzled weight image should yield an rms value which is less than 20% of the median value. In general, the [*pixfrac*]{} should be slightly larger than the scale value to allow some of the ’drop’ to spill over to adjacent pixels. Following these guidelines, we find that [*pixfrac*]{}$=$0.8 and [ *pixscale*]{}$=$0.7 provide the best resolution and PSF sampling for our dithered images. The total integration times are $\sim$ 3600 s, 2400 s, 2000 s and 360 s for the B, V, I and H$\alpha$ images, respectively. Only in a small region of overlap between the two pointings ($\sim$ 30 $\times$ 200 arcsec$^2$) the integration times are twice as those listed above. Figure \[image\] shows the mosaicked true-color image created by combining the data in the four filters. The photometric reduction of the images was performed with the DAOPHOT package [@daophot] in the IRAF environment[^2]. The instrumental magnitudes were estimated via a PSF-fitting technique. We constructed a PSF template for each of the four ACS chips (2 CCDs at 2 pointings) contributing to the final mosaicked image. To derive the PSF, we selected the most isolated and clean stars, uniformly distributed within each chip. The PSF is modeled with an analytic moffat function plus additive corrections derived from the residuals of the fit to the PSF stars. The additive corrections include the first and second order derivatives of the PSF with respect to the X and Y positions in the image. This procedure allows us to properly model the spatial variation of the PSF in the ACS/WFC field of view. The stars were detected independently in the three bands, without forcing in the shallowest frames the detection of the objects found in the deepest one. For comparison, we also ran the photometry on a list of stars detected on the sum of the B, V and I images. The luminosity functions (LFs) obtained in the various bands with the two different approaches are presented in Fig. \[lfs\]. We notice that the [*forced*]{} search pushes the detection of stars $\sim$ 0.5 magnitude deeper than the [*independent*]{} search. On the other hand, upon closer inspection the majority of the “gained" objects turn out to be spurious detections or stars with large photometric errors. Furthermore, the deeper photometry is not deep enough to detect the next features of interest in the CMD (the horizontal branch, the red clump or the asymptotic giant branch (AGB) bump, see Section 3), and thus it does not provide any additional information for our study. In the following we thus use the photometry obtained with the independent search on the three bands. Aperture photometry with PHOT, and then PSF-fitting photometry with the ALLSTAR package, were performed at the position of the objects detected in the B, V and I images. The instrumental magnitudes were measured adopting the appropriate PSF model to fit the stars according to their position in the frame. The B,V and I catalogs were then cross-correlated with the requirement of a spatial offset smaller than 1 pixel between the positions of the stars in the different frames. This led to 299,115 objects having a measured magnitude in both B and V, 402,136 objects in V and I, and 213,187 objects photometrized in all the three bands simultaneously. The conversion of the instrumental magnitudes $m_i$ to the HST VEGAMAG system was performed by following the prescriptions in @sir05. The HST VEGAMAG magnitudes are derived according to the equation: $$m = m_i + C_{ap} + C_{\inf} + ZP + C_{CTE}, \label{eq1}$$ where $m_i$ is the DAOPHOT magnitude (${\rm -2.5 \times \log(counts/exptime)}$) within a circular aperture of 2–pixel radius, and with the sky value computed in an annulus from 8 to 10 pixels; $C_{ap}$ is the aperture correction to convert the photometry from the 2 pixel to the conventional $0.5\arcsec$  radius aperture, and with the sky computed at “infinite"; we computed $C_{ap}$ from isolated stars selected in our images; $C_{\inf}$ is taken from @sir05 and is an offset to convert the magnitude from the 0.5 radius into a nominal infinite aperture; ZP is the HST VEGAMAG zeropoint for the given filter. Corrections for imperfect charge transfer efficiency (CTE) were calculated from each single expousure, and then averaged, following the formulation of [@cte] (Eq. \[eq2\]), which accounts for the time dependence of the photometric losses: $$C_{CTE}=10^A \times SKY^B \times FLUX^C \times \frac{Y}{2048} \times \frac{MJD - 52333}{365}, \label{eq2}$$ where SKY is the sky counts per pixel per exposure, FLUX is the star counts per exposure within our adopted photometry aperture (r$=$2 pixel in the resampled drizzled image, corresponding to 1.4 pixel in the original scale), Y is the number of charge transfers, and MJD is the Modified Julian Date. The coefficients of equation (\[eq2\]) were extrapolated from a r$=$3 pixel aperture to a r$=$1.4 pixel aperture (in the original scale), and are $A=1.08$, $B=-0.309$, and $C =-0.976$. The computed CTE corrections are negligible for the brightest stars, but can be as high as $\sim$ 0.1 mag for the faintest stars. We did not transform the final magnitudes to the Johnson-Cousins B, V, I system, since this would introduce additional uncertainties. However, such transformations can be done in straightforward manner using the prescriptions of @sir05. The ACS VEGAMAG magnitudes were not corrected for Galactic foreground extinction (${\rm E(B-V) = 0.019}$, [@schlegel]) and internal reddening. Concerning the internal reddening, @hill98 derived ${\rm E(B-V) \approx}$ 0.18 from the $H\alpha$/H$\beta$ ratio measured in NGC 4449 HII regions. This value can be considered an upper limit to the average internal extinction, since the nebular gas is usually associated with young star forming regions, which tend to be inherently more dusty than the regions in which older stars reside (as demonstrated explicitly for the case of the LMC; @zari). Because of both the large number of dithered exposures and the conservative approach of the independent search on the three images, our catalog is essentially free of instrumental artifacts such as cosmic rays or hot pixels. The distribution of the DAOPHOT parameters $\sigma$, $\chi^2$ and [*sharpness*]{} is shown in Fig. \[fig1I\]. The $\sigma$ parameter measures the uncertainty on the magnitude, the $\chi^2$ is the residual per degree of freedom of the PSF-fitting procedure, and the [*sharpness*]{} provides a measure of the intrinsic size of the object with respect to the PSF. Notice that the DAOPHOT/ALLSTAR package automatically rejects the objects with $\sigma > 0.55$. The obtained distributions for $\sigma$, $\chi^2$ and [*sharpness*]{} suggest that the vast majority of the detected sources are stars in the galaxy with a small contamination from stellar blends and background galaxies. There are some objects with very bright magnitudes (${\rm m_{F814W} }\la 22$) and [*sharpness*]{} $>$0.5 in Fig. \[fig1I\] (this is also observed in the F435W and F555W filters, for which we do not show the $\chi^2$ and [*sharpness*]{} distributions). Their [*sharpness*]{} values imply that they have a larger intrinsic size than the PSF, and thus may not be individual stars. By visually inspecting these objects in all the images, we recognized several candidate star clusters and background galaxies. Some of the candidate star clusters look like fairly round but extended objects; some others present a central core, and are partially resolved into individual stars in the outskirts. We detect at least 42 clusters in our data, some of which look like very massive globular clusters. The candidate clusters and the galaxies that were identified by eye were rejected from the photometric catalog. We are left with 299,014 objects in the (B, V) catalog, 402,045 objects in the (V, I) catalog, and 213,099 objects photometrized in all the three bands. We experimented with many other cuts in $\sigma$, $\chi^2$ and [*sharpness*]{}, but none of them affected the global appearance of the CMDs and the detected evolutionary features. A detailed study of the cluster properties will be presented in a forthcoming paper (Aloisi et al., in preparation). Incompleteness and blending =========================== To evaluate the role of incompleteness and blending in our data, we performed artificial star experiments on the drizzle-combined frames, following the procedure described by [@tosi01]. These tests serve to probe observational effects associated with the data reduction process, such as the accuracy of the photometric measurements, the crowding conditions, and the ability of the PSF-fitting procedure in resolving partially overlapped sources. We performed the tests using to the following procedure. We divided the frames into grids of cells of chosen width (50 pixels) and randomly added one artificial star per cell at each run. This procedure prevents the artificial stars to interfere with each other, and avoids to bias the experiments towards an artificial crowding not really present in the original frames. The position of the grid is randomly changed at each run, and after a large number of experiments the stars are uniformly distributed over the frame. In each filter, we assign to the artificial star a random input magnitude between $m_1$ and $m_2$, with $m_1$ $\approx$ 3 mag brighter than the brightest star in the CMD, and $m_2$ $\approx$ 3 mag fainter then the faintest star in the CMD. At each run, the frame is re-reduced following exactly the same procedure as for the real data. The output photometric catalog is cross-correlated with a sum of the original photometric catalog of real stars and the list of the artificial stars added into the frame. This prevents cross-correlation of artificial stars in the input list with real stars recovered in the output photometric catalog. We simulated about half a million stars for each filter. At each magnitude level, the completeness of our photometry is computed as the ratio of the number of recovered artificial stars over the number of added ones. The completeness levels in the color magnitude diagrams (see Section 4) are the product of the completeness factors in the two involved passbands. We show in Fig. \[dm\] the $\Delta m$ difference between the input and output magnitudes of the artificial stars as a function of the input magnitude, for the F435W, F555W and F814W filters. The solid lines superimposed on the artificial star distributions correspond to the mean $\Delta m$ (central line), and the $\pm 1 \sigma_m$ values around the mean. The plotted $\Delta m$ distributions provide a complete and statistically robust characterization of the photometric error as a function of magnitude, for each filter. By comparing the $\sigma_m$ with the DAOPHOT errors in Fig. \[fig1I\], it is apparent that the DAOPHOT package increasingly underestimates the actual errors toward fainter magnitudes. For instance, for a star with V$\sim$ 25.5 and I$\sim$ 24, (tip of the red giant branch, see Section 5), the mean DAOPHOT error is $\sim$ 0.05 mag in both bands, while the $\sigma_m$ from the artificial star tests is $\sim$ 0.15 and $\sim$ 0.1 in V and I, respectively. The systematic deviation from 0 of the mean $\Delta m$ indicates the increasing effect of blending, i.e. faint artificial stars recovered brighter than in input because they happen to overlap other faint objects. Color-magnitude diagrams ======================== The CMDs are shown in Figures \[cmd1\] and \[cmd2\]. We plot the ${\rm m_{F555W}}$ versus ${\rm m_{F435W}- m_{F555W}}$ CMD of the 299,014 stars matched between the B and V catalogs in Fig. \[cmd1\], and we plot the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD of the 402,045 stars matched between the V and I catalogs in Fig. \[cmd2\]. We indicate the 90 % (solid line) and 50 % (dashed line) completeness levels as derived from the artificial star experiments on the two CMDs. The average size of the photometric errors at different magnitudes, as derived from artificial star tests, is indicated as well. The two CMDs show all the evolutionary features expected at the magnitudes sampled by our data: a well defined blue plume and red plume, the red horizontal tail of the carbon stars in the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD, and a prominent red giant branch (RGB). The blue plume is located at ${\rm m_{F435W}-m_{F555W}}$ and ${\rm m_{F555W}-m_{F814W}}$ $\simeq$ $-$0.1 in the two diagrams, with the brightest stars detected at ${\rm m_{F555W}}$, ${\rm m_{F814W}}$ $\sim$ 18. It samples both stars in the main-sequence (MS) evolutionary phase and evolved stars at the hot edge of the core helium burning phase. The blue plume extends down to the faintest magnitudes in our data, at ${\rm m_{F555W} \sim 28}$. The red plume is slightly inclined with respect to the blue plume, with [$\rm m_{F555W} \la 25$]{} and colors extending from [$\rm m_{F435W}-m_{F555W} \sim 1.4$]{} to $\sim$ 1.8 in the ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$ CMD, and [$\rm m_{F814W} \la 23.5$]{} and colors extending from [$\rm m_{F555W}-m_{F814W} \sim 1.4$]{} to $\sim$ 2.2 in the ${\rm m_{F814W}}$ versus ${\rm m_{F555W}-m_{F814W}}$ CMD. It is populated by red supergiants (RSGs) at the brighter magnitudes, and AGB stars at fainter luminosities. At intermediate colors, below ${\rm m_{F555W} \sim 25}$ and ${\rm m_{F814W} \sim 23.5}$, we recognize the [*blue loops*]{} of intermediate-mass stars in the core helium burning phase. The concentration of red stars at ${\rm m_{F555W} \ga 25.5}$ and ${\rm m_{F814W} \ga 24}$, corresponds to low-mass old stars in the RGB evolutionary phase. Finally, a pronounced horizontal feature, at ${\rm m_{F814W} \sim 23.5}$, and with colors extending from ${\rm m_{F555W}-m_{F814W} \sim 1.8}$ to as much as ${\rm m_{F555W}-m_{F814W} \sim 4}$, is observed in the ${\rm m_{F814W}}$, ${\rm m_{F555W}-m_{F814W}}$ CMD. This red tail is produced by carbon stars in the thermally pulsing asymptotic giant branch (TP-AGB) phase. In order to reveal spatial differences in the stellar population of NGC 4449, we have divided the galaxy’s field of view into 28 (7 $\times$ 4) rectangular regions, as shown in Fig. \[imagegrid\]. The size of the regions ($\approx$ 55 $\times$ 55 ${\rm arcsec^2}$, corresponding to ${\rm \approx 1 \times 1 \ kpc^2}$ at the distance of NGC 4449) allows us to follow spatial variations at the kpc scale, being at the same time large enough to provide a good sampling. The ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$, and ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMDs derived for the different regions are shown in Fig. \[hessregionbv\] and \[hessregionvi\], respectively. The completeness levels plotted on the CMDs of the central column show that the photometry is deeper in the external regions than in the galaxy center, where the high crowding level makes the detection of faint objects more difficult. The errors, as estimated from the artificial star experiments, increase toward the galaxy center, as an effect of the higher crowding level and the higher background. Bright stars are mostly concentrated toward the galaxy center, and only a few of them are present at large galactocentric distances, in agreement with what was already observed in other dwarf irregular galaxies, (e.g., [@tosi01]). The external region (6,4) makes the exception to this observed global trend, showing a prominent blue plume in both the CMDs. The luminous blue stars observed in these CMDs correspond in the image of Fig. \[image\] to a symmetric structure. This structure is more clearly visible in the top right of Fig. \[spatial\], which will be discussed in Section 6 below. This structure could be due to tidal tails or spiral–like feature associated with a dwarf galaxy that is currently being disrupted. The structure is centered on a resolved cluster-like object that could be the remnant nucleus of this galaxy. Fig. \[imagegrid\] shows a blow-up of this object. Carbon stars ------------ In the ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMD of Fig. \[cmd2\], the horizontal red tail, at magnitudes brighter than the TRGB, is due to carbon-rich stars in the TP-AGB phase. Since AGB stars trace the stellar populations fromÊ$\sim$ 0.1 to several Gyrs, this well defined feature is suitable to investigate the SFH from old to intermediate ages [@cioni06]. Recently, theoretical models of TP-AGB stars have been presented by [@marigo07] for initial masses between 0.5 and 5.0 $M_{\odot}$ and for different metallicities. Their Fig. 20 shows that the position of the carbon–star tracks in the $\log L/L_{\odot}$ vs $\log T_{eff}$ plane depends on both mass and metallicity. Higher-mass stars exhibit larger luminosities for a given metallicity. On the other hand, lower metallicities imply both a wider range of masses undergoing the C-rich phase, and higher luminosities for the more massive carbon stars. From an empirical aspect, we can get the dependence of the carbon–star luminosity on age and metallicity from the work of [@batti05] (hereafter BD05). The authors provide the following relation for the dependence of the carbon–star I band magnitude on metallicity: $${\rm <M_{I,carbon}>=-4.33 +0.28 \times [Fe/H]}, \label{eq3}$$ which was derived through a least-square fit of the mean absolute I band magnitude of carbon stars in nearby galaxies with metallicities ${\rm -2<[Fe/H]<-0.5}$. According to (\[eq3\]), a drop of 1 dex in metallicity results in a decrease of $\approx$ 0.3 in the average magnitude of carbon stars. The age dependence of the carbon–star luminosity is more difficult to quantify since it requires a detailed knowledge of the SFH in galaxies. BD05 do not study such a dependence, but some qualitative considerations can be obtained from examination of their Fig. 4. The scatter of the ${\rm <M_{I,carbon}>}$ versus ${\rm [Fe/H]}$ relation is of the order of 0.1 mag (excluding AndII and AndVII from the fit), i.e. $\approx$ 20 % of the variation in ${\rm <M_{I,carbon}>}$ spanned by the data. We also notice that at fixed metallicity, galaxies with current star formation (empty dots in Fig. 1 of BD05) tend to have brighter ${\rm <M_{I,carbon}>}$ values, while galaxies with no current star formation preferentially lie below the relation. This suggests that at least part of the scatter of relation (\[eq3\]) is due to a spread in age, with younger stellar populations having brighter carbon stars than older stellar populations. This age dependence is consistent with the [@marigo07] models, where more massive (younger) carbon stars tend to be more luminous. Carbon stars were selected in NGC 4449 at ${\rm 23< m_{F814W} <24, m_{555W}-m_{F814W}>2.4}$, for each of the 28 regions shown in Fig. \[imagegrid\]. The color limit was chosen to avoid a significant contribution of RGB and oxygen-rich AGB stars. For each region, we fitted the ${\rm m_{F814W}}$-band LF with a Gaussian curve, and adopted the peak of the best-fitting Gaussian as the average ${\rm <m_{I,carbon}>}$ in that region. The errors on the results of the Gaussian fits can be approximated as $\Delta m \approx \sigma / \sqrt{N}$, where $\sigma$ is the width of the Gaussian and N is the number of stars. We experimented with different cuts in color (up to ${\rm m_{555W}-m_{F814W} >2.8}$) and different binnings of the data, and found that none of them significantly affects the final results of the analysis. The results presented in Fig. \[Clum\] were obtained by binning the stellar magnitudes in bins of 0.2 mag. The quantity ${\rm \Delta m_{I,C}}$ along the ordinate is the difference between the carbon–star magnitude measured in a specific region, and the carbon–star magnitude averaged over the whole field of view of NGC 4449. Along the abscissa is the X coordinate in pixels. From top to bottom, the panels refer to regions of decreasing Y coordinate. Fig. \[Clum\] shows that the observed carbon–star luminosity is reasonably constant over almost the whole galaxy, within the errors. The only significant variation is observed in the central regions ($ 3000 \la X \la 8000$, panel c), where the carbon stars appear to be up to $\approx$ 0.25 mag brighter than the average value. We performed Monte Carlo simulations to understand if the observed variation of the C star LF is intrinsic, i.e. due to stellar population gradients present in NGC 4449 field, or if it is an effect of the photometric error and completeness level variations across the field. We adopted an estimated [*intrinsic*]{} LF for the C stars, that we assumed to be constant all over the field. Then we investigated how the [*intrinsic*]{} LF is transformed into the [*observed*]{} one after completeness and photometric errors were applied at different galacto-centric distances. For simplicity, we assumed that the intrinsic ${\rm m_{F814}}$ distribution of the C stars is Gaussian, with parameters (${\rm m_{F814W,0}=23.64}$, $\sigma$ $\sim 0.3$) derived from the Gaussian fit to the observed LF in the most external regions ((1,1:4), (1:7,4), (7,1:4); where this notation will indicate the some of these regions). We also assumed that the C stars are not homogeneously distributed in color, but follow a power law in ${\rm m_{F555W} - m_{F814W}}$. The power law’s parameters were derived by fitting the observed C star color distribution in the same external regions. Our assumption that the observed outer LF is a good description of the intrinsic one in those regions is reasonable, since the most external ${\rm m_{F814}}$, ${\rm m_{F555W} -m_{F814W}}$ CMDs are more than 90 % complete at the average C star luminosity, and the errors $\sigma_{F814W}$ are only a few hundreds of magnitudes there (see Fig. \[hessregionvi\]). Monte Carlo extractions of (${\rm m_{F814}}$, ${\rm m_{F555W} - m_{F814W}}$) pairs were drawn from the assumed magnitude and color distributions. A completeness and a photometric error were then applied to each extracted star, for both an external and an internal region of NGC 4449. The effect of incompleteness and photometric errors on the distribution is shown in the bottom panel of Fig. \[Clf\]. In the top panel we show instead the observed LFs for an external region ((1,1:4), (1:7,4), (7,1:4)), and for the most internal one (4,3), which displays the largest shift of the distribution peak with respect to the external region. Our simulations in the bottom panel show that the C star LF is mostly unaffected by incompleteness and photometric errors in the most external regions. This result testifies that the observed LF is very close to the intrinsic one in the periphery, and that we adopted a reasonable input distribution for the Monte Carlo simulations. In the most internal region, instead, the LF is shifted toward brighter magnitudes by an amount (${\rm \Delta m_{F814W} \approx 0.25}$) comparable to the shift between the observed LFs. The resulting internal distribution also has a larger width ($\sigma \approx 0.4$) than the intrinsic one, due to the larger photometric errors. Our results show that the detected change in the C star brightness over NGC 4449 field can be largely attributed to differences in completeness between the center and the most external regions. Accounting for this effect, the average magnitude of the C stars is constant within the errors ($\approx 0.05$ mags). From the BD05 relation [^3], a change in magnitude of 0.05 corresponds to $\Delta$\[Fe/H\] $\approx 0.2$. We interpret this as an upper limit to the metallicity variation over the field of view. This is consistent with studies of metallicity gradients in other magellanic irregulars. For example, [@cole04] derive a metallicity gradient of $\approx$ $-$0.05 dex $\times$ kpc$^{-1}$ in the LMC by comparing the abundances in the inner disk and in the outer disk/spheroid [@ols], while @gro06 find no metallicity gradients from spectroscopic studies of cluster stars in the LMC. RGB stars --------- In the CMD of Fig. \[cmd2\], the morphology of the RGB, at ${\rm m_{F814W}} \ga 24$ and ${\rm m_{F555W}- m_{F814W} \ga 1}$, is connected to the properties of the stellar content older than $\sim$ 1 Gyr. Despite the poorer time resolution with increasing look-back time, and the well known age-metallicity degeneracy, some constraints on the properties of the old stellar population can be inferred from an analysis of the RGB morphology. We derived the average RGB color as a function of ${\rm m_{F814W}}$ by selecting stars with ${\rm m_{F814W}} \ga 24$, and then performing a Gaussian fit to the ${\rm m_{F555W}-m_{F814W}}$ color distribution at different magnitude bins. The peak of the Gaussian fit is as red as ${\rm m_{F555W}-m_{F814W} \approx 1.7}$ at the RGB tip, and it is ${\rm m_{F555W}-m_{F814W} \approx 1.45}$ at ${\rm m_{F814W}=25}$, one magnitude below the tip. As expected, the RGB is significantly redder than in more metal-poor star-forming galaxies that we have previously studied with HST/ACS [@alo05; @alo07]. In order to reveal the presence of age/metallicity gradients in NGC 4449, we performed a spatial analysis of the RGB morphology, following the same procedure as in Section 4.1. For each of the 28 rectangular regions identified in Fig. \[imagegrid\], we performed a Gaussian fit to the RGB ${\rm m_{F555W}- m_{F814W}}$ color distribution for bins of ${\rm \Delta m_{F814W}=0.25}$. Then we averaged for each region the colors derived in the four brightest bins, at ${\rm 24 \le m_{F555W}- m_{F814W} \le 25}$, i.e. one magnitude below the TRGB. The results of our analysis are presented in Fig. \[deltargb\]. We plot along the ordinate the difference between the RGB color in each region and the average RGB color in the total field of view of NGC 4449; along the abscissa is the X coordinate in pixels. From top to bottom, the panels refer to regions of decreasing Y coordinate. Fig. \[deltargb\] shows that the RGB is bluer in the center than in the periphery of NGC 4449, with variations up to $\approx$ 0.3 mag in the ${\rm m_{F555W}-m_{F814W}}$ color. This effect is also shown in the top panel of Fig. \[rgbcolor\], where we plotted the observed color distributions of stars with ${\rm 24 \le m_{F814} \le 25}$, for an external region ((1,1:4), (1:7,4), (7,1:4)), and for an internal region (3:5,2) of NGC 4449. The peaks of the Gaussian fits to the external and internal distributions differ by $\approx$ 0.26 mag, and their errors are very small ($\sigma / \sqrt{N} \approx$ 0.001 mag) due to the large number of stars in each distribution ($\approx 20,000$). The Gaussian fit is broader for the internal region ($\sigma \approx 0.4$) than for the external region ($\sigma \approx 0.2$). Also, the central region has a much broader tail of stars towards blue colors, due to the contamination from younger blue-loop and MS stars. The contribution from the MS $+$ blue–loop stars at the hot edge of the core He-burning phase is recognizable as a bump at V$-$I $\approx 0$. As was done for the C stars in Section 4.1, we performed Monte Carlo simulations to understand if the observed difference in the color distributions is intrinsic, or if it can be attributed to the larger crowding in the central regions of NGC 4449. We assumed an initial distribution, and drew Monte Carlo extractions from it with application of photometric errors and incompleteness. As a first guess for the initial distribution, we adopted a Gaussian with parameters ${\rm m_{F555W}-m_{F814W}=1.65}$ and $\sigma=0.15$. This has the same mean but somewhat smaller $\sigma$ than the observed color distribution in the most external regions. The simulated color distributions for the external and internal regions were generated by applying the photometric errors and the completeness levels derived from artificial star experiments in the considered regions of NGC 4449. The results are presented in the central panel of Fig. \[rgbcolor\]. As observed, the width of the simulated distribution in the internal region is larger than in the external one because of the higher photometric errors. While the peak of the external simulated distribution is the same as that of the initial distribution, the peak of the internal distribution is blueshifted by an amount of $\approx$ 0.06 mags. However, this is much less than the observed shift of $\approx$ 0.3 mags. The simulations therefore show that the completeness and the photometric error variations over NGC 4449 field of view can account only in part ($\approx$ 20 % ) for the shift between the observed internal and external distributions. Thus the observed shift must be mainly due to an intrinsic variation of the stellar population properties. As a test, we performed a new simulation starting from a Gaussian with a peak as blue as ${\rm m_{F555W} - m_{F814}=1.45}$ and with $\sigma=0.15$. The result for the internal region is shown in the bottom panel of Fig. \[rgbcolor\]. Here we plot also the observed color distribution, for comparison. Both the peaks and the widths of the simulated and observed distributions are in good agreement. The only discrepancy is observed at the bluest color, where of course we do not reproduce the tail of the MS and post-MS stars. Our simulations suggest that the peak of the color distribution toward the center is intrinsically bluer than in the periphery. Assuming that the bluer peak in the center is due to a bluer RGB, the possible interpretations are 1) younger ages; 2) lower metallicities; or 3) lower reddening. If differential reddening is present within NGC 4449, we expect the central regions to be more affected by dust extinction than the periphery, and this would cause an even redder RGB in the center. Lower metallicities in NGC 4449 center are also very unlikely, since abundance determinations in galaxies show that metallicity tends to decrease from the center outwards or to remain flat (see Section 4.1). Thus, a bluer RGB would most likely indicate a younger population in the center of NGC 4449. Alternatively, the bluer peak observed in the center could be due to “contamination" by intermediate-age blue loop stars at the red edge of their evolutionary phase. But this too would imply the presence of a younger stellar population in the center than in the periphery of NGC 4449. More quantitative results on age/metallicity in NGC 4449 gradients will be derived through fitting of synthetic CMDs, and presented in a forthcoming paper (Annibali et al. 2008, in preparation). A new distance determination ============================ The magnitude of the TRGB can be used to determine the distance of NGC 4449. The top panel of Fig. \[trgb\] shows the I-band LF of those stars in our final catalog that have V$-$I in the range $1.0$–$2.0$. Here V and I are Johnson-Cousins magnitudes, obtained from our magnitudes in the ACS filter system using the transformations of [@sir05] and applying a foreground extinction correction of ${\rm E(B-V) = 0.019}$ [@schlegel]. The TRGB is visually identifiable as the steep increase towards fainter magnitudes at ${\rm I \approx 24}$. At this magnitude, RGB stars start to contribute with a LF that increases roughly as a power law towards faint magnitudes. By contrast, the stars in the LF at brighter magnitudes are exclusively red supergiants and AGB stars. The drop at magnitudes fainter than ${\rm I \approx 25.5}$ is due to incompleteness. To determine the TRGB magnitude we used the software and methodology developed by one of us (R.P.v.d.M) and described in detail in [@cioni00]. A discontinuity produces a peak in all of the higher-order derivatives of the LF. We use a so-called Savitzky-Golay filter on the binned LF to obtain the first- and second-order derivatives. These are shown in the middle and bottom panel of Fig. \[trgb\], respectively. Peaks are indeed visible at the expected position of the TRGB. We fit these with Gaussians and find that the first derivative has a peak at ${\rm I_1 = 24.09}$, while the second derivative has a peak at ${\rm I_2 = 23.88}$. The reason for the difference between these magnitudes is the presence of photometric errors and binning in the analysis. This smooths out the underlying discontinuity. As shown in Fig. A.1 of [@cioni00], this causes the first derivative to overestimate the magnitude of the TRGB, and the second derivative to underestimate the magnitude of the TRGB. These biases can be explicitly corrected for as in Fig. A.2 of [@cioni00], using a simple model for the true underlying LF and the measured width of the Gaussian peaks in the first- and second-order derivatives of the LF. After application of this correction we obtain the final estimates ${\rm I_{TRGB,1} = 23.99}$ and ${\rm I_{TRGB,2} =24.00}$ for the underlying TRGB magnitude, based on the first and second derivatives, respectively. The good agreement between these independent estimates shows that the systematic errors in the method are small, in agreement with [@cioni00] who adopted a systematic error ${\rm \Delta I_{TRGB} = \pm 0.02}$. The additional systematic error introduced by the uncertainties in photometric zeropoints, transformations, and aperture corrections [@sir05] is ${\rm \Delta I_{TRGB} = \pm 0.03}$. The random error on ${\rm \Delta I_{TRGB}}$ is very small due to the large number of stars detected in NGC 4449. It can be estimated using bootstrap techniques to be ${\rm \Delta I_{TRGB} = \pm 0.01}$. Our estimate of the TRGB magnitude, ${\rm I_{TRGB} = 24.00 \pm 0.01}$ (random) $\pm 0.04$ (systematic), can be compared to the absolute magnitude of the TRGB, which was calibrated as a function of metallicity by, e.g., [@bella04]. Adopting ${\rm [M/H] = -0.52}$ for NGC 4449 (based on the oxygen abundance given in Section 1) their calibration (top panel of their Fig. 5) predicts ${\rm M_{I, TRGB} = -3.91}$. Comparison of different studies suggests that the systematic uncertainty in this prediction is $\sim 0.15$. This takes into account also the possibility that the RGB star metallicity is actually lower than that of the HII regions (see discussion in Section 6). The implied distance modulus for NGC 4449 is therefore ${\rm (m-M)_0 = 27.91 \pm 0.15}$, where we have added all sources of uncertainty in quadrature. This corresponds to ${\rm D = 3.82 \pm 0.27}$ Mpc. An alternative method for estimating the distance of NGC 4449 is through the average I-band magnitude of the carbon stars. Fig. \[agbdist\] shows the I-band luminosity function of those stars in our final catalog that have V$-$I in the range $2.2$–$3.0$. There is a well-defined peak, due to the horizontal “finger” of carbon stars seen in the CMD of Fig. \[cmd2\]. A Gaussian fit to the peak yields ${\rm I_{carbon} = 23.59 \pm 0.01}$, corrected for foreground extinction. Adopting ${\rm [Fe/H] = -0.52}$ for NGC 4449 (based on the oxygen abundance given in Section 1, and assuming for simplicity that oxygen traces the iron content), the BD05 calibration in (\[eq3\]) predicts $M_{\rm I, carbon} =-4.48$. The systematic uncertainty in this prediction is difficult to quantify. This is because carbon–star magnitudes are not well understood on the basis of stellar evolution theory (by contrast to the TRGB), because the dependence on stellar age or star formation history is poorly quantified, and because only a few empirical studies exist. We therefore adopt an uncertainty of $\sim 0.2$ mag, consistent with the discussion of Section 4.1 The implied distance modulus for NGC 4449 is then ${\rm (m-M)_0 = 28.07 \pm 0.20}$, where we have added all sources of uncertainty in quadrature. This corresponds to ${\rm D = 4.11 \pm 0.38}$ Mpc. This is in agreement with the TRGB result, given the uncertainties. We can compare our result of ${\rm D=3.82 \pm 0.27}$ Mpc to the previous estimate of NGC 4449 distance by [@ka03], who inferred ${\rm D=4.2 \pm 0.5}$ Mpc. This result was derived by applying the TRGB-luminosity method to HST/WFPC2 data. The extinction-corrected magnitude at which they detected the TRGB is ${\rm I_{TRGB} = 24.07 \pm 0.26}$, which is consistent with our somewhat lower value of ${\rm I_{TRGB} = 24.00 \pm 0.01}$ (random) $\pm 0.04$ (systematic). Their somewhat larger distance is also due to the different value adopted for the absolute magnitude of the TRGB. They adopt ${\rm M_{I, TRGB} =-4.05}$ (as appropriate for metal-poor systems), while we adopt ${\rm M_{I, TRGB} = -3.91}$ from the more recent [@bella04] calibration (which takes into account the metallicity dependence of the TRGB luminosity). Comparison with models ====================== For a direct interpretation of the CMDs in terms of the stellar evolutionary phases, we have superimposed stellar evolutionary tracks for different metallicities on the ${\rm m_{F555W}}$, ${\rm m_{F435W}- m_{F555W}}$, and ${\rm m_{F814W}}$, ${\rm m_{F555W}- m_{F814W}}$ CMDs (Figs. \[cmdtracks\] and \[rgbtracks\]). The tracks at Z$=$0.008, Z$=$0.004, and Z$=$0.0004 are the Padua stellar evolutionary tracks (Fagotto et al. 1994a, 1994b) transformed into the ACS Vegamag system by applying the @origlia code, and corrected for Galactic extinction (${\rm E(B-V)=0.019}$, @schlegel) and distance modulus (${\rm (m-M)_0 = 27.91}$, see Section 5). The Z$=$0.001 tracks were obtained from the Padua tracks through interpolation in metallicity [@ang06]. The metallicity range covered by the plotted tracks can account for the different populations potentially present in NGC 4449. The Z$=$0.008 and Z$=$0.004 tracks are suitable to account for the more metal rich population, given that the abundances derived in the nucleus and disk of NGC 4449 are almost one third of the solar value (see Section 1). The plotted tracks are for masses in the range 0.9–40 ${\rm M_{\odot}}$. Lower mass stars would not have the time to reach visible phases within a Hubble time at the distance of NGC4449 (in these sets a 0.8 ${\rm M_{\odot}}$ star reaches the TRGB in 19 Gyr). The tracks are divided into three groups, namely [*low-mass*]{} stars (${\rm M \le M_{HeF}}$), [*intermediate-mass*]{} stars (${\rm M_{HeF} < M \le M_{up}}$) and [*high-mass*]{} stars (${\rm M > M_{up}}$). The subdivision is made according to the critical mass at which the ignition of the central fuel (either helium or carbon) starts quietly depending on the level of core electron-degeneracy. In the adopted tracks, the value of ${\rm M_{HeF}}$ depends slightly on metallicity, being equal to 1.7, 1.8 and 1.9 ${\rm M_{\odot}}$ for Z$=$0.0004, Z$=$0.004 and Z$=$0.008, respectively. The value of ${\rm M_{up}}$ is between 5 and 6 ${\rm M_{\odot}}$. For low-mass stars, we have displayed in Figs. \[cmdtracks\] and \[rgbtracks\] only the phases up to the TRGB in order to avoid excessive confusion. At the distance of NGC 4449, the MS is sampled only for stars with ${\rm M \ga 3 M_{\odot}}$, and corresponds to the vertical lines whose ${\rm m_{F435W}- m_{F555W}}$, ${\rm m_{F555W}- m_{F814W}}$ colors go from $\sim$ $-$0.3 to $\sim$ 0 from the 40 ${\rm M_{\odot}}$ to the 3 ${\rm M_{\odot}}$ track. The turnoff is recognizable as a small blue hook on each evolutionary track of Fig. \[cmdtracks\]. The almost horizontal blue loops correspond to the later core helium-burning phase for intermediate- and high- mass stars, while the bright vertical lines in the red portion of the CMD describe the AGB phase. For low-mass stars, our data sample only the brighter red sequences of the RGB, all terminating at the TRGB with approximately the same luminosity. The models considered here do not include the horizontal feature of the carbon stars in the TP-AGB phase, recognizable in the ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD at ${\rm m_{F555W}- m_{F814W} \ga 2}$, but see @marigo07 for new appropriate models. The comparison of the CMDs with the tracks in Fig. \[cmdtracks\] shows that the blue plume is populated by high- and intermediate-mass stars on the MS, and high-mass stars at the hot edge of the core helium-burning phase. The red plume samples bright red SGs and AGB stars. The faint red sequence at ${\rm m_{F814W} \ga 24}$, featured in Fig. \[rgbtracks\], is due to low-mass stars in the RGB phase. The presence of high-, intermediate- and low-mass stars in the CMD testifies that young, intermediate-age and old stars (several Gyrs) are present at the same time in NGC 4449. In particular, the fact that we sample stars as massive as 40 ${\rm M_{\odot}}$ implies that the SF was active 5 Myr ago. Furthermore, the absence of significant gaps in the CMD suggests that the SF has been mostly continuous over the last 1 Gyr. The time resolution gets significantly poorer for higher look-back times because of both the intrinsic degeneracy of the tracks and the large photometric errors at faint magnitudes; thus small interruptions in the SF can be easily hidden in the CMD for ages older than $\ga 1$ Gyr. The metallicity ${\rm [M/H] =-0.52}$ derived in Section 1 for NGC 4449 corresponds to a metal fraction Z$=$0.005, adopting Z$_{\odot}=$ 0.017 [@sun98]. However this value refers to abundance determinations in HII regions, thus it is likely to reflect the metallicity of the youngest stars. Fig. \[cmdtracks\] and  \[rgbtracks\] show that the Z$=$0.004 tracks are in good agreement with all the phases of the empirical CMDs. The Z$=$0.0004 and Z$=$0.001 tracks are too blue to account for the observed RGB feature. So, if there is no significant extinction intrinsic to NGC 4449, then stars with such low metallicities do not account for a significant fraction of the stellar population in NGC 4449. Allowing for an age spread from 1 Gyr to a Hubble time, the Z$=$0.004 tracks are in good agreement with both the color and the width of the observed RGB. @hill98 derived an internal reddening ${\rm E(B-V) \approx}$ 0.18 from the $H\alpha$/H$\beta$ ratio measured in NGC 4449 HII regions. Adopting this value, the RGB is consistent with metallicities as low as Z$=$0.001. However, this extinction value is appropriate for the young star forming regions, and we expect a lower extinction in the regions where older stars reside (as demonstrated for the LMC by @zari). Either way, the Z$=$0.008 models seem slightly too red, especially in the AGB phase. Although there are some uncertainties in the models of this phase, we do believe that this discrepancy indicates that metallicities higher than Z$=$0.008 are ruled out for this galaxy. By superimposing the stellar tracks on the empirical CMDs, we can attempt a selection of the photometrized stars according to their age. In Fig. \[cmdsel\] we show the four regions selected on the ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD that correspond to different stellar masses, namely M${\rm \ge 20 \ M_{\odot}}$, ${\rm 5 M_{\odot} \le M < 20 M_{\odot}}$, ${\rm 1.8 M_{\odot} < M < 5 M_{\odot}}$, and ${\rm M \le 1.8 M_{\odot}}$. These regions roughly define the loci of [*very young*]{} stars, with ages $\la$ 10 Myr, [*young*]{} stars, with ${\rm 10 \ Myr \la age \la 100 \ Myr}$, [*intermediate-age*]{} stars, with ages between $\sim$ 100 Myr and 1 Gyr, and [*old*]{} stars, with ages $>$ 1 Gyr. The spatial distribution of the four groups of stars is shown in Fig. \[spatial\]. We notice that old stars are homogeneously distributed over the galaxy, except for the central regions, where the high crowding level makes their detection more difficult. As we approach younger ages, the distribution becomes more and more concentrated. Stars with ages between 10 and 100 Myr are highly clustered in an [*S shaped*]{} structure centered on the galaxy nucleus. This could be a bar, which is a common feature among Magellanic irregular galaxies. Stars with ages between 10 and 100 Myr also clearly outline the symmetric structure around the resolved cluster-like object (see Fig. \[imagegrid\]) that was already discussed in Section 4. The same structure is visible to a lesser extent in the spatial distribution of stars with ages 100 Myr – 1 Gyr. The resolved cluster-like object itself has a clear RGB (not shown here), and it is therefore not a young super star cluster. Very young stars, with ages younger than 10 Myr, are strongly clustered and detected only in the very central regions of NGC 4449. Such young stars should be able to ionize the surrounding interstellar medium and produce HII regions. Indeed, Fig. \[halpha\], where we plotted the F658N (H$\alpha$) image together with the positions of the stars younger than 10 Myr, shows that this is the case, with a strong correlation between the position of the stars younger than 10 Myr and the HII regions. This indicates that most of the emission is due to photoionization rather than to shocks, due to, e.g., supernovae explosions. Conclusions =========== We have acquired HST/ACS imaging in the F435W (B), F555W (V), F814W (I) and F658N (${\rm H\alpha}$) filters of the Magellanic starburst galaxy NGC 4449 in order to infer its star formation history and understand the properties of the observed global starburst. In this paper we present the B, V and I photometry of the resolved stars. We detect 299,014 objects in the (B,V) CMD, 402,045 objects in the (V,I) CMD, and 213,099 objects with a measured magnitude in all the three bands. The derived CMDs span a magnitude range of $\approx$ 10 mag, and sample both the young and the old resolved stellar population in NGC 4449. We also detected several candidate clusters (at least $\approx$ 40, some of which look like very massive globular clusters) and background galaxies in our images. We derived a new distance from the TRGB method. The TRGB is detected at a Johnson-Cousins I magnitude of ${\rm I_{\rm TRGB} = 24.00 \pm 0.04}$. At the metallicity of NGC 4449, the TRGB is expected at an absolute magnitude of ${\rm M_{\rm I, TRGB} =-3.91}$, with a systematic error of $\sim 0.15$ mag. This provides a distance modulus of ${\rm (m-M)_0 = 27.91 \pm 0.15}$, i.e. a distance of $3.82 \pm 0.27$ Mpc. We used also the alternative method of the carbon–star luminosity, and found a distance of $4.11 \pm 0.38$ Mpc, which is consistent with the result from the TRGB method. Our distance determinations are consistent within the errors with the value of D$=4.2 \pm 0.5$ Mpc previously provided by Karachentsev et al. (2003). In the CMDs of NGC 4449 we observe a well defined blue plume (MS and post-MS stars) and red plume (red SGs and AGB stars), the horizontal tail of the carbon stars in the TP-AGB phase (in the I, V$-$I CMD), and a prominent RGB. The presence of all these evolutionary features implies ages up to at least 1 Gyr and possibly as old as a Hubble time. The comparison of the observed CMDs with the Padua stellar evolutionary tracks, corrected for the derived distance modulus and foreground extinction, shows that stars as massive as 40 ${\rm M_{\odot}}$ are present in NGC 4449. Such high masses imply that the star formation was active 5 Myr ago, and possibly it is still ongoing. The absence of significant gaps in the CMDs suggests also that the star formation has been mostly continuous over the last 1 Gyr. However, interruptions in the star formation can be easily hidden in the CMD for ages older than $\ga 1$ Gyr, because of the intrinsic degeneracy of the tracks and the large photometric errors at faint magnitudes. The presence of a prominent RGB testifies that NGC 4449 hosts a population possibly as old as several Gyrs or more. However, the color-age degeneracy of the tracks with increasing look-back time (at a given metallicity, large age differences correspond to small color variation in the RGB), and the well known age-metallicity degeneracy, prevent us from establishing an exact age for the galaxy. We will derive a better age estimate with the synthetic CMD method, when we’ll study the detailed SFH (Annibali et al. 2008, in preparation). Abundance estimates in NGC 4449 HII regions provide 12 + log(O/H)$ \approx 8.31$ [@talent; @mar97], which corresponds to \[O/H\] $= -0.52$ if we assume a solar abundance of 12 + log(O/H)$_{\odot} =$ 8.83 [@sun98]. These measures are biased toward regions where the interstellar medium has been significantly reprocessed, and thus are likely to reflect the metallicity of the youngest generation of stars. We expect lower metallicity for the oldest stars in NGC 4449. Interestingly, though, the Z$=$0.004 stellar evolutionary tracks (the closest in the Padua set to the metallicity of NGC 4449), corresponding to ${\rm \log(Z/Z_{\odot}) = -0.63}$ if we adopt ${\rm Z_{\odot}=0.017}$ [@sun98], seem to be in very good agreement with all the features observed in the empirical CMDs, if we assume that there is not significant extinction intrinsic to NGC 4449. In particular, the next lower metallicity tracks (at ${\rm Z=0.001}$) are definitively too blue to account for the observed RGB colors, implying that the bulk of the stellar population older than 1 Gyr was already enriched in metals. We investigated the presence of age and metallicity gradients in NGC 4449. To this purpose, we divided the total galaxy’s field of view into 28 rectangular regions of $\approx$ 1 kpc$^2$ area, and derived the CMDs for the different regions. The CMD morphology presents a significant spatial dependence: while the RGB is detected over the whole field of view of the galaxy, the blue plume, red plume and blue-loop stars are present only in the more central regions, indicating that the stellar population is younger in the center than in the periphery of NGC 4449. We also studied the spatial behavior of the carbon–star luminosity and of the RGB color. Once the effect of incompleteness and photometric errors is taken into account, the average magnitude of the C stars turns out to be constant within the errors ($\approx$ 0.05 mags). This gives an upper limit of $\approx$ 0.2 dex on the metallicity variation over the field of view of NGC 4449. On the other hand, we find that the RGB is intrinsically bluer in the center than in the periphery of the galaxy. Bluer RGB colors can be due to younger and/or more metal poor stellar populations. However, as spectroscopic-based abundance determinations in galaxies show that metallicity tends to decrease from the center outwards, or to remain constant, we interpret this as the result of a younger, and not more metal poor, stellar population in the center of NGC 4449. With the help of the Padua tracks, we identified in the observed ${\rm m_{F814W}, m_{F555W}- m_{F814W}}$ CMD four different zones corresponding to different stellar masses, namely M${\rm \ge 20 M_{\odot}}$, ${\rm 5 M_{\odot} \le M < 20 M_{\odot}}$, ${\rm 1.8 M_{\odot} \le M < 5 M_{\odot}}$, and ${\rm M < 1.8 M_{\odot}}$. These regions roughly define the loci of stars with age $\la$ 10 Myr, ${\rm 10 \ Myr \la age \la 100 \ Myr}$, 100 Myr $<$ age $<$ 1 Gyr, and age $>$ 1 Gyr. Low-mass old stars are homogeneously distributed over the galaxy’s field of view, with the exception of the central regions, where the high crowding level makes their detection more difficult. As we approach younger ages, the spatial distribution of the stars becomes more and more clustered. Intermediate-age stars (100 Myr $<$ age $<$ 1 Gyr) are mostly found within $\approx$ 1 kpc from the center. Stars with ages between 100 Myr and 10 Myr are found in an [*S–shaped*]{} structure centered on the galaxy nucleus, and extending in the North-South direction up to 1 kpc away from the center. This could be a bar, which is a common feature among Magellanic irregular galaxies. Stars younger than 10 Myr are very rare, and found only in the galaxy nucleus and in the North arm of the [*S shape*]{} structure. The comparison with the H$\alpha$ image shows a tight correlation between the position of the stars younger than 10 Myr and the HII regions, indicating that we have identified the very massive and luminous stars that ionize the surrounding interstellar medium. One of the many star clusters visible in our image is of particular interest. This cluster on the West side of the galaxy is surrounded by a symmetric structure that is particularly well outlined by stars with ages in the range of 10–100 Myr (see Fig. \[spatial\]). This structure could be due to tidal tails or spiral–like feature associated with a dwarf galaxy that is currently being disrupted by NGC 4449. The cluster could be the remnant nucleus of this galaxy. It is resolved into red stars, and it has a significant ellipticity (see Fig. \[imagegrid\]). This is reminiscent of the star cluster $\omega$ Cen in our own Milky Way, which has also been suggested to be the remnant nucleus of a disrupted dwarf galaxy [@omegacen]. More details about this object will be presented in a forthcoming paper (Aloisi et al. 2008, in preparation). Quantitative information on the star formation history of NGC 4449 is fundamental in order to understand the connection between the global starburst observed and processes such as merging, accretion and interaction. The star formation history of NGC 4449 will be derived in a forthcoming paper through the synthetic CMD method, which is based on stellar evolutionary tracks, and is able to fully account for the effect of observational uncertainties, such as photometric errors, blending and incompleteness of the observations. Support for proposal \#10585 was provided by NASA through a grant from STScI, which is operated by AURA, Inc., under NASA contract NAS 5-26555. We thank Livia Origlia for providing the photometric conversion tables to the ACS Vegamag system. 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[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by AURA, Inc., under cooperative agreement with the National Science Foundation [^3]: BD05 use the Johnson-Cousins I-band. This is very similar to ${\rm m_{F814W}}$ [@sir05], and the small difference can be ignored for the purpose of the differential argument presented here.

ArXiv

--- abstract: 'Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\left\{ a_{n}:\,n\leq N\right\} $ its truncations, and let $\alpha\in\left[0,1\right]$. We prove that if the additive energy $E\left(A_{N}\right)$ of $A_{N}$ is in $\Omega\left(N^{3}\right)$, then the sequence $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ of fractional parts of $\alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $\alpha$ in the sense of Lebesgue measure. Conversely, it is known that $E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)$, for some fixed $\varepsilon>0$, implies that $\left(\left\langle \alpha a_{n}\right\rangle \right)_{n}$ has PPC for almost every $\alpha$. This note makes a contribution to investigating the energy threshold for $E\left(A_{N}\right)$ to imply this metric distribution property. We establish, in particular, that there exist sequences $\left(a_{n}\right)_{n}$ with $$E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log\left(\log N\right)}\right)$$ such that the set of $\alpha$ for which $\left(\alpha a_{n}\right)_{n}$ does not have PPC is of full Lebesgue measure. Moreover, we show that for any fixed $\varepsilon>0$ there are sequences $\left(a_{n}\right)_{n}$ with $E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\left(\log\log N\right)^{1+\varepsilon}}\right)$ satisfying that the set of $\alpha$ for which the sequence $\left(\bigl\langle\alpha a_{n}\bigr\rangle\right)_{n}$ does not have PPC is of full Hausdorff dimension.' author: - 'Thomas Lachmann[^1], and Niclas Technau[^2]' title: On Exceptional Sets in the Metric Poissonian Pair Correlations problem --- Introduction ============ The theory of uniform distribution modulo $1$ dates back, at least, to the seminal paper of Weyl [@Weyl:; @=0000DCber; @die; @Gleichverteilung; @von; @Zahlen; @mod.; @Eins]. Weyl showed, inter alia, that for any fixed irrational $\alpha\in\mathbb{R}$ and integer $d\geq1$ the sequences $\left(\bigl\langle\alpha n^{d}\bigr\rangle\right)_{n}$ are uniformly distributed modulo $1$. However, in recent years various authors have been investigating a more subtle distribution property of such sequences - namely, whether the asymptotic distribution of the pair correlations has a property which is called Poissonian, and defined as follows: Let $\left\Vert \cdot\right\Vert $ denote the distance to the nearest integer. A sequence $\left(\theta_{n}\right)_{n}$ in $\left[0,1\right]$ is said to have (asymptotically) Poissonian pair correlations, if for each $s\geq0$ the pair correlation function[^3] $$R_{2}\left(\left[-s,s\right],\left(\theta_{n}\right)_{n},N\right)\coloneqq\frac{1}{N}\#\left\{ 1\leq i\neq j\leq N:\,\left\Vert \theta_{i}-\theta_{j}\right\Vert \leq\frac{s}{N}\right\} \label{eq: definition of the Pair Correlation Counting function}$$ tends to $2s$ as $N\rightarrow\infty$. Moreover, let $\left(a_{n}\right)_{n}$ denote a strictly increasing sequence of positive integers. If no confusion can arise, we write $$R\left(\left[-s,s\right],\alpha,N\right)\coloneqq R_{2}\left(\left[-s,s\right],\left(\alpha a_{n}\right)_{n},N\right)$$ and say that a sequence $\left(a_{n}\right)_{n}$ has metric Poissonian pair correlations if $\left(\alpha a_{n}\right)_{n}$ has Poissonian pair correlations for almost all $\alpha\in\left[0,1\right]$ in the sense of Lebesgue measure. It is known that if a sequence $\left(\theta_{n}\right)_{n}$ has Poissonian pair correlations, then it is uniformly distributed modulo $1$, cf. [@Aistleitner; @Lachmann; @Pausinger:; @Pair; @correlations; @and; @equidistribution; @Larcher; @Grepstad:; @On; @pair; @correlation; @and; @discrepancy]. Yet, the sequences $\left(\left\langle \alpha n^{d}\right\rangle \right)_{n}$ do *not* have Poissonian pair correlations for *any* $\alpha\in\mathbb{R}$ if $d=1$. For $d\geq2$, Rudnick and Sarnak [@Rudnick; @Sarnak:; @The; @pair; @correlation; @function; @of; @fractional; @parts; @of; @polynomials] proved that $\left(n^{d}\right)_{n}$ has metric Poissonian pair correlations (metric PPC). For alternative proofs, we refer the reader to Heath-Brown and the work of Marklof and Strömbergsson [@Marklof; @Str=0000F6mbergsson:; @Equidistribution; @of; @Kronecker; @sequences; @along; @closed; @horocycles].[^4] Given these results, it is natural to investigate which properties of a sequence of integers $\left(a_{n}\right)_{n}$ implies the metric PPC of $\left(a_{n}\right)_{n}$. Partial answers are known, e.g. it follows from work of Boca and Zaharescu [@Boca; @Zaharescu:; @Pair; @correlation; @of; @values; @of; @rational; @functions; @(mod; @p)] that $\left(P\left(n\right)\right)_{n}$ has metric PPC if $P$ is any polynomial with integer coefficients of degree at least two. An interesting general result in this direction is due to Aistleitner, Larcher, and Lewko [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] who used a Fourier analytic approach combined with a bound on GCD sums of Bondarenko and Seip [@Bondarenko; @Seip:; @GCD; @sums; @and; @complete; @sets; @of; @square-free; @numbers] to relate the metric PPC of $\left(a_{n}\right)_{n}$ with its combinatoric properties. For stating it, let $\left(a_{n}\right)_{n}$ denote henceforth a strictly increasing sequence of positive integers and denote the set of the first $N$ elements of $\left(a_{n}\right)_{n}$ by $A_{N}$. Moreover, define the additive energy $E\left(I\right)$ of a finite set integers $I$ via $$E\left(I\right)\coloneqq\sum_{\underset{a+b=c+d}{a,b,c,d\in I}}1.$$ In the following, let $\mathcal{O}$ and $o$ denote the standard Landau symbols/O-notation.\ \ A main finding of [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] is the implication that if the truncations $A_{N}$ satisfy $$E\left(A_{N}\right)=\mathcal{O}\left(N^{3-\varepsilon}\right)\label{eq: Aistleitner bound}$$ for some fixed $\varepsilon>0$, then $\left(a_{n}\right)_{n}$ has metric PPC. Note that $\left(\#I\right)^{2}\leq E\left(I\right)\leq\left(\#I\right)^{3}$ where $\#I$ denotes the cardinality of $I\subset\mathbb{Z}$. Roughly speaking, a set $I$ has large additive energy if and only if it contains a “large” arithmetic progression like structure. Indeed, if $\left(a_{n}\right)_{n}$ is a geometric progression or of the form $\left(n^{d}\right)_{n}$ for $d\geq2,$ then (\[eq: Aistleitner bound\]) is satisfied. Furthermore, note that the metric PPC property may be seen as a sort of pseudorandomness; in fact, for a given sequence of $\left[0,1\right]$-uniformly distributed, and independent random variables $\left(\theta_{n}\right)_{n}$, one has $$\lim_{N\rightarrow\infty}R\left(\left[-s,s\right],\left(\theta_{n}\right)_{n},N\right)=2s\label{eq: counting function asymtotically Poissonian}$$ for every $s\geq0$ almost surely.\ \ Wondering about the optimal bound for the additive energy of the truncations $A_{N}$ to imply the metric PPC property of $\left(a_{n}\right)_{n}$, the two following questions were raised in [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] where we use the convention that $f=\Omega\left(g\right)$ means for $f,g:\mathbb{N}\rightarrow\mathbb{R}$ there is a constant $c>0$ such that $g\left(n\right)>cf\left(n\right)$ holds for infinitely many $n$. Is it possible for a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $E\left(A_{N}\right)=\Omega\left(N^{3}\right)$ to have metric PPC? Do all increasing strictly sequences $\left(a_{n}\right)_{n}$ of positive integers with $E\left(A_{N}\right)=o\left(N^{3}\right)$ have metric PPC? Both questions were answered in the negative by Bourgain whose proofs can be found in [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] as an appendix, without giving an estimate on the measure of the set that was used to answer Question 1, and without a quantitative bound on $E\left(A_{N}\right)$ appearing in the negation of Question 2. However, a quantitative analysis, as noted in [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian], shows that the sequence Bourgain constructed for Question 2 satisfies $$E\left(A_{N}\right)=\mathcal{O}_{\varepsilon}\left(\frac{N^{3}}{\left(\log\log N\right)^{\frac{1}{4}+\varepsilon}}\right)\label{eq: Bourgains bound for the sequence of non PPC}$$ for any fixed $\varepsilon>0$. Moreover, Nair posed the problem[^5] whether the sequence of prime numbers $\left(p_{n}\right)_{n}$, ordered by increasing value, has metric PPC. Recently, Walker [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian] answered this question in the negative. Thereby he gave a significantly better bound than (\[eq: Bourgains bound for the sequence of non PPC\]) for the additive energy $E\left(A_{n}\right)$ for a sequence $\left(a_{n}\right)_{n}$ not having metric PPC - since the additive energy of the truncations of $\left(p_{n}\right)_{n}$ is in $\Theta\bigl(\left(\log N\right)^{-1}N^{3}\bigr)$ where $f=\Theta\left(g\right)$, for functions $f,g$, means that $f=\mathcal{O}\left(g\right)$ and $g=\mathcal{O}\left(f\right)$ holds. The main objective of our work is to improve upon these answers to Questions 1, and Question 2. For a given sequence $\left(a_{n}\right)_{n}$, we denote by $\NPPC\left(\left(a_{n}\right)_{n}\right)$ the (“exceptional”) set of all $\alpha\in\left(0,1\right)$ such that the pair correlation function (\[eq: definition of the Pair Correlation Counting function\]) does not tend to $2s$, as $N$ tends to infinity, for some $s\geq0$. \[thm: Bourgain’s Result concerning the measure of the set of counterexamples\]Suppose $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If $E(A_{N})=\Omega\left(N^{3}\right)$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has positive Lebesgue measure. We prove the following sharpening. \[thm: full measure of set of counterexamples\]Suppose $\left(a_{n}\right)_{n}$ is a strictly increasing sequence of positive integers. If $E(A_{N})=\Omega\left(N^{3}\right)$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure. Moreover, we lower the known energy threshold, and estimate the Hausdorff dimension of the exceptional set from below. For stating our second main theorem, we denote by $\mathbb{R}_{>x}$ the set of real numbers exceeding a given $x\in\mathbb{R}$, and recall that for a function $g:\mathbb{R}_{>1}\rightarrow\mathbb{R}_{>0}$ the lower order of infinity $\lambda\left(g\right)$ is defined by $$\lambda\left(g\right)\coloneqq\liminf_{x\rightarrow\infty}\frac{\log g\left(x\right)}{\log x}.$$ This notion arises naturally in the context of Hausdorff dimensions. Roughly speaking, it quantifies the (lower) asymptotic growth rate of a function. \[thm: lowering the known Energy threshold\]Let $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>2}$ be a function increasing monotonically to $\infty$, and satisfying $f\left(x\right)=\mathcal{O}\bigl(\left(\log x\right)^{-\nicefrac{7}{3}}x^{\nicefrac{1}{3}}\bigr)$. Then, there exists a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $E(A_{N})=\Theta\bigl((f\left(N\right))^{-1}N^{3}\bigr)$ such that if $$\sum_{n\geq1}\frac{1}{nf(n)}\label{eq: divergence of the reciprocal of (f(n) times n)}$$ diverges, then for Lebesgue almost all $\alpha\in\left[0,1\right]$ $$\limsup_{N\rightarrow\infty}R\left(\left[-s,s\right],\alpha,N\right)=\infty\label{eq: divergence of the Pair Correlation Function}$$ holds for any $s>0$; additionally, if (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges and $\sup\left\{ f\left(2x\right)/f\left(x\right):\,x\geq x_{0}\right\} $ is strictly less than $2$ for some $x_{0}>0$, then $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has Hausdorff dimension at least $\left(1+\lambda\right)^{-1}$ where $\lambda$ is the lower order of infinity of $f$. We record an immediate consequence of Theorem \[thm: lowering the known Energy threshold\] by using the convention that the $r$-folded iterated logarithm is denoted by $\log_{r}\left(x\right)$, i.e. $\log_{r}\left(x\right)\coloneqq\log_{r-1}\left(\log\left(x\right)\right)$ and $\log_{1}\left(x\right)\coloneqq\log\left(x\right)$. \[cor: order of magnitude for the additive energy of the sequence of counter examples\]Let $r$ be a positive integer. Then, there is a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $$E\left(A_{N}\right)=\Theta\left(\frac{N^{3}}{\log\left(N\right)\log_{2}\left(N\right)\ldots\log_{r}\left(N\right)}\right)$$ such that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure. Moreover, for any $\varepsilon>0$ there is a strictly increasing sequence $\left(a_{n}\right)_{n}$ of positive integers with $$E\left(A_{N}\right)=\Theta\left(\frac{\left(\log_{r}\left(N\right)\right)^{-\varepsilon}N^{3}}{\log\left(N\right)\log_{2}\left(N\right)\ldots\log_{r}\left(N\right)}\right)$$ such that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Hausdorff dimension. The proof of Theorem \[thm: lowering the known Energy threshold\] connects the metric PPC property to the notion of “optimal regular systems” from Diophantine approximation. It uses, among other things, a Khintchine-type theorem due to Beresnevich. Furthermore, despite leading to better bounds, the nature of the sequences underpinning Theorem \[thm: lowering the known Energy threshold\] is much simpler than the nature of those sequences previously constructed by Bourgain [@Aistleitner; @Larcher; @Lewko:; @Additive; @Energy; @and; @the; @Hausdorff; @Dimension; @of; @the; @Exceptional; @Set; @in; @Metric; @Pair; @Correlation; @Problems] (who used, inter alia, large deviations inequalities form a probability theory), or the sequence of prime numbers studied by Walker [@Walker:; @The; @Primes; @are; @not; @Metric; @Poissonian] (who relied on estimates, derived by the circle-method, on the exceptional set in Goldbach-like problems).\ \ In the converse direction, there has been remarkable progress, due to a work of Bloom, Chow, Gafni, Walker - who improved under the assumption that the sequence is not “too sparse” the power saving bound (\[eq: Aistleitner bound\]) to a saving of a little more than the square of a logarithm. More precisely, their result is as follows. \[thm: Bloom, Chow, Gafni, Walker theorem\]Let $\left(a_{n}\right)_{n}$ be a strictly increasing sequence of positive integers. Suppose there is an $\varepsilon>0$ and a $C=C\left(\varepsilon\right)>0$ such that $$E\left(A_{N}\right)=\mathcal{O}_{\varepsilon}\left(\frac{N^{3}}{\left(\log N\right)^{2+\varepsilon}}\right),\qquad\delta\left(N\right)\geq\frac{C}{\left(\log N\right)^{2+2\varepsilon}}$$ where $\delta\left(N\right)\coloneqq N^{-1}\#\left(A_{N}\cap\left\{ 1,\ldots,N\right\} \right)$. Then, $\left(a_{n}\right)_{n}$ has metric PPC. First main theorem ================== Let us give an outline of the proof of Theorem \[thm: full measure of set of counterexamples\]. For doing so, we begin by sketching the reasoning of Theorem A: As it turns out, except for a set of neglectable measure, the counting function in (\[eq: definition of the Pair Correlation Counting function\]) can be written as a function that admits a non-trivial estimate for its $L^{1}$-mean value. The $L^{1}$-mean value is infinitely often too small on sets whose measure is uniformly bounded from below. Thus, there exists a sequence of set $\left(\Omega_{r}\right)_{r}$ of $\alpha\in\left[0,1\right]$ such that $R\left(\left[-s,s\right],\alpha,N\right)$ is too small for every $\alpha\in\Omega_{r}$ for having PPC and Theorem A follows. Our reasoning for proving Theorem \[thm: full measure of set of counterexamples\] is building upon this argument of Bourgain while we introduce new ideas to construct a sequence of sets $\left(\Omega_{r}\right)_{r}$ that are “quasi (asymptotically) independent” - meaning that for every fixed $t$ the relation $\lambda(\Omega_{r}\cap\Omega_{t})\leq\lambda(\Omega_{r})\lambda(\Omega_{t})+o\left(1\right)$ holds as $r\rightarrow\infty$. Roughly speaking, applying a suitable version of the Borel-Cantelli lemma, combined with a sufficiently careful treatment of the $o\left(1\right)$ term, will then yield Theorem \[thm: full measure of set of counterexamples\]. However, before proceeding with the details of the proof we collect in the next paragraph some tools from additive combinatorics that are needed. Preliminaries ------------- We start with a well-know result relating, in a quantitative manner, the additive energy of a set of integers with the existence of a (relatively) dense subset with small difference set where the difference set $B-B\coloneqq\left\{ b-b':\,b,b'\in B\right\} $ for a set $B\subseteq\mathbb{R}$. \[Balog-Szem=0000E9redi-Gowers\]Let $A\subseteq\mathbb{Z}$ be a finite set of integers. For any $c>0$ there exist $c_{1},c_{2}>0$ depending only on $c$ such that the following holds. If $E(A)\geq c\left(\#A\right)^{3}$, then there is a subset $B\subseteq A$ such that 1. $\#B\geq c_{1}\#A,$ 2. $\#\left(B-B\right)\leq c_{2}\#A.$ Moreover, we recall that for $\delta>0$ and $d\in\mathbb{Z}$ the set $$B\left(d,\delta\right)\coloneqq\left\{ \alpha\in\left[0,1\right]:\,\left\Vert d\alpha\right\Vert \leq\delta\right\}$$ is called Bohr set. These appear frequently in additive combinatorics. The following two simple observation will be useful. \[lem: upper estimate for measure of Omega\_varepsilon,n\]Let $B\subseteq\mathbb{Z}$ be a finite set of integers. Then, $$\lambda\Biggl(\Biggl\{\alpha\in\left[0,1\right]:\underset{d\in\left(B-B\right)\setminus\left\{ 0\right\} }{\min}\left\Vert d\alpha\right\Vert <\frac{\varepsilon}{\#\left(B-B\right)}\Biggr\}\Biggr)\leq2\varepsilon$$ for every $\varepsilon\in(0,1)$ where $\lambda$ is the Lebesgue measure. By observing that the set under consideration is contained in $$\bigcup_{\underset{m\not=n}{m,n\in B}}B\left(m-n,\frac{\varepsilon}{\#\left(B-B\right)}\right),$$ and $\lambda\left(B\left(m-n,\frac{\varepsilon}{\#\left(B-B\right)}\right)\right)=\frac{2\varepsilon}{\#\left(B-B\right)}$, the claim follows at once. \[lem: Omega\_n has only finitely many connected components\]Suppose $A$ is a finite intersection of Bohr sets, and $B$ is a finite union of Bohr sets. Then, $A\setminus B$ is the union of finitely many intervals. Furthermore, we shall use the Borel-Cantelli lemma in a version due to Erdős-Rényi. \[lem: Erdos Renyi version of Borel Cantelli\]Let $\left(A_{n}\right)_{n}$ be a sequence of Lebesgue measurable sets in $\left[0,1\right]$ satisfying $$\sum_{n\geq1}\lambda\left(A_{n}\right)=\infty.$$ Then, $$\lambda\left(\limsup_{n\rightarrow\infty}A_{n}\right)\geq\limsup_{N\rightarrow\infty}\frac{\left(\sum_{n\leq N}\lambda\left(A_{n}\right)\right)^{2}}{\sum_{m,n\leq N}\lambda\left(A_{n}\cap A_{m}\right)}.$$ Moreover, let us explain the main steps in the proof of Theorem \[thm: full measure of set of counterexamples\]. Let $\varepsilon\coloneqq\varepsilon\left(j\right)\coloneqq\frac{1}{10^{j}}c_{1}^{2}$ be for $j\in\mathbb{N}$ where the constant $c_{1}$ is specified later-on, and fix $j$ for now. In the first part of the argument, we show how a sequence - that is constructed in the second part of the argument - with the following crucial (but technical) properties implies the claim. For every fixed $j$, we find a corresponding $s=s(j)$ and construct a sequence $\left(\Omega_{r}\right)_{r}$ of exceptional values $\alpha$ satisfying the following properties: 1. \[enu:Pair correlations functions too small on exceptional set\]\[enu:First property of exceptional sets\]For all $\alpha\in\Omega_{r}$, the pair correlation function admits the upper bound $$R\left(\left[-s,s\right],\alpha,N\right)\leq2\tilde{c}s\label{eq: Pair correlations function too small on exceptional set}$$ for some absolute constant $\tilde{c}\in\left(0,1\right)$, depending on $\left(a_{n}\right)$ only. 2. \[enu:exceptional sets get upper asymptotically independent\]For all integers $r>t\ge1$, the relation $$\lambda\left(\Omega_{r}\cap\Omega_{t}\right)\leq\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{t}\right)+2\varepsilon\lambda\left(\Omega_{t}\right)+\mathcal{O}\left(r^{-2}\right)\label{eq: exceptional sets get upper asymptotically independent}$$ holds. 3. \[enu:Each exceptional set has only finitely many connected components\]Each $\Omega_{r}$ is the union of finitely many intervals (hence measurable). 4. \[enu: absolute lower bound for the measure of Omega\]\[enu: last property of exceptional sets\]For all $r\geq1$, the measure $\lambda\left(\Omega_{r}\right)$ is uniformly bounded from below by $$\lambda\left(\Omega_{r}\right)\geq\frac{c_{1}^{2}}{8}.\label{eq: absolute lower bound for the measure of Omega}$$ Proof of Theorem \[thm: full measure of set of counterexamples\] ---------------------------------------------------------------- 1\. Suppose there is $\left(\Omega_{r}\right)_{r}$ satisfying \[enu:Pair correlations functions too small on exceptional set\]-\[enu: absolute lower bound for the measure of Omega\]. Then, by using (\[eq: exceptional sets get upper asymptotically independent\]), we get $$\begin{aligned} \sum_{r,t\leq N}\lambda\left(\Omega_{r}\cap\Omega_{t}\right) & \le2\sum_{2\leq t\leq N}\,\sum_{1\leq r<t}\left(\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{t}\right)\right)+2\varepsilon N^{2}+\mathcal{O}\left(N\right)\\ & \leq\left(\sum_{t\leq N}\lambda\left(\Omega_{t}\right)\right)^{2}+2\varepsilon N^{2}+\mathcal{O}\left(N\right).\end{aligned}$$ By recalling that $\Omega_{r}=\Omega_{r}\left(\varepsilon\right)=\Omega_{r}\left(j\right)$, we let $\Omega(j)\coloneqq\limsup_{r\rightarrow\infty}\Omega_{r}$. By using the inequality above in combination with Lemma \[lem: Erdos Renyi version of Borel Cantelli\] and the bound (\[eq: absolute lower bound for the measure of Omega\]), we obtain that the set $\Omega(j)$ has measure at least $$\begin{aligned} \limsup_{N\rightarrow\infty}\frac{\left(\sum_{r\leq N}\lambda\left(\Omega_{r}\right)\right)^{2}}{\sum_{r,t\leq N}\lambda\left(\Omega_{r}\cap\Omega_{t}\right)} & \geq\limsup_{N\rightarrow\infty}\frac{1}{1+\frac{4\varepsilon N^{2}}{\left(\sum_{r\leq N}\lambda\left(\Omega_{r}\right)\right)^{2}}}\\ & \geq\limsup_{N\rightarrow\infty}\frac{1}{1+\frac{256}{c_{1}^{2}}\varepsilon}=\frac{1}{1+\frac{256}{c_{1}^{2}}\varepsilon}.\end{aligned}$$ Note that due to (\[eq: Pair correlations function too small on exceptional set\]) every $\alpha\in\Omega\left(j\right)$ does not have PCC. Now, letting $j\rightarrow\infty$ proves the assertion.\ 2. For constructing $\left(\Omega_{r}\right)_{r}$ with the required properties, let $c>0$ such that $E\left(A_{N}\right)>cN^{3}$ for infinitely many integers $N$. By choosing an appropriate subsequence $\left(N_{i}\right)_{i}$ and omitting the subscript $i$ for ease of notation, $E\left(A_{N}\right)>cN^{3}$ holds for every $N$ occurring in this proof. Moreover, let $c_{1},c_{2}$ and $B_{N}$ be as in Lemma \[Balog-Szem=0000E9redi-Gowers\], corresponding to the $c$ just mentioned. Arguing inductively, while postponing the base step,[^6] we assume that for $1\leq r<R$, and $s=\frac{\varepsilon}{2c_{2}}$ there are sets $\left(\Omega_{r}\right)_{1\leq r<R}$ that satisfy the properties \[enu:First property of exceptional sets\]-\[enu: last property of exceptional sets\] for all distinct integers $1\leq r,t<R$. Let $N\geq R$. Lemma \[lem: upper estimate for measure of Omega\_varepsilon,n\] implies that the set $\Omega_{\varepsilon,N}\subseteq[0,1]$ of all $\alpha\in\left[0,1\right]$ satisfying $\left\Vert \left(r-t\right)\alpha\right\Vert <N^{-1}s$ for some distinct $r,t\in B_{N}$ has measure at most $2\varepsilon$. Setting $$\mathcal{D}_{N}:=\left\{ \left(r,t\right)\in\left(A_{N}\times A_{N}\right)\setminus\left(B_{N}\times B_{N}\right):\,r\not=t\right\} ,$$ we get for $\alpha\notin\Omega_{\varepsilon,N}$ that $$R\left(\left[-s,s\right],\alpha,N\right)=\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert <N^{-1}s\right\} .$$ Let $\ell_{R}$ denote the length of the smallest subinterval of $\Omega_{r}$ for $1\leq r<R$, and define $C\left(\Omega_{r}\right)$ to be the set of subintervals of $\Omega_{r}$. Note that $\ell_{R}>0$, and $\max_{1\leq r<R}\#C\left(\Omega_{r}\right)<\infty$. We divide $\left[0,1\right)$ into $$P\coloneqq\left\lfloor 1+2\ell_{R}^{-1}R^{2}\max_{1\leq r<R}\#C\left(\Omega_{r}\right)\right\rfloor$$ parts $\mathcal{P}_{i}$ of equal lengths, i.e. $\mathcal{P}_{i}\coloneqq\left[\frac{i}{P},\frac{i+1}{P}\right)$ where $i=0,\ldots,P-1$. After writing $$\begin{aligned} & \frac{1}{N}\underset{\mathcal{P}_{i}}{\int}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \text{d}\alpha\label{eq: counting function integrated on an atom}\\ & =\frac{1}{N}\underset{\left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}}{\sum}\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha,\nonumber \end{aligned}$$ we split the sum into two parts: one part containing differences $\left|r-t\right|>R^{k}P$, and a second part containing differences $\left|r-t\right|\leq R^{k}P$ where $$k\coloneqq\left\lfloor \frac{1}{\log2}\log\frac{20}{c_{1}^{2}\left(1-2^{-1}c_{1}^{2}\right)s}\right\rfloor +1.$$ Letting $\mathbf{1}_{B}$ denote the characteristic function of $X\subseteq\left[0,1\right]$, the Cauchy-Schwarz inequality implies $$\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq\sqrt{\frac{1}{P}\frac{2s}{N}}.$$ Since for any $x>0$ there are at most $2xN$ choices of $\left(r,t\right)\in\mathcal{D}_{N}$ such that $\left|r-t\right|\leq x$, we obtain $$\frac{1}{N}\underset{\underset{\left|r-t\right|\leq PR^{k}}{\left(r,t\right)\in\mathcal{D}_{N}}}{\sum}\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq2PR^{k}\sqrt{\frac{1}{P}\frac{2s}{N}}$$ which is $\leq P^{-1}R^{-k}$ if $N$ is sufficiently large. Moreover, for any $\left|r-t\right|>PR^{k}$ we observe that $$\underset{\mathcal{P}_{i}}{\int}\mathbf{1}_{\left[-\frac{s}{N},\frac{s}{N}\right]}\left(\left\Vert \left(r-t\right)\alpha\right\Vert \right)\text{d}\alpha\leq\frac{2s}{PN}+\frac{4}{PR^{k}N}$$ and $\#\mathcal{D}_{N}\leq N^{2}-\bigl(\#B_{N}\bigr)^{2}\leq\tilde{c}N^{2}$ where $\tilde{c}\coloneqq1-c_{1}^{2}$.Therefore, the mean value (\[eq: counting function integrated on an atom\]) on $\mathcal{P}_{i}$ of the counting function $R$ is bounded from above by $$\begin{aligned} \frac{1}{N}\left(\#\mathcal{D}_{N}\right)^{2}\left(\frac{2s}{PN}+\frac{4}{PR^{k}N}\right)+\frac{1}{PR^{k}}\leq\frac{2\tilde{c}s}{P}+\frac{5}{PR^{k}}.\end{aligned}$$ Hence, it follows that the measure of the set $\Delta_{N}\left(i\right)$ of $\alpha\in\mathcal{P}_{i}$ with $$\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \leq2\left(1-\frac{c_{1}^{2}}{2}\right)s\label{eq: modified counting too small for being Poissonian}$$ admits, by the choice of $k$, the lower bound $$\lambda\left(\Delta_{N}\left(i\right)\right)\geq\frac{1}{P}-\frac{1}{P}\frac{2\tilde{c}s+5R^{-k}}{2\left(1-\frac{c_{1}^{2}}{2}\right)s}\geq\frac{1}{P}\left(\frac{c_{1}^{2}}{2}-\frac{c_{1}^{2}}{8}\right).\label{eq: absolute lower bound for Omega_N on partition}$$ Note that $\Delta_{N}\left(i\right)$ is the union of finitely many intervals, due to Lemma \[lem: Omega\_n has only finitely many connected components\]. So, we may take $\Delta_{N}'\left(i\right)\subset\Delta_{N}\left(i\right)$ being a finite union of intervals such that $\lambda\left(\Delta_{N}'\left(i\right)\right)$ equals the lower bound in (\[eq: absolute lower bound for Omega\_N on partition\]). Let $$\Omega_{R}\coloneqq\Omega_{R}\left(N\right)\coloneqq\Delta_{N}\setminus\Omega_{\varepsilon,N}\qquad\mathrm{where}\qquad\Delta_{N}\coloneqq\bigcup_{i=0}^{P-1}\Delta_{N}'\left(i\right).$$ We are going to show now that $\Omega_{R}$ satisfies the properties \[enu:First property of exceptional sets\] - \[enu: last property of exceptional sets\]. Now, $\Omega_{R}$ satisfies property \[enu: absolute lower bound for the measure of Omega\] with $r=R$ since $$\lambda\left(\Omega_{R}\right)\geq\lambda\left(\Delta_{N}\right)-\lambda\left(\Omega_{\varepsilon,N}\right)=\frac{c_{1}^{2}}{2}-\frac{c_{1}^{2}}{8}-2\varepsilon\geq\frac{c_{1}^{2}}{8}.$$ Furthermore, $\Omega_{R}$ satisfies property \[enu:Pair correlations functions too small on exceptional set\] by construction and also property \[enu:Each exceptional set has only finitely many connected components\] since all sets involved in the construction of $\Omega_{R}$ were a finite union of intervals. Let $1\leq r<R$, and $I$ be a subinterval of $\Omega_{r}$. Then, $$\begin{aligned} \lambda\left(I\cap\Delta_{N}\right) & =\sum_{i:\mathcal{P}_{i}\cap I\neq\emptyset}\lambda\left(\mathcal{P}_{i}\cap I\cap\Delta_{N}\right)\\ & \leq\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\mathcal{P}_{i}\cap\Delta_{N}\right)\\ & \leq\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\Delta_{N}'\left(i\right)\right).\end{aligned}$$ By summing over all subintervals $I\in C\left(\Omega_{r}\right)$, we obtain that $$\begin{aligned} \lambda\left(\Omega_{r}\cap\Delta_{N}\right) & \leq\sum_{I\in C\left(\Omega_{r}\right)}\left(\frac{2}{P}+\sum_{i:\mathcal{P}_{i}\subsetneq I}\lambda\left(\Delta_{N}'\left(i\right)\right)\right)\\ & \leq\frac{1}{R^{2}}+\sum_{I\in C\left(\Omega_{r}\right)}P\lambda\left(I\right)\frac{\lambda\left(\Omega_{N}\right)}{P}\\ & =\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{N}\right)+\frac{1}{R^{2}}.\end{aligned}$$ We deduce property \[enu:exceptional sets get upper asymptotically independent\] from this estimate and Lemma \[lem: upper estimate for measure of Omega\_varepsilon,n\] via $$\begin{aligned} \lambda\left(\Omega_{r}\cap\Omega_{R}\right) & \leq\lambda\left(\Omega_{r}\cap\Delta_{N}\right)\\ & \leq\lambda\left(\Omega_{r}\right)\left(\lambda\left(\Omega_{N}\right)-\lambda\left(\Omega_{\varepsilon,N}\right)\right)+\frac{1}{R^{2}}+\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{\varepsilon,N}\right)\\ & \leq\lambda\left(\Omega_{r}\right)\lambda\left(\Omega_{R}\right)+2\varepsilon\lambda\left(\Omega_{r}\right)+\mathcal{O}\left(R^{-2}\right)\end{aligned}$$ This concludes the induction step. The only part missing now is the base step of the induction. For realizing it, let $N$ denote the smallest integer $m$ with $E\left(A_{m}\right)>cm^{3}$. We replace $\mathcal{P}_{i}$ in (\[eq: counting function integrated on an atom\]) by $\left[0,1\right]$ to directly derive $$\int_{0}^{1}\frac{1}{N}\#\left\{ \left(r,t\right)\in\mathfrak{\mathcal{D}}_{N}:\,\left\Vert \left(r-t\right)\alpha\right\Vert \leq N^{-1}s\right\} \mathrm{d}\alpha\leq2\tilde{c}s,$$ and conclude that the set $\Omega_{1}'$ of $\alpha\in\left[0,1\right]$ satisfying (\[eq: modified counting too small for being Poissonian\]) has a measure at least $\frac{c_{1}^{2}}{2}$. Thus, $\Omega_{1}\coloneqq\Omega_{1}'\setminus\Omega_{N,\varepsilon}$ has measure at least as large as the right hand side of (\[eq: absolute lower bound for the measure of Omega\]). For property (\[eq: exceptional sets get upper asymptotically independent\]) is nothing to check and that $\Omega_{1}$ is a finite union of intervals follows from Lemma \[lem: Omega\_n has only finitely many connected components\] by observing that $$\Omega_{1}'=\bigcap_{d_{1},\ldots,d_{L\left(N\right)}}\left(B\left(d_{1},N^{-1}s\right)^{C}\cup\ldots\cup B\left(d_{L\left(N\right)},N^{-1}s\right)^{C}\right)$$ where the intersection runs through any set of $L\left(N\right)=\left\lfloor N2\tilde{c}s\right\rfloor $ tuples of differences $d_{i}=r_{i}-t_{i}\neq0$ of components of $\left(r_{i},t_{i}\right)\in\mathcal{D}_{N}$ for $i=1,\ldots,L\left(N\right)$. Thus, the proof is complete. Second main theorem =================== The sequences $\left(a_{n}\right)_{n}$ enunciated in Theorem \[thm: lowering the known Energy threshold\] are constructed in two steps. In the first step, we concatenate (finite) blocks, with suitable lengths, of arithmetic progressions to form a set $P_{A}$. In the second step, we concatenate (finite) blocks, with suitable lengths, of geometric progressions to form a set $P_{G}$ and then define $a_{n}$ to be the $n$-th element of $P_{A}\cup P_{G}$. On the one hand, the arithmetic progression like part $P_{A}$ serves to ensure, due to considerations from metric Diophantine approximation, the divergence property (\[eq: divergence of the Pair Correlation Function\]) on a set with full measure or controllable Hausdorff dimension; on the other hand, the geometric progression like part $P_{G}$ lowers the additive energy, as much as it can. For doing so, a geometric block will appear exactly before and after an arithmetic block, and have much more elements.\ \ For writing the construction precisely down, we introduce some notation. Let henceforth $\left\lfloor x\right\rfloor $ denote the greatest integer $m$ that is at most $x\in\mathbb{R}$. Suppose trough-out this section that $f$ is as in Theorem \[thm: lowering the known Energy threshold\]. We set $P_{A}^{\left(1\right)}$ to be the empty set while $P_{G}^{\left(1\right)}\coloneqq\left\{ 1,2\right\} $. Moreover, for $j\geq2$ we let $P_{A}^{\left(j\right)}$ denote the set of $\bigl\lfloor2^{j}\bigl(f(2^{j})\bigr)^{-\beta}\bigr\rfloor$ consecutive integers that start with $C_{j}=2\max\bigl\{ P_{G}^{(j-1)}\bigr\}$, and $P_{G}^{\left(j\right)}$ is such that the difference set $P_{G}^{\left(j\right)}-2C_{j}$ is the geometric progression $2^{i}$ for $1\leq i\leq\bigl\lfloor\bigl(f(2^{j})\bigr)^{-\gamma}2^{j}\bigl(1-\bigl(f(2^{j})\bigr)^{\gamma-\beta}\bigr)\bigr\rfloor$ where $0<\gamma<\beta<\nicefrac{3}{4}$ are parameters[^7] to be chosen later-on. In this notation, we take $$P_{A}\coloneqq\bigcup_{j\geq1}P_{A}^{\left(j\right)},\qquad P_{G}\coloneqq\bigcup_{j\geq1}P_{G}^{\left(j\right)},$$ and denote by $a_{n}$ the $n$-th smallest element in $P_{A}\cup P_{G}$. For $d\in\mathbb{Z}$ and finite sets of integers $X,Y$, we abbreviate the number of representation of $d$ as a difference of an $x\in X$ and a $y\in Y$ by $\text{rep}_{X,Y}(d)\coloneqq\#\{(x,y)\in X\times Y:\,x-y=d\}$; observe that $$E\left(X\right)=\sum_{d\in\mathbb{Z}}\left(\mathrm{rep}_{X,X}\left(d\right)\right)^{2},\label{eq: additive Energy in terms of number of representation}$$ and $$R\left(\left[-s,s\right],\alpha,N\right)=\frac{1}{N}\underset{d\neq0}{\sum}\text{rep}_{A_{N},A_{N}}(d)\mathbf{1}_{\left[0,\frac{s}{N}\right]}\left(\left\Vert \alpha d\right\Vert \right).\label{eq: lower bound for counting function fo the pair correlations}$$ Preliminaries ------------- We begin to determine the order of magnitude of $E\left(A_{N}\right)$ for the truncations $A_{N}$ of the sequence constructed above. Since the cardinality of elements in the union of the blocks $P_{G}^{\left(j\right)},P_{A}^{\left(j\right)}$ has about exponential growth, it is reasonable to expect $E\left(A_{N}\right)$ to be of the same order of magnitude as the additive energy of the last block $P_{G}^{\left(J\right)}\cup P_{A}^{\left(J\right)}$ that is fully contained in $A_{N}$ - note that $J=J\left(N\right)$; i.e. to expect the magnitude of $E\bigl(P_{G}^{\left(J\right)}\cup P_{A}^{\left(J\right)}\bigr)$ which is roughly equal to $E\bigl(P_{A}^{\left(J\right)}\bigr)$. The following proposition verifies this heuristic considerations. \[prop: additive energy of good-guy-bad-guy sequence\]Let $\left(a_{n}\right)_{n}$ be as in the beginning of Section 3, and $f$ be as in one of the two assertions in Theorem \[thm: lowering the known Energy threshold\]. Then, $E\left(A_{N}\right)=\Theta\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. For the proof of Proposition \[prop: additive energy of good-guy-bad-guy sequence\], we need the next technical lemma. \[lem: auxiliary lemma for calculating additive energy\]Let $\FJ\coloneqq2^{j}\bigl(f\bigl(2^{j}\bigr)\bigr)^{-\delta}$, for $j\geq1$ and fixed $\delta\in\left(0,1\right)$, where $f$ is as in Proposition \[prop: additive energy of good-guy-bad-guy sequence\]. Then, $\sum_{i\leq j}F_{i}=\mathcal{O}\bigl(F_{j}\bigr)$ and $$\sum_{d\in\mathbb{Z}}\biggl(\sum_{j,i\leq J}\mathrm{rep}_{P_{G}^{\left(j\right)},P_{A}^{\left(i\right)}}\left(d\right)\biggr)^{2}=\mathcal{O}\left(J^{6}2^{2J}\right).$$ Suppose that $f\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{3}}\left(\log x\right)^{-\nicefrac{7}{3}}\bigr)$ is such that (\[eq: divergence of the reciprocal of (f(n) times n)\]) diverges. Because $$\sum_{j\leq J+1}\frac{1}{f\bigl(2^{j}\bigr)}\geq\sum_{k\leq2^{J}}\frac{1}{kf\left(k\right)}$$ diverges as $J\rightarrow\infty$ and $\left(f\bigl(2^{j}\bigr)/f\bigl(2^{j+1}\bigr)\right)_{j}$ is non-decreasing, we conclude that $\lim_{j\rightarrow\infty}\bigl(f\bigl(2^{j}\bigr)/f\bigl(2^{j+1}\bigr)\bigr)=1$. Therefore, there is an $i_{0}$ such that the estimate $\bigl(f\bigl(2^{i}\bigr)\bigr)^{-1}f\bigl(2^{i+h}\bigr)<\bigl(\nicefrac{3}{2}\bigr)^{\frac{h}{\delta}}$ holds for any $i\geq i_{0}$ and $h\in\mathbb{N}$. Hence, $$\frac{1}{F_{j}}\sum_{i\leq j}F_{i}\leq o\left(1\right)+\sum_{i_{0}\leq i\leq j}2^{i-j}\left(\frac{3}{2}\right)^{j-i}=\mathcal{O}\bigl(1\bigr).$$ If $f$ is such that (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges and $f\left(2x\right)\leq\left(2-\varepsilon\right)f\left(x\right)$ for $x$ large enough, then we obtain by a similar argument that $\sum_{i\leq j}F_{i}$ is in $\mathcal{O}\bigl(F_{j}\bigr)$. Furthermore, $\mathrm{rep}_{P_{G}^{\left(j\right)},P_{A}^{\left(i\right)}}\left(d\right)=\mathcal{O}\left(i\right)$, for every $j\geq1$, and non-vanishing for $\mathcal{O}\bigl(2^{2j}\bigr)$ values of $d$ which implies the last claim. We can now prove the proposition. Let $\FJ=2^{j}\bigl(f\bigl(2^{j}\bigr)\bigr)^{-\beta}$, $N\geq1$ be large and denote by $J=J\left(N\right)\geq0$ the greatest integer $j$ such that $P_{G}^{\left(j-1\right)}\subseteq A_{N}$. Since $$E\bigl(A_{N}\bigr)\geq E\bigl(P_{A}^{\left(J-1\right)}\bigr)=\Omega\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr),$$ it remains to show that $E\bigl(A_{N}\bigr)=\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. By exploiting (\[eq: additive Energy in terms of number of representation\]), $$E\bigl(A_{N}\bigr)\leq\sum_{d\in\mathbb{Z}}\bigl(\mathrm{rep}_{A_{T_{J}},A_{T_{J}}}\left(d\right)\bigr)^{2}\quad\text{where}\quad T_{J}\coloneqq\#\bigcup_{j\leq J}\left(P_{A}^{\left(j\right)}\cup P_{G}^{\left(j\right)}\right).$$ Moreover, $\mathrm{rep}_{A_{T_{J}},A_{T_{J}}}\left(d\right)=S_{1}\left(d\right)+S_{2}\left(d\right)$ where $S_{1}\left(d\right)$ abbreviates the mixed sum $\sum_{i,j\leq J}\bigl(\mathrm{rep}_{P_{A}^{\left(j\right)},P_{G}^{\left(i\right)}}\left(d\right)+\mathrm{rep}_{P_{G}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\bigr)$ and $S_{2}\left(d\right)$ abbreviates the sum $\sum_{i,j\leq J}\bigl(\mathrm{rep}_{P_{G}^{\left(i\right)},P_{G}^{\left(j\right)}}\left(d\right)+\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\bigr)$. Using that for any real numbers $a,b$ the inequality $\left(a+b\right)^{2}\leq2\bigl(a^{2}+b^{2}\bigr)$ holds, we obtain $$E\bigl(A_{N}\bigr)=\mathcal{O}\biggl(\sum_{d\in\mathbb{Z}}\bigl(S_{1}\left(d\right)\bigr)^{2}+\sum_{d\in\mathbb{Z}}\bigl(S_{2}\left(d\right)\bigr)^{2}\biggr).$$ Lemma \[lem: auxiliary lemma for calculating additive energy\] implies that $\sum_{d\in\mathbb{Z}}\bigl(S_{2}\left(d\right)\bigr)^{2}=\mathcal{O}\bigl(\left(\log N\right)^{6}N^{2}\bigr)$ due to $J=\mathcal{O}\left(\log N\right)$. Moreover, we note that $\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)$ is non-vanishing for at most $4F_{J}$ values of $d$ as $i,j\leq J$. Since $\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)\leq F_{\min\left(i,j\right)}$ holds, we deduce that $$\sum_{i,j\leq J}\mathrm{rep}_{P_{A}^{\left(i\right)},P_{A}^{\left(j\right)}}\left(d\right)=\mathcal{O}\biggl(\sum_{j\leq J}\sum_{i\leq j}F_{i}\biggr).$$ By Lemma \[lem: auxiliary lemma for calculating additive energy\], the right hand side is in $\mathcal{O}\bigl(F_{J}\bigr)$. Since $\mathrm{rep}_{P_{G}^{\left(i\right)},P_{G}^{\left(j\right)}}\left(d\right)\leq1$, where $i,j\leq J$, is non-vanishing for at most $\mathcal{O}\bigl(T_{J}^{2}\bigr)=\mathcal{O}\left(N^{2}\right)$ values of $d$, we obtain that $$\sum_{d\in\mathbb{Z}}\bigl(S_{1}\left(d\right)\bigr)^{2}=\mathcal{O}\bigl(F_{J}^{3}+\left(\log N\right)^{6}N^{2}\bigr)$$ which is in $\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. Hence, $E\bigl(A_{N}\bigr)=\mathcal{O}\bigl(N^{3}\bigl(f\bigl(N\bigr)\bigr){}^{-3\left(\beta-\gamma\right)}\bigr)$. For estimating the measure or the Hausdorff dimension of $\NPPC\left(\left(a_{n}\right)_{n}\right)$ from below, we recall the notion of an optimal regular system. This notion, roughly speaking, describes sequences of real numbers that are exceptionally well distributed in any subinterval, in a uniform sense, of a fixed interval. Let $J$ be a bounded real interval, and $S=\left(\alpha_{i}\right)_{i}$ a sequence of distinct real numbers. $S$ is called an optimal regular system in $J$ if there exist constants $c_{1},\,c_{2},\,c_{3}>0$ - depending on $S$ and $J$ only - such that for any $I\subseteq J$ there is an index $Q_{0}=Q_{0}\left(S,I\right)$ such that for any $Q\geq Q_{0}$ there are indices $$c_{1}Q\leq i_{1}<i_{2}<\ldots<i_{t}\leq Q\label{eq: property of c1 in optimal regular system definition}$$ satisfying $\alpha_{i_{h}}\in I$ for $h=1,\ldots,t$, and $$\left|\alpha_{i_{h}}-\alpha_{i_{\ell}}\right|\geq\frac{c_{2}}{Q}\label{eq: property of c2 in optimal regular system definition}$$ for $1\leq h\neq\ell\leq t$, and $$c_{3}\lambda\left(I\right)Q\leq t\leq\lambda\left(I\right)Q.\label{eq: property of c3 in optimal regular system definition}$$ Moreover, we need the following result(s) due to Beresnevich which may be thought of as a far reaching generalization of Khintchine’s theorem, and Jarník-Besicovitch theorem in Diophantine approximation. \[thm: Khintchine a la Victor\]Suppose $\psi:\mathbb{\mathbb{R}}_{>0}\rightarrow\mathbb{R}_{>0}$ is a continuous, non-increasing function, and $S=\bigl(\alpha_{i}\bigr)_{i}$ an optimal regular system in $\left(0,1\right)$. Let $\mathcal{K}_{S}\left(\psi\right)$ denote the set of $\xi$ in $\left(0,1\right)$ such that $\left|\xi-\alpha_{i}\right|<\psi\left(i\right)$ holds for infinitely many $i$. If $$\sum_{n\geq1}\psi\left(n\right)\label{eq: sum over psi values}$$ diverges, then $\mathcal{K}_{S}\left(\psi\right)$ has full measure.\ Conversely, if (\[eq: sum over psi values\]) converges, then $\mathcal{K}_{S}\left(\psi\right)$ has measure zero and the Hausdorff dimension equals the reciprocal of the lower order of $\frac{1}{\psi}$ at infinity. For a rational $\alpha=\frac{p}{q}$, where $p,q\in\mathbb{Z}$, $q\neq0$, we denote by $H\left(\alpha\right)$ its (naive) height, i.e. $H\left(\alpha\right)\coloneqq\max\left\{ \left|p\right|,\left|q\right|\right\} $. It is well-known that the set of rational numbers in $\left(0,1\right)$, ordered in classes by increasing height and in each class ordered by numerically values, gives rise to an optimal regular system in $\left(0,1\right)$. The following lemma says, roughly speaking, that this assertion remains true for the set of rationals in $\left(0,1\right)$ whose denominators are members of a special sequence that is not too sparse in the natural numbers. The proof can be given by modifying the proof of the classical case, compare [@Bugeaud:; @Approximation; @by; @algebraic; @numbers Prop. 5.3]; however, we shall give the details for making this article more self-contained. \[lem: optimal regular system\]Let $\vartheta:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>1}$ be monotonically increasing to infinity with $\vartheta\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{4}}\bigr)$ and $\vartheta\left(2^{j+1}\right)/\vartheta\left(2^{j}\right)\rightarrow1$ as $j\rightarrow\infty$. For each $j\in\mathbb{N}$, we let $$B_{j}\coloneqq\frac{2^{j}}{f\left(2^{j}\right)\sqrt{\vartheta\left(2^{j}\right)}},\qquad b_{j}\coloneqq\frac{2}{3}B_{j}.$$ Let $S=\bigl(\alpha_{i}\bigr)_{i}$ denote a sequence running through all rationals in $\left(0,1\right)$ whose denominators are in $M\coloneqq\bigcup_{j\geq1}\bigl\{ n\in\mathbb{N}:\,b_{j}\leq n\leq B_{j}\bigr\}$ such that $i\mapsto H\bigl(\alpha_{i}\bigr)$ is non-decreasing. Then, $S$ is an optimal regular system in $\left(0,1\right)$. Let $X\geq2$. There are strictly less than $2X^{2}$ rational numbers in $\left(0,1\right)$ with height bounded by $X$. We take $J=J\left(X\right)$ to be the largest integer $j\geq1$ such that $B_{j}\leq X$. Then, for $X$ large enough, there are at least $$\begin{aligned} \sum_{j\leq J}\sum_{b_{j}\leq q\leq B_{j}}\varphi\left(q\right) & \geq\sum_{j\leq J}\left(\frac{1}{3\pi^{2}}B_{j}^{2}+\mathcal{O}\left(B_{j}\log B_{j}\right)\right)\\ & \geq\frac{1}{6\pi^{2}}\frac{2^{2J}}{f^{2}\left(2^{J}\right)\vartheta\left(2^{J}\right)}+\mathcal{O}\left(J2^{J}\right)\\ & >\left(\frac{X}{5\pi}\right)^{2}\end{aligned}$$ distinct such rationals in $\left(0,1\right)$ with height not exceeding $X$. Hence, we obtain $\frac{\sqrt{i}}{2}\leq H\left(\alpha_{i}\right)\leq\sqrt{25\pi^{2}\left(i+1\right)}+1$ for $i$ sufficiently large. Let $Q\in\mathbb{N}$, $I\subseteq\left[0,1\right]$ be a non-empty interval, and let $F$ denote the set of $\xi\in I$ satisfying the inequality $\left\Vert q\xi\right\Vert <Q^{-1}$ with some $1\leq q\leq\frac{1}{1000}Q$. Note that $F$ has measure at most $$\sum_{q\leq\frac{1}{1000}Q}\left(\frac{2}{qQ}q\lambda\left(I\right)+\frac{2}{qQ}\right)=\frac{1}{500}\lambda\left(I\right)+\mathcal{O}\left(\frac{\log Q}{Q}\right)<\frac{1}{400}\lambda\left(I\right)$$ for $Q\geq Q_{0}$ where $Q_{0}=Q_{0}\left(S,I\right)$ is sufficiently large. Let $\bigl\{\nicefrac{p_{j}}{q_{j}}\bigr\}_{1\leq j\leq t}$ be the set of all rationals $\nicefrac{p_{j}}{q_{j}}\in\left(0,1\right)$ with $q_{j}\in M$, $\frac{1}{1000}Q<q_{j}<Q$ that satisfy $$\left|\frac{p_{j}}{q_{j}}-\frac{p_{j'}}{q_{j'}}\right|>\frac{2000}{Q^{2}}$$ whenever $1\leq j\neq j'\leq t$. Observe that for $J$ as above with $X=Q$ sufficiently large, it follows that $$\left\{ q\in M:\,b_{J}\leq q\leq B_{J}\right\} \subseteq\left\{ \frac{Q}{1000},\frac{Q}{1000}+1,\ldots,Q\right\}$$ holds and there are hence at least $\frac{1}{3\pi^{2}}B_{J}^{2}+\mathcal{O}\left(B_{J}\log B_{J}\right)>\frac{1}{400}Q^{2}$ choices of $\nicefrac{p_{j}}{q_{j}}\in\left(0,1\right)$ with $q_{j}\in M$ and $\frac{1}{1000}Q<q_{j}<Q$. Due to $\lambda\left(I\setminus F\right)>\frac{399}{400}\lambda\left(I\right)$, we conclude $t\geq400\frac{Q^{2}}{4000}\frac{399}{400}\lambda\left(I\right)$. Thus, taking $c_{1}\coloneqq\nicefrac{1}{1000}$, $c_{2}\coloneqq2000$, and $c_{3}\coloneqq\frac{399}{4000}$ in (\[eq: property of c1 in optimal regular system definition\]), (\[eq: property of c2 in optimal regular system definition\]) and (\[eq: property of c3 in optimal regular system definition\]), respectively, $S$ is shown to be an optimal regular system. Now we can proceed to the proof of Theorem \[thm: lowering the known Energy threshold\]. Proof of Theorem \[thm: lowering the known Energy threshold\] ------------------------------------------------------------- We argue in two steps depending on whether or not the series (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges. Proposition (\[prop: additive energy of good-guy-bad-guy sequence\]) implies the announced $\Theta$-bounds on the additive energy of $A_{N}$, in both cases.\ \ (i) Suppose (\[eq: divergence of the reciprocal of (f(n) times n)\]) diverges, and fix $s>0$. Let $\vartheta:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>1}$ be monotonically increasing to infinity with $\vartheta\left(x\right)=\mathcal{O}\left(x^{\nicefrac{1}{4}}\right)$ such that $$\psi\left(n\right)\coloneqq\frac{1}{nf\left(n\right)\vartheta\left(n\right)}\label{eq: specification of the appxomation function-1}$$ satisfies the divergence condition (\[eq: sum over psi values\]). Thus, $\vartheta\left(2^{j}\right)/\vartheta\left(2^{j-1}\right)\rightarrow1$ as $j\rightarrow\infty$. Hence, $S=\left(\alpha_{i}\right)_{i}$ from Lemma \[lem: optimal regular system\] is an optimal regular system. Furthermore, if $\alpha_{i}=\frac{m}{n}$, then $i\geq cn^{2}$ holds true with a constant $c=c\left(f,\vartheta\right)>0$ due to $b_{J}\leq n\leq B_{J}$, for some integer $J$, and $$\sum_{j\leq J-1}\sum_{b_{j}\leq m\leq B_{j}}\varphi\left(m\right)=\Theta\bigl(B_{J}^{2}\bigr).$$ Therefore, $\psi\left(i\right)\leq c^{-1}n^{-2}\bigl(f\bigl(cn^{2}\bigr)\vartheta\bigl(cn^{2}\bigr)\bigr)^{-1}$. The growth assumption on $f$ and the growth bound $\vartheta\left(x\right)=\mathcal{O}\bigl(x^{\nicefrac{1}{4}}\bigr)$ yields that if $j$ is large enough, then $b_{j}\leq n\leq B_{j}$ implies $cn^{2}>2^{j}$ and hence we obtain $\psi\left(i\right)\leq c^{-1}n^{-2}\bigl(f\bigl(2^{j}\bigr)\vartheta\bigl(2^{j}\bigr)\bigr)^{-1}$. Combining these considerations, we have established that $$n\psi\left(i\right)=\mathcal{O}\left(2^{-j}\left(\vartheta\bigl(2^{j}\bigr)\right)^{-\nicefrac{1}{2}}\right).$$ Moreover, for a function $g:\mathbb{N}\rightarrow\mathbb{R}_{>0}$, we let $E_{g}$ denote the set of $\alpha\in\left(0,1\right)$ such that for infinitely many $j$ there is some $n$ with $b_{j}\leq n\leq B_{j}$ satisfying $\left\Vert n\alpha\right\Vert =\mathcal{O}\left(2^{-j}g\left(j\right)\right)$. Set $h\left(j\right)\coloneqq\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{-\nicefrac{1}{2}}$. Applying Theorem \[thm: Khintchine a la Victor\] with $\psi$ as in (\[eq: specification of the appxomation function-1\]), implies that $E_{h}$ has full measure. Therefore, for any $\alpha\in E_{h}$ we get $$\left\Vert n\alpha\right\Vert \leq n\left|\alpha-\alpha_{i}\right|=\mathcal{O}\Bigl(2^{-j}\left(\vartheta\bigl(2^{j}\bigr)\right)^{-\nicefrac{1}{2}}\Bigr)\label{eq: good approximation to alpha in terms of psi-1}$$ for infinitely many $j$. Now if $b_{j}\leq n\leq B_{j}$ for $j$ sufficiently large and $n,\alpha$ as in (\[eq: good approximation to alpha in terms of psi-1\]), then it follows that by taking any integer $m\leq\left(f\left(2^{j}\right)\right)^{\gamma}\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{\frac{1}{3}}$ also the multiples $$nm\leq2^{j}\left(f\left(2^{j}\right)\right)^{\gamma-1}\left(\vartheta\left(2^{j}\right)\right)^{-\nicefrac{1}{6}}$$ satisfy that $\mathbf{1}_{\left[0,sT_{j}\right]}\left(\left\Vert \alpha(mn)\right\Vert \right)=1$ where $T_{j}=\mathcal{O}\bigl(2^{j}\left(f\left(2^{j}\right)\right)^{-\gamma}\bigr)$ is as in the Proof of Proposition \[prop: additive energy of good-guy-bad-guy sequence\]. If additionally $\gamma-1\geq-\beta$ holds, then we obtain that $\text{rep}_{A_{T_{j}},A_{T_{j}}}(mn)\geq\nicefrac{1}{2}2^{j}\left(f\left(2^{j}\right)\right)^{-\beta}$ holds for $j$ sufficiently large. By (\[eq: lower bound for counting function fo the pair correlations\]), we obtain $$R\bigl(\left[-s,s\right],\alpha,T_{j}\bigr)\geq C\left(f\left(2^{j}\right)\right)^{2\gamma-\beta}\bigl(\vartheta\bigl(2^{j}\bigr)\bigr)^{\nicefrac{1}{3}}$$ for infinitely many $j$ where $C>0$ is some constant. For the optimal choice of the parameters $\beta,\gamma>0$, we are therefore led to find the maximal $\beta$ such that $2\gamma-\beta\geq0$ and $\gamma-1\geq-\beta$ is satisfied. The (unique) solution is $\beta=\nicefrac{2}{3}$ and $\gamma=\nicefrac{1}{3}$. Hence, (\[eq: divergence of the Pair Correlation Function\]) follows for $\alpha\in E_{h}$.\ \ (ii) Suppose the series (\[eq: divergence of the reciprocal of (f(n) times n)\]) converges. We keep the same sequence as in step (i) while taking $\vartheta\left(x\right)=1+\log\left(x\right)$, as we may. The arguments of step (i) show that any $\alpha\in E_{h}$, where $h\left(j\right)=j^{-\nicefrac{1}{2}}$, satisfies (\[eq: divergence of the Pair Correlation Function\]); now the conclusion is that $E_{h}$ has Hausdorff dimension equal to the reciprocal of $$\liminf_{x\rightarrow\infty}\frac{-\log\left(\psi\left(x\right)\right)}{\log x}=1+\liminf_{x\rightarrow\infty}\frac{\log f\left(x\right)}{\log x}.$$ Thus, the proof is complete. #### Concluding remarks {#concluding-remarks .unnumbered} We would like to mention two open problems related to this article. The first problem concerns extensions of Theorem \[thm: full measure of set of counterexamples\]. Let $\left(a_{n}\right)_{n}$ be an increasing sequence of positive integers with $E\left(A_{N}\right)=\Omega\left(N^{3}\right)$. Has the complement of $\NPPC\left(\left(a_{n}\right)_{n}\right)$ Hausdorff dimension zero; or is it, in fact, empty? The second problem is related to Corollary \[cor: order of magnitude for the additive energy of the sequence of counter examples\]. How large has $E\left(A_{N}\right)$ to be for ensuring that $\NPPC\left(\left(a_{n}\right)_{n}\right)$ has full Lebesgue measure? #### Acknowledgements {#acknowledgements .unnumbered} Both authors would like to express their gratitude towards C. Aistleitner for introducing us to the topic of this article, and valuable discussions. #### Addresses\ {#addresses .unnumbered} Thomas Lachmann, 5010 Institut für Analysis und Zahlentheorie 8010 Graz, Steyrergasse 30/II email: [email protected]\ \ Niclas Technau, 5010 Institut für Analysis und Zahlentheorie 8010 Graz, Steyrergasse 30/II, email: [email protected] [10]{} C. Aistleitner, T. Lachmann, F. Pausinger: *Pair correlations and equidistribution,* [arXiv:1702.07365](https://arxiv.org/abs/1612.05495), J. of Num. Th., to appear. C. Aistleitner, G. Larcher and M. Lewko: *Additive Energy and the Hausdorff Dimension of the Exceptional Set in Metric Pair Correlation Problems,* [arXiv:1606.03591 ](https://arxiv.org/abs/1606.03591), Israel J. Math., to appear. F. P. Boca and A. Zaharescu: *Pair correlation of values of rational functions (mod p)*, Duke Math. J., 105(2):267307, 2000. A. Bondarenko and K. Seip: *GCD sums and complete sets of square-free numbers*, Bull. Lond. Math. Soc., 47(1):2941, 2015. T. Bloom, S. Chow, A. Gafni and A. Walker: Additive Energy and the Metric Poissonian Property, Work in Progress Y. Bugeaud: *Approximation by algebraic numbers,* Vol. 160, Cambridge University Press, 2004. D. R. Heath-Brown: *Pair correlation for fractional parts of $n^{2}$*, Math. Proc. Cambridge Philos. Soc., 148(3):385407, 2010. G. Larcher and S. Grepstad: *On pair correlation and discrepancy, [arXiv:1612.08008](https://arxiv.org/abs/1612.08008)*, Arch. Math., to appear. J. Marklof and A. Strömbergsson: *Equidistribution of Kronecker sequences along closed horocycles*, Geom. Funct. Anal., 13(6):12391280, 2003. Z. Rudnick and P. Sarnak: *The pair correlation function of fractional parts of polynomials*, Comm. Math. Phys., 194(1):6170, 1998. Z. Rudnick, P. Sarnak, and A. Zaharescu: *The distribution of spacings between the fractional parts of $n^{2}$*, Invent. Math., 145(1):3757, 2001. Z. Rudnick and A. Zaharescu: *A metric result on the pair correlation of fractional parts of sequences*, Acta Arith., 89(3):283293, 1999. T. Tao and V. Vu: *Additive combinatorics*, Vol. 105, Cambridge University Press, 2006. J. L. Truelsen: *Divisor problems and the pair correlation for the fractional parts of $n^{2}$*, Int. Math. Res. Not., (16):31443183, 2010. A. Walker: *The Primes are not Metric Poissonian,* [arXiv:1702.07365](https://arxiv.org/abs/1702.07365) H. Weyl*: Über die Gleichverteilung von Zahlen mod. Eins*, Math. Ann., 77(3):313352, 1916. [^1]: The first author is supported by the Austrian Science Fund (FWF): Y-901. [^2]: The second author is supported by the Austrian Science Fund (FWF) projects: W1230, and (for part of the time) by Y-901. [^3]: The subscript $2$ in $R_{2}$ indicates that relations of second order, i.e. pair correlations, are counted. [^4]: It is worthwhile to mention that the case $d=2$ is of particular interest for its connection to mathematical physics, see [@Rudnick; @Sarnak:; @The; @pair; @correlation; @function; @of; @fractional; @parts; @of; @polynomials] for further references. [^5]: This problem was posed at the problem session of the ELAZ conference in 2016. [^6]: The bases step uses simplified versions of the arguments exploited in the induction step, and will therefore be postponed. [^7]: No particular importance should be attached to requiring $\beta<\nicefrac{3}{4}$, or using “dyadic steps lengths $2^{j}$”. Doing so is for simplifying the technical details only - eventually, it will turn out that $\beta=\nicefrac{2}{3}=2\gamma$ is the optimal choice of parameters in this approach. For proving this to the reader, we leave $\gamma,\beta$ undetermined till the end of this section.

ArXiv

--- address: | Laboratoire d’Astrophysique, UMR 5572, Observatoire Midi-Pyrénées\ 14 avenue E.-Belin, F-31400 Toulouse, France author: - 'G. GOLSE, J.-P. KNEIB and G. SOUCAIL' title: CONSTRAINING THE COSMOLOGICAL PARAMETERS FROM GRAVITATIONAL LENSES WITH SEVERAL FAMILIES OF IMAGES --- Introduction ============ Recent works on constraining the cosmological parameters using the CMB and the high redshift supernovae seem to converge to a new “standard cosmological model” favouring a flat universe with $\Omega_m\sim 0.3$ and $\Omega_\lambda\sim 0.7$: White [@White] and references therein. However these results are still uncertain and depend on some physical assumptions, so the flat $\Omega_m=1$ model is still possible (Le Dour [*et al.*]{} [@LeDour]). It is therefore important to explore other independent techniques to constrain these cosmological parameters. In cluster gravitational lensing, the existence of multiple images – with known redshifts – given by the same source allows to calibrate in an absolute way the total cluster mass deduced from the lens model. The great improvement in the mass modeling of cluster-lenses that includes the cluster galaxies halos (Kneib [*et al.*]{} [@Kneib96], Natarajan & Kneib [@Natarajan]) leads to the hope that clusters can also be used to constrain the geometry of the Universe, through the ratio of angular size distances, which only depends on the redshifts of the lens and the sources, and on the cosmological parameters. The observations of cluster-lenses containing large number of multiple images lead Link & Pierce [@Link] (hereafter LP98) to investigate this expectation. They considered a simple cluster potential and on-axis sources, so that images appear as Einstein rings. The ratio of such rings is then independent of the cluster potential and depends only on $\Omega_m$ and $\Omega_\lambda$, assuming known redshifts for the sources. According to them, this would allow marginal discrimination between extreme cosmological cases. But real gravitational lens systems are more complex concerning not only the potential but also off-axis positions of sources. They conclude that this method is ill-suited for application to real systems. We have re-analyzed this problem building up on the modeling technique developed by us. As demonstrated below, we reach a rather different conclusion showing that it is possible to constrain $\Omega_m$ and $\Omega_\lambda$ using the positions of multiple images at different redshifts and some physically motivated lens models. Troughout this paper we have assumed $H_0=65$ km s$^{-1}$ Mpc$^{-1}$, however the proposed method is independant of the value of $H_0$. Influence of $\Omega_m$ and $\Omega_\lambda$ on the images formation ==================================================================== Angular size distances ratio term --------------------------------- In the lens equation: $\mathbf{\theta_{S}}= \mathbf{\theta_{I}} - \displaystyle{\frac{2}{c^2}\frac{D_{OL}D_{LS}}{D_{OS}}} \mathbf\nabla \phi_\theta(\mathbf{\theta_{I}}) $, the dependence on $\Omega_m$ and $\Omega_\lambda$ is solely contained in the term $F=\displaystyle{{D_{OL}}{D_{LS}}/{D_{OS}}}$. For a given lens plane, $F(z_s)$ increases rapidly up to a certain redshift and then stalls, with significant differences for various values of the cosmological parameters (see Fig. \[F\_zs\]). Thus in order to constrain the actual shape of $F(z_s)$ several families of multiple images are needed, ideally with their redshifts regularly distributed in $F(z_s)$ to maximize the range in the $F$ variation. If we consider fixed redshifts for both the lens and the sources, at least 2 multiple images are needed to derive cosmological constraints. In that case $F$ has only an influence on the modulus of $\mathbf{\theta_{I}}-\mathbf{\theta_{S}}$. So taking the ratio of two different $F$ terms provides the intrinsic dependence on cosmological scenarios, independently of $H_0$. A typical configuration leads to the Fig. \[F\_zs\] plot. The discrepancy between the different cosmological parameters is not very large, less than 3% between an EdS model and a flat low matter density one. The figure also illustrates the expected degeneracy of the method, also confirmed by weak lensing analyzes, with a continuous distribution of background sources ([*e.g.*]{} Lombardi & Bertin [@Lombardi] ). Relative influence of the different parameters ---------------------------------------------- We now look at the relative influence of the different parameters, including the lens parameters, to derive expected error bars on $\Omega_m$ and $\Omega_\lambda$. To model the potential we choose the mass density distribution proposed by Hjorth & Kneib [@Hjorth], characterized by a core radius, $a$, and a cut-off radius $s\gg a$. We can then get the expression of the deviation angle modulus $D_{\theta_{I}}=\parallel\mathbf{\theta_{I}}-\mathbf{\theta_{S}}\parallel$. For 2 families of multiple images, the relevant quantity becomes the ratio of 2 deviation angles for 2 images $\theta_{I1}$ and $\theta_{I2}$ belonging to 2 different families at redshifts $z_{s1}$ and $z_{s2}$. Let’s define $R_{\theta_{I1},\theta_{I2}}=\displaystyle{\frac{D_{\theta_{I1}}}{D_{\theta_{I2}}}}$. With several families, the problem is highly constrained because a single potential must reproduce the whole set of images. In practice we calculate $\displaystyle{\frac{dR_{\theta_{I1},\theta_{I2}}}{R_{\theta_{I1},\theta_{I2}}}}$ versus the different parameters it depends on. We chose a typical configuration to get a numerical evaluation of the errors on the cosmological parameters: $z_l=0.3$, $z_{s1}=0.7$, $z_{s2}=2$, $\displaystyle{\frac{\theta_{I2}}{\theta_{I1}}}=2$, $\displaystyle{\frac{\theta_{s}}{\theta_{a}}}=10$ ($\theta_a=a/D_{OL}$,$\theta_s=s/D_{OL}$) and we assume $\Omega_m=0.3$ and $\Omega_\lambda=0.7$. We then obtain the following orders of magnitudes for the different contributions : = 0.57 + 0.74 + 0.17 + 0.4( - ) - 0.1 - 0.06 - 0.015 + 0.02 As expected, even with 2 families of multiple images the influence of the cosmological parameters is of the second order. The precise value of the redshifts is quite fundamental, therefore a spectroscopic determination ($dz=0.001$) is essential. The position of the (flux-weighted) centers of the images are also important. With HST observations we assume $d\theta_I=0.1$”. So even if the problem is less dependent on the core and cut-off radii (in other word the mass profile), they will represent the main sources of error. Taking $d\theta_a/\theta_a= d\theta_s/\theta_s= 20$ %, we then derive the errors $d\Omega_m$ and $d\Omega_{\lambda}$ from the above relation in the flat low matter density we chose. We did this computation for different sets of cosmological models. Indeed the errors we will obtain with this method change significantly with respect to $\Omega_m$ and $\Omega_\lambda$. All other things being equal apart from the cosmological parameters, we plot $d\Omega_m$ and $d\Omega_\lambda$ for a continuous set of universe models (Fig. \[erreurs\]). For instance in the 2 popular cosmological scenarios, we have : $\Omega_m=0.3\pm0.24 $ $\Omega_\lambda=0.7\pm0.5$ or $\Omega_m=1\pm0.33$ $\Omega_\lambda=0\pm1.2$ As this can be easily understood from the Fig. \[F\_zs\] degeneracy plot, the method is in general far more sensitive to the matter density than to the cosmological constant, for which the error bars are larger. However the results we could obtain this way are as precise as the ones given by other constraints. But these errors are just typical; provided spectroscopic and HST observations, they depend mostly on the particular cluster and the potential model chosen to describe it. They could be quite tightened with a precise model, and by increasing the number of clusters with multiple images. \[simul\]Constraint on $(\Omega_m,\Omega_\lambda)$ from strong lensing ====================================================================== Method and algorithm for numerical simulations ----------------------------------------------- We consider basically the potential introduced in section 2.2. After considering the lens equation, fixing arbitrary values $(\Omega_m^0$,$\Omega_\lambda^0)$ and a cluster lens redshift $z_l$, our code can determine the images of a source galaxy at a redshift $z_s$. Then taking as single observables these sets of images as well as the different redshifts, we can recover some parameters (the more important ones being $\sigma_0$, $\theta_a$ or $\theta_s$) of the potential we left free for each point of a grid $(\Omega_m$,$\Omega_\lambda)$. The likelihood of the result is obtained via a $\chi^2$-minimization, where the $\chi^2$ is computed in the source plane as follows : \^2= The subscript $i$ refers to the families and the subscript $j$ to the images of a family. There is a total of $\sum_{i=1}^n n^i=N$ images. $\theta_{Sj}^i$ is the source found for the image $\theta_{Ij}^i$ in the inversion. $\theta_{SG}^i$ is the barycenter of the $\theta_{Sj}^i$ (belonging to a same family). Finally if $\sigma_{Ii}^{j}$ is the error on the position of the center of $\theta_{Ij}^i$ and $A_j^i$ the amplification for this image, then $\sigma_{Si}^{j}=\sigma_{Ii}^{j}/\sqrt{A_j^i}$. Numerical simulations in a typical configuration ------------------------------------------------ To recover the parameters of the potential ([*i.e.*]{} $\sigma_0$, $\theta_a$, $\theta_s$ and adjusted lens parameters), we generated 3 families of images with regularly distributed source redshifts. For starting values $(\Omega_m^0,\Omega_\lambda^0)=(0.3,0.7)$ we obtained confidence levels shown in Fig. \[3fam\]. The method puts forward a good constraint, better on $\Omega_m$ than on $\Omega_\lambda$, and the degeneracy is the expected one (see Fig. \[F\_zs\]). Concerning the free parameters, we also recovered in a rather good way the potential, the variations being $\Delta\sigma_0\sim150$ km/s, $\Delta\theta_a\sim3$”and $\Delta\theta_s\sim20$”. This is an “ideal” case, of course, because we tried to recover the same type of potential we used to generate the images, the morphology of the cluster being quite regular and the redshift range of the sources being wide enough to check each part of the $F$ curve. Such a simple approach can be applied to regular clusters like MS2137-23, which shows at least 3 families of multiple images including a radial one, see Fig. \[MS2137\]. But the spectroscopic redshifts are still currently missing for this cluster. Conclusions & prospects ======================= Following the work of LP98, we discussed a method to obtain informations on the cosmological parameters $\Omega_m$ and $\Omega_\lambda$ while reconstructing the lens gravitational potential of clusters with multiple image systems at different redshifts. This technique gives degenerate constraints, $\Omega_m$ and $\Omega_\lambda$ being negatively correlated, with a better constraint of the matter density. With a single cluster in a typical lensing configuration we can expect the following error bars : $\Omega_m=0.3{\pm 0.24}$, $\Omega_\lambda=0.7{\pm 0.5}$. To perform that, several general conditions must be fulfilled: $\ast$ a cluster with a rather regular morphology, so that a few parameters are needed to describe the gravitational potential ; this is not so restrictive because we saw that a bimodal cluster can also provide a constraint, $\ast$ “numerous” systems of multiple images, probing each part of the cluster, $\ast$ a good spatial resolution image (HST observations) of the cluster and arcs to compute relatively precise – 0.1” – (flux weighted) images positions, $\ast$ spectroscopic precision for the different redshifts that should be also regularly distributed from $z_l$ to high values – this requires deep spectroscopy on 8-10m class telescopes due to the faintness of the multiple images . It is important to notice that one cluster could provide one constraint on the geometry of the whole universe. And it is possible to combine data from different clusters to tighten the error bars. Combining the study of about 10 clusters would lead to meaningful constraints. The dashed lines confidence levels in the Fig. \[3fam\] are the result of a numerical simulation made with 10 [*identical*]{} clusters. Actually the degeneracy depends only on the different redshifts involved that we will have various sets of when applying the method to real configurations. This should lead to a more reduced area of allowed cosmological parameters. We are encouraged by more and more known observations including systems with multiple sources and we plan to apply this technique to clusters like MS2137-23, MS0440+02, A370, AC114 and A1689. References {#references .unnumbered} ========== [99]{} J. Hjorth and J.-P. Kneib, J.-P. Kneib, R. Ellis, I. Smail, W. Couch & R. Sharples, M. Le Dour, M. Douspis, J. Bartlett & A. Blanchard, R. Link and M. Pierce, M. Lombardi and G. Bertin, P. Nararajan & J.-P. Kneib, M. White,

ArXiv

--- abstract: 'Using magnetocapacitance data in tilted magnetic fields, we directly determine the chemical potential jump in a strongly correlated two-dimensional electron system in silicon when the filling factor traverses the spin and the cyclotron gaps. The data yield an effective $g$ factor that is close to its value in bulk silicon and does not depend on filling factor. The cyclotron splitting corresponds to the effective mass that is strongly enhanced at low electron densities.' author: - 'V. S. Khrapai, A. A. Shashkin, and V. T. Dolgopolov' title: | Direct measurements of the spin and the cyclotron gaps\ in a 2D electron system in silicon --- A two-dimensional (2D) electron system in silicon metal-oxide-semiconductor field-effect transistors (MOSFETs) is remarkable due to strong electron-electron interactions. The Coulomb energy overpowers both the Fermi energy and the cyclotron energy in accessible magnetic fields. The Landau-level-based considerations of many-body gaps [@ando; @yang], which are valid in the weakly interacting limit, cannot be directly applied to this strongly correlated electron system. In a perpendicular magnetic field, the gaps for charge-carrying excitations in the spectrum should originate from cyclotron, spin, and valley splittings and be related to a change of at least one of the following quantum numbers: Landau level, spin, and valley indices. However, the gap correspondence to a particular single-particle splitting is not obvious [@brener], and the origin of the excitations is unclear. In a recent theory [@iordan], the strongly interacting limit has been studied, and it has been predicted that in contrast to the single-particle picture, the many-body gap to create a charge-carrying (iso)spin texture excitation at integer filling factor is determined by the cyclotron energy. This is also in contrast to the square-root magnetic field dependence of the gap expected in the weakly interacting limit [@ando; @yang]. A standard experimental method for determining the gap value in the spectrum of the 2D electron system in a quantizing magnetic field is activation energy measurements at the minima of the longitudinal resistance [@englert; @klein; @dol88; @usher]. Its disadvantage is that it yields a mobility gap which may be different from the gap in the spectrum. In Si MOSFETs, the activation energy as a function of magnetic field was reported to be close to half of the single-particle cyclotron energy for filling factor $\nu=4$, while decreasing progressively for the higher $\nu$ cyclotron gaps [@englert; @klein; @dol88]. At low electron densities, an interplay was observed between the cyclotron and the spin gaps, manifested by the disappearance of the cyclotron ($\nu=4$, 8, and 12) minima of the longitudinal resistance [@krav00]. On the contrary, for the 2D electrons in GaAs/AlGaAs heterostructures, the activation energy at $\nu=2$ exceeded half the single-particle cyclotron energy by about 40% [@usher]. Another, direct method for determining the gap in the spectrum is measurement of the chemical potential jump across the gap [@smith85; @aristov; @valley]. It was applied to the 2D electrons in GaAs [@smith85] and gave cyclotron gap values corresponding to the band electron mass [@aristov]. Recently, the method has been used to study the valley gap at the lowest filling factors in the 2D electron system in silicon which has been found to be strongly enhanced and increase linearly with magnetic field [@valley; @rem]. The effective electron mass, $m$, and $g$ factor in Si MOSFETs have been determined lately from measurements of the parallel magnetic field of full spin polarization in this electron system and of the slope of the metallic temperature dependence of the conductivity in zero magnetic field [@gm]. It is striking that the effective mass becomes strongly enhanced with decreasing electron density, $n_s$, while the $g$ factor remains nearly constant and close to its value in bulk silicon. This result is consistent with accurate measurements of $m$ at low $n_s$ by analyzing the temperature dependence of the Shubnikov-de Haas oscillations in weak magnetic fields in the low-temperature limit [@us; @rem1]. A priori it is unknown whether or not the so-determined values $g$ and $m$ correspond to the spin and the cyclotron splittings in strong perpendicular magnetic fields. In this paper, we report the first measurements of the chemical potential jump across the spin and the cyclotron gaps in a 2D electron system in silicon in tilted magnetic fields using a magnetocapacitance technique. We find that (i) the $g$ factor is close to its value in bulk silicon and does not change with filling factor, in contrast to the strong dependence of the valley gap on $\nu$; and (ii) the cyclotron splitting is determined by the effective mass that is strongly enhanced at low electron densities. We also verify the systematics of the gaps in that the measured $\nu=4$, 8, and 12 cyclotron gap decreases with parallel magnetic field component by the same amount as the $\nu=2$, 6, and 10 spin gap increases. Measurements were made in an Oxford dilution refrigerator with a base temperature of $\approx 30$ mK on high-mobility (100)-silicon MOSFETs (with a peak mobility close to 2 m$^2$/Vs at 4.2 K) having the Corbino geometry with diameters 250 and 660 $\mu$m. The gate voltage was modulated with a small ac voltage 15 mV at frequencies in the range 2.5 – 25 Hz and the imaginary current component was measured with high precision using a current-voltage converter and a lock-in amplifier. Care was taken to reach the low frequency limit where the magnetocapacitance, $C(B)$, is not distorted by lateral transport effects. A dip in the magnetocapacitance at integer filling factor is directly related to a jump, $\Delta$, of the chemical potential across a corresponding gap in the spectrum of the 2D electron system, and therefore we determine $\Delta$ by integrating $C(B)$ over the dip in the low temperature limit where the magnetocapacitance saturates and becomes independent of temperature [@valley]. Typical magnetocapacitance traces taken at different electron densities, temperatures, and tilt angles are displayed in Fig. \[fig1\] near the filling factor $\nu=hcn_s/eB_\perp=4$ and $\nu=6$. The magnetocapacitance shows narrow minima at integer $\nu$ which are separated by broad maxima, the oscillation pattern reflecting the modulation of the thermodynamic density of states, $D$, in quantizing magnetic fields: $1/C=1/C_0+1/Ae^2D$ (where $C_0$ is the geometric capacitance between the gate and the 2D electrons, and $A$ is the sample area) [@smith85]. As the magnetic field is increased, the maximum $C$ approaches the geometric capacitance indicated by the dashed lines in Fig. \[fig1\]. Since the magnetocapacitance $C(B)<C_0$ around each maximum is almost independent of magnetic field, this results in asymmetric minima of $C(B)$, the asymmetry being more pronounced for $\nu=4$, 8, and 12. The chemical potential jump at integer $\nu=\nu_0$ is determined by the area of the dip in $C(B)$: $$\Delta=\frac{Ae^3\nu_0}{hcC_0}\int_{\text{dip}}\frac{C_{\text{ref}}- C}{C}dB_\perp, \label{Delta}$$ where $C_{\text{ref}}$ is a step function that is defined by two reference levels corresponding to the capacitance values at the low and high field edges of the dip as shown by the dotted line in Fig. \[fig1\]. The so-determined $\Delta$ is smaller than the level splitting by the level width. The last is extracted from the data by substituting $(C_0-C_{\text{ref}})/C$ for the integrand in Eq. (\[Delta\]) and integrating for the case of resolved levels between the magnetic fields $B_1=hcn_s/e(\nu_0+1/2)$ and $B_2=hcn_s/e(\nu_0-1/2)$. Tilting the magnetic field allows us to verify the systematics of the gaps in the spectrum and probe the lowest-energy charge-carrying excitations. As the thickness of the 2D electron system in Si MOSFETs is small compared to the magnetic length in accessible fields, the parallel field couples largely to the electrons’ spins while the orbital effects are suppressed [@simonian]. Therefore, the variation of a gap with $B_\parallel$ should reflect the change in the excitation energy as the Zeeman splitting, $g\mu_BB$, is increased: the excitation energy change is determined by the difference between the spin projections onto magnetic field for the ground and the lowest excited states. Within single-particle picture, e.g., one can expect that with increasing $B_\parallel$ at fixed $B_\perp$, the spin gap will increase, the valley gap will stay constant, and the cyclotron gap, which is given by the difference between the cyclotron splitting and the sum of the spin and the valley splittings, will decrease. In contrast, for spin textures (so-called skyrmions), the dependence of the excitation energy on $B_\parallel$ should be much stronger compared to the single-particle Zeeman splitting [@yang]. In Fig. \[fig2\](a), we show the value of the chemical potential jump, $\Delta_s$, across the $\nu=2$ and $\nu=6$ gaps as a function of magnetic field for different tilt angles. It is insensitive to both filling factor and tilt angle, as expected for spin gaps. The data are best described by a proportional increase of the gap with the magnetic field with a slope corresponding to an effective $g$ factor $g\approx 1.75$. The so-determined value obviously gives a lower boundary for the $g$ factor because both the valley splitting at odd $\nu$ and the level width are disregarded. In Fig. \[fig2\](b), we show how the $\nu=6$ gap changes with $B_\parallel$ at different values of the perpendicular field component. It is noteworthy that the level width contribution, which is indicated by systematic error bars, depends weakly on parallel field, and the valley splitting has been verified to be independent of $B_\parallel$. This, therefore, allows more accurate determination of the $g$ factor as shown by the solid line in Fig. \[fig2\](b). Its slope yields $g\approx 2.6$, which is in agreement with the data obtained for the $\nu=2$ and $\nu=10$ gaps. The fact that this value is close to the $g$ factor $g=2$ in bulk silicon points to the single spin-flip origin of the excitations for the $\nu=2$, 6, and 10 gaps. Unlike spin gaps, the chemical potential jump, $\Delta_c$, across the $\nu=4$, 8, and 12 gaps decreases with parallel magnetic field component, as already seen from Fig. \[fig1\]. In Fig. \[fig3\](a), we compare the behaviors of the $\nu=6$ and $\nu=4$ gaps with $B_\parallel$ at fixed perpendicular field component. For $B_\perp$ between 2.7 and 6.6 T, the absolute values of the slopes of these dependences are equal, within experimental uncertainty, to each other so that the sum of the gaps is approximately constant even if the level width contribution is taken into account. These results lead to two important consequences: (i) the $\nu=4$, 8, and 12 gaps are cyclotron ones, the conventional systematics of the gaps remaining valid in the studied electron density range down to $1.5\times 10^{11}$ cm$^{-2}$; and (ii) the $g$ factor does not vary with filling factor $\nu$. Although our value of $g\approx 2.6$ is in agreement with the previously measured ones [@englert; @gm; @us], we do not confirm the conclusion on oscillations of the $g$ factor with $\nu$ based on activation energy measurements and made in line with theoretical predictions [@ando; @afs] under the assumption of $B_\parallel$-independent level width [@englert]. In Fig. \[fig3\](b), we compare the data for the chemical potential jump across the $\nu=4$, 8, and 12 gaps in perpendicular and tilted magnetic fields including the term $g\mu_B(B-B_\perp)$ that describes the increase of the spin gap with $B_\parallel$. The data coincidence confirms that the changing spin gap is the only cause for the dependence of the cyclotron gap on parallel field component. As is evident from the figure, $\Delta_c$ is considerably smaller than the value ($\hbar\omega_c-2\mu_BB_\perp$) expected within single-particle approach ignoring both valley splitting and level width. To reduce experimental uncertainty related to the inaccurate determination of the level width, we plot in Fig. \[fig4\] the difference, $(\Delta_c-\Delta_s)/2\mu_BB$, of the normalized values of the cyclotron and the spin gaps in a perpendicular magnetic field as a function of electron density. Assuming that the cyclotron splitting is determined by the effective mass $m$, this difference corresponds to ($m_e/m-g$), where $m_e$ is the free electron mass. Using data for $m$ and $g$ obtained in both parallel [@gm] and weak [@us; @smith72] magnetic fields, we find that the value ($m_e/m-g$) is indeed consistent with our data, see Fig. \[fig4\]. The effective mass determined from our high-$n_s$ data using $g=2.6$ is equal to $m\approx 0.23m_e$, which is close to the band mass of $0.19m_e$. As long as our $g$ value is constant, the decrease of the normalized gap difference with decreasing $n_s$ reflects the behavior of the cyclotron splitting, which is in agreement with the conclusion of the strongly enhanced effective mass at low electron densities [@gm; @us]. We now discuss comparatively the results obtained for the valley and the spin gaps. According to Ref. [@valley], the enhanced valley gap at the lowest filling factors $\nu=1$ and $\nu=3$ in Si MOSFETs is comparable to the single-particle Zeeman splitting. As our data for the spin gap correspond to the single-particle Zeeman splitting, this may lead to a different systematics of the gaps in the spectrum compared to the single-particle picture. Such a possibility has been supported by a recent theory [@brener] which shows the importance of the Jahn-Teller effect for the ground state of a 2D electron system in bivalley (100)-Si MOSFETs in quantizing magnetic fields. At $\nu=2$, due to static and dynamic lattice deformations, the valley degeneracy is predicted to lift off giving rise to a complicated phase diagram including three phases: spin-singlet, canted antiferromagnet, and ferromagnet. In our experiment, over the studied range of magnetic fields down to 3 T, we observe at $\nu=2$ a spin-ferromagnetic ground state only. This gives an estimate of the strength of the suggested mechanism for the valley splitting enhancement. The fact that we do not observe oscillations of the $g$ factor as a function of $\nu$ is not too surprising, because our value of $g$ is close to the $g$ factor in bulk silicon so that those oscillations may be small. At the same time, our data for the $g$ factor allow us to arrive at a conclusion that at $\nu=2$, the valley gap is small compared to the spin gap. Therefore, the valley splitting does oscillate with filling factor [@ando], the conclusion being valid, at least, for the strongly enhanced gaps at $\nu=1$ and $\nu=3$. We stress that this effect occurs in the strongly correlated electron system, which is beyond the conventional theory of exchange-enhanced gaps [@ando]. Let us finally discuss the results obtained for the cyclotron gap. The data of Fig. \[fig4\] indicate unequivocally that the origin of the small $\Delta_c$ value in Fig. \[fig3\](b) is not related to valley splitting and level width. Instead, it is renormalization of the effective mass and $g$ factor due to electron-electron interactions: the observed decrease of the gap difference with decreasing $n_s$ in Fig. \[fig4\] as well as the systematics of the gaps are in agreement with both the decrease of the ratio of the cyclotron and the spin gaps with decreasing $n_s$ [@krav00] and the sharp increase of the effective mass at low electron densities [@gm; @us]. Needless to say that the conventional theory [@ando] yields an opposite sign of the interaction effect on the cyclotron splitting. We gratefully acknowledge discussions with I. L. Aleiner, S. V. Iordanskii, A. Kashuba, and S. V. Kravchenko. This work was supported by the RFBR, the Russian Ministry of Sciences, and the Programme “The State Support of Leading Scientific Schools”. V.T.D. acknowledges support of A. von Humboldt foundation via Forschungspreis. [apssamp]{} T. Ando and Y. Uemura, J. Phys. Soc. Jpn. [**37**]{}, 1044 (1974); Yu. A. Bychkov, S. V. Iordanskii, and G. M. Eliashberg, JETP Lett. [**33**]{}, 143 (1981); C. Kallin and B. I. Halperin, Phys. Rev. B [**30**]{}, 5655 (1984); A. P. Smith, A. H. MacDonald, and G. Gumbs, Phys. Rev. B [**45**]{}, 8829 (1992). K. Yang, K. Moon, L. Zheng, A. H. MacDonald, S. M. Girvin, D. Yoshioka, and S.-C. Zhang, Phys. Rev. Lett. [**72**]{}, 732 (1994); K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDonald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Phys. Rev. B [**51**]{}, 5138 (1995); L. Brey, H. A. Fertig, R. Cote, and A. H. MacDonald, Phys. Rev. B [**54**]{}, 16888 (1996). S. Brener, S. V. Iordansky, and A. Kashuba, cond-mat/0211214. S. V. Iordanskii and A. Kashuba, JETP Lett. [**75**]{}, 348 (2002). T. H. Englert and K. von Klitzing, Surf. Sci., 70 (1978). N. Kleinmichel, Diploma Thesis, TU München (1984). V. T. Dolgopolov, N. B. Zhitenev, and A. A. Shashkin, Sov. Phys. JETP [**68**]{}, 1471 (1988). A. Usher, R. J. Nicholas, J. J. Harris, and C. T. Foxon, Phys. Rev. B [**41**]{}, 1129 (1992). S. V. Kravchenko, A. A. Shashkin, D. A. Bloore, and T. M. Klapwijk, Solid State Commun. [**116**]{}, 495 (2000). T. P. Smith, B. B. Goldberg, P. J. Stiles, and M. Heiblum, Phys. Rev. B [**32**]{}, 2696 (1985). V. T. Dolgopolov, A. A. Shashkin, A. V. Aristov, D. Schmerek, W. Hansen, J. P. Kotthaus, and M. Holland, Phys. Rev.Lett. [**79**]{}, 729 (1997). V. S. Khrapai, A. A. Shashkin, and V. T. Dolgopolov, Phys. Rev. B (2003, in press); cond-mat/0202505. An attempt to extract the gap value from the chemical potential jump measured in Si MOSFETs was made by V. M. Pudalov [*et al.*]{} (Sov. Phys. JETP [**62**]{}, 1079 (1985)) based on a sophisticated model which did not allow reasonably accurate determination of the gaps. A. A. Shashkin, S. V. Kravchenko, V. T. Dolgopolov, and T. M. Klapwijk, Phys. Rev. Lett. [**87**]{}, 086801 (2001); Phys.Rev. B [**66**]{}, 073303 (2002); S. V. Kravchenko, A. A. Shashkin, and V. T. Dolgopolov, Phys. Rev. Lett. [**89**]{}, 219701 (2002). A. A. Shashkin [*et al.*]{}, cond-mat/0301187. An evaluation of the effective mass at low electron densities was made by V. M. Pudalov [*et al.*]{} (Phys. Rev. Lett., 196404 (2002)) in the high-temperature limit of Shubnikov-de Haas oscillations because of electron overheating in their experiment. D. Simonian, S. V. Kravchenko, M. P. Sarachik, and V. M. Pudalov, Phys. Rev. Lett. [**79**]{}, 2304 (1997). T. Ando, A. B. Fowler, and F. Stern, Rev. Mod. Phys., 437 (1982). F. F. Fang and P. J. Stiles, Phys. Rev. [**174**]{}, 823 (1968); J. L. Smith and P. J. Stiles, Phys. Rev. Lett., 102 (1972).

ArXiv

--- abstract: 'In this paper, we extend considerably the global existence results of entropy-weak solutions related to compressible Navier-Stokes system with density dependent viscosities obtained, independently (using different strategies), by Vasseur-Yu \[[*Inventiones mathematicae*]{} (2016) and arXiv:1501.06803 (2015)\] and by Li-Xin \[arXiv:1504.06826 (2015)\]. More precisely we are able to consider a physical symmetric viscous stress tensor $\sigma = 2 \mu(\rho) \,{\mathbb D}({{ u}}) + \bigl(\lambda(\rho) {\rm div} {{ u}}- P(\rho)\bigr) \, {\rm Id}$ where ${\mathbb D}({{ u}}) = [\nabla {{ u}}+ \nabla^T {{ u}}]/2$ with a shear and bulk viscosities (respectively $\mu(\rho)$ and $\lambda(\rho)$) satisfying the BD relation $\lambda(\rho)=2(\mu''(\rho)\rho - \mu(\rho))$ and a pressure law $P(\rho)=a\rho^\gamma$ (with $a>0$ a given constant) for any adiabatic constant $\gamma>1$. The nonlinear shear viscosity $\mu(\rho)$ satisfies some lower and upper bounds for low and high densities (our mathematical result includes the case $\mu(\rho)= \mu\rho^\alpha$ with $2/3 < \alpha < 4$ and $\mu>0$ constant). This provides an answer to a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by F. Rousset in the Bourbaki 69ème année, 2016–2017, no 1135.' address: - 'LAMA UMR5127 CNRS, Université Savoie Mont-Blanc, France' - 'Department of Mathematics, The University of Texas at Austin.' - 'Department of Mathematics, University of Florida.' author: - Didier Bresch - 'Alexis F. Vasseur' - Cheng Yu title: 'Global Existence of Entropy-Weak Solutions to the Compressible Navier-Stokes Equations with Non-Linear Density Dependent Viscosities' --- Introduction ============ When a fluid is governed by the barotropic compressible Navier-Stokes equations, the existence of global weak solutions, in the sense of J. [Leray]{} (see [@Le]), in space dimension greater than two remained for a long time without answer, because of the weak control of the divergence of the velocity field which may provide the possibility for the density to vanish (vacuum state) even if initially this is not the case. There exists a huge literature on this question, in the case of constant shear viscosity $\mu$ and constant bulk viscosity $\lambda$. Before 1993, many authors such as Hoff [@Hoff87], Jiang-Zhang [@JZ], Kazhikhov–Shelukhin [@KS], Serre [@S], Veigant–Kazhikhov [@VK] (to cite just some of them) have obtained partial answers: We can cite, for instance, the works in dimension 1 in 1986 by Serre [@S], the one by Hoff [@Hoff87] in 1987, and the one in the spherical case in 2001 by Jiang-Zhang [@JZ]. The first rigorous approach of this problem in its generality is due in 1993 by P.–L. Lions [@Lions] when the pressure law in terms of the density is given by $P(\rho)=a \rho^\gamma$ where $a$ and $\gamma$ are two strictly positive constants. He has presented in 1998 a complete theory for $P(\rho)=a \rho^\gamma$ with $\gamma\ge 3d/(d+2)$ (where $d$ is the space dimension) allowing to obtain the result of global existence of weak solutions à la Leray in dimension $d=2$ and $3$ and for general initial data belonging to the energy space. His result has been then extended in 2001 to the case $P(\rho)= a \rho^\gamma$ with $\gamma>d/2$ by Feireisl-Novotny-Petzeltova [@FNP] introducing an appropriated method of truncation. Note also in 2014 the paper by Plotnikov-Weigant [@PW] in dimension 2 for the linear pressure law that means $\gamma =1$. In 2002, Feireisl [@F04] has also proved it is possible to consider a pressure $P(\rho)$ law non-monotone on a compact set $[0,\rho_*]$ (with $\rho_*$ constant) and monotone elsewhere. This has been relaxed in 2018 by Bresch-Jabin [@BJ] allowing to consider real non-monotone pressure laws. They have also proved that it is possible to consider some constant anisotropic viscosities. The Lions theory has also been extended recently by Vasseur-Wen-Yu [@VWY] to pressure laws depending on two phases (see also Mastese $\&$ [*al.*]{} [@MaMiMuNoPoZa], Novotny [@No] and Novotny-Pokorny [@NoPo]). The method introduced by Bresch-Jabin in [@BJ] has also been recently developped in the bifluid framework by Bresch-Mucha-Zatorska in [@BrMuZa]. When the shear and the bulk viscosities (respectively $\mu$ and $\lambda$) are assumed to depend on the density $\rho$, the mathematical framework is completely different. It has been discussed, mathematically, initially in a paper by Bernardi-Pironneau [@BP] related to viscous shallow-water equations and by P.–L. Lions [@Lions] in his second volume related to mathematics and fluid mechanics. The main ingredient in the constant case which is the compactness in space of the effective flux $F= (2\mu+\lambda) {\rm div} u - P(\rho)$ is no longer true for density dependent viscosities. In space dimension greater than one, a real breakthrough has been realized with a series of papers by Bresch-Desjardins [@BD; @BD2006; @BrDeFormula; @BrDeSpringer], (started in 2003 with Lin [@BDL] in the context of Navier-Stokes-Korteweg with linear shear viscosity case) who have identified an information related to the gradient of a function of the density if the viscosities satisfy what is called the Bresch-Desjardins constraint. This information is usually called the BD entropy in the literature with the introduction of the concept of entropy-weak solutions. Using such extra information, they obtained the global existence of entropy-weak solutions in the presence of appropriate drag terms or singular pressure close to vacuum. Concerning the one-dimensional in space case or the spherical case, many important results have been obtained for instance by Burtea-Haspot [@BuHa], Ducomet-Necasova-Vasseur [@DNV], Constantin-Drivas-Nguyen-Pasqualottos [@CoDrNgPa], Guo-Jiu-Xin [@GJX], Haspot [@Haspot], Jiang-Xin-Zhang [@JXZ], Jiang-Zhang [@JZ], Kanel [@Kan], Li-Li-Xin [@LiLiXi], Mellet-Vasseur [@MV2], Shelukhin [@S] without such kind of additional terms. Stability and construction of approximate solutions in space dimension two or three have been investigated during more than fifteen years with a first important stability result without drag terms or singular pressure by Mellet-Vasseur [@MV]. Several important works for instance by Bresch-Desjardins [@BD; @BD2006; @BrDeFormula; @BrDeSpringer] and Bresch-Desjardins-Lin [@BDL], Bresch-Desjardins-Zatorska [@BDZ], Li-Xin [@LiXi], Mellet-Vasseur [@MV], Mucha-Pokorny-Zatorska [@MuPoZa], Vasseur-Yu [@VY-1; @VY], and Zatorska [@Z] have also been written trying to find a way to construct approximate solutions. Recently a real breakthrough has been done in two important papers by Li-Xin [@LiXi] and Vasseur-Yu [@VY]: Using two different ways, they got the global existence of entropy-weak solutions for the compressible paper when $\mu(\rho)=\rho$ and $\lambda(\rho)=0$. Note that in the last paper [@LiXi] by Li-Xin, they also consider more general viscosities satisfying the BD relation but with a non-symmetric stress diffusion ($\sigma = \mu(\rho)\nabla u + (\lambda(\rho){\rm div} u - P(\rho)) {\rm Id}$) and more restrictive conditions on the shear $\mu(\rho)$ viscosity and bulk viscosity $\lambda(\rho)$ and on the pressure law $P(\rho)$ compared to the present paper. The objective of this current paper is to extend the existence results of global entropy-weak solutions obtained independently (using different strategies) by Vasseur-Yu [@VY] and Lin-Xin [@LiXi] to answer a longstanding mathematical question on compressible Navier-Stokes equations with density dependent viscosities as mentioned for instance by Rousset [@Ro]. More precisely extending and coupling carefully the two-velocities framework by Bresch-Desjardins-Zatorska [@BDZ] with the generalization of the quantum Böhm identity found by Bresch-Couderc-Noble-Vila [@BCNV] (proving a generalization of the dissipation inequality used by Jüngel [@J] for Navier-Stokes-Quantum system and established by Jüngel-Matthes in [@JuMa]) and with the renormalized solutions introduced in Lacroix-Violet and Vasseur [@LaVa], we can get global existence of entropy-weak solutions to the following Navier-Stokes equations: $$\label{NS equation} \begin{split} &\rho_t+{{\rm div}}(\rho{{ u}})=0\\ &(\rho{{ u}})_t+{{\rm div}}(\rho{{ u}}\otimes{{ u}})+\nabla P(\rho) - 2 {\rm div}\bigl(\sqrt{\mu(\rho)} \mathbb{S}_\mu + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb S_\mu) {\rm Id} \bigr)=0, \end{split}$$ where $$\sqrt{\mu(\rho)} \mathbb{S}_\mu = \mu(\rho) {\mathbb D}({{ u}})$$ with data $$\label{initial data} \rho|_{t=0}=\rho_0(x)\ge 0,\;\;\;\;\;\rho{{ u}}|_{t=0}={{ m}}_0(x)=\rho_0{{ u}}_0,$$ and where $P(\rho) =a \rho^{\gamma}$ denotes the pressure with the two constants $a>0$ and $\gamma >1$, $\rho$ is the density of fluid, ${{ u}}$ stands for the velocity of fluid, $\mathbb{D}{{ u}}=[\nabla{{ u}}+\nabla^T{{ u}}]/2$ is the strain tensor. As usually, we consider $${{ u}}_0= \frac{m_0}{\rho_0} \hbox{ when } \rho_0\not=0 \hbox{ and }{{ u}}_0 = 0 \hbox{ elsewhere}, \qquad \frac{|m_0|^2}{\rho_0} = 0 \hbox{ a.e. on } \{x\in \Omega: \rho_0(x) = 0\}.$$ We remark the following identity $$2 {\rm div}\bigl(\sqrt{\mu(\rho)} \mathbb{S}_\mu + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb S_\mu) {\rm Id} \bigr)=-2{{\rm div}}(\mu(\rho)\mathbb{D}{{ u}})-\nabla(\lambda(\rho){{\rm div}}{{ u}}).$$ The viscosity coefficients $\mu=\mu(\rho)$ and $\lambda=\lambda(\rho)$ satisfy the Bresch-Desjardins relation introduced in [@BrDeFormula] $$\label{BD relationship} \lambda(\rho)=2(\rho\mu'(\rho)-\mu(\rho)).$$ The relation between the stress tensor $\mathbb{S}_\mu$ and the triple $(\mu(\rho)/\sqrt\rho, \sqrt \rho {{ u}}, \sqrt\rho {{ v}})$ where ${{ v}}= 2 \nabla s(\rho)$ with $s'(\rho)= \mu'(\rho)/\rho$ will be proved in the following way: The matrix $\mathbb{S}_\mu$ is the symetric part of a matrix value function $\mathbb{T}_\mu$ namely $$\label{Smu} \mathbb{S}_\mu = \frac{(\mathbb{T}_\mu + \mathbb{T}_\mu^t)}{2}$$ where $\mathbb{T}_\mu$ is defined through $$\label{Tmu} \begin{split} \sqrt{\mu(\rho)} \mathbb{T}_\mu = \nabla (\sqrt\rho {{ u}}\, \frac{\mu(\rho)}{\sqrt\rho}) - \sqrt\rho {{ u}}\otimes \sqrt\rho \nabla s(\rho) \end{split}$$ with $$\label{s} s'(\rho) = \mu'(\rho) /\rho,$$ and $$\label{Tmu1} \begin{split} \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb T_\mu) {\rm Id} = \Bigl[ {\rm div}(\frac{\lambda(\rho)}{\mu(\rho)} \sqrt\rho {{ u}}\, \frac{\mu(\rho)}{\sqrt\rho}) - \sqrt\rho {{ u}}\cdot \sqrt\rho \,\nabla s(\rho) \, \frac{\rho \mu''(\rho)}{\mu'(\rho)}\Bigr] {\rm Id}. \end{split}$$ For the sake of simplicity, we will consider the case of periodic boundary conditions in three dimension in space namely ${\Omega}=\mathbb{T}^3$. In the whole paper, we assume: $$\label{regmu} \mu \in C^0({\mathbb R_+}; \, {\mathbb R_+})\cap C^2({\mathbb R}_+^*; \,{\mathbb R}),$$ where $\mathbb R_+=[0,\infty) \text{ and } \mathbb R_+^*=(0,\infty).$ We also assume that there exists two positive numbers $\alpha_1,\alpha_2$ such that $$\label{mu estimate} \begin{array}{l} \displaystyle{ \frac{2}{3}<\alpha_1<\alpha_2<4, }\\[0.3cm] \displaystyle{\mathrm{for \ any } \ \rho>0, \qquad 0<\frac{1}{\alpha_2}\rho \mu'(\rho)\leq \mu(\rho)\leq \frac{1}{\alpha_1}\rho \mu'(\rho), } \end{array}$$ and there exists a constant $C>0$ such that $$\label{mu estimate1} \left|\frac{\rho \mu''(\rho)}{\mu'(\rho)}\right| \le C < +\infty.$$ Note that if $\mu(\rho)$ and $\lambda(\rho)$ satisfying and , then $$\lambda(\rho) + 2\mu(\rho)/3 \ge 0$$ and thanks to $$\mu(0)= \lambda(0) = 0.$$ Note that the hypothesis – allow a shear viscosity of the form $\mu(\rho)=\mu \rho^{\alpha}$ with $\mu>0$ a constant where $2/3<\alpha<4$ and a bulk viscosity satisfying the BD relation: $\lambda(\rho)= 2(\mu'(\rho)\rho - \mu(\rho))$. [**Remark.**]{} In [@VY] and [@LiXi] the case $\mu(\rho)=\mu\rho$ and $\lambda(\rho)=0$ is considered, and in [@LiXi] more general cases have been considered but with a non-symmetric viscous term in the three-dimensional in space case, namely $- {{\rm div}}(\mu(\rho)\nabla {{ u}}) - \nabla (\lambda(\rho){{\rm div}}{{ u}})$. In [@LiXi] the viscosities $\mu(\rho)$ and $\lambda(\rho)$ satisfy with $\mu(\rho) = \mu \rho^\alpha$ where $\alpha \in [3/4,2)$ and with the following assumption on the value $\gamma$ for the pressure $p(\rho)=a\rho^\gamma$: $$\hbox{ If } \alpha\in [3/4,1], \qquad \gamma \in (1,6\alpha-3)$$ and $$\hbox{ if } \alpha \in (1,2), \qquad \gamma\in [2\alpha-1,3\alpha-1].$$ The main result of our paper reads as follows: \[main result\] Let $\mu(\rho)$ verify – and $\mu$ and $\lambda$ verify . Let us assume the initial data satisfy $$\label{initial energy} \begin{split} & \int_{{\Omega}}\left(\frac{1}{2}\rho_0|{{ u}}_0+ 2\kappa \nabla s(\rho_0)|^2 +\kappa(1-\kappa)\rho_0\frac{|2\nabla s(\rho_0)|^2}{2}\right) \, dx \\ & \hskip6cm + \int_{{\Omega}}\left(a\frac{\rho_0^{\gamma}}{\gamma-1} + \mu(\rho_0)\right)\,dx\leq C <+\infty. \end{split}$$ with $k\in (0,1)$ given. Let $T$ be given such that $0<T<+\infty$, then, for any $\gamma>1$, there exist a renormalized solution to - as defined in Definition \[def\_renormalise\_u\]. Moreover, this renormalized solution with initial data satisfying is a weak solution to - in the sense of Definition \[defweak\]. Our result may be considered as an improvement of [@LiXi] for two reasons: First it takes into account a physical symmetric viscous tensor and secondly, it extends the range of coefficients $\alpha$ and $\gamma$. The method is based on the consideration of an approximated system with an extra pressure quantity, appropriate non-linear drag terms and appropriate capillarity terms. This generalizes the Quantum-Navier-Stokes system with quadratic drag terms considered in [@VY-1; @VY]. First we prove that weak solutions of the approximate solution are renormalized solutions of the system, in the sense of [@LaVa]. Then we pass to the limit with respect to $r_2,r_1, r_0, r, \delta$ to get renormalized solutions of the compressible Navier-Stokes system. The final step concerns the proof that a renormalized solution of the compressible Navier-Stokes system is a global weak solution of the compressible Navier–Stokes system. Note that, thanks to the technique of renormalized solution introduced in [@LaVa], it is not necessary to derive the Mellet-Vasseur type inequality in this paper: This allows us to cover the all range $\gamma>1$. [*First Step.*]{} Motivated by the work of [@LaVa], the first step is to establish the existence of global $\kappa$ entropy weak solution to the following approximation $$\label{last level approximation} \begin{split} &\rho_t+{{\rm div}}(\rho{{ u}})=0\\ &(\rho{{ u}})_t+{{\rm div}}(\rho{{ u}}\otimes{{ u}})+\nabla P(\rho) + \nabla P_\delta(\rho) \\ &\hskip3cm- 2 {\rm div}\Bigl(\sqrt{\mu(\rho)} \mathbb{S}_\mu + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb S_\mu) {\rm Id}\Bigr) \\ & \hskip3cm- 2 r {\rm div}\Bigl(\sqrt{\mu(\rho)} \mathbb{S}_r + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb S_r) {\rm Id}\Bigr) \\ & \hskip7cm + r_0{{ u}}+r_1\frac{\rho}{\mu'(\rho)}|{{ u}}|^2{{ u}}+r_2\rho|{{ u}}|{{ u}}= 0 \end{split}$$ where the barotorpic pressure law and the extra pressure term are respectively $$P(\rho)= a\rho^\gamma, \qquad P_\delta (\rho)= \delta \rho^{10} \hbox{ with } \delta>0.$$ The matrix $\mathbb{S}_\mu$ is defined in and $\mathbb{T}_\mu$ is given in- . The matrix $\mathbb{S}_r$ is compatible in the following sense: $$\label{eq_quantic} \begin{split} r\sqrt{\mu(\rho)} \mathbb{S}_r = 2r \Bigl[2 \sqrt{\mu(\rho)} \nabla\nabla Z(\rho) - \nabla (\sqrt{\mu(\rho)} \nabla Z(\rho))\Bigr], \end{split}$$ where $$\label{ZZ} \displaystyle Z(\rho) = \int_0^\rho [(\mu(s))^{1/2} \mu'(s)]/s \, ds, \qquad \displaystyle k(\rho) = \int_0^\rho [{\lambda(s)\mu'(s)}]/{\mu(s)^{3/2}} ds$$ and $$\label{eq_quantic11} \begin{split} r\frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr}(\sqrt{\mu(\rho)} \mathbb S_r) {\rm Id} = r(\frac{\lambda(\rho)}{\sqrt{\mu(\rho)}}+ \frac{1}{2} k(\rho))\Delta Z(\rho) {\rm Id} - \frac{r}{2}{\rm div} [ k(\rho)\nabla Z(\rho)] {\rm Id}. \end{split}$$ [**Remark.**]{} Note that the previous system is the generalization of the quantum viscous Navier-Stokes system considered by Lacroix-Violet and Vasseur in [@LaVa] (see also the interesting papers by Antonelli-Spirito [@AnSp1; @AnSp2] and by Carles-Carrapatoso-Hillairet [@CaCaHi]). Indeed if we consider $\mu(\rho)=\rho$ and $\lambda(\rho)=0$, we can write $\sqrt{\mu(\rho)} \mathbb S_r$ as $$\sqrt{\mu(\rho)} \mathbb{S}_r =4 \sqrt{\rho} \Bigl[ \nabla\nabla \sqrt\rho - 4 (\nabla \rho^{1/4} \otimes \nabla \rho^{1/4}) \Bigr],$$ using $Z(\rho) = 2\sqrt\rho.$ The Navier–Stokes equations for quantum fluids was also considered by A. J" ungel in [@J]. As the first step generalizing [@VY], we prove the following result. \[main result 1\] Let $\mu(\rho)$ verifies – and $\lambda(\rho)$ is given by . If $r_0>0$, then we assume also that ${\rm inf}_{s \in [0,+\infty)} \mu'(s)=\epsilon_1 >0$. Assume that $r_1$ is small enough compared to $r$, $r_2$ is small enough compared to $\delta$, and that the initial values verify $$\label{Initial conditions} \begin{split} & \int_\Omega \rho_0\left(\frac{|{{ u}}_0+2\kappa\nabla s(\rho_0)|^2}{2}+(\kappa (1-\kappa)+r)\frac{|2\nabla s(\rho_0)|^2}{2}\right) \, dx\\ & \hskip4cm + \int_\Omega \bigl(a \frac{\rho_0^\gamma}{\gamma-1}+ \mu(\rho_0) + \delta \frac{\rho_0^{10}}{9}+\frac{r_0}{\varepsilon_1}|(\ln \rho_0)_-|\bigr)\,dx < + \infty, \end{split}$$ for a fixed $\kappa\in (0,1)$. Then there exists a $\kappa$ entropy weak solution $(\rho,{{ u}}, \mathbb T_\mu, \mathbb S_r)$ to – satisfying the initial conditions , in the sense that $(\rho,{{ u}}, \mathbb T_\mu, \mathbb S_r)$ satisfies the mass and momentum equations in a weak form, and satisfies the compatibility formula in the sense of definition \[defweak\]. In addition, it verifies the following estimates: $$\label{priori estimates} \begin{split} &\|\sqrt{\rho}\, ({{ u}}+2\kappa \nabla s(\rho))\|^2_{L^{\infty}(0,T;L^2({\Omega}))}\leq C, \quad\quad\quad\quad\quad a \|\rho\|^\gamma_{L^{\infty}(0,T;L^{\gamma}({\Omega}))}\leq C, \\&\|\mathbb T_\mu\|^2_{L^2(0,T;L^2({\Omega}))}\leq C, \quad\quad\quad (\kappa(1-\kappa)+r)\|\sqrt\rho \nabla s(\rho)\|^2_{L^{\infty}(0,T;L^2({\Omega}))}\leq C, \\& \kappa\|\sqrt{\mu'(\rho)\rho^{\gamma-2}}\nabla\rho\|^2_{L^2(0,T;L^2({\Omega}))}\leq C, \end{split}$$ and $$\label{priorie estimate2} \begin{split} \\&\delta\|\rho\|^{10}_{L^{\infty}(0,T;L^{10}({\Omega}))}\leq C,\quad\quad\;\;\;\quad\quad\quad\quad\delta\|\sqrt{\mu'(\rho)\rho^{8}}\nabla\rho\|^2_{L^2(0,T;L^2({\Omega}))}\leq C, \\&r_2\|(\frac{\rho}{\mu'(\rho_n)})^{\frac{1}{4}}{{ u}}\|^4_{L^4(0,T;L^4({\Omega}))}\leq C, \quad\quad\quad r_1\|\rho^{\frac{1}{3}}|{{ u}}|\|^3_{L^3(0,T;L^3({\Omega}))}\leq C, \\&r_0\|{{ u}}\|^2_{L^2(0,T;L^2({\Omega}))}\leq C, \quad\quad\quad\quad\quad\quad\quad r \|\mathbb S_r\|^2_{L^2(0,T;L^2({\Omega}))} \leq C. \end{split}$$ Note that the bounds provide the following control on the velocity field $$\|\sqrt{\rho}\, {{ u}}\|^2_{L^{\infty}(0,T;L^2({\Omega}))}\leq C.$$ Moreover let $$\displaystyle Z (\rho)= \int_0^\rho \frac{\sqrt{\mu(s)}\mu'(s)}{s}\, ds\;\;\text{and }\; \displaystyle Z_1(\rho) = \int_0^\rho \frac{\mu'(s)}{(\mu(s))^{1/4} s^{1/2}} \, ds,$$ we have the extra control $$\label{J inequality for sequence} r \left[\int_0^T\int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx\,dt +\int_0^T\int_{{\Omega}} |\nabla Z_1(\rho)|^4\,dx\,dt\right] \leq C,$$ and $$\label{priori mu} \begin{split} &\|\mu(\rho)\|_{L^\infty(0,T;W^{1,1}(\Omega))} + \|\mu(\rho){{ u}}\|_{L^\infty(0,TL^{3/2}({\Omega}))\cap L^2(0,T;W^{1,1} ({\Omega}))} \leq C,\\ & \|\partial_t \mu(\rho)\|_{L^{\infty}(0,T;W^{-1,1}({\Omega}))}\leq C, \\ & \|Z(\rho)\|_{L^\infty(0,T;L^{1+}(\Omega))} + \|Z_1(\rho)\|_{L^\infty(0,T;L^{1+}(\Omega))} \leq C, \end{split}$$ where $C>0$ is a constant which depends only on the initial data. [**Sketch of proof for Theorem \[main result 1\].**]{} To show Theorem \[main result 1\], we need to build the smooth solution to an approximation associated to . Here, we adapt the ideas developed in [@BDZ] to construct this approximation. More precisely, we consider an augmented version of the system which will be more appropriate to construct approximate solutions. Let us explain the idea. 0.1cm [*First step: the augmented system.*]{} Defining a new velocity field generalizing the one introduced in the BD entropy estimate namely $$\w={{ u}}+ 2\kappa\nabla s(\rho)$$ and a drift velocity ${{ v}}=2 \nabla s(\rho)$ and ${s}(\rho)$ defined in . Assuming to have a smooth solution of with damping terms, it ca[v]{}own that $(\rho,\w,{v})$ satisfies the following system of equations $$\rho_t + {\rm div}(\rho \w) - 2\kappa \Delta \mu(\rho) = 0$$ and $$\begin{split} \\&(\rho\w)_t+{{\rm div}}(\rho{{ u}}\otimes\w)-2(1-\kappa){{\rm div}}(\mu(\rho)\mathbb{D}\, \w) -2\kappa{{\rm div}}(\mu(\rho)\mathbf{A}(w)) \\&- (1-\kappa) \nabla(\lambda(\rho) {{\rm div}}(w-\kappa {v})) +\nabla\rho^{\gamma}+\delta\nabla\rho^{10} +4(1-\kappa)\kappa {{\rm div}}(\mu(\rho)\nabla^2{s}(\rho)) \\&=- r_0 (w-2\kappa \nabla s(\rho)) - r_1 \rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho))\\& - r_2 \frac{\rho}{\mu'(\rho)} |\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho)) +r\rho\nabla\left(\sqrt{K(\rho)}\D(\int_0^{\rho}\sqrt{K(s)}\,ds)\right), \end{split}$$ and $$\begin{split} & (\rho{v})_t+{{\rm div}}(\rho{{ u}}\otimes{v})-2\kappa{{\rm div}}(\mu(\rho)\nabla {v}) + 2{{\rm div}}(\mu(\rho)\nabla^t\w) + \nabla(\lambda(\rho){{\rm div}}(\w-\kappa {v}))=0, \end{split}$$ where $${v}= 2 \nabla {s}(\rho), \qquad \w={{ u}}+\kappa {v}$$ and $$K(\rho) = 4 (\mu'(\rho))^2 / \rho .$$ This is the augmented version for which we will show that there exists global weak solutions, adding an hyperdiffusivity $\varepsilon_2[ \Delta^{2s}\w -{{\rm div}}((1+|\nabla w|^2)\nabla w)]$ on the equation satisfied by $w$, and passing to the limit $\varepsilon_2$ goes to zero. [**Important remark.**]{} Note that recently Bresch-Couderc-Noble-Vila [@BCNV] showed the following interesting relation $$\rho\nabla\left(\sqrt{K(\rho)}\D(\int_0^{\rho}\sqrt{K(s)}\,ds)\right) ={{\rm div}}(F(\rho)\nabla^2 \psi(\rho))+ \nabla\left((F'(\rho)\rho- F(\rho))\D\psi(\rho)\right),$$ with $F'(\rho)=\sqrt{K(\rho)\rho}$ and $\sqrt\rho \psi'(\rho) = \sqrt{K(\rho)}.$ Thus choosing $$F(\rho)=2\,\mu(\rho) \hbox{ and therefore } F'(\rho)\rho- F(\rho)=\lambda(\rho),$$ this gives $\psi(\rho) = 2 {s}(\rho)$ and thus $$\label{BCNV relationship} \rho\nabla\left(\sqrt{K(\rho)}\D(\int_0^{\rho}\sqrt{K(s)}\,ds)\right)= 2 {{\rm div}}\Bigl(\mu(\rho)\nabla^2\bigl(2 {s}(\rho)\bigr)\Bigr) +\nabla\Bigl(\lambda(\rho)\D\bigl(2{s}(\rho)\bigr)\Bigr).$$ This identity will play a crucial role in the proof. It defines the appropriate capillarity term to consider in the approximate system. Other identities will be used to define the weak solution for the Navier-Stokes-Korteweg system and to pass to the limit in it namely $$\label{rel} \begin{split} & 2\mu(\rho)\nabla^2(2{\mathbf s}(\rho)) + \lambda(\rho) \Delta (2{\mathbf s}(\rho)) = 4 \Bigl[2 \sqrt{\mu(\rho)} \nabla\nabla Z(\rho) - \nabla (\sqrt{\mu(\rho)} \nabla Z(\rho)\Bigr] \\ & \hskip3cm + (\frac{2\lambda(\rho)}{\sqrt{\mu(\rho)}}+ k(\rho))\Delta Z(\rho)\, {\rm Id} - {\rm div} [ k(\rho)\nabla Z(\rho)]\, {\rm Id}. \end{split}$$ where $\displaystyle Z(\rho) = \int_0^\rho [(\mu(s))^{1/2} \mu'(s)]/s \, ds$ and $\displaystyle k(\rho) = \int_0^\rho \frac{\lambda(s)\mu'(s)}{\mu(s)^{3/2}} ds.$ Note that the case considered in [@LaVa; @VY-1; @VY] is related $\mu(\rho) = \rho$ and $K(\rho) = 4/\rho$ which corresponds to the quantum Navier-Stokes system. Note that two very interesting papers have been written by Antonelli-Spirito in [@AnSp0; @AnSp] considering Navier-Stokes-Korteweg systems without such relation between the shear viscosity and the capillary coefficient. The additional pressure $\delta\rho^{10}$ is used in thanks to $3\alpha_2-2\leq 10$. [*Second Step and main result concerning the compressible Navier-Stokes system.*]{} To prove global existence of weak solutions of the compressible Navier-Stokes equations, we follow the strategy introduced in [@LaVa; @VY]. To do so, first we approximate the viscosity $\mu$ by a viscosity $\mu_{\varepsilon_1}$ such that $\inf_{s\in [0,+\infty)} \mu_{\varepsilon_1}'(s)\ge \varepsilon_1 >0$. Then we use Theorem \[main result 1\] to construct a $\kappa$ entropy weak solution to the approximate system . We then show that this $\kappa$ entropy weak solution is a renormalized solution of in the sense introduced in [@LaVa]. More precisely we prove the following theorem: \[renorm\] Let $\mu(\rho)$ verifies –, $\lambda(\rho)$ given by . If $r_0>0$, then we assume also that ${\rm inf}_{s \in [0,+\infty)} \mu'(s)=\epsilon_1 >0$. Assume that $r_1$ is small enough compared to $r$ and $r_2$ is small enough compared to $\delta$, the initial values verify and $$\label{Initial conditions} \begin{split} & \int_\Omega \left(\rho_0\left(\frac{|{{ u}}_0+ 2\kappa \nabla s(\rho_0)|^2}{2}+(\kappa (1-\kappa)+r)\frac{|2\nabla s(\rho_0)|^2}{2}\right) \right)\, dx\\ &\hskip4cm +\int_\Omega \left(a \frac{\rho_0^\gamma}{\gamma-1}+ \mu(\rho_0) +\delta \frac{\rho^{10}}{9}+\frac{r_0}{\varepsilon_1}|(\ln \rho_0)_-|\right)\,dx <+\infty. \end{split}$$ Then the $\kappa$ entropy weak solutions is a renormalized solution of in the sense of Definition \[def\_renormalise\_u\]. We then pass to the limit with respect to the parameters $r,r_0,r_1,r_2$ and $\delta$ to recover a renormalized weak solution of the compressible Navier-Stokes equations and prove our main theorem. 0.3cm **Definitions**. Following [@LaVa] (based on the work in [@VY]), we will show the existence of renormalized solutions in ${{ u}}$. Then, we will show that this renormalized solution is a weak solution. The renormalization provides weak stability of the advection terms $\rho {{ u}}\otimes {{ u}}$ together and $\rho {{ u}}\otimes {{ v}}$. Let us first define the renormalized solution: 0.3cm \[def\_renormalise\_u\] Consider $\mu>0$, $3\lambda +2 \mu>0$, $r_0\geq0$, $r_1\geq0$, $r_2\ge 0$ and $r\geq0$. We say that $({{\sqrt{\rho}}},{{\sqrt{\rho}}}{{ u}})$ is a renormalized weak solution in ${{ u}}$, if it verifies -, and for any function ${{\varphi}}\in W^{2,\infty}({{\mathbb R}}^d)$ with $\varphi(s)s \in L^{\infty}({{\mathbb R}}^d)$, there exists three measures $R_{{{\varphi}}}, \overline{R}^1_{{\varphi}}, \overline{R}^2_{{\varphi}}\in \mathcal{M}({{\mathbb R}}^+\times{\Omega})$, with $$\|R_{{{\varphi}}}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})}+ \|\overline{R}^1_{{{\varphi}}}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})} + \|\overline{R}^2_{{{\varphi}}}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})} \leq C \|{{\varphi}}''\|_{L^\infty({{\mathbb R}})},$$ where the constant $C$ depends only on the solution $({{\sqrt{\rho}}},{{\sqrt{\rho}}}{{ u}})$, and for any function $\psi\in {C^\infty_c({{\mathbb R}}^+\times{\Omega})}$, $$\begin{aligned} &&\int_0^T \int_{\Omega}\left(\rho \psi_t + \sqrt \rho \sqrt \rho {{ u}}\cdot \nabla\psi \right)dx\, dt=0,\\ &&\int_0^T \int_{\Omega}\bigl( \rho {{\varphi}}({{ u}}) \psi_t + \rho {{\varphi}}({{ u}})\otimes {{ u}}:\nabla \psi \bigr) \> dx\, dt\\ && \hskip.2cm - \int_0^T \int_\Omega \left( 2 (\sqrt{\mu(\rho)} {{\mathbb S}_\mu}+ \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)}\mathbb S_\mu) {\rm Id}) \, {{\varphi}}'({{ u}}) \right)\cdot \nabla\psi \, dx dt \\ && \hskip.2cm - \, r \int_0^T \int_\Omega \left(2(\sqrt{\mu(\rho)} \mathbb S_r + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)}\mathbb S_r) {\rm Id}\bigr)\, {{\varphi}}'({{ u}}) \right) \cdot \nabla\psi \, dx dt\\ &&\hskip7cm +F(\rho,{{ u}})\, {{\varphi}}'({{ u}}) \psi \, dx\, dt=\left \langle R_{{{\varphi}}}, \psi\right\rangle, \\ && \int_0^T \int_{\Omega}(\mu(\rho) \psi_t + \frac{\mu(\rho)}{\sqrt \rho} \sqrt \rho {{ u}}\cdot \nabla \psi) \, dx dt - \int_0^T \int_\Omega \frac{\lambda(\rho)}{2\mu(\rho)} {\mathrm Tr} (\sqrt{\mu(\rho)} {{\mathbb T}_\mu}) \psi \, dx dt = 0,\end{aligned}$$ where ${{\mathbb S}_\mu}$ is given in and $\mathbb{T}_\mu$ is given in . The matrix $\mathbb S_r$ is compatible in , , and . The vector valued function $F$ is given by $$\label{eq_F} \begin{split} F(\rho,{{ u}}) & = \sqrt{\frac{P'(\rho) \rho }{\mu'(\rho)}} \nabla \int_0^\rho \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds \\ & \hskip.5cm + \delta\sqrt{\frac{P_\delta'(\rho) \rho }{\mu'(\rho)}} \nabla \int_0^\rho \sqrt{\frac{P_\delta'(s)\mu'(s)}{s}}\, ds -r_0 {{ u}}- r_1 \rho|{{ u}}|{{ u}}-\frac{r_2}{\mu'(\rho)}\rho|{{ u}}|^2{{ u}}. \end{split}$$ For every $i,j,k$ between 1 and $d$: $$\label{eq_viscous_renormaliseAAA} \sqrt{\mu(\rho)}{{\varphi}}_i'({{ u}})[{{\mathbb T}_\mu}]_{jk}= \partial_j(\mu(\rho)\rho{{\varphi}}'_i({{ u}}){{ u}}_k) -{{\sqrt{\rho}}}\ u_k{{\varphi}}'_i({{ u}}) \sqrt\rho \partial_j s(\rho)+ \overline{R}^1_{{\varphi}},$$ $$\label{eq_kortweg_renormalise} r{{\varphi}}_i'({{ u}})[\nabla(\sqrt{\mu(\rho)} \nabla Z(\rho))]_{jk}= r\partial_j(\sqrt{\mu(\rho)} {{\varphi}}'_i({{ u}})\partial_k Z(\rho))+ \overline{R}^2_{{\varphi}},$$ and $$\|\overline{R}^1_{{\varphi}}\|_{\mathcal{M}({{\mathbb R}}^+\times{\Omega})} + \|\overline{R}^2_{{\varphi}}\|_{\mathcal{M}({{\mathbb R}}^+\times{\Omega})} + \|R_{{\varphi}}\|_{\mathcal{M}({{\mathbb R}}^+\times{\Omega})} \leq C\|{{\varphi}}''\|_{L^\infty}.$$ and for any $\overline{\psi}\in C^\infty_c({\Omega})$: $$\begin{aligned} &&\lim_{t\to0}\int_{\Omega}\rho(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}\rho_0(x)\overline{\psi}(x)\,dx,\\ &&\lim_{t\to0}\int_{\Omega}\rho(t,x){{ u}}(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}m_0 (x)\overline{\psi}(x)\,dx,\\ && \lim_{t\to0}\int_{\Omega}\mu(\rho)(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}\mu(\rho_0)(x)\overline{\psi}(x)\,dx\end{aligned}$$ We define a global weak solution of the approximate system or the compressible Navier-Stokes equation (when $r=r_0=r_1=r_2=\delta=0$) as follows \[defweak\] Let ${\mathbb S}_\mu$ the symmetric part of $\mathbb {T}_\mu$ in $L^2((0,T)\times {\Omega})$ verifying – and $\mathbb{S}_r$ the capillary quantity in $L^2((0,T)\times {\Omega})$ given by –. Let us denote $P(\rho) = a \rho^\gamma$ and $P_\delta (\rho) = \delta \rho^{10}$. We say that $(\rho,{{ u}})$ is a weak solution to –, if it satisfies the [*a priori*]{} estimates – and for any function $\psi \in {\mathcal C}_c^\infty ((0,T)\times \Omega)$ verifying $$\begin{split} & \int_0^T \int_\Omega (\rho \partial_t \psi + \rho {{ u}}\cdot \nabla \psi) \, dxdt= 0, \\ &\int_0^T \int_\Omega (\rho {{ u}}\partial_t \psi + \rho {{ u}}\otimes {{ u}}: \nabla \psi )\, dx dt \\ & \hskip1.5cm - \int_0^T \int_\Omega 2 ( \sqrt{\mu(\rho)} \mathbb{S}_\mu + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu) {\rm Id}) \cdot \nabla\psi \, dx dt \\ & \hskip1.5cm - r \int_0^T \int_\Omega 2 ( \sqrt{\mu(\rho)} \mathbb{S}_r + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_r) {\rm Id}) \cdot\nabla\psi \, dx dt \\ & \hskip7cm + F(\rho,{{ u}}) \, \psi \, dx dt = 0,\\ & \int_0^\infty \int_{\Omega}\left(\mu(\rho) \psi_t + \frac{\mu(\rho)}{\sqrt \rho} \sqrt \rho {{ u}}\cdot \nabla \psi\right) dx \, dt \\ &\hskip5cm - \int_0^T \int_\Omega \frac{\lambda(\rho)}{2\mu(\rho)} {\mathrm Tr} (\sqrt{\mu(\rho)}\mathbb T_\mu) \psi \, dx dt = 0, \end{split}$$ with $F$ given through and for any $\overline \psi \in {\mathcal C}_c^\infty({\Omega})$: $$\begin{aligned} &&\lim_{t\to0}\int_{\Omega}\rho(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}\rho_0(x)\overline{\psi}(x)\,dx,\\ &&\lim_{t\to0}\int_{\Omega}\rho(t,x){{ u}}(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}m_0 (x)\overline{\psi}(x)\,dx,\\ && \lim_{t\to0}\int_{\Omega}\mu(\rho)(t,x)\overline{\psi}(x)\,dx=\int_{\Omega}\mu(\rho_0)(x)\overline{\psi}(x)\,dx.\end{aligned}$$ [**Remark.**]{} As mentioned in [@BrGiLa], the equation on $\mu(\rho)$ is important: By taking $\psi= {\rm div} \varphi$ for all $\varphi \in {\mathcal C}_0^\infty$, we can write the equation satisfied by $\nabla \mu(\rho)$ namely $$\label{grad} \begin{split} \partial_t \nabla\mu(\rho) + {\rm div}(\nabla\mu(\rho) \otimes {{ u}}) & = {\rm div}(\nabla\mu(\rho) \otimes {{ u}}) - \nabla {\rm div} (\mu(\rho) {{ u}}) \\ & \hskip3cm - \nabla\bigl( \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)}{\mathbb T}_\mu)\Bigr) \\ & = - {\rm div}(\sqrt{\mu(\rho)} {}^t{\mathbb T}_\mu) - \nabla\bigl( \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)}{\mathbb T}_\mu)\Bigr). \\ \end{split}$$ This will justify in some sense the two-velocities formulation introduced in [@BDZ] with the extra velocity linked to $\nabla\mu(\rho)$. The first level of approximation procedure ========================================== The goal of this section is to construct a sequence of approximated solutions satisfying the compactness structure to prove Theorem \[main result 1\] namely the existence of weak solutions of the approximation system with capillarity and drag terms. Here we present the first level of approximation procedure. 1\. The continuity equation $$\label{approximation of the continuity equation} \begin{split}&\rho_t+{{\rm div}}(\rho[\w]_{\varepsilon_3})=2\kappa{{\rm div}}\left([\mu'(\rho)]_{\varepsilon_4}\nabla\rho\right), \end{split}$$ with modified initial data $$\rho(0,x)=\rho_0\in C^{2+\nu}(\bar{{\Omega}}), \quad0<\underline{\rho}\leq \rho_0(x)\leq \bar{\rho}.$$ Here $\varepsilon_3$ and $\varepsilon_4$ denote the standard regularizations by mollification with respect to space and time. This is a parabolic equation recalling that in this part ${\rm Inf}_{[0,+\infty)} \mu'(s) >0$. Thus, we can apply the standard theory of parabolic equation to solve it when $\w$ is given smooth enough. In fact, the exact same equation was solved in paper [@BDZ]. In particular, we are able to get the following bound on the density at this level approximation $$\label{low and upper bound on density} 0<\underline{\rho}\leq \rho(t,x)\leq\bar{\rho}<+\infty.$$ 2\. The momentum equation with drag terms is replaced by its Faedo-Galerkin approximation with the additional regularizing term $\varepsilon_2[ \Delta^{2s}\w -{{\rm div}}((1+|\nabla w|^2)\nabla w)]$ where $s\ge 2$ $$\begin{split} \label{approximation of the momentum equation} &\int_{{\Omega}}\rho\w\cdot\psi\,dx-\int_0^t\int_{{\Omega}}\left(\rho([\w]_{\varepsilon_3}-2\kappa\frac{[\mu'(\rho)]_{\varepsilon_4}}{\rho}\nabla\rho)\otimes\w\right):\nabla\psi\,dx\,dt \\&+2(1-\kappa)\int_0^t\int_{{\Omega}}\mu(\rho)\mathbb{D}\w:\nabla\psi\,dx\,dt+2\kappa\int_0^t\int_{{\Omega}}\mu(\rho)\mathbf{A}(w):\nabla\psi\,dx\,dt \\&+(1-\kappa)\int_0^t\int_{{\Omega}}\lambda(\rho){{\rm div}}\w{{\rm div}}\psi\,dx\,dt-2\kappa(1-\kappa)\int_0^t\int_{{\Omega}}\mu(\rho)\nabla{{ v}}:\nabla\psi\,dx\,dt \\&-\kappa(1-\kappa)\int_0^t\int_{{\Omega}}\lambda(\rho){{\rm div}}{{ v}}{{\rm div}}\psi\,dx\,dt-\int_0^t\int_{{\Omega}}\rho^{\gamma}{{\rm div}}\psi\,dx\,dt -\delta\int_0^t\int_{{\Omega}}\rho^{10}{{\rm div}}\psi\,dx\,dt \\&+\varepsilon_2\int_0^t\int_{{\Omega}}\left( \Delta^s\w\cdot\Delta^s\psi+(1+|\nabla \w|^2)\nabla\w:\nabla \psi\right)\,dx\,dt= -\int_0^t\int_{{\Omega}} r_0 (\w-2\kappa\nabla{s}(\rho))\cdot\psi\,dx\,dt \\ & -r_1\int_0^t\int_{{\Omega}} \rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho))\cdot\psi\,dx\,dt \\&-r_2\int_0^t\int_{{\Omega}}\frac{\rho}{\mu'(\rho)}|\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho))\cdot\psi\,dx\,dt \\&-r\int_0^t\int_{{\Omega}} \sqrt{K(\rho)}\D(\int_0^{\rho}\sqrt{K(s)}\,ds){{\rm div}}(\rho\psi)\,dx\,dt+\int_{{\Omega}}\rho_0\w_0\cdot\psi\,dx \end{split}$$ satisfied for any $t>0$ and any test function $\psi\in C([0,T],X_n)$, where $\lambda (\rho)= 2(\mu'(\rho)\rho-\mu(\rho))$, and ${s}'(\rho)= \mu'(\rho) /\rho $, and $X_n=\text{span}\{e_i\}_{i=1}^{n}$ is an orthonormal basis in $W^{1,2}({\Omega})$ with $e_i\in C^{\infty}({\Omega})$ for any integers $i>0$. 3\. The Faedo-Galerkin approximation for the equation on the drift velocity ${v}$ reads $$\begin{split} \label{artificial equation} &\int_{{\Omega}}\rho{{ v}}\cdot\phi\,dx-\int_0^t\int_{{\Omega}}(\rho ([\w]_{\varepsilon_3}-2\kappa\frac{[\mu'(\rho)]_{\varepsilon_4}}{\rho} \nabla\rho)\otimes{{ v}}):\nabla\phi\,dx\,dt \\&+2\kappa\int_0^t\int_{{\Omega}}\mu(\rho)\nabla{{ v}}:\nabla\phi\,dx\,dt + \kappa\int_0^t\int_{{\Omega}} \lambda(\rho){{\rm div}}{{ v}}\, {{\rm div}}\phi\,dx\,dt \\&-\int_0^t\int_{{\Omega}} \lambda(\rho){{\rm div}}\w{{\rm div}}\phi\,dx\,dt +2\int_0^t\int_{{\Omega}}\mu(\rho)\nabla^T\w:\nabla\phi\,dx\,dt =\int_{{\Omega}}\rho_0{{ v}}_0\cdot\phi\,dx \end{split}$$ satisfied for any $t>0$ and any test function $\phi\in C([0,T],Y_n)$, where $Y_n=\text{span}\{b_i\}_{i=1}^n$ and $\{b_i\}_{i=1}^{\infty}$ is an orthonormal basis in $W^{1,2}({\Omega})$ with $b_i\in C^{\infty}({\Omega})$ for any integers $i>0.$\ The above full approximation is similar to the ones in [@BDZ]. We can repeat the same argument as their paper to obtain the local existence of solutions to the Galerkin approximation. In order to extend the local solution to the global one, the uniform bounds are necessary so that the corresponding procedure can be iterated. The energy estimate if the solution is regular enough. ------------------------------------------------------ For any fixed $n>0,$ choosing test functions $\psi=\w, \,\phi={{ v}}$ in and , we find that $(\rho,\w,{{ v}})$ satisfies the following $\kappa-$entropy equality $$\label{entropy for first level approximation} \begin{split} &\int_{{\Omega}}\left(\rho\left(\frac{|\w|^2}{2}+(1-\kappa)\kappa\frac{|{{ v}}|^2}{2}\right)+\frac{\rho^{\gamma}}{\gamma-1}+\delta\frac{\rho^{10}}{9}\right)\,dx +2(1-\kappa)\int_0^t \int_{{\Omega}}\mu(\rho)|\mathbb{D}\w-\kappa\nabla{{ v}}|^2\,dx\,dt \\ &+ (1-\kappa)\int_0^t\int_{{\Omega}} \lambda(\rho)({{\rm div}}\w-\kappa{{\rm div}}{{ v}})^2\,dx\,dt+ +2\kappa\int_0^t\int_{{\Omega}}\frac{\mu'(\rho)p'(\rho)}{\rho}|\nabla\rho|^2\,dx\,dt \\&+2\kappa\int_0^t\int_{{\Omega}}\mu(\rho)|A\w|^2\,dx\,dt+\varepsilon_2\int_0^t\int_{{\Omega}}\left( |\Delta^s\w|^2+(1+|\nabla \w|^2)|\nabla \w|^2\right)\,dx\,dt \\&+ r \int_0^t \int_{{\Omega}} \sqrt{K(\rho)} \Delta (\int_0^\rho \sqrt{K(s)}\, ds) {\rm div}(\rho w) \,dx\,dt +20\kappa\int_0^t\int_{{\Omega}}\mu'(\rho)\rho^8|\nabla\rho|^2\,dx\,dt \\& + r_0 \int_0^t \int_{\Omega}(w-2\kappa\nabla{s}(\rho))\cdot w \, dx\,dt + r_1 \int_0^t\int_{{\Omega}} \rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho))\cdot\w\,dx\,dt \\&+ r_2 \int_0^t\int_{{\Omega}}\frac{\rho}{\mu '(\rho)}|\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho))\cdot\w\,dx\,dt \\&= \int_{{\Omega}}\left(\rho_0\left(\frac{|\w_0|^2}{2}+(1-\kappa)\kappa\frac{|{{ v}}_0|^2}{2}\right)+\frac{\rho_0^{\gamma}}{\gamma-1}+\delta\frac{\rho_0^{10}}{9}\right)\,dx-\int_0^T\int_{{\Omega}}\rho^{\gamma}{{\rm div}}([\w]_{\varepsilon_3}-\w)\,dx\,dt \\&-\delta\int_0^T\int_{{\Omega}}\rho^{10}{{\rm div}}([\w]_{\varepsilon_3}-\w)\,dx\,dt, \end{split}$$ where ${s}'= {\mu'(\rho)}/{\rho}$ and $p(\rho)=\rho^{\gamma}.$ Compared to the calculations made in [@BDZ], we have to take care of the capillary term and then to take care of the drag terms showing that they can be controlled using that $\int_{s\in [0,T]} \mu'(s) \ge \varepsilon_1$ for the linear drag, using the extra pressure term $\delta \rho^{10}$ for the quadratic drag term and using the capillary term $r \rho \nabla(\sqrt{K(\rho)} \Delta (\int_0^\rho \sqrt{K(s)})$ for the cubic drag term. To do so, let us provide some properties on the capillary term and rewrite the terms coming from the drag quantities. ### Some properties on the capillary term Using the mass equation, the capillary term in the entropy estimates reads $$\begin{split} &\int_\Omega \sqrt{K(\rho)} \Delta(\int_0^\rho \sqrt{K(s)} \, ds)\, {{\rm div}}(\rho w) = \frac{r}{2} \frac{d}{dt}\int_\Omega |\nabla \int_0^\rho \sqrt{K(s)} \, ds|^2 \\ & + 2\kappa \int_{{\Omega}} \sqrt{K(\rho)} \Delta(\int_0^\rho \sqrt{K(s)} \, ds)\, \Delta \mu(\rho) = I_1 + I_2 . \end{split}$$ In fact, we write term $I_1$ as follows $$\frac{r}{2} \frac{d}{dt}\int_\Omega |\nabla \int_0^\rho \sqrt{K(s)} \, ds|^2 =\frac{r}{2} \frac{d}{dt}\int_\Omega \rho|\nabla{s}(\rho)|^2\,dx.$$ By , we have $$\begin{split} I_2 &= \int_{{\Omega}} \sqrt{K(\rho)} \Delta(\int_0^\rho \sqrt{K(s)} \, ds)\, \Delta \mu(\rho) \\&= - \int_{{\Omega}} \rho \nabla \Bigl( \sqrt{K(\rho)} \Delta(\int_0^\rho \sqrt{K(s)} \, ds)\Bigr) \cdot \nabla {s}(\rho) \\& = \int_{{\Omega}} 2 \mu(\rho) |2\nabla^2 {s}(\rho)|^2 + \lambda(\rho)|2\Delta {s}(\rho)|^2. \end{split}$$ [*Control of norms using $I_2$.*]{} Let us first recall that since $$\lambda(\rho) = 2(\mu'(\rho)\rho- \mu(\rho)) > -2\mu(\rho)/3,$$ there exists $\eta >0$ such that $$2 \int_0^T\int_{{\Omega}}\mu(\rho)|\nabla^2{s}(\rho)|^2\,dx\,dt + \int_0^T\int_{{\Omega}}\lambda(\rho)|\Delta{s}(\rho)|^2\,dx\,dt$$ $$\hskip3cm \ge \eta \Bigl[ 2 \int_0^T\int_{{\Omega}}\mu(\rho)|\nabla^2{s}(\rho)|^2\,dx\,dt + \frac{1}{3}\int_0^T\int_{{\Omega}}\mu(\rho)|\Delta{s}(\rho)|^2\,dx\,dt \Bigr].$$ As the second term in the right-hand side is positive, lower bound on the quantity $$\label{kor} \int_0^T\int_{{\Omega}}\mu(\rho)|\nabla^2{s}(\rho)|^2\,dx\,dt$$ will provide the same lower bound on $I_2$. Let us now precise the norms which are controlled by . To do so, we need to rely on the following lemma on the density. In this lemma, we prove a more general entropy dissipation inequality than the one introduced by Jüngel in [@J] and more general than those by Jüngel-Matthes in [@JuMa]. \[Lemma on jungel type inequality\] Let $\mu'(\rho)\rho<k\mu(\rho)$ for $2/3<k<4$ and $${s}(\rho)= \int_0^\rho \frac{\mu'(s)}{s} \, ds, \qquad Z(\rho) =\int_0^\rho \frac{\sqrt{\mu(s)}}{s}\mu'(s)\, ds, \qquad Z_1(\rho) = \int_0^\rho \frac{\mu'(s)}{(\mu(s))^{1/4}s^{1/2}} \, ds.$$ [i)]{} Assume $\rho>0$ and $\rho\in L^2(0,T;H^2(\Omega))$ then there exists $\varepsilon(k) >0$, such that we have the following estimate $$\int_0^T\int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx\,dt+\varepsilon(k)\int_0^T\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^{3}}|\nabla Z(\rho)|^4\,dx\,dt \leq \frac{C}{\varepsilon(k)} \int_0^T\int_{{\Omega}}\mu(\rho)|\nabla^2{s}(\rho)|^2\,dx\,dt,$$ where $C$ is a universal positive constant. [ii)]{} Consider a sequence of smooth densities $\rho_n>0$ such that $Z(\rho_n)$ and $Z_1(\rho_n)$ converge strongly in $L^1((0,T)\times\Omega)$ respectively to $Z(\rho)$ and $Z_1(\rho)$ and $\sqrt{\mu(\rho_n)} \nabla^2 {\mathbf s}(\rho_n)$ is uniformly bounded in $L^2((0,T)\times\Omega)$. Then $$\int_0^T\int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx\,dt+\varepsilon(k)\int_0^T\int_{{\Omega}}|\nabla Z_1(\rho)|^4\,dx\,dt \leq C < +\infty$$ The case of $Z=2\sqrt{\rho}$ for the inequality was proved in [@J], which is critical to derive the uniform bound on approximated velocity in $L^2(0,T;L^2({\Omega}))$ in [@VY-1; @VY]. The above lemma will play a similar role in this paper. Let us first prove the part i). Note that $Z'(\rho)=\frac{\sqrt{\mu(\rho)}}{\rho}\mu'(\rho)$, we get the following calculation: $$\begin{split} \label{key-1} \sqrt{\mu(\rho)}\nabla^2s(\rho)&=\sqrt{\mu(\rho)}\nabla(\frac{\nabla\mu(\rho)}{\rho})=\sqrt{\mu(\rho)}\nabla\left(\frac{1}{\sqrt{\mu(\rho)}}\nabla Z(\rho)\right) \\&=\nabla^2 Z(\rho)-\frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}}\otimes\nabla\sqrt{\mu(\rho)} \\&=\nabla^2 Z(\rho) -\frac{\rho\nabla Z(\rho)\otimes \nabla Z(\rho)}{2 \mu(\rho)^{\frac{3}{2}}}. \end{split}$$ Thus, we have $$\begin{split} \label{key-2} \int_{{\Omega}}\mu(\rho)|\nabla^2s(\rho)|^2\,dx&=\int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx +\frac{1}{4}\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx \\&- \int_{{\Omega}}\frac{\rho}{\mu(\rho)^{\frac{3}{2}}}\nabla^2Z(\rho) :(\nabla Z(\rho)\otimes \nabla Z(\rho))\,dx. \end{split}$$ By integration by parts, the cross product term reads as follows $$\begin{split} \label{key-3}&-\int_{{\Omega}}\frac{\rho}{\mu(\rho)^{\frac{3}{2}}}\nabla^2Z(\rho):(\nabla Z(\rho)\otimes \nabla Z(\rho))\,dx \\ & = -\int_{{\Omega}}\frac{\rho\sqrt{\mu(\rho)}}{\mu(\rho)}\nabla^2Z(\rho):(\frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}}\otimes \frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}})\,dx \\&=\int_{{\Omega}}\frac{\rho}{\mu(\rho)}\sqrt{\mu(\rho)}\nabla Z(\rho)\cdot{{\rm div}}(\frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}}\otimes \frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}})\,dx \\ & \hskip1cm +\int_{{\Omega}}\nabla(\frac{\rho}{\sqrt{\mu(\rho)}})\otimes\nabla Z(\rho):\frac{\nabla Z(\rho)\otimes \nabla Z(\rho)}{\mu(\rho)}\,dx \\&=I_1+I_2. \end{split}$$ To this end, we are able to control $I_1$ directly, $$\label{key-I1}\begin{split} |I_1|&\leq \varepsilon\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx + \frac{C}{\varepsilon} \int_{{\Omega}}\mu(\rho)|\nabla(\frac{\nabla Z(\rho)}{\sqrt{\mu(\rho)}})|^2\,dx \\&\leq \varepsilon\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx + \frac{C}{\varepsilon}\int_{{\Omega}}\mu(\rho)|\nabla^2 s(\rho)|^2\,dx, \end{split}$$ where $C$ is a universal positive constant. We calculate $I_2$ to have $$\label{key-I2} \begin{split} I_2&=\int_{{\Omega}}\nabla(\frac{\rho}{\sqrt{\mu(\rho)}})\otimes\nabla Z(\rho):\frac{\nabla Z(\rho)\otimes \nabla Z(\rho)}{\mu(\rho)}\,dx \\&=\int_{{\Omega}}\frac{\nabla\rho\otimes\nabla Z(\rho)}{\mu(\rho)^{\frac{3}{2}}}:\left(\nabla Z(\rho)\otimes \nabla Z(\rho)\right)\,dx \\ &\hskip2cm -\int_{{\Omega}}\frac{\rho}{\mu(\rho)^2}\nabla\sqrt{\mu(\rho)}\otimes \nabla Z(\rho):\left(\nabla Z(\rho)\otimes \nabla Z(\rho)\right)\,dx \\&=\int_{{\Omega}}\frac{\rho}{\mu(\rho)^2\mu(\rho)'}|\nabla Z(\rho)|^4\,dx-\frac{1}{2}\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx. \end{split}$$ Relying on -, we have $$\begin{split}& \int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx+\int_{{\Omega}}\frac{\rho}{\mu(\rho)^2\mu'(\rho)}|\nabla Z(\rho)|^4\,dx -(\frac{1}{4}+\varepsilon)\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx \\&\leq \frac{C}{\varepsilon} \int_{{\Omega}}\mu(\rho)|\nabla^2 s(\rho)|^2\,dx. \end{split}$$ Since $k_1\mu'(s) s\leq \mu(s),$ we have $$\frac{s}{\mu^2(s)\mu'(s)}-(\frac{1}{4}+\varepsilon)\frac{s^2}{\mu(s)^3}\geq (k_1-\frac{1}{4}-\varepsilon)\frac{s^2}{\mu(s)^3}>\varepsilon\frac{s^2}{\mu(s)^3},$$ where we choose $k_1>\frac{1}{4}$. This implies $$\int_{{\Omega}}|\nabla^2Z(\rho)|^2\,dx+\varepsilon\int_{{\Omega}}\frac{\rho^2}{\mu(\rho)^3}|\nabla Z(\rho)|^4\,dx \leq \frac{C}{\varepsilon} \int_{{\Omega}}\mu(\rho)|\nabla^2 s(\rho)|^2\,dx.$$ This ends the proof of part i). Concerning part ii), it suffices to pass to the limit in the inequality proved previously using the lower semi continuity on the left-hand side. ### Drag terms control. We have to discuss three kind of drag terms: Linear drag term, quadratic drag term and finally cubic drag term. [a) *Linear drag terms.*]{} As in previous works [@BD; @VY-1; @Z], we need to choose a linear drag with constant coefficient $$\begin{split} \label{11additional velocity term-control} &r_0\int_0^t\int_{{\Omega}}(\w-2\kappa\nabla{s}(\rho))\cdot\w\,dx\,dt =r_0\int_0^t\int_{{\Omega}}|\w-2\kappa\nabla{s}(\rho)|^2\,dx\,dt \\& +r_0\int_0^t\int_{{\Omega}}(\w-2\kappa\nabla{s}(\rho)) \cdot(2\kappa\nabla{s}(\rho))\,dx\,dt. \end{split}$$ The second term on the right side of reads $$\begin{split}r_0\int_0^t\int_{{\Omega}}(\w-2\kappa\nabla{s}(\rho))&\cdot(2\kappa\nabla{s}(\rho))\,dx\,dt =r_0\int_0^t\int_{{\Omega}}\rho(\w-2\kappa\nabla{s}(\rho))\cdot\frac{2\kappa\nabla{s}(\rho)}{\rho}\,dx\,dt \\& =r_0\int_0^t\int_{{\Omega}}\rho(\w-2\kappa\nabla{s}(\rho))\cdot2\kappa\nabla g(\rho)\,dx\,dt\\& =r_0\int_0^t\int_{{\Omega}}\rho_t g(\rho)\,dx\,dt, \end{split}$$ where $g'(\rho)= \frac{s'(\rho)}{\rho}=\frac{\mu'(\rho)}{\rho^2}$ and $g(\rho)=\int_1^\rho\frac{\mu'(r)}{r^2}\,dr.$ Letting $$G(\rho)=\int_1^\rho\int_1^r\frac{\mu'(\zeta)}{\zeta^2}\,d\zeta\,dr,$$ then $$r_0\int_{{\Omega}}\rho_t g(\rho)\,dx= r_0\frac{\partial}{\partial_t}\int_{{\Omega}}G(\rho)\,dx,$$ which implies $$r_0\int_0^t\int_{{\Omega}}\rho_t g(\rho)\,dx\,dt= r_0\int_{{\Omega}}G(\rho)\,dx.$$ Meanwhile, since $\lim_{\zeta\to 0}\mu'(\zeta)=\varepsilon_1>0$, for any $|\zeta|<\epsilon$ and any small number $\epsilon>0$, we have $\mu'(\zeta)\geq \frac{\varepsilon_1}{2}.$ Thus, we have further estimate on $G(\rho)$ as follows $$\begin{split} G(\rho)=\int_1^\rho\int_1^r\frac{\mu'(\zeta)}{\zeta^2}\,d\zeta\,dr &\geq \frac{\varepsilon_1}{2}\int_1^\rho(1-\frac{1}{r})\,dr \\&= \frac{\varepsilon_1}{2}(\rho-1-\ln\rho) \\&\geq -\frac{\varepsilon_1}{4}(\ln\rho)_{-}, \end{split}$$ for any $\rho\leq \epsilon$. Similarly, we can show that $$G(\rho)\leq 4\varepsilon_1(\ln\rho)_{+}$$ for any $\rho\leq \epsilon$. For given number $\epsilon_0>0$, if $\rho\geq \epsilon_0$, then we have $$0\leq G(\rho)\leq C\int_1^\rho\int_1^r\mu'(\zeta)\,d\zeta\,dr\leq C\mu(\rho)\rho.$$ [b) *Quadratic drag term.*]{} We use the same argument as in [@BDZ] to handle this term. The quadratic drag term gives $$\begin{split} \label{drag term control} &r_1\int_0^t\int_{{\Omega}} \rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho))\cdot\w\,dx\,dt \\&=r_1\int_0^t\int_{{\Omega}} \rho |\w-2\kappa\nabla{s}(\rho)|^3\,dx\,dt \\&\quad\quad\quad\quad+r_1\int_0^t\int_{{\Omega}} \rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho))\cdot(2\kappa\nabla{s}(\rho))\,dx\,dt. \end{split}$$ The second drag term of the right–hand side can be controlled as follows $$\label{the second term of drag term control} \begin{split} &r_1\left|\int_0^t\int_{{\Omega}}\rho|\w-2\kappa\nabla{s}(\rho)|(\w-2\kappa\nabla{s}(\rho)) \cdot(2\kappa\nabla{s}(\rho))\,dx\,dt\right| \\&\leq r_1\int_0^t\int_{{\Omega}}\mu(\rho)|{{ u}}||\mathbb{D}{{ u}}|\,dx\,dt\\ &\leq \frac{1}{2}\int_0^t\int_{{\Omega}}\mu(\rho)|\mathbb{D}{{ u}}|^2\,dx\,dt +\frac{r_1^2}{2}\int_0^t\int_{{\Omega}}\mu(\rho)|{{ u}}|^2\,dx\,dt, \end{split}$$ and $$\|\sqrt{\mu(\rho)}|{{ u}}|\|_{L^2(0,T;L^2({\Omega}))}\leq C\|\rho^{\frac{1}{3}}|{{ u}}|\|_{L^3(0,T;L^3({\Omega}))}\|\frac{\sqrt{\mu(\rho)}}{\rho^{\frac{1}{3}}}\|_{L^6(0,T;L^6({\Omega}))}.$$ Note that $$\begin{split} &\int_0^t\int_{{\Omega}}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt=\int_0^t\int_{0\leq \rho\leq 1}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt+ \int_0^t\int_{\rho\geq 1}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt \\&\leq C\int_0^t\int_{0\leq \rho\leq 1}\mu(\rho)(\mu'(\rho))^2\,dx\,dt+ \int_0^t\int_{\rho\geq 1}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt \\&\leq C+\int_0^t\int_{\rho\geq 1}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt. \end{split}$$ From , for any $\rho\geq 1$, we have $$c'\rho^{\alpha_1} \leq \mu(\rho)\leq c\rho^{\alpha_2},$$ where $2/3<\alpha_1\leq \alpha_2<4.$ This yields to $$\label{control preasuer}\int_0^t\int_{\rho\geq 1}\frac{\mu(\rho)^3}{\rho^2}\,dx\,dt\leq c\int_0^t\int_{\rho\geq 1}\rho^{3\,\alpha_2-2}\,dx\,dt \leq c \int_0^t \int_{{\Omega}}\rho^{10}\,dx$$ for any time $t>0.$ [c) *Cubic drag term.*]{} The non-linear cubic drag term gives $$\begin{split} \label{drag term control} &r_2\int_0^t\int_{{\Omega}} \frac{\rho}{\mu'(\rho) }|\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho))\cdot\w\,dx\,dt \\&=r_2\int_0^t\int_{{\Omega}} \frac{\rho}{\mu'(\rho) } |\w-2\kappa\nabla{s}(\rho)|^4\,dx\,dt \\&\quad\quad\quad\quad+r_2\int_0^t\int_{{\Omega}} \frac{\rho}{\mu'(\rho) }|\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho))\cdot(2\kappa\nabla{s}(\rho))\,dx\,dt. \end{split}$$ The novelty now is to show that we control the second drag term of the right–hand side using the Korteweg-type information on the left-hand side $$\label{the second term of drag term control} \begin{split} &r_2\int_0^t\int_{{\Omega}}\frac{\rho}{\mu'(\rho) }|\w-2\kappa\nabla{s}(\rho)|^2(\w-2\kappa\nabla{s}(\rho))\cdot(2\kappa\nabla{s}(\rho))\,dx\,dt \\&\le r_2 \Bigl( \frac{3}{4} \int_0^t \int_{{\Omega}} \frac{\rho}{\mu'(\rho)} |w-2\kappa \nabla {s}(\rho)|^4 + \frac{(2\kappa)^4}{4} \int_0^t \int_{{\Omega}} \frac{\rho}{\mu'(\rho)} |\nabla {s}(\rho)|^4 \Bigr). \end{split}$$ Remark that the first term in the right-hand side may be absorbed using the first term in . Let us now prove that if $r_1$ small enough, the second term in the right-hand side may be absorbed by the term coming from the capillary quantity in the energy. From Lemma \[Lemma on jungel type inequality\], we have $$\int_0^t\int_{{\Omega}}\frac{\rho^2}{\mu^{3}(\rho)}|\nabla Z(\rho)|^4\,dx\,dt=\int_0^t\int_{{\Omega}}\frac{1}{\mu(\rho)\rho^2}|\nabla\mu(\rho)|^4\,dx\,dt.$$ It remains to check that $$\int_0^t \int_{{\Omega}} \frac{\rho}{\mu'(\rho)} |\nabla {s}(\rho)|^4= \int_0^t\int_{{\Omega}}\frac{1}{\mu'(\rho)\rho^3}|\nabla\mu(\rho)|^4\,dx\,dt\leq C\int_0^t\int_{{\Omega}}\frac{1}{\mu(\rho)\rho^2}|\nabla\mu(\rho)|^4\,dx\,dt.$$ This concludes assuming $r_1$ small enough compared to $r$. ### The $\kappa$-entropy estimate. Using the previous calculations, assuming $r_2$ small enough compared to $r$, and denoting $$E[\rho,u+2\kappa \nabla \mathbf{s(\rho)}, \nabla \mathbf{s(\rho)}] = \int_{\Omega}\rho\left(\frac{|{{ u}}+2\kappa\nabla{s}(\rho)|^2}{2} + (1-\kappa)\kappa\frac{|\nabla{s}(\rho)|^2}{2}\right)+ \frac{\rho^{\gamma}}{\gamma-1}+\frac{\delta\rho^{10}}{9}+G(\rho),$$ we get the following $\kappa$-entropy estimate $$\label{entropy obtained} \begin{split} & E[\rho,u+2\kappa \nabla \mathbf{s(\rho)}, \nabla \mathbf{s(\rho)}](t) +r_0\int_0^t\int_{{\Omega}}|{{ u}}|^2\,dx\,dt \\&+\frac{r}{2}\int_{{\Omega}}|\nabla\int_0^{\rho}\sqrt{K(s)}\,ds|^2 \,dx +2(1-\kappa)\int_0^t\int_{{\Omega}}\mu(\rho)|\mathbb{D}{{ u}}|^2\,dx\,dt+ 20\kappa\int_0^t\int_{{\Omega}}\mu'(\rho)\rho^8|\nabla\rho|^2\,dx\,dt \\ &+2(1-\kappa)\int_0^t\int_{{\Omega}}(\mu'(\rho)\rho-\mu(\rho))({{\rm div}}{{ u}})^2\,dx\,dt+2\kappa\int_0^t\int_{{\Omega}}\mu(\rho)|A({{ u}}+2\kappa\nabla{s}(\rho))|^2\,dx\,dt \\&+2\kappa\int_0^t\int_{{\Omega}}\frac{\mu'(\rho)p'(\rho)}{\rho}|\nabla\rho|^2\,dx\,dt + r_1 \int_0^t \int_{{\Omega}} \rho|{{ u}}|^3\, dx\,dt + \frac{r_2}{4} \int_0^t \int_{{\Omega}} \frac{\rho}{\mu'(\rho)} |{{ u}}|^4 \, dx\, dt\\ & + \kappa r\int_0^t\int_{{\Omega}}\mu(\rho)|2 \nabla^2{s}(\rho)|^2\,dx\,dt + \frac{1}{2}\kappa r\int_0^t\int_{{\Omega}}\lambda(\rho)|2\Delta{s}(\rho)|^2\,dx\,dt \\&\leq \int_{{\Omega}}\left(\rho_0\left(\frac{|\w_0|^2}{2}+(1-\kappa)\kappa\frac{|{{ v}}_0|^2}{2}\right)+\frac{\rho_0^{\gamma}}{\gamma-1}+ \frac{\delta\rho_0^{10}}{9}+\frac{r}{2}|\nabla\int_0^{\rho_0} \sqrt{K(s)} \, ds|^2+G(\rho_0)\right)\,dx\\ & + C \frac{r_1}{\delta} \int_\Omega E[\rho,u+2\kappa \nabla \mathbf{s(\rho)}, \nabla \mathbf{s(\rho)}] dx \, dt . \end{split}$$ It suffices now to remark that $$\nonumber \begin{split} & \int_0^t\int_{\Omega}\mu(\rho) | \mathbb{D}{{ u}}|^2 + \int_0^t \int_{\Omega}(\mu'(\rho)\rho - \rho) |{\rm div} {{ u}}|^2 \\ & = \int_0^t\int_{\Omega}\mu(\rho) | \mathbb{D}{{ u}}-\frac{1}{3} {\rm div} {{ u}}\, {\rm Id}|^2 \, dx dt + \int_0^t \int_{\Omega}(\mu'(\rho)\rho - \mu(\rho) + \frac{1}{3}\mu(\rho)) |{\rm div} {{ u}}|^2 . \end{split}$$ Note that $\alpha_1>2/3$, there exists $\varepsilon>0$ such that $$\mu'(\rho)\rho - \frac{2}{3}\mu(\rho) > \varepsilon \mu(\rho).$$ Such information and the control of $\sqrt{\mu(\rho)} |A(u)+2\kappa\nabla {\mathbf s}(\rho)|$ in $L^2(0,T;L^2({\Omega}))$ allow us, using the Grönwall Lemma and the constraints on the parameters, to get the uniform estimates –. Now we can show . First, we have $$\nabla \mu(\rho) = \frac{\nabla \mu(\rho)}{\sqrt \rho} \sqrt \rho \in L^\infty(0,T;L^1(\Omega)),$$ due to the mass conservation and the uniform control on $\nabla\mu(\rho)/\sqrt\rho$ given in . Let us now write the equation satisfied by $\mu(\rho)$ namely $$\partial_t\mu(\rho) + {\rm div}(\mu(\rho) {{ u}}) + \frac{ \lambda(\rho)}{2} {\rm div} {{ u}}= 0.$$ Recalling that $\lambda(\rho) = 2( \mu'(\rho)\rho - \mu(\rho))$ and the hypothesis on $\mu(\rho)$, we get $$\frac{d}{dt} \int_\Omega \mu(\rho) \le C \, \bigl(\int_\Omega |\lambda(\rho)||{\rm div} {{ u}}|^2 + \int_\Omega \mu(\rho)\bigr),$$ and therefore $$\mu(\rho) \in L^\infty(0,T;L^1(\Omega)),$$ if $\mu(\rho_0) \in L^1(\Omega)$ due to the fact that $\sqrt{|\lambda(\rho)|}{\rm div} {{ u}}\in L^2(0,T;L^2(\Omega)).$ Now, we observe that $\mu(\rho)/\sqrt{\rho}$ is smaller than $1$ for $\rho\leq 1$ because $\alpha_1 > 2/3$, and smaller than $\mu(\rho)$ for $\rho_n>1$, then $$\frac{\mu(\rho)}{\sqrt{\rho}} \in L^\infty(L^1).$$ Meanwhile, thanks to , we have $$|\nabla( \mu(\rho)/\sqrt{\rho})|\leq \left|\frac{\nabla\mu(\rho)}{\sqrt{\rho}}\right|+\frac{\mu(\rho)}{2\rho\sqrt{\rho}}|\nabla\rho|\leq \left(1+\frac{1}{\alpha_1}\right)\left|\frac{\nabla\mu(\rho)}{\sqrt{\rho}}\right|.$$ By , $\nabla(\mu(\rho)/\sqrt{\rho})$ is bounded in $L^\infty(0,T;L^2(\Omega))$ and finally $\mu(\rho)/\sqrt{\rho}$ is bounded in $L^\infty(0,T;(L^6(\Omega))$. Thus, we have that $$\mu(\rho) {{ u}}= \frac{\mu(\rho)}{\sqrt\rho} \sqrt \rho{{ u}},$$ is uniformly bounded in $ L^\infty(0,T;L^{3/2}({\Omega})).$ Let us come back to the equation satisfied by $\mu(\rho)$ which reads $$\partial_t \mu(\rho) + {\rm div}(\mu(\rho) {{ u}}) + \frac{\lambda(\rho)}{2}{\rm div} {{ u}}= 0.$$ Recalling that $\lambda(\rho) {\rm div} {{ u}}\in L^\infty(0,T;L^1({\Omega}))$, then we get the conclusion on $\partial_t \mu(\rho)$. Let us now to prove that $$Z(\rho)= \displaystyle \int_0^{\rho_n} \frac{\sqrt{\mu(s)} \mu'(s)}{s} ds \in L^{1+}((0,T)\times {\Omega}) \hbox{ uniformly.}$$ Note first that $$0 \le \frac{\sqrt{\mu(s)} \mu'(s)}{s} \le \alpha_2 \frac{\mu(s)^{3/2}}{s^2} \le c_2 \alpha_2(s^{3\alpha_1/2-2} 1_{s\le 1} + \frac{\mu(s)^{3/2-}}{s^{2-}} 1_{s\ge 1}).$$ There exists $ \varepsilon>0 \hbox{ such that } \alpha_1 > 2/3+ \varepsilon,$ thus $$0 \le \frac{\sqrt{\mu(s)} \mu'(s)}{s} \le c_2 \alpha_2 ( s^{\varepsilon -1}1_{s\le 1} + \frac{\mu(s)^{3/2-}}{s^{2-}} 1_{s\ge 1}).$$ Note that $\mu'(s) > 0$ for $s>0$ and the definition of $Z(\rho)$, we get $$0\le Z(\rho) \le C (\rho^\varepsilon + \mu(\rho)^{3/2-})$$ with $C$ independent of $n$. Thus $Z(\rho) \in L^{\infty}(0,T; L^{1+}({\Omega}))$ uniformly with respect to $n$. Bound on $Z_1(\rho)$ follows the similar lines. Compactness Lemmas. ------------------- In this subsection, we provide general compactness lemmas which will be used several times in this paper. [*Some uniform compactness.*]{} \[compactuniforme\] Assume we have a sequence $\{\rho_n\}_{n\in \mathbb N}$ satisfying the estimates in Theorem \[main result 1\], uniformly with respect to $n$. Then, there exists a function $\rho \in L^\infty(0,T;L^\gamma({\Omega}))$ such that, up to a subsequence, $$\mu(\rho_n) \to \mu(\rho) \hbox{ in } {\mathcal C}([0,T]; L^{3/2}({\Omega}) \hbox{ weak}),$$ and $$\rho_n \to \rho \hbox{ a.e. in } (0,T)\times {\Omega}.$$ Moreover $$\rho_n \to \rho \hbox{ in } L^{(4\gamma/3)^+}((0,T)\times \Omega),$$ $$\sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \rightharpoonup \sqrt{\frac{P'(\rho)\rho}{\mu'(\rho)}} \nabla \displaystyle \Bigl(\int_0^{\rho} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \hbox{ in } L^{1}((0,T)\times {\Omega})$$ and $$\sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \in L^{1+}((0,T)\times\Omega).$$ If $\delta_n>0$ is such that $\delta_n\to \delta\geq 0$, then $$\delta_n\rho_n^{10}\to \delta\rho^{10}\quad\text{ in } L^{\frac{4}{3}}((0,T)\times{\Omega}).$$ [**Proof.**]{} From the estimate on $\mu(\rho_n)$ and Aubin-Lions lemma, up to a subsequence, we have $$\mu(\rho_n) \to \mu(\rho) \hbox{ in } {\mathcal C}([0,T]; L^{3/2}({\Omega}) \hbox{ weak})$$ and therefore using that $\mu'(s)>0$ on $(0,+\infty)$ with $\mu(0)=0$, we get the conclusion on $\rho_n$. Let us now recall that $$\label{muineq} \frac{\alpha_1}{\rho_n} \le \frac{\mu'(\rho_n)}{\mu(\rho)} \le \frac{\alpha_2}{\rho_n}$$ and therefore $$c_1 \rho_n^{\alpha_2} \le \mu(\rho_n) \le c_2 \rho_n^{\alpha_1} \qquad \hbox{ for } \rho_n \le 1,$$ and $$c_1 \rho_n^{\alpha_1} \le \mu(\rho_n) \le c_2 \rho_n^{\alpha_2} \qquad \hbox{ for } \rho\ge 1.$$ with $c_1$ and $c_2$ independent on $n$. Note that $$\label{estimpressure} \sqrt{\frac{p'(\rho_n)\mu'(\rho_n)}{\rho_n}}\nabla \rho_n \in L^\infty(0,T;L^2({\Omega})) \hbox{ uniformly.}$$ Let us prove that there exists $\varepsilon$ such that $$I_0= \displaystyle \int_0^T\int_\Omega \rho_n^{\frac{4\gamma}{3}+\varepsilon} < C$$ with $C$ independent on $n$ and the parameters. We first remark that it suffices to look at it when $\rho_n \ge 1$ and to remark there exists $\varepsilon$ such that $\varepsilon \le (\gamma-1)/3.$ Let us take such parameter then $$\int_0^T\int_\Omega \rho_n^{\frac{4\gamma}{3}+\varepsilon} 1_{\rho \ge 1} \le \int_0^T\int_\Omega \rho_n^{\frac{2\gamma}{3} + \gamma - \frac{1}{3}} 1_{\rho \ge 1} \le \int_0^T \int_{\Omega}\rho_n^{\frac{2\gamma}{3} + \gamma + \alpha_1 -1} 1_{\rho \ge 1}$$ recalling that $\alpha_1 >2/3.$ Following [@LiXi], it remains to prove that $$\displaystyle I_1= \int_0^T\int_{\Omega}\bigl[\rho_n^{[5\gamma + 3(\alpha_1-1)]/3} \, 1_{\rho \ge 1} \bigr] <+\infty$$ uniformly. Denoting $$I_2 = \int_0^T\int_{\Omega}\bigl[\rho^{[5\gamma + 3(\alpha_2-1)]/3} \, 1_{\rho \le 1} \bigr]$$ and using the bounds on $\mu(\rho_n)$ in terms of power functions in $\rho$, which are different if $\rho_n \ge 1$ or $\rho_n\le 1$, we can write: $$I_1 \le I_1 + I_2 \le C_a \int_0^T \int_{\Omega}\rho_n^{2\gamma/3} P'(\rho_n) \,\mu(\rho_n) \le C_a \int_0^T \|\rho_n^\gamma\|^{2/3}_{L^1({\Omega})}\|P'(\rho_n)\mu(\rho_n)\|_{L^3({\Omega})}$$ where $C$ does not depend on $n$. Using the Poincaré-Wirtinger inequality, one obtains that $$\begin{split}\|P'(\rho_n)\mu(\rho_n)\|_{L^3({\Omega})} &= \|\sqrt{P'(\rho_n) \mu(\rho_n)}\|_{L^6({\Omega})}^2 \\&\le \|\sqrt{P'(\rho_n)\mu(\rho_n)}\|_{L^1({\Omega})} + \|\nabla \bigl[\sqrt{P'(\rho_n)\mu(\rho_n)}\bigr]\|_{L^2({\Omega})}^2. \end{split}$$ Let us now check that the two terms are uniformly bounded in time. First we caculate $$\nabla \bigl[\sqrt{P'(\rho_n)\mu(\rho_n)}\bigr] = \frac{P''(\rho_n) \mu(\rho_n) + P'(\rho_n)\mu'(\rho_n)}{\sqrt{P'(\rho_n)\mu(\rho_n)} }\nabla \rho_n$$ and using , we can check that $$\frac{P''(\rho_n) \mu(\rho_n) + P'(\rho_n)\mu'(\rho_n)}{\sqrt{P'(\rho_n)\mu(\rho_n)} } \le \sqrt{\frac{P'(\rho_n)\mu'(\rho_n)}{\rho_n}}.$$ Therefore, using , uniformly with respect to $n$, we get $$\sup_{t\in [0,T]} \|\nabla \bigl[\sqrt{P'(\rho_n)\mu(\rho_n)}\bigr]\|_{L^2({\Omega})}^2 < + \infty.$$ Let us now check that uniformly with respect to $n$ $$\label{AAAestimm} \sup_{t\in [0,T]} \|\sqrt{P'(\rho_n)\mu(\rho_n)}\|_{L^1({\Omega})} < + \infty.$$ Using the bounds on $\mu(\rho_n)$, we have $$\int_{\Omega}\sqrt{P'(\rho_n)\mu(\rho_n)} \le C \int_{\Omega}\Bigl[\rho_n^{(\gamma-1+\alpha_1)/2} 1_{\rho_n \le 1} + \rho_n^{(\gamma-1+\alpha_2)/2} 1_{\rho_n \ge 1} \Bigr]$$ with $C$ independent on $n$. Recalling that $\alpha_1 \ge 2/3$ and $\alpha_2 < 4$, we can check that $$\int_{\Omega}\sqrt{P'(\rho_n)\mu(\rho_n)} \le C \int_{\Omega}\Bigl[\rho_n^{\gamma/3} + \rho_n^{\frac{\gamma}{2}}\rho_n^{\frac{3}{2}} \Bigr],$$ and therefore using that $\rho_n^\gamma \in L^\infty(0,T;L^1({\Omega}))$ and $\rho_n\in L^{\infty}(0,T;L^{10}({\Omega}))$, we get . This ends the proof of the convergence of $\rho_n$ to $\rho$ in $L^{(4\gamma/3)^+}((0,T)\times \Omega$. Let us now focus on the convergence of $$\label{weak convergence of product} \sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr).$$ First let us recall that $$\nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \in L^\infty(0,T;L^2(\Omega)) \hbox{ uniformly}.$$ Let us now prove that $$\label{estimm} \sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \in L^{2+}((0,T)\times \Omega).$$ Recall first that $\alpha_1 >\frac{2}{3}$, we just have to consider $\rho_n \ge 1$. We write $$\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)} 1_{\rho_n\ge 1} \le C \rho_n^{\gamma - \alpha_1 +1} 1_{\rho_n\ge 1} \le C \rho_n^{\gamma +1/3} 1_{\rho_n\ge 1} \le C \rho_n^{\frac{4\gamma}{3}} 1_{\rho_n \ge 1}.$$ We can use the fact that $\rho_n^{(4\gamma/3)^+} \in L^1((0,T)\times \Omega)$ uniformly to conclude on . Thanks to $$\sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \to \sqrt{\frac{P'(\rho)\rho}{\mu'(\rho)}} \hbox{ in } L^2((0,T)\times \Omega)$$ and $$\nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \to \nabla \displaystyle \Bigl(\int_0^{\rho} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr) \hbox{ weakly in } L^2((0,T)\times \Omega),$$ we have the weak convergence of in $ L^{1}((0,T)\times {\Omega})$. We now investigate limits on ${{ u}}$ independent of the parameters. We need to differentiate the case with hyper-viscosity ${\varepsilon}_2>0$, from the case without. In the case with hyper-viscosity, the estimate depends on ${\varepsilon}_1$ because of the drag force $r_1$, while the estimate in the case ${\varepsilon}_2=0$ is independent of all the other parameters. This is why we will consider the limit ${\varepsilon}_2$ converges to 0 first. \[lem u\] Assume that ${\varepsilon}_1>0$ is fixed. Then, there exists a constant $C>0$ depending on ${\varepsilon}_1$ and $C_{in}$, but independent of all the other parameters (as long as they are bounded), such that for any initial values $(\rho_0, \sqrt{\rho_0}u_0)$ verifying (\[Initial conditions\]) for $C_{in}>0$ we have $$\begin{aligned} &&\|\partial_t(\rho {{ u}})\|_{L^{1+}(0,T;W^{-s,2}(\Omega))}\leq C,\\ &&\|\nabla(\rho {{ u}})\|_{L^2(0,T;L^1(\Omega))}\leq C.\end{aligned}$$ Assume now that ${\varepsilon}_2=0$. Let $\Phi:{{\mathbb R}}^+\to {{\mathbb R}}$ be a smooth function, positive for $\rho>0$, such that $$\begin{aligned} &&\Phi(\rho)+|\Phi'(\rho)|\leq C e^{-\frac{1}{\rho}}, \qquad \mathrm{for} \ \rho\leq 1,\\ &&\Phi(\rho)+|\Phi'(\rho)|\leq C e^{-\rho}, \qquad \mathrm{for} \ \rho\geq 2.\end{aligned}$$ Assume that the initial values $(\rho_0, \sqrt{\rho_0}u_0)$ verify (\[Initial conditions\]) for a fixed $C_{in}>0$. Then, there exists a constant $C>0$ independent of ${\varepsilon}_1, r_0, r_1, r_2, \delta$ (as long as they are bounded), such that $$\begin{aligned} &&\|\partial_t\left[\Phi(\rho) {{ u}}\right]\|_{L^{1+}(0,T;W^{-2,1}(\Omega))}\leq C,\\ &&\|\nabla\left[\Phi(\rho) {{ u}}\right]\|_{L^2(0,T;L^1(\Omega))}\leq C.\end{aligned}$$ We split the proof into the two cases. 0.3cm [**Case 1:**]{} Assume that ${\varepsilon}_1>0$. From the equation on $\rho u$ and the [*a priori* ]{} estimates, we find directly that $$\|\partial_t (\rho {{ u}})\|_{L^{1+}(0,T;W^{-s,2}(\Omega))}\leq C+ r_1^{1/4} \frac{\|\rho\|^{1/4}_{L^1((0,T)\times\Omega)}}{\|\mu'(\rho)\|_{L^\infty((0,T)\times\Omega)}}\left(r_1\int_0^T\int_\Omega \rho |{{ u}}|^4\,dx\,dt \right)^{3/4}\leq C(1+1/{\varepsilon}_1).$$ We have $\mu(\rho)\geq {\varepsilon}_1 \rho$, and from (\[priori estimates\]), we have the [*a priori*]{} estimate $$\|\nabla \sqrt{\rho}\|^2_{L^\infty(0,T;L^2(\Omega))}\leq \frac{C}{{\varepsilon}_1}.$$ Hence $$\begin{aligned} \|\nabla(\rho {{ u}})\|_{L^2(0,T;L^1(\Omega))} && \leq \left\|\frac{\rho}{\sqrt{\mu}(\rho)}\right\|_{L^\infty(0,T;L^2(\Omega))} \left\|\sqrt{\mu}(\rho)\nabla u\right\|_{L^2(0,T;L^2(\Omega)))} \\ && \> +2\|\nabla \sqrt{\rho}\|_{L^\infty(0,T;L^2(\Omega))} \|\sqrt{\rho} {{ u}}\|_{L^\infty(0,T;L^2(\Omega))}\\ &&\> \leq C.\end{aligned}$$ 0.3cm [**Case 2**]{}: Assume now that ${\varepsilon}_2=0$. Multiplying the equation on $(\rho u)$ by $\Phi(\rho)/\rho$, we get, as for the renormalization, that $$\|\partial_t\left[\Phi(\rho){{ u}}\right]\|_{L^{1+}(0,T;W^{-2,1}(\Omega))}\leq C.$$ Note that $$\begin{aligned} && \|\nabla\left[\Phi(\rho) {{ u}}\right]\|_{L^2(0,T;L^1(\Omega))}\leq \left\|\frac{\Phi(\rho)}{\sqrt{\mu}(\rho)}\right\|_{L^\infty} \left\|\sqrt{\mu}(\rho)\nabla {{ u}}\right\|_{L^2(L^2)}\\ &&\qquad\qquad +2\| \frac{\Phi'(\rho)}{\mu'(\rho)}\|_{L^\infty((0,T)\times\Omega)} \|\mu'(\rho)\nabla \sqrt{\rho}\|_{L^\infty(0,T;L^2(\Omega))} \|\sqrt{\rho} {{ u}}\|_{L^\infty(0,T;L^2(\Omega))}\\ &&\qquad\qquad \leq C.\end{aligned}$$ \[Compactnesstool1\] Assume either that ${\varepsilon}_{2,n}=0$, or ${\varepsilon}_{1,n}={\varepsilon}_1>0$. Let $(\rho_n,\sqrt{\rho_n} {{ u}}_n)$ be a sequence of solutions for a family of bounded parameters with uniformly bounded initial values verifying (\[Initial conditions\]) with a fixed $C_{in}$. Assume that there exists $\alpha>0$, and a smooth function $h:{{\mathbb R}}^+\times{{\mathbb R}}^3\to{{\mathbb R}}$ such that $\rho_n^\alpha$ is uniformly bounded in $L^p((0,T)\times\Omega)$ and $h(\rho_n,{{ u}}_n)$ is uniformly bounded in $L^q((0,T)\times \Omega)$, with $$\frac{1}{p}+\frac{1}{q}<1.$$ Then, up to a subsequence, $\rho_n$ converges to a function $\rho$ strongly in $L^1$, $\sqrt{\rho_n}{{ u}}_n$ converges weakly to a function $q$ in $L^2$. We define ${{ u}}=q/\sqrt{\rho}$ whenever $\rho\neq 0$, and ${{ u}}=0$ on the vacuum where $\rho=0$. Then $\rho_n^\alpha h(\rho_n,{{ u}}_n)$ converges strongly in $L^1$ to $\rho^\alpha h(\rho, {{ u}})$. Thanks to the uniform bound on the kinetic energy $\int \rho_n |{{ u}}_n|^2$, and to Lemma \[compactuniforme\], up to a subsequence, $\rho_n$ converges strongly in $L^1((0,T)\times \Omega)$ to a function $\rho$, and $\sqrt{\rho_n} {{ u}}_n$ converges weakly in $L^2((0,T)\times \Omega)$ to a function $q$. 0.3cm We want to show that, up to a subsequence, ${{ u}}_n {\bf 1}_{\{\rho>0\}}$ converges almost every where to $u {\bf 1}_{\{\rho>0\}}$. We consider the two cases. First, if ${\varepsilon}_{1,n}={\varepsilon}_1>0$, then from Lemma \[lem u\] and the Aubin-Lions Lemma, $\rho_n {{ u}}_n$ converges strongly in $C^0(0,T; L^1(\Omega))$ to $\sqrt{\rho} q=\rho {{ u}}$. Up to a subsequence, both $\rho_n$ and $\rho_n {{ u}}_n$ converges almost everywhere to, respectively, $\rho$ and $\rho {{ u}}$. For almost every $(t,x) \in \{\rho>0\}$, for $n$ big enough, $\rho_n(t,x)>0$, so ${{ u}}_n=\rho_n {{ u}}_n/\rho_n$ at this point converges $u$. If ${\varepsilon}_{2,n}=0$ we use the second part of Lemma \[lem u\] and thanks to the Aubin-Lions Lemma, $\Phi(\rho_n){{ u}}_n$ converges strongly in $C^0(0,T; L^1(\Omega))$ to $\Phi(\rho) {{ u}}$. We still have, up to a subsequence, both $\rho_n$ and $\Phi(\rho_n) {{ u}}_n$ converging almost everywhere to, respectively, $\rho$ and $\phi(\rho) {{ u}}$ (we used the fact that $\Phi(r)/\sqrt{r}=0$ at $r=0$). Since $\Phi(r)\neq 0$ for $r\neq0$, for almost every $(t,x) \in \{\rho>0\}$, for $n$ big enough, $\Phi(\rho_n)(t,x)>0$, so $u_n=\Phi(\rho_n) {{ u}}_n/\Phi(\rho_n)$ at this point converges ${{ u}}$. 0.3cm Note that $$\rho_n^\alpha h(\rho_n,{{ u}}_n) =\rho_n^\alpha h(\rho_n,{{ u}}_n) {\bf 1}_{\{\rho>0\}}+\rho_n^\alpha h(\rho_n,{{ u}}_n) {\bf 1}_{\{\rho=0\}}.$$ The first term converges almost everywhere to $\rho^\alpha h(\rho,{{ u}}) {\bf 1}_{\{\rho>0\}}$, and therefore to $\rho^\alpha h(\rho,{{ u}}) $ in $L^1$ by the Lebesgue’s theorem. The second part can be estimated as follows $$\|\rho_n^\alpha h(\rho_n,{{ u}}_n) {\bf 1}_{\{\rho=0\}}\|_{{L^1}}\leq \|h(\rho_n,{{ u}}_n)\|_{L^q}\|\rho_n^\alpha {\bf 1}_{\{\rho=0\}}\|_{{L^{p-{\varepsilon}}}}.$$ But $\rho_n^\alpha {\bf 1}_{\{\rho=0\}}$ converges almost everywhere to 0, by the Lebesgue’s theorem, the last term converges to 0. [*Some compactness when the parameters are fixed.*]{} For any positive fixed $\delta$, $r_0$, $r_1$, $r_2$ and $r$, to recover a weak solution to , we only need to handle the compactness of the terms $$r\rho_n\nabla\left(\sqrt{K(\rho_n)}\D(\int_0^{\rho_n}\sqrt{K(s)}\,ds)\right)$$ and $$\frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n.$$ Indeed due to the term $r_0\rho_n|{{ u}}_n|{{ u}}_n$ and the fact that $\inf_{s\in [0,+\infty)}\mu'(s) >\varepsilon_1>0$, one obtains the compactness for all other terms in the same way as in [@BDZ; @MV]. [*Capillarity term.*]{} To pass to the limits in $$r\rho_n\nabla\left(\sqrt{K(\rho_n)}\D(\int_0^{\rho_n}\sqrt{K(s)}\,ds)\right),$$ we use the identity $$\begin{split} &\rho\nabla\left(\sqrt{K(\rho_n)}\D(\int_0^{\rho_n}\sqrt{K(s)}\,ds)\right) \\ & \hskip3cm = 4 \Bigl[2{\rm div}(\sqrt{\mu(\rho_n)} \nabla\nabla Z(\rho_n)) - \Delta (\sqrt{\mu(\rho_n)} \nabla Z(\rho_n)\Bigr]\\ &\hskip4cm + \Bigl[ \nabla \bigl[(\frac{2\lambda(\rho_n)}{\sqrt{\mu(\rho_n)}} + k(\rho_n))\Delta Z(\rho_n)\bigr] - \nabla {\rm div} [ k(\rho_n)\nabla Z(\rho_n)] \Bigr] \end{split}$$ where $\displaystyle Z(\rho_n) = \int_0^{\rho_n} [(\mu(s))^{1/2} \mu'(s)]/s \, ds$ and $\displaystyle k(\rho_n) = \int_0^{\rho_n} \frac{\lambda(s)\mu'(s)}{\mu(s)^{3/2}} ds.$ It allows us to rewrite the weak form coming for the capillarity term as follows $$\begin{split} &\int_0^t\int_{\Omega}\sqrt{K(\rho_n)} \Delta (\int_0^{\rho_n} \sqrt{K(s)}\, ds) {\rm div} (\rho_n \psi) \, dx\, dt \\&= 4 \int_0^t\int_{\Omega}\bigl(2\sqrt{\mu(\rho_n)} \nabla\nabla Z(\rho_n): \nabla \psi + \sqrt{\mu(\rho_n)}\nabla Z(\rho_n)\cdot \Delta \psi\bigr) \\ & \hskip1cm + \int_0^t\int_\Omega \bigl(\frac{2\lambda(\rho_n)}{\sqrt{\mu(\rho_n)}} + k(\rho_n))\Delta Z(\rho_n) \, {\rm div} \psi + k(\rho_n) \nabla Z(\rho_n) . \nabla {\rm div} \psi\bigr) \\ &=A_1+A_2. \end{split}$$ In fact, with Lemma \[compactuniforme\] at hand, we are able to have compactness of $A_1$ and $A_2$ easily. Concerning $A_1$, we know that $$\sqrt{\mu(\rho_n)} \to \sqrt{\mu(\rho)} \hbox{ in } L^p((0,T); L^q({\Omega})) \hbox{ for all } p<+\infty \hbox{ and } q<3.$$ Note that $\nabla\nabla Z(\rho_n)$ is uniformly bounded in $L^2(0,T;L^2({\Omega}))$, we have $\nabla Z(\rho_n)$ is uniformly bounded in $L^2(0,T;L^6({\Omega}))$, because $\int_{\Omega}\nabla Z(\rho_n) = 0$ due to the periodic condition. Thus we have following weak convergence $$\int_{{\Omega}}\sqrt{\mu(\rho_n)}\nabla Z(\rho_n)\cdot \Delta\psi\,dx \to \int_{{\Omega}}\sqrt{\mu}\nabla Z\cdot \Delta\psi\,dx,$$ and $$\int_{{\Omega}}\sqrt{\mu(\rho_n)}\nabla \nabla Z(\rho_n)\nabla\psi\,dx \to \int_{{\Omega}}\sqrt{\mu}\nabla \nabla Z:\nabla\psi\,dx,$$ thanks to Lemma \[compactuniforme\]. We conclude that $Z=Z(\rho)$, thanks to the bound on $Z(\rho_n)$ and the strong convergence on $\rho_n$. Thus using the compactness on $\rho_n$, the passage to the limit in $A_1$ is done. Concerning $A_2$, we just have to look at the coefficients $$\displaystyle k(\rho_n)= \int_0^{\rho_n} \lambda(s)\mu'(s)/\mu(s)^{3/2} \, ds, \qquad j(\rho_n)= {2\lambda(\rho_n)}/{\sqrt{\mu(\rho_n)}}.$$ Recalling the assumptions on $\mu(s)$ and the relation $\lambda(s) = 2 (\mu'(s)s -\mu(s))$, we have $$2(\alpha_1- 1) \mu(s) \le \lambda(s) \le 2(\alpha_2-1) \mu(s),$$ and $$\frac{\alpha_1}{\sqrt{\mu(s)}s} \le \frac{\mu'(s)}{\mu(s)^{3/2}} \le \frac{\alpha_2}{\sqrt{\mu(s)}s}.$$ This means that the coefficients $k(\rho_n)$ and $j(\rho_n)$ are comparable to $\sqrt{\mu(\rho_n)}$. Using the compactness of the density $\rho_n$ and the informations on $\mu(\rho_n)$ given in Corollary \[compactuniforme\], we conclude the compactness of $A_2$ doing as for $A_1$. [*Cubic non-linear drag term.*]{} We will use Lemma \[Compactnesstool1\] to show the compactness of $$\frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n.$$ More precisely, we write $$\label{decompdrag} \frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n=\rho_n^{\frac{1}{6}}\sqrt{\frac{\rho_n}{\mu'(\rho_n)}}|{{ u}}_n|^2\rho_n^{\frac{1}{3}}|{{ u}}_n|\frac{1}{\sqrt{\mu'(\rho_n)}} = \rho_n^{1/6} h(\rho_n,|{{ u}}_n|),$$ By Lemma \[compactuniforme\], there exists $\varepsilon>0$ such that $\rho_n^{\frac{1}{6}}$ is uniformly bounded in $L^{\infty}(0,T;L^{6\gamma+\varepsilon}({\Omega}))$ and $\rho_n\to \rho\text{ a.e.}$, so $$\label{comp1} \rho_n^{\frac{1}{6}}\to\rho^{\frac{1}{6}}\quad\text{ in } L^{6\gamma+\varepsilon}((0,T)\times {\Omega})).$$ Note that $\sqrt{\frac{\rho_n}{\mu'(\rho_n)}}|{{ u}}_n|^2$ is uniformly bounded in $L^2(0,T;L^2({\Omega}))$, and $\inf_{s\in [0,+\infty)} \mu'(s) \ge \varepsilon_1 >0$, $\rho_n^{\frac{1}{3}}|{{ u}}_n|\frac{1}{\sqrt{\mu'(\rho_n)}}$ is uniformly bounded in $L^3(0,T;L^3({\Omega}))$, thus $$\label{comp2} h(\rho_n,|{{ u}}_n|) = \sqrt{\frac{\rho_n}{\mu'(\rho_n)}}|{{ u}}_n|^2\rho_n^{\frac{1}{3}}|{{ u}}_n|\frac{1}{\sqrt{\mu'(\rho_n)}} \in L^{\frac{6}{5}}(0,T;L^{\frac{6}{5}}({\Omega})) \hbox{ uniformly.}$$ By Lemma \[Compactnesstool1\] and –, we deduce that $$\int_0^t\int_{{\Omega}}\frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n\,dx\,dt\to \int_0^t\int_{{\Omega}}\frac{\rho}{\mu'(\rho)}|{{ u}}|^2{{ u}}\,dx\,dt. \, \square$$ Relying on the compactness stated in this section and the compactness in [@MV], we are able to follow the argument in [@BDZ] to show Theorem \[main result 1\]. Thanks to term $r_0\rho_n|{{ u}}_n|{{ u}}_n$, we have $$\int_0^T\int_{{\Omega}}r_0\rho_n|{{ u}}_n|^4\,dx\,dt\leq C.$$ This gives us that $$\sqrt{\rho_n}{{ u}}_n\to\sqrt{\rho}{{ u}}\; \text{ strongly in } L^2(0,T;L^2({\Omega})).$$ With above compactness of this section, we are able to pass to the limits for recovering a weak solution. In fact, to recover a weak solution to , we have to pass to the limits as the order of $\varepsilon_4\to 0$, $n\to\infty,$ $\varepsilon_3\to0$ and $\varepsilon\to 0$ respectively. In particular, when passing to the limit $\varepsilon_3$ tends to zero, we also need to handle the identification of ${{ v}}$ with $2\nabla{s}(\rho)$. Following the same argument in [@BDZ], one shows that ${{ v}}$ and $2\nabla{s}(\rho)$ satisfy the same moment equation. By the regularity and compactness of solutions, we can show the uniqueness of solutions. By the uniqueness, we have ${{ v}}=2\nabla{s}(\rho)$. This ends the proof of Theorem \[main result 1\]. From weak solutions to renormalized solutions to the approximation ================================================================== This section is dedicated to show that a weak solution is a renormalized solution for our last level of approximation namely to show Theorem \[renorm\]. First, we introduce a new function $$[f(t,x)]_\varepsilon =f*\eta_{\varepsilon}(t,x),\text{ for any\ \ } t>\varepsilon,\quad\text{ and }\;[f(t,x)]_\varepsilon^x =f*\eta_{\varepsilon}(x)$$ where $$\eta_{\varepsilon}(t,x)=\frac{1}{\varepsilon^{d+1}}\eta(\frac{t}{\varepsilon},\frac{x}{\varepsilon}),\quad\text{ and } \eta_{\varepsilon}(x)=\frac{1}{\varepsilon^{d}}\eta(\frac{x}{\varepsilon}),$$ with $\eta$ a smooth nonnegative even function compactly supported in the space time ball of radius 1, and with integral equal to 1. In this section, we will rely on the following two lemmas to proceed our ideas. Let $\partial$ be a partial derivative in one direction (space or time) in these two lemmas. The first one is the commutator lemma of DiPerna and Lions, see [@Lions]. \[Lions’s lemma\] Let $f\in W^{1,p}({{\mathbb R}}^N\times{{\mathbb R}}^{+}),\,g\in L^{q}({{\mathbb R}}^N\times{{\mathbb R}}^{+})$ with $1\leq p,q\leq \infty$, and $\frac{1}{p}+\frac{1}{q}\leq 1$. Then, we have $$\| [\partial(fg)]_\varepsilon -\partial(f([g]_{\varepsilon}))\|_{L^{r}({{\mathbb R}}^N\times {{\mathbb R}}^+)}\leq C\|f\|_{W^{1,p}({{\mathbb R}}^N\times{{\mathbb R}}^{+})}\|g\|_{L^{q}({{\mathbb R}}^N\times{{\mathbb R}}^{+})}$$ for some $C\geq 0$ independent of $\varepsilon$, $f$ and $g$, $r$ is determined by $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}.$ In addition, $$[\partial(fg)]_{\varepsilon}-\partial(f([g]_{\varepsilon}))\to0\;\;\text{ in }\,L^{r}({{\mathbb R}}^N\times{{\mathbb R}}^{+})$$ as $\varepsilon \to 0$ if $r<\infty.$ Moreover, in the same way if $f\in W^{1,p}({{\mathbb R}}^N),\,g\in L^{q}({{\mathbb R}}^N)$ with $1\leq p,q\leq \infty$, and $\frac{1}{p}+\frac{1}{q}\leq 1$. Then, we have $$\| [\partial(fg)]^x_\varepsilon -\partial(f([g]^x_{\varepsilon}))\|_{L^{r}({{\mathbb R}}^N)}\leq C\|f\|_{W^{1,p}({{\mathbb R}}^N)}\|g\|_{L^{q}({{\mathbb R}}^N)}$$ for some $C\geq 0$ independent of $\varepsilon$, $f$ and $g$, $r$ is determined by $\frac{1}{r}=\frac{1}{p}+\frac{1}{q}.$ In addition, $$[\partial(fg)]^x_{\varepsilon}-\partial(f([g]^x_{\varepsilon}))\to0\;\;\text{ in }\,L^{r}({{\mathbb R}}^N)$$ as $\varepsilon \to 0$ if $r<\infty.$ We also need another very standard lemma as follows. \[standard lemma\] If $f\in L^p({\Omega}\times{{\mathbb R}}^{+})$ and $g\in L^q({\Omega}\times{{\mathbb R}}^{+})$ with $\frac{1}{p}+\frac{1}{q}=1$ and $H\in W^{1,\infty}({{\mathbb R}})$, then $$\begin{split} &\int_0^T\int_{{\Omega}} [f]_\varepsilon g\,dx\,dt=\int_0^T\int_{{\Omega}}f [g]_\varepsilon \,dx\,dt, \\&\lim_{\varepsilon\to 0 } \int_0^T\int_{{\Omega}} [f]_\varepsilon g\,dx\,dt=\int_0^T\int_{{\Omega}}f g\,dx\,dt, \\&\partial [f]_\varepsilon =[\partial f]_\varepsilon, \\&\lim_{\varepsilon\to 0}\|H([f]_\varepsilon)-H(f)\|_{L^s_{loc}}({\Omega}\times{{\mathbb R}}^+)=0,\quad\text{for any }\, 1\leq s<\infty. \end{split}$$ We define a nonnegative cut-off functions $\phi_m$ for any fixed positive $m$ as follows. $$\label{cutoff function} \phi_m(y)\begin{cases}= 0, \;\;\;\;\;\quad\quad\quad\quad\text{ if }0\leq y\leq \frac{1}{2m}, \\ =2my-1,\;\;\;\;\;\quad\text{ if } \frac{1}{2m}\leq y\leq \frac{1}{m}, \\ =1,\,\;\;\;\;\quad\quad\;\quad\quad\text{ if } \frac{1}{m}\leq y\leq m, \\=2-\frac{y}{m},\,\;\;\;\;\quad\quad\text{ if } m\leq y\leq 2m, \\=0,\,\;\;\;\;\quad\quad\;\quad\quad\text{ if } y\geq 2m. \end{cases}$$ It enables to define an approximated velocity for the density bounded away from zero and bounded away from infinity. It is crucial to process our procedure, since the gradient approximated velocity is bounded in $L^2((0,T)\times {\Omega})$. In particular, we introduce ${{ u}}_m={{ u}}\phi_m(\rho)$ for any fixed $m>0$. Thus, we can show $\nabla{{ u}}_m$ is bounded in $L^2(0,T;L^2({\Omega}))$ due to . In fact, $$\begin{split} \nabla{{ u}}_m&=\phi_m'(\rho){{ u}}\otimes\nabla\rho+\phi_m(\rho)\frac{1}{\sqrt{\mu(\rho)}}{{\mathbb T}_\mu}\\&=\big(\phi_m'(\rho)\frac{(\mu(\rho)\rho)^{1/4}}{(\mu'(\rho))^{\frac{3}{4}}}\big) \big((\frac{\rho}{\mu'(\rho)})^{\frac{1}{4}}{{ u}}\big)\otimes \big(\frac{\mu'(\rho)}{\rho^{\frac{1}{2}}\mu(\rho)^{\frac{1}{4}}}\nabla\rho\big) +\phi_m(\rho)\frac{1}{\sqrt{\mu(\rho)}}{{\mathbb T}_\mu}. \end{split}$$ Similarly to [@LaVa], thanks to the cut-off function and for $m$ fixed, $\phi_m'(\rho){(\mu(\rho)\rho)^{\frac{1}{4}}}/{(\mu'(\rho))^{\frac{3}{4}}}$ and $\phi_m(\rho)/\sqrt{\mu(\rho)}$ are bounded. Then $\nabla{{ u}}_m$ is bounded in $L^2((0,T)\times \Omega)$ using the estimates with $r>0$ and $r_2>0$, and hence for $\varphi \in W^{2,+\infty}({{\mathbb R}})$, we get $\nabla\varphi'(({{ u}}_m)_j)$ is bounded in $L^2((0,T)\times \Omega)$ for $j=1,2,3$. The following estimates are necessary. We state them in the lemma as follows. \[estimate of approximation\] There exists a constant $C>0$ depending only on the fixed solution $(\sqrt{\rho},\sqrt{\rho}{{ u}})$, and $C_m$ depending also on $m$ such that $$\begin{split}&\|\rho\|_{L^{\infty}(0,T;L^{10}(\Omega))} +\|\rho{{ u}}\|_{L^3(0,T;L^{\frac{5}{2}}({\Omega}))} + \|\rho|{{ u}}|^2\|_{L^{2}(0,T; L^{\frac{10}{7}}({\Omega}))} \\& +\|\sqrt{\mu}\big(|{{\mathbb S}_\mu}|+r|{{\mathbb S}_r}|\big)\|_{L^{2}(0,T; L^{\frac{10}{7}}({\Omega}))} + \|\frac{\lambda(\rho)}{\mu(\rho)}\|_{L^{\infty}((0,T)\times {\Omega})} \\& + \|\sqrt{\frac{P'(\rho_n)\rho_n}{\mu'(\rho_n)}} \nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P'(s)\mu'(s)}{s}}\, ds\Bigr)\|_{L^{1+}((0,T)\times\Omega)} \\ & + \|\sqrt{\frac{P_\delta'(\rho_n)\rho_n}{\mu'(\rho_n)}} \nabla \displaystyle \Bigl(\int_0^{\rho_n} \sqrt{\frac{P_\delta'(s)\mu'(s)}{s}}\, ds\Bigr)\|_{L^{1+}((0,T) \times\Omega)} +\|r_0{{ u}}\|_{L^2((0,T)\times \Omega)}\leq C, \end{split}$$ and $$\|\nabla\phi_m(\rho)\|_{L^4((0,T)\times \Omega}+\|\partial_t\phi_m(\rho)\|_{L^2((0,T\times\Omega))}\leq C_m.$$ By , we have $\rho \in L^{\infty}(0,T;L^{10}({\Omega}))$. Now we have $\nabla\sqrt{\rho}\in L^{\infty}(0,T;L^2({\Omega}))$ because $\mu'(s) \ge \varepsilon_1$ and $\mu'(\rho) \nabla \rho /\sqrt \rho \in L^\infty((0,T);L^2({\Omega}))$. Note that $$\rho{{ u}}=\rho^{\frac{2}{3}}\rho^{\frac{1}{3}}{{ u}},$$ $\rho^{\frac{2}{3}}\in L^{\infty}(0,T;L^{15}({\Omega}))$ and $\rho^{\frac{1}{3}}{{ u}}\in L^{3}(0,T;L^3({\Omega}))$, $\rho{{ u}}$ is bounded in $L^{3}(0,T;L^{\frac{5}{2}}({\Omega}))$. By , we have $(\frac{\rho}{\mu'(\rho)})^{1/2}|{{ u}}|^2\in L^2((0,T)\times {\Omega})$. Note that $$\rho|{{ u}}|^2= (\rho\mu'(\rho))^{1/2} (\frac{\rho}{\mu'(\rho)})^{1/2}|{{ u}}|^2,$$ it is bounded in $L^{2}(0,T;L^{\frac{10}{7}}({\Omega}))$, where we used facts that $\mu(\rho) \in L^\infty(0,T;L^{5/2}(\Omega))$ (recalling that for $\rho \ge 1$ we have $\mu(\rho)\le c\rho^4$ and $\rho \in L^\infty(0,T;L^{10}(\Omega))$) and $\mu'(\rho) \rho \le \alpha_2 \mu(\rho)$. Similarly, we get $\sqrt{\mu}(|{{\mathbb S}_\mu}|+r|{{\mathbb S}_r}|) \in L^2(0,T;L^{10/7}(\Omega))$ by . The $L^\infty((0,T)\times {\Omega})$ bound for $\lambda(\rho)/\mu(\rho)$ may be obtained easily due to and . Concerning the estimates related to the pressures, we just have to look at the proof in Lemma \[compactuniforme\]. Note that $$\begin{split} &\nabla\phi_m(\rho)=\phi_m'(\rho) \nabla\rho= \phi_m'(\rho)\frac{\rho^{1/2} \mu(\rho)^{1/4}}{\mu'(\rho)} [\frac{\mu'(\rho)}{\rho^{1/2}\mu(\rho)^{1/4}}\nabla\rho] \end{split}$$ by , we conclude that $\nabla\phi_m(\rho)$ is bounded in $L^4((0,T)\times{\Omega})$. It suffices to recall that thanks to the cut-off function $\phi_m$, we have $\phi_m'(\rho) \rho^{1/2} \mu(\rho)^{1/4}/\mu'(\rho)$ bounded in $L^{\infty}((0,T)\times {\Omega})$. Similarly, we write $$\begin{split} \partial _t\phi_m(\rho)&=\phi_m'(\rho)\partial_t \rho=-\phi'_m(\rho){{\rm div}}(\rho{{ u}}) \\&=-\phi_m'(\rho)\frac{\rho}{\sqrt{\mu}}\mathrm{Tr} ({{\mathbb T}_\mu}) - \big(\phi_m'(\rho)\frac{(\mu(\rho)\rho)^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{3}{4}}}\big) \big(\frac{\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{1}{4}}}{{ u}}\big)\cdot \big(\frac{\mu'(\rho)}{\rho^{1/2} \mu(\rho)^{1/4}} \nabla \rho \big) ,\end{split}$$ which provides $\partial_t\phi_m(\rho)$ bounded in $L^2(0,T;L^2({\Omega}))$ thanks to , and . and using the cut-off function property to bound the extra quantiies in $L^\infty((0,T)\times{\Omega})$ as previously. \[Lemma of renormalized approxiamtion\] The $\kappa$-entropic weak solution constructed in Theorem \[main result 1\] is a renormalized solution, in particular, we have $$\label{limit for m large} \begin{split} & \int_0^T\int_{{\Omega}}\big(\rho\varphi({{ u}})\psi_t+ (\rho \varphi({{ u}})\otimes {{ u}}) \nabla\psi\big)\\ & - \int_0^T\int_{\Omega}\nabla\psi \varphi'({{ u}})\big[2\bigl(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r \, {{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)}{\rm Tr}(\sqrt{\mu(\rho)} {{\mathbb S}_\mu}+ r \sqrt{\mu(\rho)} {{\mathbb S}_r}) {\rm Id} \big]\\ & -\int_0^T \int_{\Omega}\psi\varphi''({{ u}}){{\mathbb T}_\mu}\big[2\bigl(({{\mathbb S}_\mu}+ r \, {{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)}{\rm Tr}({{\mathbb S}_\mu}+ r {{\mathbb S}_r}) {\rm Id} \big] \\ & + \int_0^T \int_{\Omega}\psi\varphi'({{ u}})F(\rho,{{ u}})\big)\,dx\,dt=0, \end{split}$$ where $$\label{eq_viscous_renormalise} \begin{split} & \sqrt{\mu(\rho)}\varphi_i'({{ u}})[{{\mathbb T}_\mu}]_{jk}= \partial_j(\mu\varphi'_i({{ u}}){{ u}}_k)-{{\sqrt{\rho}}}{{ u}}_k\varphi'_i({{ u}})\frac{\nabla\mu}{\sqrt{\rho}}+ \bar{R}^1_\varphi, \\ &\sqrt{\mu(\rho)} \varphi_i'({{ u}}) [\mathbb S_r]_{jk} = 2 \sqrt{\mu(\rho)} \varphi_i'({{ u}}) \partial_j \partial_k Z(\rho) - 2 \partial_j (\sqrt{\mu(\rho)} \partial_k Z(\rho) \varphi_i'({{ u}})) + \bar{R}^2_\varphi \\ &\frac{\lambda(\rho)}{2\mu(\rho)} \varphi_i'({{ u}}) {\rm Tr} (\sqrt{\mu(\rho)} \mathbb T_\mu) = {\rm div} \bigl(\frac{\lambda(\rho)}{\mu(\rho)} \sqrt\rho {{ u}}\frac{\mu(\rho)}{\sqrt\rho} \varphi'({{ u}}) \bigr) \\ & \hskip4.4cm - \sqrt \rho u \cdot \sqrt\rho \nabla s(\rho)\frac{\rho \mu''(\rho)}{\mu(\rho)} \varphi'({{ u}})+ \bar{R}^3_\varphi \\ & \frac{\lambda(\rho)}{\mu(\rho)} \varphi'({{ u}}) {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_r) = \varphi_i'({{ u}}) \bigl(\frac{\lambda(\rho)}{\sqrt{\mu(\rho)}} + \frac{1}{2} k(\rho) \bigr) \Delta Z(\rho) \\ & \hskip4.4cm - \frac{1}{2} {\rm div}(k(\rho) \varphi'_i({{ u}}) \nabla Z(\rho)) + \bar{R}^4_\varphi \end{split}$$ where $$\begin{split} &\bar{R}^1_{\varphi}=\varphi''_i({{ u}}){{\mathbb T}_\mu}\sqrt{\mu(\rho)}{{ u}}\\ &\bar{R}^2_\varphi = 2 \varphi_i''(u) \mathbb T_\mu \nabla Z(\rho)\\ &\bar{R}^3_\varphi = - \varphi_i''(u) \mathbb T_\mu\cdot \sqrt{\mu(\rho)} {{ u}}\frac{\lambda(\rho)}{\mu(\rho)} \\ & \bar{R}^4_\varphi = \frac{k(\rho)}{2 \sqrt{\mu(\rho)}} \varphi_i''({{ u}}) \mathbb T_\mu \cdot \nabla Z(\rho) \end{split}$$ We choose a function $\Bigl[\phi_m'([\rho]_\varepsilon)\psi\Bigr]_\varepsilon$ as a test function for the continuity equation with $\psi\in C_c^{\infty}((0,T)\times{\Omega})$. Using Lemma \[standard lemma\], we have $$\begin{split} \label{weak formulation for mass with varepsilon} 0&=\int_0^T\int_{{\Omega}}\big(\partial_t\Bigl[\phi_m'([\rho]_\varepsilon)\psi\Bigr]_\varepsilon \rho +\rho{{ u}}\cdot\nabla\Bigl[\phi_m'([\rho]_\varepsilon)\psi\Bigr]_\varepsilon\big)\,dx\,dt \\&=-\int_0^T\int_{{\Omega}}\big(\phi_m'([\rho]_\varepsilon)\psi \, \partial_t [\rho]_\varepsilon +{{\rm div}}([\rho{{ u}}]_{\varepsilon}) \phi_m'([\rho]_\varepsilon)\psi\big)\,dx\,dt \\&=\int_0^T\int_{{\Omega}}\left(\psi_t\phi_m([\rho]_\varepsilon) -\psi\phi'_m([\rho]_\varepsilon) \bigl[\frac{\rho}{\sqrt{\mu(\rho)}}\mathrm{Tr} ({{\mathbb T}_\mu})+2 \sqrt{\rho}{{ u}}\cdot\nabla\sqrt{\rho}\bigr]_\varepsilon\right)\,dx\,dt. \end{split}$$ Using Lemma \[estimate of approximation\] and Lemma \[standard lemma\], and passing into the limit as $\varepsilon$ goes to zero, from , we get: $$\begin{split} \label{modified continuity equation} 0&=\int_0^T\int_{{\Omega}}\big(\psi_t\phi_m(\rho)-\psi\phi'_m(\rho)[\frac{\rho}{\sqrt{\mu}}\mathrm{Tr} ({{\mathbb T}_\mu})+2\sqrt{\rho}{{ u}}\cdot\nabla\sqrt{\rho}]\big)\,dx\,dt \\&=\int_0^T\int_{{\Omega}}\big(\psi_t\phi_m(\rho) -\psi \bigl[\phi'_m(\rho)\frac{\rho}{\sqrt{\mu}}\mathrm{Tr} ({{\mathbb T}_\mu})+{{ u}}\cdot\nabla\phi_m(\rho)\bigr]\big)\,dx\,dt, \end{split}$$ thanks to $\psi\nabla\phi_m(\rho)\in L^4((0,T)\times {\Omega})$, ${{ u}}\in L^2((0,T)\times {\Omega})$, and $\psi $ compactly supported. Similarly, we can choose $[\psi\phi_m(\rho)]_\varepsilon$ as a test function for the momentum equation. In particular, we have the following lemma. \[Lemma for limits-first two terms\] $$\int_0^T\int_{{\Omega}} [\psi\phi_m(\rho)]_\varepsilon \big(\partial_t (\rho {{ u}}) +{{\rm div}}(\rho{{ u}}\otimes{{ u}})\big)\,dx\,dt$$ tends to $$-\int_0^T\int_{{\Omega}}\psi_t\rho{{ u}}_m+\nabla\psi\cdot(\rho{{ u}}\otimes{{ u}}_m +\psi(\partial_t\phi_m(\rho)+{{ u}}\cdot\nabla\phi_m(\rho))\rho{{ u}}\,dx\,dt$$ as $\varepsilon\to 0.$ By Lemma \[Lions’s lemma\], we can show that $$\begin{split} & \int_0^T\int_{{\Omega}} [\psi\phi_m(\rho)]_\varepsilon \partial_t(\rho {{ u}})\,dx\,dt\to -\int_0^T\int_{{\Omega}}\partial_t\psi \rho{{ u}}_m +\psi\partial_t\phi_m(\rho) \rho {{ u}}\,dx\,dt. \end{split}$$ For the second term, we have $$\begin{split} &\int_0^T\int_{{\Omega}} \bigl[\psi\phi_m(\rho)\bigr]_\varepsilon {{\rm div}}(\rho{{ u}}\otimes{{ u}})\,dx\,dt =\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[ {{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]_\varepsilon\,dx\,dt\\ &=\big(\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) \bigl[ {{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]_\varepsilon\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[{{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]^x_\varepsilon\,dx\,dt\big) \\&+\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[ {{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]^x_\varepsilon\,dx\,dt \\&=R_1+R_2, \end{split}$$ where $[f (t,x)]_\varepsilon =f(t,x)*\eta_{\varepsilon}(t,x)$ and $[f(t,x)]_\varepsilon^x =f*\eta_{\varepsilon}(x)$ with $\varepsilon>0$ a small enough number. We write $R_1$ in the following way $$\begin{split} R_1&=\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[{{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]_\varepsilon\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[{{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]_\varepsilon^x\,dx\,dt \\&=\int_0^T\int_{{\Omega}}\psi\nabla\phi_m(\rho):\bigl[\rho{{ u}}\otimes{{ u}}\bigr]_\varepsilon\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\nabla\phi_m(\rho):\bigl[\rho{{ u}}\otimes{{ u}}\big]^x_\varepsilon\,dx\,dt. \end{split}$$ Thanks to Lemma \[estimate of approximation\], $\rho|{{ u}}|^2 \in L^{2}(0,T; L^{10/7}(\Omega))$ and $\psi\nabla\phi_m(\rho)\in L^4((0,T)\times \Omega)$, we conclude that $R_1\to 0$ as $\varepsilon\to0.$ Meanwhile, we can apply Lemma \[Lions’s lemma\] to $R_2$ directly, thus $$\begin{split}& \int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[ {{\rm div}}(\rho{{ u}}\otimes{{ u}})\big]^x_\varepsilon \,dx\,dt \\&=\big(\int_0^T\int_{{\Omega}}\psi\phi_m(\rho)\bigl[{{\rm div}}(\rho{{ u}}\otimes{{ u}})\bigr]^x_\varepsilon\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) {{\rm div}}(\rho{{ u}}\otimes [{{ u}}]^x_\varepsilon)\,dx\,dt\big) \\&+\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) {{\rm div}}(\rho{{ u}}\otimes [{{ u}}]^x_\varepsilon)\,dx\,dt \\&=R_{21}+R_{22}. \end{split}$$ By Lemma \[Lions’s lemma\], we have $R_{21}\to 0$ as $\varepsilon\to 0$. The term $R_{22}$ will be calculated in the following way, $$\begin{split} &\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) {{\rm div}}(\rho{{ u}}\otimes [{{ u}}]^x_\varepsilon)\,dx\,dt \\&=\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) {{\rm div}}(\rho{{ u}}) [{{ u}}]^x_\varepsilon\,dx\,dt +\int_0^T\int_{{\Omega}}\psi\phi_m(\rho) \rho{{ u}}\cdot \nabla [{{ u}}]^x_\varepsilon\,dx\,dt \\&=\int_0^T\int_{{\Omega}}\psi {{\rm div}}(\rho{{ u}})[{{ u}}_m]^x_\varepsilon\,dx\,dt+\int_0^T\int_{{\Omega}}\psi\rho{{ u}}\nabla(\phi_m(\rho) [{{ u}}]^x_\varepsilon)\,dx\,dt- \\&\int_0^T\int_{{\Omega}}\psi [{{ u}}]_\varepsilon^x \cdot\nabla\phi_m(\rho)\rho{{ u}}\,dx\,dt \\&=-\int_0^T\int_{{\Omega}}\nabla\psi\rho{{ u}}\otimes [{{ u}}_m]_\varepsilon^x\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\cdot [{{ u}}]_\varepsilon^x \nabla\phi_m(\rho)\rho{{ u}}\,dx\,dt, \end{split}$$ which tends to $$-\int_0^T\int_{{\Omega}}\nabla\psi\rho{{ u}}\otimes {{ u}}_m\,dx\,dt -\int_0^T\int_{{\Omega}}\psi\cdot {{ u}}\nabla\phi_m(\rho)\rho{{ u}}\,dx\,dt,$$ as $\varepsilon \to 0$. For the other terms in the momentum equation, we can follow the same way as above method for to have $$\label{modified momentum equation} \begin{split}&\int_0^T\int_{{\Omega}}\big(\psi_t\rho{{ u}}_m+\nabla\psi\cdot(\rho{{ u}}\otimes{{ u}}_m - 2\phi_m(\rho) (\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) {\rm Id} )) \\ & + \int_0^T\int_{\Omega}\psi(\partial_t\phi_m(\rho)+{{ u}}\cdot\nabla\phi_m(\rho))\rho{{ u}}\\&- \int_0^T\int_{\Omega}2 \psi( \sqrt{\mu(\rho)}({{\mathbb S}_\mu}+{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu + r \mathbb S_r) {\rm Id} )\nabla\phi_m(\rho)+\psi\phi_m(\rho) F(\rho,{{ u}})\big)\,dx\,dt \\&=0. \end{split}$$ Thanks to , we have $$\label{modified momentum equation} \begin{split}&\int_0^T\int_{{\Omega}}\big(\psi_t\rho{{ u}}_m+\nabla\psi\cdot(\rho{{ u}}\otimes{{ u}}_m - 2\phi_m(\rho)(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} (\mathbb S_\mu+ r \mathbb S_r) ) {\rm Id} )\\ & - \int_0^T\int_{\Omega}\psi \phi'_m(\rho)\frac{\rho}{\sqrt{\mu(\rho)}}\mathrm{Tr} ({{\mathbb T}_\mu})\rho{{ u}}-\psi\phi_m(\rho) F(\rho,{{ u}}) \\&-\int_0^T\int_{\Omega}2 \psi(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} (\mathbb S_\mu+ r \mathbb S_r) ) {\rm Id}) \nabla\phi_m(\rho)\big)\,dx\,dt=0. \end{split}$$ The goal of this subsection is to derive the formulation of renormalized solution following the idea in [@LaVa]. We choose the function $\bigl[\psi\varphi'([{{ u}}_m]_\varepsilon)\bigr]_\varepsilon$ as a test function in . As the same argument of Lemma \[Lemma for limits-first two terms\], we can show that $$\begin{split}& \int_0^T\int_{{\Omega}}\big(\partial_t\bigl[\psi\varphi'([{{ u}}_m]_\varepsilon)\bigr]_\varepsilon\, \rho{{ u}}_m +\nabla\bigl[\psi\varphi'([{{ u}}_m]_\varepsilon)\bigr]_\varepsilon:(\rho{{ u}}\otimes{{ u}}_m)\big)\,dx\,dt \\&\to \int_0^T\int_{{\Omega}}\big(\rho\varphi({{ u}}_m)\psi_t+\rho{{ u}}\otimes\varphi({{ u}}_m)\nabla\psi\big)\,dx\,dt, \end{split}$$ and $$\begin{split}&\int_0^T\int_{{\Omega}}\nabla\bigl[\psi\varphi'([{{ u}}_m]_\varepsilon)\bigr]_\varepsilon \big(-2 \phi_m(\rho)(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) ) {\rm Id} \big) \\ & +\bigl[\psi\varphi'([{{ u}}_m]_\varepsilon)\bigr]_\varepsilon \big(-\phi'_m(\rho)\frac{\rho}{\sqrt{\mu(\rho)}}\mathrm{Tr} ({{\mathbb T}_\mu})\rho{{ u}}\\&-2(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} (\mathbb S_\mu+ r \mathbb S_r) {\rm Id}) )\nabla\phi_m(\rho)+\phi_m(\rho) F(\rho,{{ u}})\big)\,dx\,dt \\&\to \int_0^T\int_{{\Omega}}\nabla(\psi\varphi'({{ u}}_m))\ \big(-2\phi_m(\rho)(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} (\mathbb S_\mu+ r \mathbb S_r) ) {\rm Id} )\big)\\ & +\psi\varphi'({{ u}}_m) \big(-\phi'_m(\rho)\frac{\rho}{\sqrt{\mu(\rho)}}\mathrm{Tr} ({{\mathbb T}_\mu})\rho{{ u}}\\&-2(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) )\nabla\phi_m(\rho)+\phi_m(\rho) F(\rho,{{ u}})\big)\,dx\,dt \end{split}$$ as $\varepsilon$ goes to zero. Putting these two limits together, we have $$\label{weak formulation with m} \begin{split}& \int_0^T\int_{{\Omega}}\big(\rho\varphi({{ u}}_m)\psi_t+\rho{{ u}}\otimes\varphi({{ u}}_m)\nabla\psi\big) \\&+\nabla\psi\varphi'({{ u}}_m) \big(-2 \phi_m(\rho)(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) )\big)\\ & +\psi\varphi''({{ u}}_m)\nabla{{ u}}_m\big(-\phi_m(\rho)2(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) + \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) )\big) \\&+\psi\varphi'({{ u}}_m) \big(-\phi'_m(\rho)\frac{\rho}{\sqrt{\mu(\rho)}}\mathrm{Tr} ({{\mathbb T}_\mu})\rho{{ u}}-2(\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) \\&+ \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} \mathbb S_\mu+ r \mathbb S_r) )\nabla\phi_m(\rho)+\phi_m(\rho) F(\rho,{{ u}})\big)\,dx\,dt=0. \end{split}$$ Now we should pass to the limit in as $m$ goes to infinity. To this end, we should keep the following convergences in mind: $$\begin{split} \label{basic convergence for m} &\phi_m(\rho)\;\text{ converges to }1, \quad \text{ for almost every} (t,x)\in {{\mathbb R}}^+\times{\Omega},\\ &{{ u}}_m\text{ converges to } {{ u}}, \quad\text{ for almost every} (t,x)\in {{\mathbb R}}^+\times{\Omega},\\ &|\rho\phi'_m(\rho)|\leq 2, \quad\text{ and converges to } 0 \text{ for almost every} (t,x)\in {{\mathbb R}}^+\times{\Omega}. \end{split}$$ We can find that $$\begin{split} &\sqrt{\mu(\rho)}\nabla{{ u}}_m=\sqrt{\mu(\rho)}\nabla(\phi_m(\rho){{ u}}) =\phi_m(\rho)\sqrt{\mu(\rho)}\nabla{{ u}}+\phi'_m(\rho)\sqrt{\mu(\rho)}{{ u}}\cdot\nabla\rho \\&=\frac{\phi_m(\rho)}{\sqrt{\mu(\rho)}}\big(\nabla(\mu(\rho){{ u}})-\sqrt{\rho}{{ u}}\cdot\frac{\nabla\mu(\rho)}{\sqrt{\rho}}\big)+ \frac{\sqrt{\rho}}{\mu(\rho)^{\frac{3}{4}}}\big(\frac{\sqrt{\mu(\rho)}}{\rho}\mu'(\rho)\nabla\rho\big) \big(\frac{\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{1}{4}}}{{ u}}\big)\big(\phi_m'(\rho)\frac{\mu(\rho)^{\frac{3}{4}}\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{3}{4}}}\big) \\&=\phi_m(\rho){{\mathbb T}_\mu}+\frac{\sqrt{\rho}}{\mu(\rho)^{\frac{3}{4}}}\big(\frac{\sqrt{\mu(\rho)}}{\rho}\mu'(\rho)\nabla\rho\big) \big(\frac{\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{1}{4}}}{{ u}}\big)\big(\phi_m'(\rho)\frac{\mu(\rho)^{\frac{3}{4}}\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{3}{4}}}\big) \\&=A_{1m}+A_{2m}. \end{split}$$ Note that $$|\phi_m'(\rho)\frac{\mu(\rho)^{\frac{3}{4}}\rho^{\frac{1}{4}}}{(\mu'(\rho))^{\frac{3}{4}}}|\leq C|\phi'_m(\rho)\rho|,$$ thus $\phi_m'(\rho){\mu(\rho)^{\frac{3}{4}}\rho^{\frac{1}{4}}}/{(\mu(\rho)')^{\frac{3}{4}}}$ converges to zero for almost every $(t,x).$ Thus, the Dominated convergence theorem yields that $A_{2m}$ converges to zero as $m\to\infty.$ Meanwhile, the Dominated convergence theorem also gives us $A_{1m}$ converges to ${{\mathbb T}_\mu}$ in $L^2_{t,x}$. Hence, with at hand, letting $m\to\infty$ in , one obtains that $$\label{limit for m large} \begin{split}& \int_0^T\int_{{\Omega}}\big(\rho\varphi({{ u}})\psi_t+\rho{{ u}}\otimes\varphi({{ u}})\nabla\psi\big) - 2 \nabla\psi\varphi'({{ u}})\big((\sqrt{\mu(\rho)}({{\mathbb S}_\mu}+ r{{\mathbb S}_r}) \\&+ \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} (\sqrt{\mu(\rho)} (\mathbb S_\mu+ r \mathbb S_r) ) {\rm Id} \big)-2\psi\varphi''({{ u}}){{\mathbb T}_\mu}(({{\mathbb S}_\mu}+r {{\mathbb S}_r}) \\ &+ \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} ((\mathbb S_\mu+ r \mathbb S_r) {\rm Id}) +\psi\varphi'({{ u}})F(\rho,{{ u}})\big)\,dx\,dt=0. \end{split}$$ From now, we denote $R_{\varphi}=2\psi\varphi''({{ u}}){{\mathbb T}_\mu}(({{\mathbb S}_\mu}+ r{{\mathbb S}_r})+ \frac{\lambda(\rho)}{2\mu(\rho)} {\rm Tr} ( (\mathbb S_\mu+ r \mathbb S_r) {\rm Id})$. This ends the proof of Theorem \[renorm\]. renormalized solutions and weak solutions ========================================== The main goal of this section is the proof of Theorem \[main result\] that obtains the existence of renormalized solutions of the Navier-Stokes equations without the additional terms, thus the existence of weak solutions of the Navier-Stokes equations. Renormalized solutions ---------------------- In this subsection, we will show the existence of renormalized solutions. To this end, we need the following lemma of stability. \[Lemma of stability of renormalized solution\] For any fixed $\alpha_1<\alpha_2$ as in and consider sequences $\delta_n$, $r_{0n}$, $r_{1n}$ and $r_{2n}$, such that $r_{i,n}\to r_{i}\geq 0$ with $i=0,1,2$ and then $\delta_n\to \delta\geq 0$. Consider a family of $\mu_n:{{\mathbb R}}^{+}\to {{\mathbb R}}^{+}$ verifying and for the fixed $\alpha_1$ and $\alpha_2$ such that $$\mu_n\to \mu\quad\text{ in }C^0({{\mathbb R}}^{+}).$$ Then, if $(\rho_n,{{ u}}_n)$ verifies -, up to a subsequence, still denoted $n$, the following convergences hold.\ 1. The sequence $\rho_n$ convergences strongly to $\rho$ in $C^0(0,T;L^p({\Omega}))$ for any $1\leq p<\gamma.$\ 2. The sequence $\mu_n(\rho_n)\, {{ u}}_n$ converges to $\mu(\rho){{ u}}$ in $L^{\infty}(0,T;L^p({\Omega})$ for $p \in [1,3/2)$.\ 3. The sequence $({{\mathbb T}_\mu})_n$ convergences to ${{\mathbb T}_\mu}$ weakly in $L^2(0,T;L^2({\Omega}))$.\ 4. For every function $H\in W^{2,\infty}(\overline{{{\mathbb R}}^d})$ and $0<\alpha<{2\gamma}/{\gamma+1}$, we have that $\rho_n^{\alpha} H({{ u}}_n)$ convergences to $\rho^{\alpha}H({{ u}})$ strongly in $L^p(0,T;{\Omega})$ for $1\leq p<\frac{2\gamma}{(\gamma+1)\alpha}.$ In particular, $\sqrt{\mu(\rho_n)}H({{ u}}_n)$ convergences to $\sqrt{\mu(\rho)}H({{ u}})$ strongly in $L^{\infty}(0,T;L^2({\Omega})).$ Using , the Aubin-Lions lemma gives us, up to a subsequence, $$\mu_n(\rho_n)\to \tilde{\mu}\quad\text{ in }\; C^0(0,T;L^q({\Omega}))$$ for any $q<\frac{3}{2}.$ But $$\sup|\mu_n-\mu|\to 0$$ as $n\to \infty.$ Thus, we have $$\label{almost anywhere for mu} \mu_n(\rho_n)\to \tilde{\mu}(t,x)\quad\text{ in }\; C^0([0,T];L^q({\Omega})),$$ so up to a subsequence, $$\mu(\rho_n)\to \tilde{\mu}(t,x)\;\;\text{a. e}.$$ Note that $\mu$ is increasing function, so it is invertible, and $\mu^{-1}$ is continuous. This implies that $\rho_n\to \rho$ a.e. with $\mu(\rho)=\tilde{\mu}(t,x).$ Together with and $\rho_n$ is uniformly bounded in $L^{\infty}(0,T;L^{\gamma}({\Omega}))$, thus we get part 1. Note that $$\nabla\frac{\mu(\rho_n)}{\sqrt{\rho_n}}=\frac{\sqrt{\rho_n}\nabla \mu(\rho_n)}{\rho_n}-\frac{\mu(\rho_n)\nabla\rho_n}{2\rho\sqrt{\rho_n}},$$ thus $$\left|\nabla\frac{\mu(\rho_n)}{\sqrt{\rho_n}}\right|\leq C\left|\sqrt{\rho_n}\right|\left|\frac{\nabla\mu(\rho_n)}{\sqrt{\rho_n}}\right|,$$ so $\nabla\frac{\mu(\rho_n)}{\sqrt{\rho_n}}$ is bounded in $L^{\infty}(0,T;L^2({\Omega}))$, thanks to . Using , we have $\frac{\mu(\rho_n)}{\sqrt{\rho_n}}$ is bounded in $L^{\infty}(0,T;W^{1,2}({\Omega}))$, thus it is uniformly bounded in $L^{\infty}(0,T;L^6({\Omega}))$. On the other hand, $\sqrt{\rho_n}{{ u}}_n$ is uniformly bounded in $L^{\infty}(0,T;L^2({\Omega}))$. From Lemma \[Compactnesstool1\], we have $$\mu(\rho_n){{ u}}_n=\frac{\mu(\rho_n)}{\sqrt{\rho_n}} \sqrt{\rho_n}{{ u}}_n\to \mu(\rho){{ u}}\;\;\text{ in }\; L^{\infty}(0,T;L^q({\Omega}))$$ for any $1\leq q<\frac{3}{2}.$ Since $({{\mathbb T}_\mu})_n$ is bounded in $L^2(0,T;L^2({\Omega}))$, and so, up to a sequence, convergences weakly in $L^2(0,T;L^2({\Omega}))$ to a function ${{\mathbb T}_\mu}$. Using Lemma \[Compactnesstool1\], this gives part 4. With Lemma \[Lemma of stability of renormalized solution\], we are able to recover the renormalized solutions of Navier-Stokes equations without any additional term by letting $n\to\infty$ in . We state this result in the following Lemma. In this lemma, we fix $\mu$ such that $\varepsilon_1>0$. \[Lemma of existence for ren\] For any fixed $\varepsilon_1>0$, there exists a renormalized solution $(\sqrt{\rho},\sqrt{\rho}{{ u}})$ to the initial value problem -. We can use Lemma \[Lemma of stability of renormalized solution\] to pass to the limits for the extra terms. We will have to follow this order: let $r_2$ goes to zero, then $r_1$ tends to zero, after that $r_0, \delta, r$ go to zero together. – If $r_2= r_2(n) \to 0$, we just write $$r_2\frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n=r_2^{\frac{1}{4}}\big(\frac{\rho_n}{\mu'(\rho_n)}\big)^{\frac{1}{4}}\big(\frac{\rho_n}{\mu'(\rho_n)}\big)^{\frac{3}{4}}|{{ u}}_n|^2{{ u}}_n,$$ and $\mu'(\rho_n)\geq \varepsilon_1 >0,$ so $\big(\frac{\rho_n}{\mu'(\rho_n)}\big)^{\frac{1}{4}}\leq C|\rho_n|^{\frac{1}{4}}$, thus, $$r_2\frac{\rho_n}{\mu'(\rho_n)}|{{ u}}_n|^2{{ u}}_n\to 0 \hbox{ in } L^{\frac{4}{3}}(0,T;L^{\frac{6}{5}}({\Omega})).$$ – For $r_1=r(n)\to 0$, $$|r_1\rho_n|{{ u}}_n|{{ u}}_n|\leq r^{\frac{1}{3}}\rho_n^{\frac{1}{3}}r^{\frac{2}{3}}\rho_n^{\frac{2}{3}}|{{ u}}_n|^2,$$ which convergences to zero in $L^{\frac{3}{2}}(0,T;L^{\frac{9}{7}}({\Omega}))$ using the drag term control in the energy and the information on the pressure law $P(\rho) = a \rho^\gamma$. – For $r_0 = r_0(n) \to 0$, it is easy to conclude that $$r_0 {{ u}}_{n} \to 0 \hbox{ in } L^2((0,T)\times \Omega).$$ – We now consider the limit $r\to 0$ of the term $$r\rho_n\nabla\left(\sqrt{K(\rho_n)}\D(\int_0^{\rho_n}\sqrt{K(s)}\,ds)\right).$$ Note the following identity $$\label{BCNV relation} \rho_n\nabla\left(\sqrt{K(\rho_n)}\D(\int_0^{\rho_n}\sqrt{K(s)}\,ds)\right)= 2 {{\rm div}}\Bigl(\mu(\rho_n)\nabla^2\bigl(2 {s}(\rho_n)\bigr)\Bigr) +\nabla\Bigl(\lambda(\rho_n)\D\bigl(2{s}(\rho_n)\bigr)\Bigr),$$ we only need to focus on ${{\rm div}}\Bigl(\mu(\rho_n)\nabla^2\bigl(2 {s}(\rho_n)\bigr)\Bigr)$ since the same argument holds for the other term. Since $$\begin{split} r\int_{{\Omega}}{{\rm div}}\Bigl(\mu(\rho_n)&\nabla^2\bigl(2 {s}(\rho_n)\bigr)\Bigr)\psi\,dx \\&=r\int_{{\Omega}}\frac{\rho_n}{\mu_n}\nabla Z(\rho_n) \otimes \nabla Z(\rho_n)\nabla\psi\,dx +r\int_{{\Omega}}\mu_n\nabla{s}(\rho_n)\Delta\psi\,dx\\&= r\int_{{\Omega}}\frac{\rho_n}{\mu_n}\nabla Z(\rho_n) \otimes \nabla Z(\rho_n)\nabla\psi\,dx+r\int_{{\Omega}}\sqrt{\mu_n}\nabla Z(\rho_n)\Delta\psi\,dx, \end{split}$$ the first term can be controlled as $$\begin{split} &\big|r\int_{{\Omega}}\sqrt{\mu_n}\nabla Z(\rho_n)\Delta\psi\,dx\big|\leq Cr^{\frac{1}{2}}\|\sqrt{\mu(\rho_n)}\|_{L^2(0,T;L^2({\Omega}))}\|\sqrt{r}\nabla Z(\rho_n)\|_{L^2(0,T;L^2({\Omega}))}\to 0, \end{split}$$ thanks to and ; and the second term as $$\begin{split} &\big|\int_{{\Omega}}\frac{\rho_n}{\mu_n}\nabla Z(\rho_n)\otimes \nabla Z(\rho_n)\nabla\psi\,dx\big|\leq \sqrt{r}\sqrt{r}\int_{{\Omega}}\sqrt{\mu(\rho_n)}\frac{\rho_n}{\mu(\rho_n)^{\frac{3}{2}}}|\nabla Z(\rho_n)|^2|\nabla\psi|\,dx \\&\leq C\|\sqrt{r}\frac{\rho_n}{\mu(\rho_n)^{\frac{3}{2}}}|\nabla Z(\rho_n)|^2\|_{L^2(0,T;L^2({\Omega}))}\|\sqrt{\mu(\rho_n)}\|_{L^2(0,T;L^2({\Omega}))}r^{\frac{1}{2}}\to 0. \end{split}$$ – Concerning the quantity $\delta \rho^{10}$, thanks to $\mu'_{\varepsilon_1}(\rho)\geq \varepsilon_1>0,$ $\sqrt{\delta}|\nabla\rho^{5}|$ is uniformly bounded in $L^2(0,T;L^2({\Omega}))$. This gives us that $\delta^{\frac{1}{30}}\rho$ is uniformly bounded in $L^{10}(0,T;L^{30}({\Omega})).$ Thus, we have $$\left| \int_0^T\int_{{\Omega}}\delta\rho^{10}\nabla\psi\,dx\,dt\right| \leq C(\psi) \delta^{\frac{2}{3}}\|\delta^{\frac{1}{3}}\rho^{10}\|_{L^1(0,T;L^3({\Omega}))}\to 0$$ as $\delta\to0.$ With Lemma \[Lemma of stability of renormalized solution\] at hand, we are ready to recover the renormalized solutions to -. By part 1 and part 2 of Lemma \[Lemma of stability of renormalized solution\], we are able to pass to the limits on the continuity equation. Thanks to part 4 of Lemma \[Lemma of stability of renormalized solution\], $$\sqrt{\mu(\rho_n)}\varphi'({{ u}}_n)\to \sqrt{\mu(\rho)}\varphi'({{ u}}) \quad\text{ in }\;\; L^{\infty}(0,T;L^2({\Omega})).$$ With the help of Lemma \[compactuniforme\], we can pass to the limit on pressure, thus we can recover the renormalized solutions. Recover weak solutions from renormalized solutions -------------------------------------------------- In this part, we can recover the weak solutions from the renormalized solutions constructed in Lemma \[Lemma of existence for ren\]. Now we show that Lemma \[Lemma of existence for ren\] is valid without the condition $\varepsilon_1>0$. For such a $\mu$, we construct a sequence $\mu_n$ converging to $\mu$ in $C^0({{\mathbb R}}^+)$ and such that $\varepsilon_{1n}=\inf \mu_n'>0$. Lemma \[Lemma of stability of renormalized solution\] shows that, up to a subsequence, $$\rho_n\to\rho\;\;\text{ in }\; C^0(0,T;L^p({\Omega}))$$ and $$\rho_n{{ u}}_n\to\rho{{ u}}\;\;\text{ in } L^{\infty}(0,T;L^{\frac{p+1}{2p}}({\Omega}))$$ for any $1\leq p<\gamma,$ where $(\rho,\sqrt{\rho}{{ u}})$ is a renormalized solution to . Now, we want to show that this renormalized solution is also a weak solution in the sense of Definition 1.2. To this end, we introduce a non-negative smooth function $\Phi:{{\mathbb R}}\to{{\mathbb R}}$ such that it has a compact support and $\Phi(s)=1$ for any $-1\leq s\leq1.$ Let $\tilde{\Phi}(s)=\int_0^s\Phi(r)\,dr$, we define $$\varphi_n(y)=n\tilde{\Phi}(\frac{y_1}{n})\Phi(\frac{y_2}{n})....\Phi(\frac{y_N}{n})$$ for any $y=(y_1,y_2,....,y_N)\in {{\mathbb R}}^N$. Note that $\varphi_n$ is bounded in $W^{2,\infty}({{\mathbb R}}^N)$ for any fixed $n>0$, $\varphi_n(y)$ converges everywhere to $y_1$ as $n$ goes to infinity, $\varphi_n'$ is uniformly bounded in $n$ and converges everywhere to unit vector $(1,0,....0)$, and $$\|\varphi_n''\|_{L^{\infty}}\leq \frac{C}{n}\to 0$$ as $n$ goes to infinity. This allows us to control the measures in Definition \[def\_renormalise\_u\] as follows $$\|R_{{{\varphi}}_n}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})}+ \|\overline{R}^1_{{{\varphi}}_n}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})} + \|\overline{R}^2_{{{\varphi}}_n}\|_{ \mathcal{M}({{\mathbb R}}^+\times{\Omega})} \leq C \|{{\varphi}}''_n\|_{L^\infty({{\mathbb R}})}\to 0$$ as $n$ goes to infinity. Using this function $\varphi_n$ in the equation of Definition \[def\_renormalise\_u\], the Lebesgue’s Theorem gives us the equation on $\rho{{ u}}_1$ in Definition 1.2 by passing limits as $n$ goes to infinity. In this way, we are able to get full vector equation on $\rho{{ u}}$ by permuting the directions. Applying the Lebesgue’s dominated convergence Theorem, one obtains by passing to limit in with $i=1$ and the function $\varphi_n$. 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ArXiv

--- abstract: 'We study cryptography based on operator theory, and propose quantum no-key (QNK) protocols from the perspective of operator theory, then present a framework of QNK protocols. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Then we defined the information-theoretical security of QNK protocols and the security of identification keys. Two kinds of QNK protocols are also proposed. The first scheme is constructed based on unitary transformation, and the other is constructed based on two multiplicative commutative sets.' address: - 'State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China' - 'Data Communication Science and Technology Research Institute, Beijing 100191, China' author: - Li Yang - Min Liang title: 'Cryptography based on operator theory (): quantum no-key protocols' --- quantum cryptography ,quantum no-key protocol ,man-in-the-middle attack,information-theoretical security Introduction ============ The earliest group of quantum message oriented protocols is suggested in [@Boykin00; @Ambainis00; @Nayak07], which can be regarded as a quantum version of one-time pad, the sender and the receiver must preshare secretly a classical key. Later, a public-key encryption scheme of quantum message is proposed [@Yang03]. Recently, this kind of public-key cryptosystems has been developed [@Yang10]. Here we consider another technique to securely transmit quantum message, so called quantum no-key (QNK) protocol. No-key protocol was first proposed by Shamir [@Menezes97]. It is a wonderful idea to transmit classical messages secretly in public channel, independent of the idea of public-key cryptosystem and that of secret-key cryptosystem. However, the protocol presented is computationally secure, cannot resists a man-in-the-middle(MIM) attack. [@Yangli02a; @Yangli02] develop a quantum from of no-key protocol based on single-photon rotations, which can be used to transmit classical and quantum messages secretly. It can be seen that the security of the QNK protocol is based on the laws of quantum mechanics, so it is beyond computational hypothesis. Ref. [@Yangli03] proposed a protocol based on quantum computing of Boolean functions. This protocols is constructed with inherent identifications in order to prevent MIM attack. Similar to the idea of QNK protocol, Kanamori et al.[@Kanamori05] proposed a protocol for secure data communication, Kye et al.[@Kye05] proposed a quantum key distribution scheme, and Kak [@Subhash07] proposed a three-stage quantum cryptographic protocol for key agreement. Wu and Yang [@Wu09] presents a practical QNK protocol, and studied a new kind of attack named unbalance-of-information-source (UIS) attack. This kind of attack may also be effective to quantum secure direct communication protocols, such as those in [@Beige01; @Bostrom02; @Deng2003; @Deng2004]. In the paper, the theory of QNK protocols is studied, and a framework of QNK protocols is presented. Then we defined the information-theoretical security of QNK protocols, and the security of identification keys. Finally, two kinds of QNK protocols are presented. Quantum no-key protocols ======================== Framework --------- Let us consider a general framework of quantum no-key protocol, in which two ancillary states are used. Suppose Alice will send quantum message $\rho\in H_M$. The ancillary states used by Alice and Bob are $\rho_A$ and $\rho_B$, respectively. The framework of QNK protocol is described as (see Figure \[fig1\]): ![\[fig1\] A general framework of quantum no key protocol. This figure is divided into two part by a dashed line. The part above the dashed line describes Alice’s operations, and the other part describes Bob’s operations. The quantum state $\rho$ is the plain state, and $\rho_1,\rho_2,\rho_3$ represents the three cipher states transmitted between Alice and Bob. $\rho_A,\rho_B$ are two ancillary states generated randomly by Alice and Bob, respectively.](quantum-no-key){width="12cm"} 1. Alice randomly prepare a quantum state $\rho_A$, then performs $U_A$ on the quantum states $\rho_A\otimes\rho$ and gets $U_A(\rho_A\otimes\rho)U_A^\dagger$. Then she sends to Bob the first cipher state $\rho_1$, $$\rho_1=tr_A(U_A(\rho_A\otimes\rho)U_A^\dagger)\triangleq \mathcal{E}_A(\rho).$$ She retains the state $\rho_A'=tr_M(U_A(\rho_A\otimes\rho)U_A^\dagger)$. 2. Bob randomly prepares a quantum state $\rho_B$, then performs $U_B$ on the quantum states $\rho_1\otimes\rho_B$ and gets $U_B(\rho_1\otimes\rho_B)U_B^\dagger$. Then he sends to Alice the second cipher state $\rho_2$, $$\rho_2=tr_B(U_B(\rho_1\otimes\rho_B)U_B^\dagger)\triangleq \mathcal{E}_B(\rho_1).$$ He retains the state $\rho_B'=tr_M(U_B(\rho_1\otimes\rho_B)U_B^\dagger)$. 3. Alice performs $U_A'$ on $\rho_A'\otimes\rho_2$, and sends to Bob the third cipher state $\rho_3$, $$\rho_3=tr_A(U_A'(\rho_A'\otimes\rho_2)U_A'^\dagger)\triangleq\mathcal{E}_A'(\rho_2).$$ 4. Bob performs $U_B'$ on $\rho_3\otimes\rho_B'$, and gets the message $\rho\prime$, $$\rho'=e^{i\phi}\rho=tr_B(U_B'(\rho_3\otimes\rho_B')U_B'^\dagger)\triangleq\mathcal{E}_B'(\rho_3).$$ In the above protocols, the four quantum operations $\mathcal{E}_A,\mathcal{E}_B,\mathcal{E}_A',\mathcal{E}_B'$ are all trace-preserving quantum operators. This protocol is correct if and only if the four quantum operators satisfy this condition: $$\label{eqn5} \mathcal{E}_B'\circ\mathcal{E}_A'\circ\mathcal{E}_B\circ\mathcal{E}_A=e^{i\phi}\mathcal{I}.$$ [**Remark:**]{} As a special case, the unitary transformations $U_A$,$U_B$ can be chosen as bitwise controlled-unitary transformations where the message qubits act as control qubits, and $U_A'=U_A^\dagger$,$U_B'=U_B^\dagger$. In this case, $(I\otimes U_B)(U_A \otimes I)=(U_A \otimes I)(I\otimes U_B)$, and $(I\otimes U_B')(U_A' \otimes I)(I\otimes U_B)(U_A \otimes I)=I$. Natural Representation ---------------------- Trace-preserving quantum operator $\mathcal{E}$ can be written as the form of operator-sum representation $$\begin{aligned} \mathcal{E}(\rho)=\sum_i E_{i}\rho E_{i}^\dagger.\end{aligned}$$ Its natural representation [@Watrous08] is $$\begin{aligned} \label{eqn7} \overrightarrow{\mathcal{E}}(\rho)= \sum_i (E_{i}\otimes E_{i}^*)\overrightarrow{\rho} \triangleq B\overrightarrow{\rho},\end{aligned}$$ where $\overrightarrow{\rho}$ is a the column vector, and represents the vector form of the density matrix $\rho$. The quantum operations $\mathcal{E}_1$ and $\mathcal{E}_2$ are trace-preserving quantum operator, and their natural representations are denoted as $A,B$ respectively. Because trace-preserving quantum operator can be realized physically, the operators $A,B$ can be realized physically. Suppose four operators $A,B,A',B'$ represent the natural representation of the four trace-preserving quantum operators $\mathcal{E}_A,\mathcal{E}_B,\mathcal{E}_A',\mathcal{E}_B'$. Next we deduce the expressions of the natural representation $A,B,A',B'$. Let the ancillary states of Alice and Bob are $\rho_A=\rho_B=|0\rangle\langle 0|$, the orthogonal basis on the ancillary space is the set $\{e_k\}_k$. From the trace-preserving quantum operator $\mathcal{E}_A(\rho)=tr_A(U_A(|0\rangle\langle 0|\otimes\rho)U_A^\dagger)$, it can be inferred that $$\begin{aligned} \mathcal{E}_A(\rho)&=&\sum_k\langle e_k|U_A(|0\rangle\langle 0|\otimes\rho)U_A^\dagger|e_k\rangle \\ &=&\sum_k E_k\rho E_k^\dagger,\end{aligned}$$ where $E_k\equiv\langle e_k|U_A|0\rangle$ is an operator acting on the message space $H_M$. Thus, according to Eq.(\[eqn7\]), one know that the natural representation of trace-preserving quantum operator $\mathcal{E}_A$ is as follows $$A=\sum_k E_k\otimes E_k^*=\sum_k \langle e_k|U_A|0\rangle\otimes\langle e_k|U_A^*|0\rangle.$$ Similarly, one can present the natural representations of the other three trace-preserving quantum operators, for example $B=\sum_k \langle e_k|U_B|0\rangle\otimes\langle e_k|U_B^*|0\rangle$. From the above analysis, the four transformations to the quantum message $\rho$ in the quantum no-key protocol can be described in the form of natural representation. $$\begin{aligned} \overrightarrow{\rho}&\longrightarrow&\overrightarrow{\rho_1}=A\overrightarrow{\rho}\\ &\longrightarrow&\overrightarrow{\rho_2}=B\overrightarrow{\rho_1}\\ &\longrightarrow&\overrightarrow{\rho_3}=A'\overrightarrow{\rho_2}\\ &\longrightarrow&\overrightarrow{\rho}'=B'\overrightarrow{\rho_3}\end{aligned}$$ Quantum no-key protocol can be described in the form of trace-preserving quantum operators or natural representation, and the two forms are equivalent. In the following sections, we use the natural representation, and the operators $A$ and $B$ are both natural representations of trace-preserving quantum operators. $\mathcal{E}_1\circ\mathcal{E}_2=\mathcal{E}_2\circ\mathcal{E}_1$ if and only if $AB=BA$. The additive commutator $A,B$ satisfy $$AB=BA+K;$$ The multiplicative commutator $A,B$ satisfy $$BA=e^{i\lambda}AB, \lambda\neq 0,$$ or be written as $B^{-1}A^{-1}BA=e^{i\lambda}I, \lambda\neq 0$. When $K=\lambda=0$, the additive commutator equals to multiplicative commutator. The multiplicative commutator $BA=e^{i\lambda}AB$ is just additive commutator when $\lambda=0$; Multiplicative commutator is different from additive commutator in only a global phase $e^{i\lambda}$ when $\lambda \neq 0$, and they have no difference in physical implementation. [**Theorem 1**]{}: The multiplicative commutators $A,B$ are unitary matrices. If $BA=e^{i\lambda}AB$, where $\lambda\neq 0$, then the sum of the eigenvalues of the operators $A,B$ are zero, respectively. [**Proof**]{} It means to prove: if $A=\sum_j e^{i\varphi_j}|a_j\rangle\langle a_j|$, then $\sum_j e^{i\varphi_j}=0$; if $B=\sum_j e^{i\phi_j}|b_j\rangle\langle b_j|$, then $\sum_j e^{i\phi_j}=0$. The proof is as follows. From $BA=e^{i\phi_{AB}}AB$ where $A,B$ are both unitary, one know that $A=e^{i\phi_{AB}}B^\dagger AB$. Because $A$ is a unitary transformation, it has spectral decomposition $A=\sum_j\lambda_j|j\rangle\langle j|$, and each eigenvalue $\lambda_j$ can be written as $e^{i\phi_j}$. Thus, $\sum_j\lambda_j|j\rangle\langle j|=\sum_je^{i\phi_{AB}}\lambda_jB^\dagger |j\rangle\langle j|B$. Then $$\label{eqn8} \lambda_k=\sum_je^{i\phi_{AB}}\lambda_j\langle k|B^\dagger|j\rangle\langle j|B|k\rangle=\sum_je^{i\phi_{AB}}\lambda_j|\langle j|B|k\rangle|^2.$$ Let the $j$-th row and $k$-th column of $B$ is $b_{jk}$. The two side of Eq.(\[eqn8\]) is added with the variable $k$, and obtains $\sum_k\lambda_k=e^{i\phi_{AB}}\sum_j\lambda_j\sum_k|b_{jk}|^2=e^{i\phi_{AB}}\sum_j\lambda_j$. Thus, $\sum_j\lambda_j(e^{i\phi_{AB}}-1)=0$. If $\phi_{AB}\neq 0$, then $\sum_j\lambda_j=0$, that means the sum of the eigenvalues of $A$ is 0. Similarly, one can prove that the sum of the eigenvalues of $B$ is 0.$~\hfill{}\Box$ In the next part of this paper, we consider the case that $A'=A^{-1},B'=B^{-1}$. In this case, Eq.(\[eqn5\]) can also be expressed as $B^{-1}A^{-1}BA=e^{i\phi}I$. It can be seen that $A,B$ are multiplicative commutator. In order to identify personal identification in the protocols, Alice and Bob must preshare a secret key $k,i$, and the multiplicative commutator $A,B$ should satisfy: $$B^{-1}A^{-1}BA=e^{i\phi}N_k(i).$$ Alice and Bob preshare secret identification key $k,i$, where $k\in\mathcal{K}$. From the value of $k$, a operator $N_k$ can be obtained and a set $I(k)$ can be constructed, and the secret key $i\in I(k)$. From $k,i$, we can get a set $L(k,i)$ and a set of operators satisfying $$S_k(i)=\{A_l^{(i)}(k)|l\in L(k,i)\}\cup \{B_l^{(i)}(k)|l\in L(k,i)\}.$$ The set of operators $S_k(i)$ should satisfy the condition: for any two elements $A_{l_1}^{(i)}(k),B_{l_2}^{(i)}(k)\in S_k(i)$, it holds that $$\label{eqn1} [B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k) =e^{i\phi(l_1,l_2)}N_k(i).$$ Alice and Bob communicates according to $k,i$ and the set $S_k(i)$. The process is as follows 1. Alice randomly selects $l_1\in L(k,i)$, and performs an operator $A_{l_1}^{(i)}(k)$ on quantum message $\overrightarrow{\rho}$, then obtains a state $\overrightarrow{\rho_1}$. She sends it to Bob. 2. Bob randomly selects $l_2\in L(k,i)$, and performs an operator $B_{l_2}^{(i)}(k)$ on quantum state $\overrightarrow{\rho_1}$, then obtains a state $\overrightarrow{\rho_2}$. He sends it to Alice. 3. According to $l_1$, Alice performs an operator $[A_{l_1}^{(i)}(k)]^{-1}$ on quantum state $\overrightarrow{\rho_2}$, then obtains a state $\overrightarrow{\rho_3}$. She sends it to Bob. 4. According to $l_2$, Bob performs an operator $[B_{l_2}^{(i)}(k)]^{-1}$ on quantum state $\overrightarrow{\rho_3}$, then obtains a state $\overrightarrow{\rho_4}$. From the condition satisfied by $S_k(i)$, it is deduced that $$\overrightarrow{\rho_4}=[B_{l_2}^{(i)}(k)]^{-1} [A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k) \overrightarrow{\rho} = e^{i\phi(l_1,l_2)}N_k(i) \overrightarrow{\rho}.$$ Thus, the quantum state obtained by Bob in the end is $N_k(i) \overrightarrow{\rho}$. He can recovery the quantum message $\overrightarrow{\rho}$ by performing the inverse transformation of $N_k(i)$. Through the value of $l_1$ is unknown by Bob, Bob randomly selects a value of $l_1'\in L(k,i)$, then it can satisfy the relation in Eq.(\[eqn1\])(only a little difference on total phase). Thus, the inverse transformation of $N_k(i)$ can also be replace by $$N_k^{-1}(i)=[A_{l_1'}^{(i)}(k)]^{-1}[B_{l_2}^{(i)}(k)]^{-1}A_{l_1'}^{(i)}(k)B_{l_2}^{(i)}(k), \forall l_1'\in L(k,i).$$ Because the operators $A_l^{(i)}(k)$, $B_l^{(i)}(k)$ are all trace-preserving quantum operators and can be implemented physically, the quantum operator $N_k^{-1}$ is physically implementable. Thus, Bob can recovery the quantum message $\overrightarrow{\rho}$ by performing $N_k^{-1}(i)$ on the state $\overrightarrow{\rho_4}$. In order to enhance the security of this protocol, we can select a group of similarity transformations $\{T_i|i\in L(k,i)\}$. The set $S_k(i)$ in the above protocol is replaced with another set $$\tilde{S}_k(i)=\{T_iA_l^{(i)}(k)T_i^{-1}|l\in L(k,i)\}\cup \{T_iB_l^{(i)}(k)T_i^{-1}|l\in L(k,i)\}.$$ Similarly, $N_k(i)$ is replaced with $T_iN_k(i)T_i^{-1}$, but it is still denoted as $N_k(i)$ for convenient. Then according to Eq.(\[eqn1\]), we know $$T_i[B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k)T_i^{-1}=e^{i\phi(l_1,l_2)}N_k(i).$$ From the following relation $$\begin{aligned} & T_i[B_{l_2}^{(i)}(k)]^{-1}[A_{l_1}^{(i)}(k)]^{-1} B_{l_2}^{(i)}(k) A_{l_1}^{(i)}(k)T_i^{-1} \nonumber\\ =& [T_iB_{l_2}^{(i)}(k)T_i^{-1}]^{-1}[T_iA_{l_1}^{(i)}(k)T_i^{-1}]^{-1} T_iB_{l_2}^{(i)}(k)T_i^{-1} T_iA_{l_1}^{(i)}(k)T_i^{-1},\end{aligned}$$ it is inferred that: any two elements in $\tilde{S}_k(i)$ such as $T_iA_{l_1}^{(i)}(k)T_i^{-1},T_iB_{l_2}^{(i)}(k)T_i^{-1}$ satisfy the following relation $$[T_iB_{l_2}^{(i)}(k)T_i^{-1}]^{-1}[T_iA_{l_1}^{(i)}(k)T_i^{-1}]^{-1} T_iB_{l_2}^{(i)}(k)T_i^{-1} T_iA_{l_1}^{(i)}(k)T_i^{-1}=e^{i\phi(l_1,l_2)}N_k(i).$$ Security -------- In QNK protocol, there are 3 times of transmitting quantum ciphers through public quantum channel. Denote $\rho_i$ as the $i$-th quantum cipher with respect to the attackers, where $i=1,2,3$. [**Definition 1:**]{} QNK protocol is information-theoretically secure, if the three quantum ciphers $\rho_1,\rho_2,\rho_3$ satisfy the following condition: for any positive polynomial $p(.)$, and all sufficiently big number $n$, it holds that $$D\left(\rho_i,\rho_j\right)<\frac{1}{p(n)},\forall i,j\in\{1,2,3\},$$ where $\rho_1,\rho_2,\rho_3$ are all $n$-qubit ciphers. In fact, this definition equals to the following definition. [**Definition 2:**]{} QNK protocol is information-theoretically secure, if the three quantum ciphers $\rho_1,\rho_2,\rho_3$ satisfy the following condition: there exists a quantum state $\tau$, such that for any positive polynomial $p(.)$ and all sufficiently big number $n$, it holds that $$D\left(\rho_i,\tau\right)<\frac{1}{p(n)},\forall i\in\{1,2,3\},$$ where $\rho_1,\rho_2,\rho_3$ are all $n$-qubit ciphers. In one hand, if QNK protocol satisfies Definition 2, it can be deduced that $D\left(\rho_i,\rho_j\right)<D\left(\rho_i,\tau\right)+D\left(\rho_j,\tau\right)<\frac{2}{p(n)},\forall i,j\in\{1,2,3\}$, then the protocol satisfies Definition 1; In the other hand, if QNK protocol satisfies Definition 1, and let $\tau=\rho_1$, we know that $D\left(\rho_i,\tau\right)=D\left(\rho_i,\rho_1\right)<\frac{1}{p(n)},\forall i\in\{1,2,3\}$, then the protocol satisfies Definition 2. Thus, the two definitions are equivalent to each other. Here, $\rho_i$ is the $i$-th quantum cipher with respect to the attackers, where $i=1,2,3$. In the QNK protocol, $\overrightarrow{\rho_1}$ is obtained by performing quantum transformation $A_{l_1}^{(i)}(k)$ on quantum state $\overrightarrow{\rho}$. However, with regard to the attackers, the random number $l_1$ and authentication key $i,k$ used by Alice cannot be obtained, so $$\overrightarrow{\rho_1}=\sum_{i,k,l_1}p_i p_k p_{l_1} A_{l_1}^{(i)}(k)\overrightarrow{\rho},$$ where $p_i, p_k$ are the probability of selecting the authentication key $i,k$, and $p_{l_1}$ is the probability of the local number $l_1$ being selected by Alice. Similarly, the attackers cannot obtain the random number $l_2$ and authentication key $i,k$ used by Bob, so $$\overrightarrow{\rho_2}=\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\overrightarrow{\rho},$$ where $p_{l_2}$ is the probability of the local number $l_2$ being selected by Bob. $$\overrightarrow{\rho_3}=\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\overrightarrow{\rho}.$$ [**Definition 3:**]{} QNK protocol is information-theoretically secure, if the three operators $\sum_{i,k,l_1}p_i p_k p_{l_1}A_{l_1}^{(i)}(k)$, $\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)$, and\ $\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)$ satisfy the condition: for any positive polynomial $p(.)$, and all sufficiently large number $n$, it holds that $$\begin{aligned} &&||\sum_{i,k,l_1}p_i p_k p_{l_1} A_{l_1}^{(i)}(k)-\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)},\\ &&||\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}\left(B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)-(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)\right)||_{\diamondsuit}<\frac{1}{p(n)},\\ &&||\sum_{i,k,l_1}p_i p_k p_{l_1}A_{l_1}^{(i)}(k)-\sum_{i,k,l_1,l_2}p_i p_k p_{l_1} p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)A_{l_1}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)}.\end{aligned}$$ where the notation $||*||_{\diamondsuit}$ represents diamond norm. It is obvious that a sufficient condition for information-theoretical security is as follow: $$\begin{aligned} &&||I-\sum_{l_2}p_{l_2}B_{l_2}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)},\forall i,k,\\ &&||I-A_{l_1}^{(i)}(k)^{-1}||_{\diamondsuit}<\frac{1}{p(n)}, \forall i,k,l_1, \\ &&||I-\sum_{l_2}p_{l_2}(A_{l_1}^{(i)}(k))^{-1}B_{l_2}^{(i)}(k)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i,k,l_1.\end{aligned}$$ For the QNK protocol which uses authentication key, when considering its security, besides analyzing the security of quantum message, we also should analyze the security of the authentication key. Here we present a definition of the security of authentication key in QNK protocol. [**Definition 4:**]{} The authentication key in QNK protocol is secure, if for any positive polynomial $p(.)$, and all sufficiently large number $n$, it holds that $$\begin{aligned} &&||\sum_{l_1}A_{l_1}^{(i_1)}(k_1)-\sum_{l_1}A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\\ &&||\sum_{l_1,l_2}B_{l_2}^{(i_1)}(k_1)A_{l_1}^{(i_1)}(k_1)-\sum_{l_1,l_2}B_{l_2}^{(i_2)}(k_2)A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\\ &&||\sum_{l_1,l_2}(A_{l_1}^{(i_1)}(k_1))^{-1}B_{l_2}^{(i_1)}(k_1)A_{l_1}^{(i_1)}(k_1)-\sum_{l_1,l_2}(A_{l_1}^{(i_2)}(k_2))^{-1}B_{l_2}^{(i_2)}(k_2)A_{l_1}^{(i_2)}(k_2)||_{\diamondsuit}\\ &&<\frac{1}{p(n)}, \forall i_1,i_2,k_1,k_2.\end{aligned}$$ Two schemes of quantum no-key protocols ======================================= First scheme ------------ Unitary transformation is a special kind of trace-preserving quantum operator. Here we assume the quantum operators used by Alice and Bob in QNK protocols are all unitary. Let $N_k(i)=I$. The sets of operators $\{A_l|l\in\{1,2,\ldots,n_A\}\}$ and $\{B_k|k\in\{1,2,\ldots,n_B\}\}$ are natural representations of unitary operator used by Alice and Bob. The two set satisfy the relation: $$\label{eqn4} B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}=e^{i\phi(l_1,l_2)}I.$$ The above formula can also be written as $B_{l_2}A_{l_1}=e^{i\phi(l_1,l_2)}A_{l_1}B_{l_2}$. $A_l,B_k$ are natural representations of unitary operators. So we can assume $A_l=E_l\otimes E_l^*,B_k=F_k\otimes F_k^*$, where $E_l, F_k$ are unitary transformations, and $l\in\{1,2,\ldots,n_A\}$, $k\in\{1,2,\ldots,n_B\}$. Then $$\begin{aligned} B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}&=&(F_{l_2}\otimes F_{l_2}^*)^{-1}(E_{l_1}\otimes E_{l_1}^*)^{-1}(F_{l_2}\otimes F_{l_2}^*)(E_{l_1}\otimes E_{l_1}^*)\nonumber\\ &=& (F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1})\otimes(F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1})^*\nonumber\\ &\triangleq&V_{l_1,l_2}\otimes V_{l_1,l_2}^*,\end{aligned}$$ where $V_{l_1,l_2}=F_{l_2}^{-1}E_{l_1}^{-1}F_{l_2}E_{l_1}$ is unitary transformation. According to Eq.(\[eqn4\]), we have $V_{l_1,l_2}\otimes V_{l_1,l_2}^*=e^{i\phi(l_1,l_2)}I_{2^{2n}}$. From the identity $I_{2^{2n}}=I_{2^n}\otimes I_{2^n}$, it has $V_{l_1,l_2}\otimes V_{l_1,l_2}^*=aI_{2^n}\otimes bI_{2^n}$, where $ab=e^{i\phi(l_1,l_2)}$, and $b=a^*$. So $e^{i\phi(l_1,l_2)}=aa^*=|a|^2$, then $e^{i\phi(l_1,l_2)}=1$. That means it is impossible to produce a global phase in Eq.(\[eqn4\]). Thus the Eq.(\[eqn4\]) is rewritten as follows $$\label{eqn6} B_{l_2}^{-1}A_{l_1}^{-1}B_{l_2}A_{l_1}=I.$$ Based on this relation, we construct a QNK scheme, which is described as follows. Firstly, two sets of operators are selected, such as $$S(A)=\{A_l|l\in\{1,2,\ldots,n_A\}\},$$ $$S(B)=\{B_k|k\in\{1,2,\ldots,n_B\}\}.$$ By using the two sets, Alice and Bob communicate following this QNK protocol. 1. Alice randomly selects a number $l_1\in\{1,2,\ldots,n_A\}$, then performs quantum operator $A_{l_1}$ on quantum message $\overrightarrow{\rho}$, and obtains the state $\overrightarrow{\rho_1}$. She sends it to Bob. 2. Bob randomly selects a number $l_2\in\{1,2,\ldots,n_B\}$, then performs quantum operator $B_{l_2}$ on quantum message $\overrightarrow{\rho_1}$, and obtains the state $\overrightarrow{\rho_2}$. He sends it to Alice. 3. According to the value of $l_1$, Alice performs quantum operator $A_{l_1}^{-1}$ (or $A_{l_1}^\dagger$) on quantum message $\overrightarrow{\rho_2}$, and obtains the state $\overrightarrow{\rho_3}$. She sends it to Bob. 4. According to the value of $l_2$, Bob performs quantum operator $B_{l_2}^{-1}$ (or $B_{l_2}^\dagger$) on quantum message $\overrightarrow{\rho_3}$, and obtains the state $\overrightarrow{\rho_4}$. According to the relation (Eq.(\[eqn6\])) of the two sets $S(A),S(B)$, we know that $$\overrightarrow{\rho_4}=B_{l_2}^{-1} A_{l_1}^{-1} B_{l_2} A_{l_1} \overrightarrow{\rho} = \overrightarrow{\rho}.$$ Thus, the quantum state obtained by Bob’s performing quantum transformation $B_{l_2}^{-1}$ is just the quantum message sent by Alice. Second scheme ------------- Denote the operation of two operators $A,B$: $(A,B)=B^{-1}A^{-1}BA$. Suppose there exists two groups of operators $S(A)=\{A_i|i=1,\cdots,n_A\}$ and $S(B)=\{B_i|i=1,\cdots,n_B\}$, which satisfy the following condition $$\label{eqn2} (A_i,B_j)=N, \forall i\in\{1,\cdots,n_A\},j\in\{1,\cdots,n_B\},$$ where $N$ is an operator that is independent of $i,j$. [**Proposition 1:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j$ $$\begin{aligned} && (A_i^{-1},B_jB_i^{-1})=I,\\ && (A_j^{-1},B_jB_i^{-1})=I,\\ && (B_i,A_jA_i^{-1})=I,\\ && (B_j,A_jA_i^{-1})=I.\end{aligned}$$ [**Proof:**]{} See the appendix.$\hfill{}\Box$ [**Proposition 2:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j,k,l$ $$(A_iA_j,B_kB_l)=(A_j^2,B_l^2).$$ [**Proof:**]{} See the appendix.$\hfill{}\Box$ According to Proposition 2, we know the relation $(A_{l_1}A_{k_1},B_{l_2}B_{k_2})=(A_{k_1}^2,B_{k_2}^2)$. Based on this relation, a QNK scheme is constructed as follows. Suppose the set of similarity transformations $\{T_i|i\in I\}$ is selected. Firstly we construct a set of operators $S_k(i)$ as follows: $$\begin{aligned} S_{k_1||k_2}(i)&=&\{T_iA_lA_{k_1}T_i^{-1}|l\in\{1,\cdots,n_A\}\} \nonumber\\ &&\cup\{T_iB_lB_{k_2}T_i^{-1}|l\in\{1,\cdots,n_B\}\}.\end{aligned}$$ In the set $S_k(i)$, any two elements $A_{l_1}^{(i)}(k_1||k_2)=T_iA_{l_1}A_{k_1}T_i^{-1}$ and $B_{l_2}^{(i)}(k_1||k_2)=T_iB_{l_2}B_{k_2}T_i^{-1}$ satisfy the following relation $$\begin{aligned} (A_{l_1}^{(i)}(k_1||k_2),B_{l_2}^{(i)}(k_1||k_2))&=& (T_iA_{l_1}A_{k_1}T_i^{-1},T_iB_{l_2}B_{k_2}T_i^{-1}) \nonumber\\ &=& T_i(A_{l_1}A_{k_1},B_{l_2}B_{k_2})T_i^{-1} \nonumber\\ &=& T_i(A_{k_1}^2,B_{k_2}^2)T_i^{-1}.\end{aligned}$$ Denote $N_{k_1||k_2}=(A_{k_1}^2,B_{k_2}^2)$, $N_{k_1||k_2}(i)=T_i N_{k_1||k_2} T_i^{-1}$, then $$(A_{l_1}^{(i)}(k_1||k_2),B_{l_2}^{(i)}(k_1||k_2))=N_{k_1||k_2}(i), ~\forall l_1,l_2.$$ Alice and Bob communicate according to $k_1||k_2,i$ ($k_1\in\{1,\cdots,n_A\},k_2\in\{1,\cdots,n_B\}$, $i\in I$) and the set of operators $S_{k_1||k_2}(i)$. The process is as follows. 1. Alice randomly selects a number $l_1\in\{1,\cdots,n_A\}$, and performs quantum transformation $A_{l_1}^{(i)}(k_1||k_2)$ on quantum message $\overrightarrow{\rho}$, then obtains $\overrightarrow{\rho_1}$. She sends it Bob. 2. Bob randomly selects a number $l_2\in \{1,\cdots,n_B\}$, and performs quantum transformation $B_{l_2}^{(i)}(k_1||k_2)$ on quantum state $\overrightarrow{\rho_1}$, then obtains $\overrightarrow{\rho_2}$. He sends it to Alice. 3. According to $l_1$, Alice performs quantum transformation $[A_{l_1}^{(i)}(k_1||k_2)]^{-1}$ on quantum state $\overrightarrow{\rho_2}$, and obtains $\overrightarrow{\rho_3}$. She sends it to Bob. 4. According to $l_2$, Bob performs quantum transformation $[B_{l_2}^{(i)}(k_1||k_2)]^{-1}$ on quantum state $\overrightarrow{\rho_3}$, and obtains $\overrightarrow{\rho_4}$. From the condition satisfied by $S_{k_1||k_2}(i)$, it can be deduced that $$\overrightarrow{\rho_4}=N_{k_1||k_2}(i)\overrightarrow{\rho}=T_i (A_{k_1}^2,B_{k_2}^2) T_i^{-1}\overrightarrow{\rho}.$$ Bob performs the inverse transformation of $N_{k_1||k_2}(i)$, which is $[N_{k_1||k_2}(i)]^{-1}=T_i (B_{k_2}^2,A_{k_1}^2) T_i^{-1}$, then the quantum message $\overrightarrow{\rho}$ is recovered. [**Proposition 3:**]{} $(A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})=(A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3})$, $\forall i_1,i_2,i_3$, $\forall j_1,j_2,j_3$. [**Proof:**]{} See the appendix.$\hfill{}\Box$ According to Proposition 3, we know the relation $(A_{l_1}A_{k_1}A_{k_2},B_{l_2}B_{k_3}B_{k_4})=(A_{k_2}A_{k_1}A_{k_2},B_{k_4}B_{k_3}B_{k_4})$. Suppose the set of similarity transformations $\{T_i|i\in I\}$ is selected. We construct the set of operators $S_k(i)$ as follows: $$\begin{aligned} S_{k_1||k_2||k_3||k_4}(i)&=&\{T_iA_lA_{k_1}A_{k_2}T_i^{-1}|l=1,\cdots,n_A\} \nonumber\\ &&\cup\{T_iB_lB_{k_3}B_{k_4}T_i^{-1}|l=1,\cdots,n_B\}.\end{aligned}$$ When the set of operators $S_{k_1||k_2||k_3||k_4}(i)$ is used in the QNK communication, $l$ is a random number selected locally, and $k_1,k_2,k_3,k_4,i$ are authentication keys preshared by Alice and Bob. If Alice selects a local random number $l_1$, then she should perform quantum transformation $T_iA_{l_1}A_{k_1}A_{k_2}T_i^{-1}$; if Bob selects a local random number $l_2$, then he should perform quantum transformation $T_iB_{l_2}B_{k_3}B_{k_4}T_i^{-1}$. The detailed process of the three interactive communication is the same as the last scheme. After the fourth step, Bob obtains a quantum state $$\overrightarrow{\rho_4}=T_i (A_{k_2}A_{k_1}A_{k_2},B_{k_4}B_{k_3}B_{k_4}) T_i^{-1}\overrightarrow{\rho}.$$ Then Bob performs quantum transformation $T_i (B_{k_4}B_{k_3}B_{k_4},A_{k_2}A_{k_1}A_{k_2}) T_i^{-1}$ and obtains the quantum message $\overrightarrow{\rho}$ sent by Alice. In general, we have the following relation $$(A_{l_1}A_{k_1}A_{k_2}\cdots A_{k_n},B_{l_2}B_{k_1'}B_{k_2'}\cdots B_{k_n'})=(A_{k_n}A_{k_1}A_{k_2}\cdots A_{k_n},B_{k_n'}B_{k_1'}B_{k_2'}\cdots B_{k_n'}).$$ According to this relation, a new QNK scheme can be constructed similarly. In the new scheme, more operators can be used by Alice and Bob. Construction of the sets $S(A)$ and $S(B)$ ------------------------------------------ Suppose there exists two groups of operators such as $S(A)=\{A_i|i=1,\cdots,n_A\}$ and $S(B)=\{B_i|i=1,\cdots,n_B\}$, and they satisfy the relation expressed in Eq.(\[eqn2\]), then $A_i^{-1}B_jA_i=B_jN,\forall j$. Thus $A_{i_1}^{-1}B_jA_{i_1}=A_{i_2}^{-1}B_jA_{i_2}$. That is $$\label{eqn3} (A_{i_2}A_{i_1}^{-1})B_j=B_j(A_{i_2}A_{i_1}^{-1}),$$ or written as $(A_{i_2}A_{i_1}^{-1},B_j)=I$ or $[A_{i_2}A_{i_1}^{-1},B_j]=0$. (Denote $[A,B]=AB-BA$) According to Eq.(\[eqn2\]) and the similar deduction, it can be inferred that $\forall i,j,k$, $$\begin{aligned} && [A_iA_j^{-1},B_k]=0, \\ && [A_iA_j^{-1},B_k^{-1}]=0, \\ && [B_iB_j^{-1},A_k]=0, \\ && [B_iB_j^{-1},A_k^{-1}]=0,\end{aligned}$$ From $[A_iA_j^{-1},B_k]=0,\forall i,j,k$, it can be deduced that $$\label{eqn3} [A_iA_j^{-1},B_kB_l^{-1}]=[A_iA_j^{-1},B_k]B_l^{-1}+B_k[A_iA_j^{-1},B_l^{-1}]=0+0=0.$$ That means $(A_iA_j^{-1},B_kB_l^{-1})=I, \forall A_i,A_j\in S(A), B_k,B_l\in S(B)$. Thus, we can extend the two sets of operators $S(A),S(B)$ to two new sets of operators $\tilde{S}(A),\tilde{S}(B)$ as follows: $$\begin{aligned} & \tilde{S}(A)=\{A_{i||j}=A_iA_j^{-1}|A_i,A_j\in S(A)\}, \\ & \tilde{S}(B)=\{B_{i||j}=B_iB_j^{-1}|B_i,B_j\in S(B)\},\end{aligned}$$ According to Eq.(\[eqn3\]), if we select one element from each of the two new sets $\tilde{S}(A),\tilde{S}(B)$, such as $A_{i||j},B_{k||l}$ (where $i,j\in\{1,\cdots,n_A\}$, $k,l\in\{1,\cdots,n_B\}$), then the two elements satisfy the relation $(A_{i||j},B_{k||l})=I$. Thus, the new sets extended from $S(A),S(B)$ can also be used in QNK protocol. When $n_A=n_B=2$, the extended sets of operators are $\tilde{S}(A)=\{A_1A_2^{-1}, \\A_2A_1^{-1},A_1A_1^{-1}=I,A_1A_1^{-1}=I\}$ and $\tilde{S}(B)=\{B_1B_2^{-1},B_2B_1^{-1},B_1B_1^{-1}=I,B_1B_1^{-1}=I\}$. Because the identity operator $I$ in the two sets is meaningless, the number of the operators in the two extended sets does not increase through the above method (The number of the operators in each set remains 2). Thus the extension is meaningless. However, the extension is meaningful when $n_A,n_B\geq 3$. Discussions and Conclusions =========================== We study the theory of quantum no-key protocols. Firstly, a framework of quantum no-key protocols is presented. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Secondly, we defined the information-theoretical security of quantum no-key protocol, and the security of authentication keys. Finally, two kinds of quantum no-key protocols are presented. In the first scheme, the quantum operators used by Alice and Bob are all unitary. In the second scheme, Alice and Bob use trace-preserving quantum operators, and the two sets of operators used by Alice and Bob are constructed based on two multiplicative commutative sets. In the paper, there exists the following questions. 1. $k,i$ are secret identification key. How to use them to identify the personal identification of Alice, Bob, or attackers. How to identify in each of the three transformations? 2. How to construct operator $N_k$ and a set $I(k)$ from a given number $k$? 3. How to construct a set $L(k,i)$ and a set of operators $S_k(i)$ from the given $k$ and $i\in I(k)$? 4. In the quantum no-key protocols, the numbers $l_1,l_2\in L(k,i)$ are selected randomly. Whether the random selection of $l_1,l_2$ can prevent from the leakage of identification keys $k\in\mathcal{K}$ and $i\in I(k)$ during the communication? Acknowledgements {#acknowledgements .unnumbered} ================ This work was supported by the National Natural Science Foundation of China (Grant No. 61173157). Appendix {#appendix .unnumbered} ======== [**Proposition 1:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j$ $$\begin{aligned} && (A_i^{-1},B_jB_i^{-1})=I,\\ && (A_j^{-1},B_jB_i^{-1})=I,\\ && (B_i,A_jA_i^{-1})=I,\\ && (B_j,A_jA_i^{-1})=I.\end{aligned}$$ [**Proof:**]{} According to the condition Eq.(\[eqn2\]), $\forall i\neq j$ $$\begin{aligned} && (A_i,B_i)=N,\\ && (A_i,B_j)=N,\\ && (A_j,B_i)=N,\\ && (A_j,B_j)=N.\end{aligned}$$ From the identity $(A_i,B_i)=N=(A_i,B_j)$, it can be deduced $B_i^{-1}A_i^{-1}B_iA_i=B_j^{-1}A_i^{-1}B_jA_i$, then $$B_i^{-1}A_i^{-1}B_i=B_j^{-1}A_i^{-1}B_j.$$ Thus, it can be known that $(B_jB_i^{-1})A_i^{-1}=A_i^{-1}(B_jB_i^{-1})$, that is $(A_i^{-1},B_jB_i^{-1})=I$. The other three relations can be obtained in the same way.$\hfill{}\Box$ [**Proposition 2:**]{} Suppose two sets of operators $S(A),S(B)$ satisfy the condition Eq.(\[eqn2\]), then the following relations hold: $\forall i,j,k,l$ $$(A_iA_j,B_kB_l)=(A_j^2,B_l^2).$$ [**Proof:**]{} In the deduction of this proof, the four identities such as $B_lA_i=A_iB_lN$ (that is $(A_i,B_l)=N$), $A_i^{-1}B_kA_i=B_kN$ (that is $(A_i,B_k)=N$), $B_k^{-1}A_j^{-1}B_k=B_l^{-1}A_j^{-1}B_l$, and $B_lN=A_j^{-1}B_lA_j$ (that is $(A_j,B_l)=N$) are used in turn. The deduction is as follows. $\forall i,j,k,l$, $$\begin{aligned} (A_iA_j,B_kB_l)&=& (B_l^{-1}B_k^{-1})(A_j^{-1}A_i^{-1})B_kB_lA_iA_j \\ &=& (B_l^{-1}B_k^{-1})A_j^{-1}A_i^{-1}B_kA_iB_lNA_j \\ &=& B_l^{-1}B_k^{-1}A_j^{-1}B_kNB_lNA_j \\ &=& B_l^{-1}B_l^{-1}A_j^{-1}B_lNB_lNA_j \\ &=& (B_l^{-1})^2 A_j^{-1}A_j^{-1}B_lA_jA_j^{-1}B_lA_jA_j \\ &=& (B_l^{-1})^2 (A_j^{-1})^2B_l^2A_j^2 \\ &=& (A_j^2,B_l^2).\end{aligned}$$ $\hfill{}\Box$ [**Proposition 3:**]{} $(A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})=(A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3})$, $\forall i_1,i_2,i_3$, $\forall j_1,j_2,j_3$. [**Proof:**]{} It can be inferred from the formula $A_i^{-1}B_kA_i=A_j^{-1}B_kA_j$ that $$\begin{aligned} A_i^{-1}B_{k_1}B_{k_2}B_{k_3}A_i &=& A_i^{-1}B_{k_1}A_iA_i^{-1}B_{k_2}A_iA_i^{-1}B_{k_3}A_i \\ &=& A_j^{-1}B_{k_1}A_jA_j^{-1}B_{k_2}A_jA_j^{-1}B_{k_3}A_j \\ &=& A_j^{-1}B_{k_1}B_{k_2}B_{k_3}A_j.\end{aligned}$$ Similarly, it can be inferred $$\begin{aligned} B_i^{-1}A_{k_1}^{-1}A_{k_2}^{-1}A_{k_3}^{-1}B_i &=& B_j^{-1}A_{k_1}^{-1}A_{k_2}^{-1}A_{k_3}^{-1}B_j.\end{aligned}$$ Thus, the following result holds $$\begin{aligned} (A_{j_1}A_{j_2}A_{j_3},B_{i_1}B_{i_2}B_{i_3})&=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_1}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_1}^{-1}B_{i_1}B_{i_2}B_{i_3}A_{j_1}A_{j_2}A_{j_3} \\ &=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_1}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_2}^{-1}B_{i_1}B_{i_2}B_{i_3}A_{j_2}A_{j_2}A_{j_3} \\ &=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_2}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_2}^{-1}B_{i_2}B_{i_2}B_{i_3}A_{j_2}A_{j_2}A_{j_3} \\ &=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_2}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_3}^{-1}B_{i_2}B_{i_2}B_{i_3}A_{j_3}A_{j_2}A_{j_3} \\ &=& B_{i_3}^{-1}B_{i_2}^{-1}B_{i_3}^{-1}A_{j_3}^{-1}A_{j_2}^{-1}A_{j_3}^{-1}B_{i_3}B_{i_2}B_{i_3}A_{j_3}A_{j_2}A_{j_3} \\ &=& (A_{j_3}A_{j_2}A_{j_3},B_{i_3}B_{i_2}B_{i_3}).\end{aligned}$$ $\hfill{}\Box$ [99]{} P. 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Yang, Quantum no-key protocol for direct and secure transmission of quantum and classical messages. arXiv: quant-ph/0309200, 28 Sep 2003. L. Yang and L. Hu, Quantum no-key protocol with inherent identification, Proc. SPIE Vol. 6305, pp. 63050J (2006). Y. Kanamori, S. M. Yoo and Mohammad, A Quantum No-Key Protocol for Secure Data Communication, 43rd ACM SE Conference, March 18-20, 2005 W. H. Kye, C. M. Kim, M. S. Kim and Y. J. Park, Quantum Key Distribution with Blind Polarization Bases, Phys.Rev.Lett. 95 (4), 2005, 040501. S. Kak, A Three-Stage Quantum Cryptography Protocol, Foundations of Physics Letters, Vol. 19, No. 3, June 2006. Y. Wu and L. Yang, Practical quantum no-key protocol with identification. IAS 2009: 540-543, IEEE Computer Society. A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, Secure communication with a publicly known key. Acta physica Polonica. A 101(3): 357-368. (see also arXiv:quant-ph/0101066). K. Boström and T. Felbinger. 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ArXiv

--- abstract: 'It is shown that signal energy is the only available degree-of-freedom ([DOF]{}) for fiber-optic transmission as the input power tends to infinity. With $n$ signal [DOFs]{}at the input, $n-1$ [DOFs]{} are asymptotically lost to signal-noise interactions. The main observation is that, nonlinearity introduces a multiplicative noise in the channel, similar to fading in wireless channels. The channel is viewed in the spherical coordinate system, where signal vector ${\underaccent{\bar}{X}}\in{\mathbb{C}}^n$ is represented in terms of its norm ${\left|{\underaccent{\bar}{X}}\right|}$ and direction $\hat{{\underaccent{\bar}{X}}}$. The multiplicative noise causes signal direction $\hat{{\underaccent{\bar}{X}}}$ to vary randomly on the surface of the unit $(2n-1)$-sphere in ${\mathbb{C}}^{n}$, in such a way that the effective area of the support of $\hat {{\underaccent{\bar}{X}}}$ does not vanish as ${\left|{\underaccent{\bar}{X}}\right|}\rightarrow\infty$. On the other hand, the surface area of the sphere is finite, so that $\hat{{\underaccent{\bar}{X}}}$ carries finite information. This observation is used to show several results. Firstly, let ${{\mathcal{C}}}({{\mathcal{P}}})$ be the capacity of a discrete-time periodic model of the optical fiber with distributed noise and frequency-dependent loss, as a function of the average input power ${{\mathcal{P}}}$. It is shown that asymptotically as ${{\mathcal{P}}}\rightarrow\infty$, ${{\mathcal{C}}}=\frac{1}{n}\log\bigl(\log{{\mathcal{P}}}\bigr)+c$, where $n$ is the dimension of the input signal space and $c$ is a bounded number. In particular, $\lim_{{{\mathcal{P}}}\rightarrow\infty}{{\mathcal{C}}}({{\mathcal{P}}})=\infty$ in finite-dimensional periodic models. Secondly, it is shown that capacity saturates to a constant in infinite-dimensional models where $n=\infty$. An expression is provided for the constant $c$, by showing that, as the input ${\left|{\underaccent{\bar}{X}}\right|}\rightarrow\infty$, the action of the discrete periodic stochastic nonlinear Schrödinger equation tends to multiplication by a random matrix (with fixed distribution, independent of input). Thus, perhaps counter-intuitively, noise simplifies the nonlinear channel at high powers to a *linear* multiple-input multiple-output fading channel. As ${{\mathcal{P}}}\rightarrow\infty$ signal-noise interactions gradually reduce the slope of the ${{\mathcal{C}}}({{\mathcal{P}}})$, to a point where increasing the input power returns diminishing gains. Nonlinear frequency-division multiplexing can be applied to approach capacity in optical networks, where linear multiplexing achieves low rates at high powers.' author: - 'Mansoor I. Yousefi' title: 'The Asymptotic Capacity of the Optical Fiber[^1] ' --- Introduction ============ Several decades since the introduction of the optical fiber, channel capacity at high powers remains a vexing conundrum. Existing achievable rates saturate at high powers because of linear multiplexing and treating the resulting interference as noise in network environments [@yousefi2012nft1; @yousefi2012nft2; @yousefi2012nft3]. Furthermore, it is difficult to estimate the capacity via numerical simulations, because channel has memory. Multi-user communication problem for (an ideal model of) optical fiber can be reduced to single-user problem using the nonlinear frequency-division multiplexing (NFDM) [@yousefi2012nft1; @yousefi2012nft3]. This addresses deterministic distortions, such as inter-channel and inter-symbol interference (signal-signal interactions). The problem is then reduced to finding the capacity of the point-to-point optical fiber set by noise. There are two effects in fiber that impact Shannon capacity in point-to-point channels. (1) Phase noise. Nonlinearity transforms additive noise to phase noise in the channel. As the amplitude of the input signal tends to infinity, the phase of the output signal tends to a uniform random variable in the zero-dispersion channel [@yousefi2011opc Section IV]. As a result, phase carries finite information in the non-dispersive fiber. (2) Multiplicative noise. Dispersion converts phase noise to amplitude noise, introducing an effect which at high powers is similar to fading in wireless channels. Importantly, the conditional entropy grows strongly with input signal. In this paper, we study the asymptotic capacity of a discrete-time periodic model of the optical fiber as the input power tends to infinity. The role of the nonlinearity in point-to-point discrete channels pertains to signal-noise interactions, captured by the conditional entropy. The main result is the following theorem, describing capacity-cost function in models with constant and non-constant loss; see Definition \[def:loss\]. Consider the discrete-time periodic model of the NLS channel described in Section \[sec:mssfm\], with non-zero dispersion. Capacity is asymptotically [rCl]{} C(P)= (P)+c, & [non-constant loss]{.nodecor},\ P+c, & [constant loss]{.nodecor}, where $n$ is dimension of the input signal space, ${{\mathcal{P}}}\rightarrow\infty$ is the average input signal power and $c{\stackrel{\Delta}{=}}c(n,{{\mathcal{P}}})<\infty$. In particular, $\lim\limits_{{{\mathcal{P}}}\rightarrow\infty} {{\mathcal{C}}}({{\mathcal{P}}})=\infty$ in finite-dimensional models. Intensity modulation and direct detection (photon counting) is nearly capacity-achieving in the limit ${{\mathcal{P}}}\rightarrow\infty$, where capacity is dominated by the first terms in ${{\mathcal{C}}}({{\mathcal{P}}})$ expressions. \[thm:main\] From the Theorem \[thm:main\] and [@yousefi2011opc Theorem 1], the asymptotic capacity of the dispersive fiber is much smaller than the asymptotic capacity of (the discrete-time model of) the zero-dispersion fiber, which is $\frac{1}{2}\log{{\mathcal{P}}}+c$, $c<\infty$. Dispersion reduces the capacity, by increasing the conditional entropy. With $n$ [DOFs]{} at the input, $n-1$ [DOFs]{} are asymptotically lost to signal-noise interactions, leaving signal energy as the only useful [DOF]{} for transmission. There are a finite number of [DOFs]{} in all computer simulations and physical systems. However, as a mathematical problem, the following Corollary holds true. Capacity saturates to a constant $c<\infty$ in infinite-dimensional models, including the continuous-time model. \[cor:inf\] The power level where signal-noise interactions begin to appreciably impact the slope of the ${{\mathcal{C}}}({{\mathcal{P}}})$ is not determined in this paper. Numerical simulations indicate that the conditional entropy does not increase with input in the nonlinear Fourier domain, for a range of power larger than the optimal power in wavelength-division multiplexing [@yousefi2016nfdm Fig. 9 (a)]. In this regime, signal-noise interactions are weak and the capacity is dominated by the (large) number $c$ in the Theorem \[thm:main\]. A numerical estimation of the capacity of the point-to-point fiber at input powers higher than those in Fig. \[fig:nfdm\] should reveal the impact of the signal-dependent noise on the asymptotic capacity. The contributions of the paper are presented as follows. The continuous-time model is discretized in Section \[sec:mssfm\]. The main ingredient is a modification of the split-step Fourier method (SSFM) that shows noise influence more directly compared with the standard SSFM. A *unit* is defined in the modified SSFM (MSSFM) model that plays an important role throughout the paper. The MSSFM and units simplify the information-theoretic analysis. Theorem \[thm:main\] and Corollary \[cor:inf\] are proved in Section \[sec:proof1\]. The main ingredient here is an appropriate partitioning of the [DOFs]{} in a suitable coordinate system, and the proof that the achievable rate of one group of [DOFs]{} is bounded in input. No assumption is made on input power in this first proof. Theorem \[thm:main\] is proved again in Section \[sec:proof2\] by considering the limit ${{\mathcal{P}}}\rightarrow\infty$, which adds further intuition. Firstly, it is shown that, as the input ${\left|{\underaccent{\bar}{X}}\right|}\rightarrow\infty$, the action of the discrete periodic stochastic nonlinear Schrödinger (NLS) equation tends to multiplication by a random matrix (with fixed probability distribution function (PDF), independent of the input). As a result, perhaps counter-intuitively, as ${\left|{\underaccent{\bar}{X}}\right|}\rightarrow\infty$ noise simplifies the nonlinear channel to a *linear* multiple-input multiple-output (non-coherent) fading channel. Secondly, the asymptotic capacity is computed, without calculating the conditional PDF of the channel, entropies, or solving the capacity optimization problem. Because of the multiplicative noise, the asymptotic rate depends only on the knowledge that whether channel random operator has any deterministic component. The conditional PDF merely modifies the bounded number $c$ in the Theorem \[thm:main\]. Note that we do not apply local analysis based on perturbation theories (valid in the low power regime). The proof of the Theorem \[thm:main\], [*e.g.*]{}, the asymptotic loss of [DOFs]{}, is based on a global analysis valid for any signal and noise; see Section \[sec:proof1\]. Notation and Preliminaries {#sec:notation} ========================== The notation in this paper is motivated by [@moser2004dbb]. Upper- and lower-case letters represent scalar random variables and their realizations, [*e.g.*]{}, $X$ and $x$. The same rule is applied to vectors, which are distinguished using underline, [*e.g.*]{}, ${\underaccent{\bar}{X}}$ for a random vector and ${\underaccent{\bar}{x}}$ for a deterministic vector. Deterministic matrices are shown by upper-case letter with a special font, [*e.g.*]{}, ${\mathsf{R}}=(r_{ij})$. Random matrices are denoted by upper-case letters with another special font, [*e.g.*]{}, ${\mathbb{M}}=(M_{ij})$. Important scalars are distinguished with calligraphic font, [*e.g.*]{}, ${{\mathcal{P}}}$ for power and ${{\mathcal{C}}}$ for capacity. The field of real and complex numbers is respectively ${\mathbb{R}}$ and ${\mathbb{C}}$. A sequence of numbers $X_1,\cdots, X_n $ is sometimes abbreviated as $X^n$, $X^0=\emptyset$. A zero-mean circularly-symmetric complex Gaussian random vector with covariance matrix ${\mathsf{K}}$ is indicated by ${\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathsf{K}}\right)}$. Uniform distribution on interval $[a,b)$ is designated as $\mathcal U(a,b)$. Throughout the paper, the asymptotic equivalence ${{\mathcal{C}}}({{\mathcal{P}}}) \sim f({{\mathcal{P}}})$, often abbreviated by saying “asymptotically,” means that $\lim_{{{\mathcal{P}}}\rightarrow\infty} {{\mathcal{C}}}({{\mathcal{P}}})/f({{\mathcal{P}}})=1$. Letter $c{\stackrel{\Delta}{=}}c(n,{{\mathcal{P}}})$ is reserved to denote a real number bounded in $n$ and ${{\mathcal{P}}}$. A sequence of independent and identically distributed () random variables $X_n$ drawn from the PDF $p_X(x)$ is presented as $X_n\sim{\text{i.i.d.}}\ p_X(x)$. The identity matrix with size $n$ is $I_n$. The Euclidean norm of a vector ${\underaccent{\bar}{x}}\in{\mathbb{C}}^n$ is [rCl]{} [|x|]{}=(|x\_1|\^2++|x\_n|\^2)\^. This gives rise to an induced norm ${\left|{\mathsf{M}}\right|}$ for matrix ${\mathsf{M}}$. We use the spherical coordinate system in the paper. Here, a vector ${\underaccent{\bar}{x}}\in{\mathbb{C}}^n$ is represented by its norm ${\left|{\underaccent{\bar}{x}}\right|}$ and direction $\hat{{\underaccent{\bar}{x}}}={\underaccent{\bar}{x}}/{\left|{\underaccent{\bar}{x}}\right|}$ (with convention $\hat{{\underaccent{\bar}{x}}}=0$ if ${\underaccent{\bar}{x}}=0$). The direction can be described by $m=2n-1$ angles. When direction is random, its entropy can be measured with respect to the spherical measure $\sigma^{m}(A)$, $A\subseteq \mathcal S^m$, where $\mathcal S^m$ is the $m-$sphere [rCl]{} S\^m={\^[m+1]{}:  [||]{}=1 }. It is shown in the Appendix \[app:one\] that the differential entropy with respect to the Lebesgue and spherical measures, denoted respectively by $h(\hat{{\underaccent{\bar}{X}}})$ and $h_{\sigma}(\hat{{\underaccent{\bar}{X}}})$, are related as [rCl]{} h(X)=h([||]{})+h\_(| [||]{})+m|X|. \[eq:sph-leb\] The entropy power of a random direction $\hat{{\underaccent{\bar}{X}}}\in{\mathbb{C}}^n$ is [rCl]{} ()=(h\_()). It represents the effective area of the support of $\hat{{\underaccent{\bar}{X}}}$ on $\mathcal S^m$. The Modified Split-Step Fourier Method {#sec:mssfm} ====================================== Signal propagation in optical fiber is described by the stochastic nonlinear Schrödinger (NLS) equation [@yousefi2012nft1 Eq. 2] [rCl]{} =L\_L(Q)+L\_N(Q)+N(t,z), \[eq:nls\] where $Q(t,z)$ is the complex envelope of the signal as a function of time $t\in{\mathbb{R}}$ and space $z\in{\mathbb{R}}^+$ and $N(t,z)$ is zero-mean circularly-symmetric complex Gaussian noise with [rCl]{} (N(t,z)N\^\*(t’,z’))=\^2\_[W]{}(t-t’)(z-z’), where $\delta_{{{\mathcal{W}}}}(x){\stackrel{\Delta}{=}}2{{\mathcal{W}}}\operatorname{sinc}(2{{\mathcal{W}}} x)$, $\operatorname{sinc}(x){\stackrel{\Delta}{=}}\sin(\pi x)/(\pi x)$, and ${{\mathcal{W}}}$ is noise bandwidth. The operator $L_L$ represents linear effects [rCl]{} L\_L(Q)= \_[k=0]{}\^j\^[k+1]{} -\_[r]{}(t,z)Q(t,z), \[eq:L-L\] where $\beta_k$ are dispersion coefficients, $\convolution$ is convolution and $\alpha_{r}$ is the residual fiber loss. The operator $L_N(Q)=j\gamma |Q|^2Q$ represents Kerr nonlinearity, where $\gamma$ is the nonlinearity parameter. The average power of the transmit signal is [rCl]{} P= \_[T]{}\_[-T/2]{}\^[T/2]{}|Q(t,0)|\^2t. \[eq:power-cont\] The residual loss in accounts for uncompensated loss and non-flat gain of the Raman amplification in distance and is generally frequency dependent. The constant loss model refers to the case where $\alpha_{r}(t,z)$ is constant in the frequency $f$, [*i.e.*]{}, $\hat{\alpha}_{r}(f,z)=\mathcal F(\alpha(t,z)){\stackrel{\Delta}{=}}\alpha_{r}(z)$, where $\mathcal F$ is the Fourier transform with respect to $t$. In realistic systems, however, loss varies over frequency, polarization or spatial models. This is the non-constant loss model. Channel filters act similar to a non-constant loss function. \[def:loss\] We discretize in space and time. Divide a fiber of length ${{\mathcal{L}}}$ into a cascade of a large number $m\rightarrow\infty$ of pieces of discrete fiber segments of length $\epsilon={{\mathcal{L}}}/m$ [@yousefi2011opc Section III. A]. A small segment can be discretized in time and modeled in several ways. An appropriate approach is given by the split-step Fourier method (SSFM). The standard SSFM splits the *deterministic* NLS equation into linear and nonlinear parts. In applying SSFM to the *stochastic* NLS equation, typically noise is added to the signal. We introduce a modified split-step Fourier method where, instead of noise addition, the nonlinear part of is solved in the presence of noise analytically. In the linear step, is solved with $L_N+N=0$. In the discrete-time model, linear step in a segment of length $\epsilon$ consists of multiplying a vector ${\underaccent{\bar}{X}}\in{\mathbb{C}}^n$ by the dispersion-loss matrix ${\mathsf{R}}=(r_{kl})$. In the constant loss model, ${\mathsf{R}}=e^{-\frac{1}{2}\alpha_{r}\epsilon}{\mathsf{U}}$, where ${\mathsf{U}}$ is a unitary matrix. In the absence of loss, ${\mathsf{R}}$ is unitary. The values of $r_{kl}$ depend on the dispersion coefficients, $\epsilon$ and $n$. In general, all entries of ${\mathsf{R}}$ are non-zero, although in a small segment, the off-diagonal elements can be very small. Matrix ${\mathsf{R}}$ is fully dispersive, [*i.e.*]{}, $r_{kl}\neq 0$, for all $k,l$. \[ass:U\] In the nonlinear step, is solved with $L_L=0$ resulting in [@mecozzi1994llh Eq. 12], [@yousefi2011opc Eq. 30]: [rCl]{} Q(t, z)=(Q(t,0)+W(t,z))e\^[j(t,z)]{}, \[eq:zd\] in which [rCl]{} (t,z)= \_0\^z|Q(t,0)+W(t,l)|\^2l, where $W(t,z)=\int_0^z N(t,l){\mathrm{d}}l$ is Wiener process. The modified nonlinear step in the MSSFM is obtained by discretizing . Divide a small segment $0\leq z\leq\epsilon$ into $L$ sub-segments of length $\mu=\epsilon/L$. Define $\Phi:{\mathbb{C}}\times{\mathbb{C}}^n\mapsto [0,\infty)$ as [rCl]{} (X,N)&=&\_[[signal-noise interactions, unknown]{.nodecor}]{} +\ &&+\_[[conditionally known]{.nodecor}]{}, \[eq:phase\] where $N_k\sim {\text{i.i.d.}}\ {\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathcal{D}}/L\right)}$, ${\mathcal{D}}=\sigma^2{{\mathcal{W}}}\epsilon/n$. The nonlinear step in a segment of length $\epsilon$ maps vector ${\underaccent{\bar}{X}}\in{\mathbb{C}}^n$ to vector ${\underaccent{\bar}{Y}}\in{\mathbb{C}}^n$, according to [rCl]{} Y\_k=(X\_k+N\_[k1]{}++N\_[kL]{})e\^[j(X\_k, N\_k)]{}, \[eq:discrete-zd\] where ${\underaccent{\bar}{N}}_k=(N_{k1},\cdots, N_{kL})^T$, $N_{ki}\sim{\text{i.i.d.}}\ {\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathcal{D}}/L\right)}$. The nonlinear step is a deterministic phase change in the SSFM. In this form, nonlinearity is entropy-preserving and does not interact with noise immediately [@yousefi2015cwit2 Lemma 2–3] — unless several steps in the SSFM are considered, which complicates the analysis. In the MSSFM, noise is introduced in a distributed manner within each nonlinear step. This shows noise influence more directly. Note that, conditioned on ${\left|Y_k\right|}$, the last term in is known. Other terms in represent signal-noise interactions. They are conditionally unknown and are responsible for capacity limitation. The MSSFM model for a fiber of length ${{\mathcal{L}}}$ consists of the cascade of linear and modified nonlinear steps (without noise addition between them). A *unit* in the MSSFM model is defined as the cascade of three segments of length $\epsilon$: A modified nonlinear step ${\underaccent{\bar}{X}}\mapsto{\underaccent{\bar}{U}}$, followed by a linear step ${\underaccent{\bar}{U}}\mapsto{\underaccent{\bar}{V}}$, followed by another modified nonlinear step ${\underaccent{\bar}{V}}\mapsto{\underaccent{\bar}{Y}}$; see Fig. \[fig:mssfm\]. A unit of length $3\epsilon$ is the smallest piece of fiber whose capacity behaves qualitatively similar to the capacity of the full model with length ${{\mathcal{L}}}$. In the Appendix \[app:in-out-mssfm\] it is shown that the input output relation ${\underaccent{\bar}{X}}\mapsto {\underaccent{\bar}{Y}}$ in one unit is given by [rCl]{} Y=MX+Z,\[eq:one-seg\] where ${\mathbb{M}}{\stackrel{\Delta}{=}}{\mathbb{M}}({\underaccent{\bar}{X}},{\mathbb{N}}^1,{\mathbb{N}}^2)$ is a random matrix with entries [rCl]{} M\_[kl]{}= r\_[kl]{}e\^[j\_k+j\_l]{}, \[eq:Mkl\] in which [rCl]{} \_l=(X\_l,N\_l\^1),\_k=(V\_k, N\_k\^2). Here ${\mathbb{N}}^1=({\underaccent{\bar}{N}}_1^1,\cdots, {\underaccent{\bar}{N}}_n^1)^T$ and ${\mathbb{N}}^2=({\underaccent{\bar}{N}}_1^2,\cdots, {\underaccent{\bar}{N}}_n^2)^T$ are $n\times L$ Gaussian ensembles with  entries drawn from ${\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathcal{D}}/L\right)}$, independent of any other random variable. The additive noise ${\underaccent{\bar}{Z}}{\stackrel{\Delta}{=}}{\underaccent{\bar}{Z}}({\underaccent{\bar}{X}},{\mathbb{N}}^1,{\mathbb{N}}^2)$ is in general non-Gaussian but bounded in ${\left|{\underaccent{\bar}{X}}\right|}$; see . Finally, vector ${\underaccent{\bar}{V}}$ is the output of the linear step in Fig. \[fig:mssfm\]. The input output relation ${\underaccent{\bar}{X}}\mapsto {\underaccent{\bar}{Y}}$ in a fiber of length ${{\mathcal{L}}}$ is obtained by composing $\bar m=m/2$ blocks ${\underaccent{\bar}{Y}}_k={\mathsf{R}}\bigl({\mathbb{M}}_k{\underaccent{\bar}{X}}_k+{\underaccent{\bar}{Z}}_k\bigl)$: [rCl]{} Y(k)=M(k)X(k)+Z(k), \[eq:m-seg\] where $k=1,2,\cdots,$ is the transmission index, $\{{\underaccent{\bar}{Z}}(k)\}_{k}$ is an  stochastic process, and [rCl]{} M(k)= \_[k=1]{}\^[|m]{} RM\_k,  Z(k)=RZ\_[|m]{}+\_[k=1]{}\^[|m-1]{} (\_[l=k+1]{}\^[|[m]{}]{} RM\_l)RZ\_k. \[eq:M-m-seg\] The power constraint is discretized to ${{\mathcal{P}}}=\frac{1}{n}{\mathsf{E}}{\left\lVert{\underaccent{\bar}{X}}\right\rVert}^2$ in the discrete-time model. Bandwidth, spectral broadening and spectral efficiency in the continuous-time model are discussed in Section \[sec:cor\]. Note that ${\mathbb{M}}({\underaccent{\bar}{X}},{\mathbb{N}}^1,{\mathbb{N}}^2)$ is a nonlinear random operator. Particularly, it depends on input. Dimension of the input space is $n$. To approximate the continuous-time model, $n\rightarrow\infty$. However, we let $n$ be arbitrary, [*e.g.*]{}, $n=5$. Dimension should not be confused with codeword length that tends to infinity. Proof of the Theorem \[thm:main\] {#sec:proof1} ================================= We first illustrate the main ideas of the proof via elementary examples. Consider the additive white Gaussian noise (AWGN) channel $Y=X+Z$, where $X\in{\mathbb{C}}$ is input, $Y\in{\mathbb{C}}$ is output and $Z\sim{\mathcal{N}_{{\mathbb{C}}}\!\left(0,1\right)}$ is noise. Applying chain rule to the mutual information [rCl]{} I(X;Y)=I(X;Y)+I(X;Y|Y), where $\angle$ denotes phase. The amplitude channel $X\mapsto |Y|$ is $${\left|Y\right|}\approx {\left|X\right|}+Z_r,$$ where $Z_r\sim{\mathcal{N}_{{\mathbb{C}}}\!\left(0,\frac{1}{2}\right)}$ and ${\left|X\right|}\gg 1$. It asymptotically contributes [rCl]{} I(X; |Y|)P+c to the capacity. Phase, on the other hand, is supported on the finite interval $[0,2\pi)$. The only way that the contribution of the phase to the capacity could tend to infinity is that, phase noise tends to zero on the circle as ${\left|X\right|}\rightarrow\infty$. Indeed, [rCl]{} Y&=&X+\^[-1]{}()\ &&X+, where $Z_r, Z_i\sim{\text{i.i.d.}}\ {\mathcal{N}_{{\mathbb{C}}}\!\left(0,\frac{1}{2}\right)}$. The output entropy is clearly bounded, $h(\angle Y|{\left|Y\right|})\leq \log 2\pi$. However, [rCl]{} h(Y|X, Y)&=&h(Z\_i)-|X|\ &&-P+c,[as]{.nodecor}P. \[eq:cond-ent-awgn\] Note that the differential entropy can be negative. The contribution of the phase to the mutual information is [rCl]{} I(X;Y|Y)P+c’. Condition implies $\operatorname{V}(\angle Y | X, {\left|Y\right|})\rightarrow 0$, [*i.e.*]{}, the effective phase noise on the unit circle asymptotically vanishes. Now consider the fading channel $Y=MX+Z$, where $X\in{\mathbb{C}}$ is input, $Y\in{\mathbb{C}}$ is output and $M, Z\sim{\text{i.i.d.}}\ {\mathcal{N}_{{\mathbb{C}}}\!\left(0,1\right)}$. To prepare for generalization to optical channel, we represent a complex scalar $X$ as ${\underaccent{\bar}{X}}=(\Re X, \Im X)^T$. Thus ${\underaccent{\bar}{Y}}={\mathbb{M}}{\underaccent{\bar}{X}}+{\underaccent{\bar}{Z}}$, where [rCl]{} = M\_r & -M\_i\ M\_i & M\_r , Z= Z\_r\ Z\_i , in which $M_{r,i},Z_{r,i}\sim{\text{i.i.d.}}\ {\mathcal{N}_{{\mathbb{C}}}\!\left(0,\frac{1}{2}\right)}$. As ${\left| {\underaccent{\bar}{X}}\right|}\rightarrow\infty$, ${\underaccent{\bar}{Y}}\approx {\mathbb{M}}{\underaccent{\bar}{X}}$, $\hat{{\underaccent{\bar}{Y}}}\approx {\mathbb{M}}\hat{{\underaccent{\bar}{X}}}/\bigl|{\mathbb{M}}\hat{{\underaccent{\bar}{X}}}\bigr|$, and randomness in $\hat{{\underaccent{\bar}{Y}}}$ does not vanish with ${\left|X\right|}$. Formally, [rCl]{} h\_(| X, [||]{})&=& h\_(| M\^[-1]{}Y,[||]{})\ &=& h\_(| M\^[-1]{}, [||]{})\ &&gt;&-, \[eq:cond-entr-fad\] where follows because ${\underaccent{\bar}{a}}={\mathbb{M}}^{-1}\hat{{\underaccent{\bar}{Y}}}$ does not determine $\hat{{\underaccent{\bar}{Y}}}$ for random ${\mathbb{M}}$: There are four random variables $M_{r,i}$ and $\hat{{\underaccent{\bar}{Y}}}_{1,2}$ for three equations ${\mathbb{M}}^{-1}\hat{{\underaccent{\bar}{Y}}}={\underaccent{\bar}{a}}$ and $|\hat{{\underaccent{\bar}{Y}}}|=1$. As a result, $I({\underaccent{\bar}{X}};\hat{{\underaccent{\bar}{Y}}}|{\left|{\underaccent{\bar}{Y}}\right|})<\infty$, and ${\left|{\underaccent{\bar}{Y}}\right|}$ is the only useful [DOF]{} at high powers, in the sense that its contribution $I({\underaccent{\bar}{X}}; {\left|{\underaccent{\bar}{Y}}\right|})$ to the mutual information $I({\underaccent{\bar}{X}}; {\underaccent{\bar}{Y}})$ tends to infinity with ${\left|{\underaccent{\bar}{X}}\right|}$. The zero-dispersion optical fiber channel is similar to the fading channel at high powers. The trivial condition [rCl]{} h((.,z) | Q(.,0), |Q(.,z)|)&gt;-, Q(.,0), is sufficient to prove that the capacity of is asymptotically the capacity of the amplitude channel, namely $\frac{1}{2}\log{{\mathcal{P}}}+c$. The intuition from the AWGN, fading and zero-dispersion channels suggests to look at the dispersive optical channel in the spherical coordinate system. The mutual information can be decomposed using the chain rule [rCl]{} I(Q(0); Q(z))&=&I([|Q(0)|]{} ; Q(z))+I((0); Q(z)|[|Q(0)|]{})\ &=&I([|Q(0)|]{} ; [|Q(z)|]{})+I([|Q(0)|]{} ; (z)|[|Q(z)|]{})\ &&+I((0); Q(z)|[|Q(0)|]{}), \[eq:I3\] where we dropped time index in $Q(t,z)$. The first term in is the rate of a single-input single-output channel which can be computed in the asymptotic limit as follows. Let ${\underaccent{\bar}{X}}$ and ${\underaccent{\bar}{Y}}$ represent discretizations of the input $Q(0,.)$ and output $Q(z,.)$. Consider first the lossless model. In this case, ${\mathbb{M}}$ is unitary and from , and [rCl]{} [||]{}\^2&=&[|+|]{}\^2\ &=&[|+\^|]{}\^2\ &=&[|+|]{}\^2, \[eq:chi-squared\] where ${\mathbb{M}}^\dag$ is the adjoint (nonlinear) operator and follows because ${\underaccent{\bar}{Z}}$ and ${\mathbb{M}}^\dag{\underaccent{\bar}{Z}}$ are identically distributed when ${\underaccent{\bar}{Z}}\sim{\mathcal{N}_{{\mathbb{C}}}\!\left(0,m{\mathcal{D}}I_n\right)}$; see Appendix \[app:in-out-mssfm\]. Thus $|{\underaccent{\bar}{Y}}|^2/(m{\mathcal{D}})$ is a non-central chi-square random variable with $2n$ degrees-of-freedom and parameter ${\left|{\underaccent{\bar}{x}}\right|}^2/(m{\mathcal{D}})$. The non-central chi-square conditional PDF $p(|{\underaccent{\bar}{y}}|^2||{\underaccent{\bar}{x}}|^2)$ can be approximated at large ${\left|{\underaccent{\bar}{x}}\right|}^2$ using the Gaussian PDF, giving the asymptotic rate [rCl]{} I([||]{}; [||]{})P+c. \[eq:I(|X|;|Y|)\] The bounded number $c$ can be computed using the exact PDF. The case $\alpha_{r}(z)\neq 0$ is similar to the lossless case. Here ${\mathbb{M}}=e^{-\frac{1}{2}\alpha_{r}{{\mathcal{L}}}}{\mathbb{U}}$, where ${\mathbb{U}}$ is a random unitary operator. Thus, ${\mathbb{M}}^\dag=e^{-\frac{1}{2}\alpha_{r}{{\mathcal{L}}}}{\mathbb{U}}^\dag$; furthermore ${\left|{\mathbb{M}}\right|}=e^{-\frac{1}{2}\alpha_r{{\mathcal{L}}}}$ is deterministic. The loss simply influences the signal power, modifying constant $c$ in . In the non-constant loss model, loss interacts with nonlinearity, dispersion and noise. Here, ${\left|{\mathbb{M}}\right|}$ is a random variable, and [rCl]{} [||]{}=[||]{}[||]{}|+|, \[eq:multi-chan\] where $\hat{{\mathbb{M}}}={\mathbb{M}}/{\left|{\mathbb{M}}\right|}$. Taking logarithm [rCl]{} [||]{}=[||]{}+[||]{}+|+|. \[eq:log-fading\] Applying Lemma \[lemm:decomposition\], we can assume ${\left|X\right|}>x_{0}$ for a suitable $x_{0}>0$ without changing the asymptotic capacity. The last term in is a bounded real random variable because [rCl]{} \_[[||]{}=1, [||]{}=1]{} |+|\^2&lt;. Thus, the logarithm transforms the channel with multiplicative noise ${\left|{\mathbb{M}}\right|}$ to the channel with additive bounded noise. The asymptotic capacity, independent of the PDF of ${\left|{\mathbb{M}}\right|}$, is [rCl]{} I([||]{}; [||]{})&& (([|X|]{}))\^2+c\ &=&P+c’. The last two terms in are upper bounded in one unit of the MSSFM using the data processing inequality [rCl]{} I(Q(0); Q(z)|[|Q(0)|]{})&& I(Q(0); Q(3)|[|Q(0)|]{}), \[eq:dp1\]\ I([|Q(0)|]{}; (z)| [|Q(z)|]{})&& I([|Q(0)|]{}; Q(3)|[|Q(3)|]{}). \[eq:dp2\] We prove that the upper bounds in – do not scale with input ${\left|Q(0)\right|}$. Let ${\underaccent{\bar}{X}},{\underaccent{\bar}{Y}}\in{\mathbb{C}}^n$ denote discretization of $Q(0,t)$ and $Q(3\epsilon,t)$. In one unit of the MSSFM [rCl]{} \_[[||]{}]{}I(; Y|[||]{})&&lt;&, \[eq:hatX-Y\]\ \_[[||]{}]{}I([||]{}; |[||]{})&&lt;&. \[eq:absX-hatY\] ![The area on the surface of the unit sphere, representing $\operatorname{V}(\hat Y)$, does not vanish as ${\left|\underline{x}\right|}\rightarrow\infty$.[]{data-label="fig:spherical-sector"}](fig2){width="20.00000%"} Consider first the lossless model, where ${\mathbb{M}}$ is a unitary operator. From Lemma \[lemm:decomposition\], as ${\left|{\underaccent{\bar}{x}}\right|}\rightarrow\infty$, the additive noise in can be ignored. Thus ${\underaccent{\bar}{Y}}={\left|{\underaccent{\bar}{Y}}\right|}\hat{{\underaccent{\bar}{Y}}}\approx{\left|{\underaccent{\bar}{X}}\right|}\hat{{\underaccent{\bar}{Y}}}$. To prove , [rCl]{} I(; Y|[||]{})&=&I(;[||]{}|[||]{})\ && I(;|[||]{})\ &=&h\_(|[||]{})-h\_(|, [||]{}). Step $(a)$ follows from the identity [rCl]{} I(X; ZY|Z)=I(X; Y|Z),Z0. \[eq:I-indentity\] We measure the entropy of $\hat Y$ with respect to the spherical probability measure $\sigma^{m}$, $m=2n-1$, on the surface of the unit sphere $S^{m}$. From the maximum entropy theorem (MET) for distributions with compact support, [rCl]{} h\_(|[||]{})A\_n, where $A_n=2\pi^n/\Gamma(n)$ is the surface area of $S^m$, in which $\Gamma(n)$ is the gamma function. We next show that the conditional entropy $h_{\sigma}(\hat{{\underaccent{\bar}{Y}}}|\hat{{\underaccent{\bar}{X}}}, {\left|{\underaccent{\bar}{x}}\right|})$ does not tend to $-\infty$ with ${\left|{\underaccent{\bar}{x}}\right|}$. The volume of the spherical sector in Fig. \[fig:spherical-sector\] vanishes if and only if the corresponding area on the surface of the sphere vanishes. This can be formalized using identity . Let ${\underaccent{\bar}{W}}=U\hat{{\underaccent{\bar}{Y}}}$, where $U\sim \mathcal U(0,1)$ independent of ${\underaccent{\bar}{X}}$ and ${\underaccent{\bar}{Y}}$. From [rCl]{} h\_(|, [||]{})=h( W|, [||]{})-h(U)-mU. \[eq:interm\] Applying chain rule to the differential entropy [rCl]{} h(W|.)&=&\_[k=1]{}\^[n]{} h(W\_k|W\^[k-1]{},.)\ &=&\_[k=1]{}\^[n]{} h(W\_k, | W\^[k-1]{},.) \[eq:chain-rule1\]\ &&+\_[k=1]{}\^[n]{} h(|W\_k| |W\^[k-1]{}, W\_[k]{},.), \[eq:chain-rule2\] where entropy is conditioned on ${\left|{\underaccent{\bar}{x}}\right|}$ and $\hat{{\underaccent{\bar}{X}}}$. For the phase entropies in , note that, from –, $\angle W_k=\angle Y_k$ contains random variable $\Phi_k$ with finite entropy, which does not appear in $W^{k-1}$. Formally, $$\angle W_k=\Phi_k+F(\Psi^n,\hat{{\underaccent{\bar}{x}}}),$$ for some function $F$, which can be determined from –. Thus [rCl]{} h(W\_k| W\^[k-1]{}, .)&=& h(\_k+F(\^n,)|W\^[k-1]{}, .)\ & &h(\_k+F(\^n,))|W\^[k-1]{},\^[k-1]{},\^[n]{}, U, .)\ &&h(\_k+F(\^n,))|\^[k-1]{},\^[n]{}, U, .)\ && h(\_k|\^[k-1]{},\^[n]{},.)\ &&gt;& -. \[eq:phase-entropies\] Step $(a)$ follows from the rule that conditioning reduces the entropy. Step $(b)$ holds because $W^{k-1}$ is a function of $\{\Phi^{k-1}, \Psi^n, U\}$. Step $(c)$ follows because $\{\Psi^n, .\}$ determines $F(\Psi^n,\hat{{\underaccent{\bar}{x}}})$. For the amplitude entropies in , we explain the argument for $n=3$: [rCl]{} W\_k =Ue\^[j\_k]{}&&(r\_[k1]{}x\_1e\^[j\_1]{}+r\_[k2]{}x\_2e\^[j\_2]{}+r\_[k3]{}x\_3e\^[j\_3]{}), \[eq:Ys\] where $1\leq k\leq 3$. Noise addition in implies ${\textnormal{Pr}}(\hat X_k= 0)=0$, $\forall k$; we thus assume $\hat x_k\neq 0$ for all $k$. It is clear that $h({\left|W_1\right|})>-\infty$. There are 5 random variables $U$, $\Phi_1$, $\Psi_{1,2,3}$ for two amplitude and phase relations in the $W_1$ equation in . Given $W_1$ and $\angle W_2$, there are 6 random variables and three equations. One could, for instance, express $\Psi_{1,2,3}$ in terms of $U$ and $\Phi_{1,2}$. This leaves free at least $U$ in $|W_2|$, giving [rCl]{} h(|W\_2|| W\_1, W\_2,.)&&h(U)+c\ &&gt;&-. The last equation for $W_3$ adds one random variable $\Phi_3$ and one equation for $\angle W_3$. Together with the equation for ${\left|W_2\right|}$, the number of free random variables, defined as the number of all random variables minus the number of equations, is 2; thus [rCl]{} h(|W\_3|| W\_1, W\_2, W\_3,.)&gt;-. In a similar way, in general, there are $n+k+1$ random variables in $W^k$ and $2k-1$ equations in $(W^{k-1}, \angle W_k)$, resulting in $n-k+2\geq 2$ free random variables. Thus [rCl]{} h(|W\_k|| W\^[k-1]{}, W\_[k]{})&gt;-,1kn. \[eq:amp-entropies\] Substituting and into –, we obtain $h({\underaccent{\bar}{W}}|.)>-\infty$. Finally, from [rCl]{} h\_(|, [||]{})&gt;-. The proof for lossy models, and , is similar. Loss changes matrix ${\mathsf{R}}$, which has no influence on our approach to proving the boundedness of terms in –. The essence of the above proof is that, as ${\left|{\underaccent{\bar}{x}}\right|}\rightarrow\infty$, the additive noise in gets smaller relative to the signal, but phase noise (and thus randomness in ${\mathbb{M}}$) does not decrease with ${\left|{\underaccent{\bar}{x}}\right|}$. Furthermore, ${\mathbb{M}}$ has enough randomness, owing to the mixing effect of the dispersion, so that all $2n-1$ angles representing signal direction in the spherical coordinate system are random variables that do not vanish with ${\left|{\underaccent{\bar}{x}}\right|}$. For some special cases of the dispersion-loss matrix ${\mathsf{R}}$, it is possible to obtain deterministic components in $\hat{{\underaccent{\bar}{Y}}}$ as ${\left|{\underaccent{\bar}{x}}\right|}\rightarrow\infty$. These are cases where mixing does not fully occur, [*e.g.*]{}, ${\mathsf{R}}=I_n$. In the MSSFM, however, ${\mathsf{R}}$ is arbitrary, due to, [*e.g.*]{}, step size $\epsilon$. Proof of the Corollary \[cor:inf\] {#sec:cor} ---------------------------------- We fix the power constraint and let $n\rightarrow\infty$ in the definition of the capacity. The logarithmic terms depending on ${{\mathcal{P}}}$ in the Theorem \[thm:main\] approach zero, so that ${{\mathcal{C}}}<\infty$. Consider now the continuous-time model . We discretize the channel in the frequency domain, according to the approach in [@yousefi2015cwit2]. As the time duration ${{\mathcal{T}}}\rightarrow\infty$ in [@yousefi2015cwit2 Section II], we obtain a discrete-time model with infinite number of [DOFs]{} (Fourier modes) in any frequency interval at $z=0$. Therefore, ${{\mathcal{C}}}<\infty$ in the corresponding discrete-time periodic model. It is shown in [@yousefi2011opc Section VIII] that, because of the spectral broadening, the capacity of the continuous-time model ${{\mathcal{C}}}_c$ can be strictly lower than the capacity of the discrete-time model ${{\mathcal{C}}}_d$. Since ${{\mathcal{C}}}_c\leq {{\mathcal{C}}}_d$, and ${{\mathcal{C}}}_d<\infty$, we obtain ${{\mathcal{C}}}_c<\infty$. We do not quantify constant $c'$ in the continuous-time model, which can be much lower than the constant $c$ in the discrete-time model, due to spectral broadening (potentially, $c'(\infty,\infty)=0$). A crude estimate, based on the Carson bandwidth rule, is given in [@yousefi2011opc Section VIII] for the zero-dispersion channel. To summarize, SE is bounded in input power in the continuous-time model with $n=\infty$ (with or without filtering). The extent of the data rate loss due to the spectral broadening ($c'$ versus $c$) remains an open problem. Random Matrix Model and the Asymptotic Capacity {#sec:proof2} =============================================== In this section it is shown that, as ${\left|{\underaccent{\bar}{X}}\right|}\rightarrow\infty$, the action of the discrete-time periodic stochastic NLS equation tends to multiplication by a random matrix (with fixed PDF, independent of the input). Noise simplifies the NLS channel to a *linear* multiple-input multiple-output non-coherent fading channel. This section also proves Theorems \[thm:main\] in an alternative intuitive way. The approach is based on the following steps. *Step 1)* In Section \[sec:decomposition\], the input signal space is partitioned into a bounded region $\mathcal R^-$ and its complement $\mathcal R^+$. It is shown that the overall rate is the interpolation of rates achievable using signals in $\mathcal R^{\pm}$. Lemma \[lemm:I&lt;infty\] is proved, showing that the contribution of $\mathcal R^-$ to the mutual information is bounded. Suitable regions $\mathcal R^{\pm}$ are chosen for the subsequent use. *Step 2)* In Section \[sec:fading-model\], it is shown that for all $q(t,0)\in \mathcal R^+$, the nonlinear operator $L_N=j\gamma|Q|^2Q$ is multiplication by a uniform phase random variable, [*i.e.*]{}, [rCl]{} L\_N(Q)= j(t,z) Q,t, z, where[^2]$\Theta(t,z)\sim{\text{i.i.d.}}\ \mathcal U(0,2\pi)$. In other words, for input signals in $\mathcal R^+$ the stochastic NLS equation is a simple linear channel with additive and multiplicative noise [rCl]{} =L\_L(Q)+j (t,z)Q+N(t,z). \[eq:multiplicative\] Discretizing , we obtain that optical fiber is a fading channel when input is in $\mathcal R^+$: [rCl]{} Y&=&X+Z,\^2P, \[eq:Y=HX+N\] in which ${\mathbb{M}}$ is a random matrix of the form [rCl]{} M = \_[k=1]{}\^m RD\_k,D\_k=(e\^[j\_[ki]{}]{}), \[eq:M-expr\] where $\Theta_{kl}\sim{\text{i.i.d.}}\ \mathcal U(0,2\pi)$ and ${\underaccent{\bar}{Z}}$ is noise [rCl]{} Z=\_[k=1]{}\^[m]{}(\_[l=1]{}\^kR D\_l)Z\_k,Z\_k\~ [\_(0,I\_n)]{}. In general, ${\mathbb{M}}$ and ${\underaccent{\bar}{Z}}$ are non-Gaussian. However, in the constant loss model, ${\underaccent{\bar}{Z}}\sim{\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathsf{K}}\right)}$ where ${\mathsf{K}}=(\sigma^2{{\mathcal{W}}}{{\mathcal{L}}}_e/n)I_n$, ${{\mathcal{L}}}_e=(1-e^{-\alpha{{\mathcal{L}}}})/\alpha$. Note that ${\mathbb{M}}$ and ${\underaccent{\bar}{Z}}$ have fixed PDFs, independent of ${\underaccent{\bar}{X}}$. Summarizing, the channel law is [rCl]{} p(y|x)= [given by the NLS equation]{.nodecor}, & xR\^-,\ p(Mx+Z|x), & xR\^+. \[eq:law\] *Step 3)* In Section \[sec:asymptotic-capacity\], the capacity of the multiplicative-noise channel is studied. Lemma \[lem:cap-Y=HX+N\] and \[lem:h(Mx)\] are proved showing that, for any ${\mathbb{M}}$ that does not have a deterministic component and is finite (see ), the asymptotic capacity is given by the Theorem \[thm:main\]. Importantly, the asymptotic rate is nearly independent of the PDF of ${\mathbb{M}}$, which impacts only the bounded number $c$. Finally, Lemma \[lem:h(M-fiber)&gt;-infty\] is proved showing that the random matrix underlying the optical fiber at high powers meets the assumptions of the Lemma \[lem:cap-Y=HX+N\]. An expression is provided for $c$, which can be evaluated, depending on the PDF of ${\mathbb{M}}$. Step 1): Rate Interpolation {#sec:decomposition} --------------------------- We begin by proving the following lemma, which is similar to the proof approach in [@agrell2015conds], where the notion of satellite constellation is introduced. Let $p({\underaccent{\bar}{y}}|{\underaccent{\bar}{x}})$, ${\underaccent{\bar}{x}}, {\underaccent{\bar}{y}} \in{\mathbb{R}}^n$, be a conditional PDF. Define [rCl]{} X= X\_1, & [with probability]{.nodecor} ,\ X\_2, & [with probability]{.nodecor} 1-, where ${\underaccent{\bar}{X}}_{1}$ and ${\underaccent{\bar}{X}}_2$ are random variables in ${\mathbb{R}}^n$ and $0\leq\lambda\leq 1$. Then [rCl]{} R\_1+(1-)R\_2RR\_1+(1-)R\_2+H(), \[eq:R1-R2-H\] where $R_1$, $R_2$ and $R$ are, respectively, mutual information of $X_1$, $X_2$ and $X$, and $H(x)=-x\log x-(1-x)\log(1-x)$ is the binary entropy function, $0\leq x\leq 1$. \[lemm:decomposition\] The PDF of the time sharing random variable ${\underaccent{\bar}{X}}$ and its output ${\underaccent{\bar}{Y}}$ are [rCl]{} p\_[X]{}(x)&=&p\_[X\_1]{}(x)+(1-)p\_[X\_2]{}(x),\ p\_[Y]{}(y)&=&p\_[Y\_1]{}(y)+(1-)p\_[Y\_2]{}(y), \[eq:py=py1+py2\] where [rCl]{} p\_[Y\_1,Y\_2]{}(y)=p(y|x)p\_[X\_1,X\_2]{}(x)x. By elementary algebra [rCl]{} I(X; Y) = I(X\_1;Y\_1)+(1-) I(X\_2,Y\_2)+I, where [rCl]{} I &=& D( p\_[Y\_1]{}(y\_1)||p\_[Y]{}(y)) +(1-) D( p\_[Y\_2]{}(y\_2)||p\_[Y]{}(y)). From , $$p_{{\underaccent{\bar}{Y}}}({\underaccent{\bar}{y}})\geq \max\left\{\lambda p_{{\underaccent{\bar}{Y}}_1}({\underaccent{\bar}{y}}), (1-\lambda) p_{{\underaccent{\bar}{Y}}_2}({\underaccent{\bar}{y}})\right\},$$ which gives $\Delta I\leq H(\lambda)$. From $\log x\leq x-1$, $D(p({\underaccent{\bar}{y}}_{1,2})||p({\underaccent{\bar}{y}}))\geq 0$, giving $\Delta I\geq 0$. Thus [rCl]{} I (X;Y) && I (X\_1;Y\_1)+(1-) I(X\_2;Y\_2)+H(),\ I(X;Y) && I (X\_1;Y\_1)+(1-) I(X\_2;Y\_2). Define [rCl]{} |R=\_[n]{}I(X;Y). With definitions in the Lemma \[lemm:decomposition\], we have $\bar R=\lambda\bar R_1+(1-\lambda)\bar R_2$. \[cor:rate-interpolation\] For the rest of the paper, we choose $\mathcal R^-$ to be an $n$-hypercube in ${\mathbb{C}}^n$ [rCl]{} R\^-\_={x\^n | |x\_k|&lt; ,1kn }, and $\mathcal R^+_{\kappa}={\mathbb{C}}^n\backslash\mathcal R^-_{\kappa}$. We drop the subscript $\kappa$ when we do not need it. The following Lemma shows that, if $\kappa<\infty$, the contribution of the signals in $\mathcal R^-_\kappa$ to the mutual information in the NLS channel is bounded. Let ${\underaccent{\bar}{X}}\in{\mathbb{C}}^n$ be a random variable supported on $\mathcal R^-_{\kappa}$ and $\kappa<\infty$. For the NLS channel [rCl]{} I(X;Y)&lt;. \[lemm:I&lt;infty\] From the MET, $h({\underaccent{\bar}{Y}})\leq \log\left|\mathcal R^-_\kappa\right|< \infty$. Let ${\underaccent{\bar}{X}}\rightarrow {\underaccent{\bar}{Z}}\rightarrow {\underaccent{\bar}{Y}}={\underaccent{\bar}{Z}}+{\underaccent{\bar}{N}}$ be a Markov chain, where ${\underaccent{\bar}{N}}$ is independent of ${\underaccent{\bar}{Z}}$ and $h({\underaccent{\bar}{N}})>-\infty$. Then $h({\underaccent{\bar}{Y}}|{\underaccent{\bar}{X}})\geq h({\underaccent{\bar}{Y}}|{\underaccent{\bar}{X}},{\underaccent{\bar}{Z}})=h({\underaccent{\bar}{Y}}|{\underaccent{\bar}{Z}})=h({\underaccent{\bar}{N}})>-\infty$. Applying this to the NLS channel with an independent noise addition in the last stage, we obtain $I({\underaccent{\bar}{X}}, {\underaccent{\bar}{Y}})<\infty$. Alternatively, from [@yousefi2015cwit2], $$\frac{1}{n}I({\underaccent{\bar}{X}}; {\underaccent{\bar}{Y}})\leq \log(1+\frac{|\mathcal R^-_\kappa|^2}{nm{\mathcal{D}}})<\infty.$$ \[lem:R-\] Step 2): Channel Model in the High Power Regime {#sec:fading-model} ----------------------------------------------- We begin with the zero-dispersion channel. Let $Q(t,0)=X=R_x\exp(j\Phi_x)$ and $Q(t,z)=Y=R_y\exp(j\Phi_y)$ be, respectively, channel input and output in . For a fixed $t$, $X$ and $Y$ are complex numbers. We have [rCl]{} \_[[|x|]{}]{}p(\_y|x)&=&\_[[|x|]{}]{}p(\_y|x,r\_y)\ &=&. Thus, the law of the zero-dispersion channel tends to the law of the following channel $$Y=Xe^{j\Theta}+Z,$$ where $\Theta\sim\mathcal U(0,2\pi)$, $Z\sim {\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathcal{D}}\right)}$, and $(X,Z,\Theta)$ are independent. \[lemm:uniform\] The condition PDF is [@yousefi2011opc Eq. 18] [rCl]{} p(r\_y,\_y|r\_x,\_x)&=&p\_0(r\_y|r\_x)\ &&+\_[m=1]{}\^(p\_m(r\_y|r\_x)e\^[jm(\_y-\_x-r\^2\_xz)]{}). Here [rCl]{} p\_m(r\_y|r\_x)&=&2r\_xb\_m(-a\_m(r\_x\^2+r\_y\^2))I\_m(2b\_mr\_xr\_y), where [rCl]{} a\_m=x\_m(x\_m),b\_m=, in which $x_m=\sqrt{jm\gamma\mathcal D}z=t_m(1+j)$, $t_m=\sqrt{\frac{1}{2}m\gamma\mathcal D}z$. Note that $p(r_y|r_x)=p_0(r_y|r_x)$. The conditional PDF of the phase is [rCl]{} p(\_y|r\_x,\_x, r\_y)&=&\ &&\ &=& \_[m=1]{}\^( D\_m(r\_x) e\^[jm(\_y-\_x- r\_x\^2z)]{})\ && +, where step $(a)$ follows from $p(r_y|r_x,\phi_x)=p(r_y|r_x)$ (see [@yousefi2011opc Fig. 6 (b)]) and [rCl]{} D\_m(r\_x)&=&\ &=&\ &&{-b\_0(x\_mx\_m-1)(r\_x\^2+r\_y\^2)}. \[eq:Dm\] The following three inequalities can be verified: [rCl]{} ||\^2&=&\ && 1,t&gt;0. \[eq:inq1\] [rCl]{} ||&& ||\ &&1. \[eq:inq2\] [rCl]{} F(t)&=&(x\_mx\_m-1)\ &=&t-1\ &&gt;&0, \[eq:inq3\] where $t{\stackrel{\Delta}{=}}t_m>0$. Using – in , we obtain $|D_m(r_x)|\leq E_m(r_x)$, where [rCl]{} E\_m(r\_x)={-F(t\_m)(r\_x\^2+r\_y\^2)}. \[eq:D&lt;1\] We have [rCl]{} \_[r\_x]{}|\_[m=1]{}\^D\_m(r\_x) e\^[jm(\_y-\_x-r\_x\^2z)]{}| && \_[r\_x]{} \_[m=1]{}\^|D\_m(r\_x)|\ && \_[r\_x]{} \_[m=1]{}\^E\_m(r\_x)\ && \_[m=1]{}\^\_[r\_x]{} E\_m(r\_x)\ && 0. Step $(a)$ follows because $E_m(r_x)\leq E_m(0)$ and $\sum E_m(0)$ is convergent; thus, by the dominated convergence theorem, $\sum E_m(r_x)$ is uniformly convergent. Step $(b)$ follows from . It follows that [rCl]{} \_[r\_x]{}p(\_y|r\_x, \_x, r\_y)=. Furthermore [rCl]{} \_[r\_x]{}p(\_y|r\_x,\_x)&=& \_[r\_x]{} p(\_y|r\_x,\_x, r\_[y’]{})p(r\_[y’]{}|r\_x,)r\_[y’]{}\ &=&. Lemma \[lemm:uniform\] generalizes to the vectorial zero-dispersion channel . Since noise is independent and identically distributed in space and time, so are the corresponding uniform phases. This is true even if $X_i$ in Fig. \[fig:mssfm\] are dependent, [*e.g.*]{}, ${\underaccent{\bar}{X}}=(x,\cdots, x)$. We now consider the dispersive model. To generalize Lemma \[lemm:uniform\] to the full model, we use the following notion [@moser2004dbb Section 2.6]. A family of PDFs $\{p_{{\underaccent{\bar}{X}}_{\theta}}({\underaccent{\bar}{x}})\}_{\theta}$, $0\leq\theta\leq\theta_0$, is said to *escape to infinity* with $\theta$ if $\lim\limits_{\theta\rightarrow\theta_0}{\textnormal{Pr}}(|{\underaccent{\bar}{X}}_{\theta}|<c)=0$ for any finite $c$. Let ${\underaccent{\bar}{X}}\in\mathcal R^+_{\kappa}$ and ${\underaccent{\bar}{Y}}$ be, respectively, the channel input and output in the dispersive model. The PDF of $Y_k$ escapes to infinity as $\kappa\rightarrow\infty$ for all $k$. \[lemm:scape\] The proof is based on induction in the MSSFM units. We make precise the intuition that, as $\kappa\rightarrow \infty$, the PDF of $|Y_k|$ spreads out, so that an ever decreasing probability is assigned to any finite interval. Consider vector ${\underaccent{\bar}{V}}$ in Fig. \[fig:mssfm\], at the end of the linear step in the first unit. Setting $W={\left|V_k\right|}$, we have [rCl]{} [[Pr]{.nodecor}]{}(W&lt;c)&=&\_[0]{}\^[c]{} p\_W(w)w\ &=& \_[0]{}\^[c]{} p\_W()t\ && c [\_[T\_]{}(t)]{}\_, \[eq:p(W&lt;c)\] where $T_{\epsilon}=\epsilon W$, $p_{T_{\epsilon}}(t)=\frac{1}{\epsilon}p_W(\frac{t}{\epsilon})$ and $\epsilon{\stackrel{\Delta}{=}}1/\kappa$. Below, we prove that ${\left\lVertp_{T_0}(t)\right\rVert}_{\infty}<\infty$. Fix $0<\delta<1$ and define the (non-empty) index set [rCl]{} I={i: |x\_i|\^[1-]{}}. The scaled random variable $T_{\epsilon}$ is [rCl]{} T\_&=&|V\_k|\ &=&|\_[l=1]{}\^n e\^[j\_l(x\_l,\_l\^1)]{}r\_[kl]{}\_l+|\ &=&| \_[lI]{} + \_[lI]{}+ |, where $\tilde{{\underaccent{\bar}{Z}}}$ is an additive noise and $\tilde{{\underaccent{\bar}{x}}}=\epsilon{\underaccent{\bar}{x}}$. As $\epsilon\rightarrow 0$, the second sum vanishes because, if $l\notin \mathcal I$, $|\tilde{x}_l|<\epsilon^\delta\rightarrow 0$. In the first sum, $|x_l|\rightarrow\infty$, thus $\Psi_l(x_l,{\underaccent{\bar}{N}}_l^1)\overset{\textnormal{a.s.}}{\rightarrow} U_l$, where $U_l\sim\mathcal U(0,2\pi)$. Therefore $T_{\epsilon}\overset{\textnormal{a.s.}}{\rightarrow} T_0$, in which [rCl]{} T\_[0]{}=|\_[l I]{}e\^[jU\_l]{}r\_[kl]{}x\_l|, \[eq:T-eps\] where $|\tilde x_l|>0$. Since the PDF of $e^{jU_l}$ is in $L^{\infty}(\mathbb T)$ on the circle $\mathbb T$, so is the conditional PDF $p_{T_0|\tilde{{\underaccent{\bar}{ X}}}}(t|\tilde{{\underaccent{\bar}{x}}})$, [*i.e.*]{}, [rCl]{} [\_[T\_0]{}(t)]{}\_&lt;. \[eq:p(t)&lt;infty\] Substituting into [rCl]{} \_[0]{}[[Pr]{.nodecor}]{}(W&lt;c)=0. \[eq:p(W&lt;c)-2\] In a similar way, can be proved for ${\underaccent{\bar}{V}}$ at the output of the linear step in the second unit, by replacing $e^{jU_l}r_{kl}$ in with $M_{kl}$, and noting that, as $\epsilon\rightarrow 0$, $\{M_{kl}\}_{l\in\mathcal I}$ tend to random variables independent of input, with a smooth PDF (without delta functions). From the Lemma \[lemm:scape\], as $\kappa\rightarrow\infty$ the probability distribution at the input of every zero-dispersion segment in the link escapes to infinity, turning the operation of the nonlinearity in that segment into multiplication by a uniform phase and independent noise addition. We thus obtain an input region $\mathcal R^+_\kappa$ for which, if ${\underaccent{\bar}{x}}\in\mathcal R^+_\kappa$, the channel is multiplication by a random matrix, as described in . The channel converts any small noise into worst-case noise in evolution. Step 3): The Asymptotic Capacity {#sec:asymptotic-capacity} -------------------------------- In this section, we obtain the asymptotic capacity of the channel . Applying Lemma \[lemm:decomposition\] to [rCl]{} |R(P)=|R\_-(P)+(1-)| R\_+(P), where $\bar R_{\pm}({{\mathcal{P}}})=\frac{1}{n}I({\underaccent{\bar}{X}};{\underaccent{\bar}{Y}})$, ${\underaccent{\bar}{X}}\in\mathcal R^{\pm}_\kappa$ and $\lambda$ is a parameter to be optimized. To shorten the analysis, we ignore the term $c=H(\lambda)/n$ in , as it does not depend on ${{\mathcal{P}}}$. We choose $\kappa$ sufficiently large, *independent of the average input power* ${{\mathcal{P}}}$. From the Lemma \[lemm:I&lt;infty\], $\sup_{{{\mathcal{P}}}}\bar R_-({{\mathcal{P}}})<\infty$. The following Lemma shows that $\bar{R}_+({{\mathcal{P}}})$ is given by the logarithmic terms in Theorem \[thm:main\] with [rCl]{} c|R\_-+(1-)\_I(;M), where $\bar R_-$ is the achievable rate at low powers. If ${\mathbb{M}}$ is Haar distributed, $c=\lambda \bar R_-$. Define $h({\mathbb{M}}){\stackrel{\Delta}{=}}h(M_{11},\cdots, M_{nn})$. Assume that [rCl]{} h(M)&gt;-, |M\_[ij]{}|\^2&lt;,1i,jn. \[eq:h(M)&gt;-infty\] Then, the asymptotic capacity of is given by the expressions stated in the Theorem \[thm:main\]. \[lem:cap-Y=HX+N\] The capacity of the multiple-input multiple-output non-coherent memoryless fading channel is studied in [@moser2004dbb; @lapidoth2003capacity]. Here, we present a short proof with a bit of approximation. Using chain rule for the mutual information [rCl]{} I(X;Y)&=&I([||]{}; Y)+I(; Y|[||]{})\ &=&I([||]{}; [||]{})+I(; Y|[||]{})+I([||]{}; |[||]{}). \[eq:I(X;Y)\] The first term in gives the logarithmic terms in Theorem \[thm:main\], as calculated in Section \[sec:proof1\]. We prove that the other terms are bounded in ${\left|{\underaccent{\bar}{X}}\right|}$. From the Lemma \[lemm:decomposition\], the additive noise in can be ignored when ${\underaccent{\bar}{X}}\in\mathcal R^+_\kappa$, so that ${\underaccent{\bar}{Y}}\approx{\mathbb{M}}{\underaccent{\bar}{X}}$. The second term in is [rCl]{} I(; Y|[||]{}) &=& I(; |X| M|[||]{})\ &=&I(; M| [||]{}), where we used identity . Note that we can not assume that ${\left|{\underaccent{\bar}{X}}\right|}$ and $\hat{{\underaccent{\bar}{X}}}$ are independent. For the output entropy [rCl]{} h(M|[||]{})&& h(M)\ &&\_[k=1]{}\^n h(\_[l=1]{}\^n M\_[kl]{}\_l )\ &&\_[k=1]{}\^n (e |\_[l=1]{}\^n M\_[kl]{}\_l|\^2)\ &&\_[k=1]{}\^n (\_[l=1]{}\^n |M\_[kl]{}|\^2)+ne\ &&n(||\^2\_F)+ne\ &&lt;&, where ${\left|{\mathbb{M}}\right|}_F=\Bigl(\sum\limits_{k,l=1}^n|M_{kl}|^2\Bigr)^{\frac{1}{2}}$ is the Frobenius norm. Step $(a)$ is obtained using the inequality $h({\underaccent{\bar}{W}})=\sum_k h(W_k|W^{k-1})\leq\sum_k h(W_k)$. Step $(b)$ is due to the MET. Cauchy-Schwarz and Jensen’s inequalities are, respectively, applied in steps $(c)$ and $(d)$. For the conditional entropy [rCl]{} h(M|[||]{}, )&=& \_ h(M|[||]{}, )\ && \_ h(M)\ &&-. \[eq:inter-6-aa\] Step $(a)$ holds because, from the Lemma \[lem:h(Mx)\], $h({\mathbb{M}}\hat{{\underaccent{\bar}{x}}})>-\infty$ for any $\hat{{\underaccent{\bar}{x}}}$. The third term in can be upper bounded using the second term by setting ${\underaccent{\bar}{X}}={\mathbb{M}}^{-1}{\underaccent{\bar}{Y}}$. We prove it alternatively. Since $\hat{{\underaccent{\bar}{Y}}}$ is compactly supported, $h_{\sigma}(\hat{{\underaccent{\bar}{Y}}}||{\left|{\underaccent{\bar}{Y}}\right|}|)\leq h_{\sigma}(\hat{{\underaccent{\bar}{Y}}})<\infty$. The conditional entropy is [rCl]{} h\_(|[||]{},[||]{})&=& h\_(|[||]{},[||]{}|M|)\ &=&h\_(|[||]{},|M|). \[eq:cond-entropy-Y=HX+N\] Applying identity to ${\mathbb{M}}\hat{{\underaccent{\bar}{X}}}$ and conditioning on ${\left|{\underaccent{\bar}{X}}\right|}$ [rCl]{} h\_(|[||]{}, |M|) &=& h(M| [||]{})-h(|M|| [||]{})\ && -(2n-1)((|M|)|[||]{}). \[eq:inter-6\] For the first term in [rCl]{} h(M|[||]{})&&h(M|[||]{}, )\ &&gt;&-, \[eq:inter-6-a\] where we used . Since $|{\mathbb{M}}\hat{{\underaccent{\bar}{X}}}|\leq{\left|{\mathbb{M}}\right|}\leq {\left|{\mathbb{M}}\right|}_F$, from the MET [rCl]{} h(|M||[||]{})&& h(|M|)\ && (2e|MX|\^2)\ && (2e[||]{}\_F\^2)\ &&lt;&. \[eq:inter-6-b\] Furthermore, [rCl]{} ((|M|\^2)|[||]{})&& (|M|\^2|[||]{})\ &&[||]{}\_F\^2\ &&lt;&. \[eq:inter-6-c\] Substituting – into and , we obtain [rCl]{} h\_(|[||]{},[||]{})&gt;-. The main ingredient in the proof of the Lemma \[lem:cap-Y=HX+N\], as well as Theorem \[thm:main\], is the following lemma. Let ${\mathbb{M}}$ be a random matrix and ${\underaccent{\bar}{x}}\in{\mathbb{C}}^n$ a non-zero deterministic vector. If ${\mathbb{M}}$ satisfies the assumptions , then [rCl]{} h(M x)&gt;-. \[lem:h(Mx)\] Since ${\underaccent{\bar}{x}}\neq 0$, at least one element of $ {\underaccent{\bar}{x}}$ is nonzero, say $ x_1\neq 0$. We switch the order of ${\mathbb{M}}$ and ${\underaccent{\bar}{x}}$ in the product ${\mathbb{M}}{\underaccent{\bar}{x}}$ as follows. Let ${\underaccent{\bar}{ M}}\in{\mathbb{C}}^{n^2}$ denote the vectorized version of ${\mathbb{M}}$, where rows are concatenated as a column vector. Define ${\underaccent{\bar}{V}}\in{\mathbb{C}}^{n^2}$ as follows: [rCl]{} V\_[k]{}= M\_[in]{}, & r=0,\ Y\_[i+r]{}=\_[l=1]{}\^n M\_[(i+r)l]{}x\_l, & r=1,\ M\_[(i+1)r]{}, & r2, \[eq:V-vec\] where $k=in+r$, $0\leq i\leq n$, $0\leq r \leq n-1$. Then ${\underaccent{\bar}{Y}}={\mathbb{M}} {\underaccent{\bar}{x}}$ is transformed to , which in matrix notation is [rCl]{} V=AM, \[eq:V=AM\] in which ${\mathsf{A}}_{n^2\times n^2}=\operatorname{diag}(\underbrace{{\mathsf{X}},\cdots,{\mathsf{X}}}_{n~\textnormal{times}})$, where the deterministic matrix ${\mathsf{X}}_{n\times n}$ is [rCl]{} X=( x\_1 & x\_2\^n\ 0 & I\_[n-1]{} ),x\_2\^n=( x\_2,, x\_n), in which $0$ is the $(n-1)\times 1$ all-zero matrix. From [rCl]{} h(V|)&=&h(M|x)+|A|\ &=&h(M)+n[|x\_1|]{}\ &=&h(M)+n[|x\_1|]{}. On the other hand, from [rCl]{} h(V|)&&h(Y, { M\_[ij]{}}\_[j2]{}|)\ &=&h(Y | )+h({ M\_[ij]{}}\_[j2]{}|x, Y)\ &=&h(Mx )+h({ M\_[ij]{}}\_[j2]{}|Y). Step $(a)$ holds because conditions $r=1$ and $r=0,1$ in include, respectively, ${\underaccent{\bar}{Y}}$ and $\{M_{ij}\}_{j\geq 2}$. Combining the last two relations [rCl]{} h(Mx)= h(M)+ n[| x\_1|]{}-h({M\_[ij]{}}\_[j2]{}|Y). If ${\mathsf{E}}|M_{ij}|^2 < \infty$, from the MET, the last term is bounded from below. Since $h({\mathbb{M}})> -\infty$ and $x_1\neq 0$, $h({\mathbb{M}}{\underaccent{\bar}{x}})>-\infty$. The random matrix ${\mathbb{M}}$ , underlying optical fiber at high powers, satisfies the assumptions of the Lemma \[lem:cap-Y=HX+N\]. \[lem:h(M-fiber)&gt;-infty\] Applying the triangle inequality to , $|M_{ij}|\leq \bigl({\mathsf{|}}{\mathsf{R}}|^m\bigr)_{ij}$, where $|{\mathsf{R}}|$ is the matrix with entries $|r_{ij}|$ and $m$ is the number of stages. We check the entropy condition in . In what follows, let $\theta_{i}\sim{\text{i.i.d.}}\ \mathcal U(0,2\pi)$. For one linear and nonlinear steps $m=1$: $$\begin{aligned} {\mathbb{M}}= \begin{pmatrix} e^{j\theta_1}r_{11} & e^{j\theta_2} r_{12}\\ e^{j\theta_1}r_{21} & e^{j\theta_2}r_{22} \end{pmatrix}.\end{aligned}$$ In this case, there are four amplitude dependencies $|M_{ij}|=|r_{ij}|$, $1\leq i,j\leq 2$, and two phase dependencies: [rCl]{} M\_[11]{}=M\_[21]{}+k,M\_[12]{}= M\_[22]{}+k,k=0,1. A dependency means that ${\mathbb{M}}$ contains a deterministic component, [*i.e.*]{}, $h({\mathbb{M}})>-\infty$. For $m=2$: $$\begin{aligned} M_{11}&=& e^{j(\theta_1+\theta_3)}r_{11}^2+ e^{j(\theta_1+\theta_4)}r_{12}r_{21}, \\ M_{12}&=& e^{j(\theta_2+\theta_3)}r_{12}\left(r_{11}+e^{j(\theta_4-\theta_3)}r_{22}\right),\\ M_{21} &=& e^{j(\theta_1+\theta_3)}r_{21}\left(r_{11}+e^{j(\theta_4-\theta_3)}r_{22}\right),\\ M_{22} &=& e^{j(\theta_2+\theta_3)}r_{21}r_{12}+ e^{j(\theta_2+\theta_4)}r_{22}^2. \end{aligned}$$ In this case too, there is a dependency $|r_{21}M_{12}|=|r_{12}M_{21}|$. For $m=3$: [rCl]{} M\_[11]{} &=& e\^[j(\_1+\_3+\_5)]{}r\_[11]{}\^3+ e\^[j(\_1+\_4+\_5)]{}r\_[11]{}r\_[12]{}r\_[21]{}\ &&+e\^[j(\_1+\_3+\_6)]{}r\_[11]{}r\_[12]{}r\_[21]{}+ e\^[j(\_1+\_4+\_6)]{}r\_[12]{}r\_[21]{}r\_[22]{},\ M\_[12]{}&=&e\^[j\_2]{}r\_[12]{}( e\^[j(\_3+\_5)]{}r\_[11]{}\^2+ + e\^[j(\_4+\_6)]{}r\_[22]{}\^2\ &&+ ),\ M\_[21]{} &=& e\^[j\_1]{}r\_[21]{}(e\^[j(\_3+\_5)]{}r\_[11]{}\^2+ e\^[j(\_4+\_6)]{}r\_[22]{}\^2\ &&+ ),\ M\_[22]{} &=& e\^[j(\_2+\_3+\_5)]{}r\_[11]{}r\_[12]{}r\_[21]{}+ e\^[j(\_2+\_4+\_5)]{}r\_[12]{}r\_[21]{}r\_[22]{}\ &&+e\^[j(\_2+\_3+\_6)]{}r\_[12]{}r\_[21]{}r\_[22]{}+ e\^[j(\_2+\_4+\_6)]{}r\_[22]{}\^3. Comparing the boxed terms, $|r_{21}M_{12}|\neq |r_{12}M_{21}|$. There are still 8 equations for 6 variables. In general, the number of entries of ${\mathbb{M}}$ is $n^2$. As $m> 2n$ steps are taken in distance, sufficient number of random variables $\theta_i$ are introduced in a matrix with fixed dimension. Since $n$ is fixed and $m$ is free, we obtain an under-determined system of polynomial equations for $x_i=\exp(j\theta_i)$ whose solution space has positive dimension. Thus an entry of ${\mathbb{M}}$ can not be determined from all other entries. The rate interpolation Lemma \[lemm:decomposition\] implies that, replacing ${\mathbb{C}}^n$ by $\mathcal R^+$ changes the asymptotic capacity by a finite number $c$. From the upper bound ${{\mathcal{C}}}\leq \log(1+{\text{SNR}})$ in [@yousefi2015cwit2] and Theorem 2.5 in [@moser2004dbb], we think that the asymptotic capacity can be achieved by an input distribution that escapes to infinity. This implies that $\lambda=0$, so that $c$ is indeed zero. We do not investigate this rigorously. Multivariate Gaussian input distribution is a poor choice for channels with multiplicative noise. Indeed, it achieves a rate bounded in power in . Log-normal input PDF for the signal norm achieves the asymptotic capacity of the non-constant loss model. Review of the Information Theory of the Optical Fiber {#sec:review} ===================================================== An information-theoretic analysis of the full model of the optical fiber does not exist. Even in the special case of the zero-dispersion, spectral efficiency is unknown. In the full model, we do not know anything about the capacity in the high power regime, let alone the spectral efficiency. The state-of-the-art is still lower bounds that are good in the nearly-linear regime. This situation calls for basic research, in order to make progress on these open problems. The present paper builds on earlier work. We acknowledge [@mecozzi1994llh Eq. 12] for the equation , [@mecozzi1994llh; @turitsyn2003ico; @yousefi2011opc] for the PDF of the zero-dispersion channel, [@yousefi2011opc] for the analysis of the zero-dispersion model, [@yousefi2015cwit2; @kramer2015upper] for noting that Shannon entropy is invariant under the flow of a broad class of deterministic partial differential equations and for highlighting the usefulness of the operator splitting (in numerical analysis) in the analysis of the NLS equation. Furthermore, we acknowledge [@agrell2015conds] for helpful insight leading to the rate interpolation Lemma \[lemm:decomposition\], [@moser2004dbb; @lapidoth2003capacity] for the study of the fading channels and Section II of [@yousefi2012nft3] for unfolding the origin of the capacity limitations in fiber — particularly the finding that signal-signal interactions are not fundamental limitations in the deterministic model if communication takes place in the right basis ([*i.e.*]{}, the nonlinear Fourier basis), which led us to the study of the remaining factor in this paper, namely the signal-noise interactions. We do not intend to survey the literature in this paper. There is a good review in [@ghozlan2015focusing Section I-A]. The achievable rates of 1- and multi-solitons is studied, respectively, in [@yousefi2012nft3; @meron2012soliton; @shevchenko2015; @zhang2016isit] and [@kaza2012; @kaza2016soliton; @buelow2016]. There is also a myriad of lower bounds that hold good in the low power regime; see, [*e.g.*]{}, [@mecozzi2012nsl; @secondini2013achievable; @dar2014new; @terekhov2016physrev; @secondini2016limitsv2; @turitsyn2016nature]. The achievable rates of the nonlinear frequency-division multiplexing for multi-user communication are presented in [@yousefi2016nfdm] for the Hermitian channel. Fig. \[fig:nfdm\] compares the NFDM and WDM rates [@yousefi2016nfdm Fig. 6]. The gap between the WDM and NFDM curve reflects signal-signal interactions. The gap between the NFDM and AWGN curve reflects signal-noise interactions. We conjecture that the NFDM rate is close to the capacity. At the power levels shown in Fig. \[fig:nfdm\], ${{\mathcal{C}}}_{\textnormal{wdm}}({{\mathcal{P}}})=\log{{\mathcal{P}}}+c$ and ${{\mathcal{C}}}_{\textnormal{nfdm}}=\log{{\mathcal{P}}}+c'$, $c<c'$. Although more gains are expected at ${{\mathcal{P}}}>-2.4$ dB, the slope of the blue curve will gradually decrease, converging, in the limit ${{\mathcal{P}}}\rightarrow\infty$, to the asymptotic form in Theorem \[thm:main\]. It is interesting to compare the extent of the signal-noise interactions in the time domain [@serena2016signalnoise] and in the nonlinear Fourier domain [@tavakkolnia2015sig Section IV. A]. Conclusions =========== The asymptotic capacity of the discrete-time periodic model of the optical fiber is characterized as a function of the input power in Theorem \[thm:main\]. With $n$ signal [DOFs]{}at the input, $n-1$ [DOFs]{} are asymptotically lost, leaving signal energy as the only available [DOF]{}  for transmission. The appropriate input distribution is a log-normal PDF for the signal norm. Signal-noise interactions limit the operation of the optical communication systems to low-to-medium powers. Acknowledgments {#acknowledgments .unnumbered} =============== The research was partially conducted when the author was at the Technische Universität München (TUM). The support of the TUM Institute for Advanced Study, funded by the German Excellence Initiative, and the support of the Alexander von Humboldt Foundation, funded by the German Federal Ministry of Education and Research, are gratefully acknowledged. The author thanks Luca Barletta for comments. Proof of the Identity {#app:one} ====================== Let ${\mathrm{d}}V({\underaccent{\bar}{x}})$ and ${\mathrm{d}}S({\underaccent{\bar}{x}})$ be the volume and surface element at point ${\underaccent{\bar}{x}}\in{\mathbb{R}}^n$ in the spherical coordinate system. Then [rCl]{} V(x)&=&|x|\^[n-1]{}V()\ &=&|x|\^[n-1]{}S()|x|. Thus the Jacobian of the transformation from the Cartesian system with coordinates ${\underaccent{\bar}{x}}$ to the spherical system with coordinates $({\left|{\underaccent{\bar}{x}}\right|}, \hat{{\underaccent{\bar}{x}}})$ is ${\left|{\underaccent{\bar}{x}}\right|}^{n-1}$. As a consequence [rCl]{} h(X)&=&h\_([||]{},)+[||]{}\^[n-1]{}\ &=& h([||]{})+h\_(|[||]{})+(n-1)[||]{}. Input Output Relation in a Unit {#app:in-out-mssfm} =============================== Define [rCl]{} D\_1=(e\^[j\_k]{}),D\_2=(e\^[j\_k]{}). The nonlinear steps in Fig. \[fig:mssfm\] in matrix notation are [rCl]{} U=D\_1(X+\^1e), Y=D\_2(V+\^2e), where ${\underaccent{\bar}{e}}\in{\mathbb{R}}^L$ is the all-one column vector. Combining the linear and nonlinear steps, we obtain with ${\mathbb{M}}={\mathbb{D}}_2{\mathsf{R}}{\mathbb{D}}_1$ and [rCl]{} Z =M\^1e+D\_2 \^2e. \[eq:additive-Z\] Clearly ${\mathbb{N}}^{1,2}{\underaccent{\bar}{e}}\sim{\mathcal{N}_{{\mathbb{C}}}\!\left(0,{\mathcal{D}}I_n\right)}$. However ${\mathbb{M}}{\mathbb{N}}^1{\underaccent{\bar}{e}}$ and ${\mathbb{D}}_2 {\mathbb{N}}^2{\underaccent{\bar}{e}}$ are generally non-Gaussian due to the signal and noise terms in $\Phi_k$ and $\Psi_l$. But, 1) in the constant loss model, if 2) $\forall k$ ${\underaccent{\bar}{x}}_k\rightarrow\infty$, then [rCl]{} Z\~[\_(0,)]{},K=(1+e\^[-\_[r]{}]{})I\_n. \[eq:noise\] In summary, $N$ variables are Gaussian; $Z$ variables are Gaussians in the asymptotic analysis of the constant loss model. [10]{} \[1\][\#1]{} url@samestyle \[2\][\#2]{} \[2\][[l@\#1=l@\#1\#2]{}]{} M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear [F]{}ourier transform, [P]{}art [I]{}: [M]{}athematical tools,” *IEEE Trans. Inf. Theory*, vol. 60, no. 7, pp. 4312–4328, Jul. 2014, [A]{}lso published at arXiv, Feb. 2012. \[Online\]. Available: <http://arxiv.org/abs/1202.3653> ——, “Information transmission using the nonlinear [F]{}ourier transform, [P]{}art [II]{}: [N]{}umerical methods,” *IEEE Trans. Inf. Theory*, vol. 60, no. 7, pp. 4329–4345, Jul. 2014, [A]{}lso published at arXiv, Apr. 2012. \[Online\]. Available: <http://arxiv.org/abs/1204.0830> ——, “Information transmission using the nonlinear [F]{}ourier transform, [P]{}art [III]{}: [S]{}pectrum modulation,” *IEEE Trans. Inf. Theory*, vol. 60, no. 7, pp. 4346–4369, Jul. 2014, [A]{}lso published at arXiv, Feb. 2013. \[Online\]. Available: <http://arxiv.org/abs/1302.2875> ——, “On the per-sample capacity of nondispersive optical fibers,” *IEEE Trans. Inf. Theory*, vol. 57, no. 11, pp. 7522–7541, Nov. 2011. M. I. Yousefi and X. Yangzhang, “Linear and nonlinear frequency-division multiplexing,” *[ar[X]{}iv:1603.04389]{.nodecor}*, pp. 1–14, Mar. 2016. \[Online\]. Available: <http://arxiv.org/abs/1603.04389> S. M. 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ArXiv

--- abstract: 'Let $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq{\operatorname{GL}}_r$ be a Levi subgroup of ${\operatorname{GL}}_r$, where $r=r_1+\cdots+r_k$, and ${\widetilde{M}}$ its metaplectic preimage in the $n$-fold metaplectic cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$. For automorphic representations $\pi_1,\dots,\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$, we construct (under a certain technical assumption, which is always satisfied when $n=2$) an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$ which can be considered as the “tensor product” of the representations $\pi_1,\dots,\pi_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic tensor product of $\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.' address: 'Shuichiro Takeda: Mathematics Department, University of Missouri, Columbia, 202 Math Sciences Building, Columbia, MO, 65211' author: - Shuichiro Takeda title: 'Metaplectic tensor products for automorphic representations of ${\widetilde{\operatorname{GL}}}(r)$' --- **Introduction** ================ Let $F$ be either a local field of characteristic 0 or a number field, and $R$ be $F$ if $F$ is local and the ring of adeles ${\mathbb{A}}$ if $F$ is global. Consider the group ${\operatorname{GL}}_r(R)$. For a partition $r=r_1+\cdots+r_k$ of $r$, one has the Levi subgroup $$M(R):={\operatorname{GL}}_{r_1}(R)\times\cdots\times{\operatorname{GL}}_{r_k}(R)\subseteq{\operatorname{GL}}_r(R).$$ Let $\pi_1,\dots,\pi_k$ be irreducible admissible (resp. automorphic) representations of ${\operatorname{GL}}_{r_1}(R),\dots,{\operatorname{GL}}_{r_k}(R)$ where $F$ is local (resp. $F$ is global). Then it is a trivial construction to obtain the representation $\pi_1\otimes\cdots\otimes\pi_k$, which is an irreducible admissible (resp. automorphic) representation of the Levi $M(R)$. Though highly trivial, this construction is of great importance in the representation theory of ${\operatorname{GL}}_r(R)$. Now if one considers the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(R)$ constructed by Kazhdan and Patterson in [@KP], the analogous construction turns out to be far from trivial. Namely for the metaplectic preimage ${\widetilde{M}}(R)$ of $M(R)$ in ${\operatorname{GL}}_r(R)$ and representations $\pi_1,\dots,\pi_k$ of the metaplectic $n$-fold covers ${\widetilde{\operatorname{GL}}}_{r_1}(R),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(R)$, one cannot construct a representation of ${\widetilde{M}}(R)$ simply by taking the tensor product $\pi_1\otimes\cdots\otimes\pi_k$. This is simply because ${\widetilde{M}}(R)$ is not the direct product of ${\widetilde{\operatorname{GL}}}_{r_1}(R),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(R)$, namely $${\widetilde{M}}(R)\ncong{\widetilde{\operatorname{GL}}}_{r_1}(R)\times\dots\times{\widetilde{\operatorname{GL}}}_{r_k}(R),$$ and even worse there is no natural map between them. When $F$ is a local field, for irreducible admissible representations $\pi_1,\dots,\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1}(F),\dots,{\widetilde{\operatorname{GL}}}_{r_k}(F)$, P. Mezo ([@Mezo]), whose work, we believe, is based on the work by Kable [@Kable2], constructed an irreducible admissible representation of the Levi ${\widetilde{M}}(F)$, which can be called the “metaplectic tensor product” of $\pi_1,\dots,\pi_k$, and characterized it uniquely up to certain character twists. (His construction will be reviewed and expanded further in Section \[S:Mezo\].) The theme of the paper is to carry out a construction analogous to Mezo’s when $F$ is a number field, and our main theorem is Let $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ be a Levi subgroup of ${\operatorname{GL}}_r$, and let $\pi_1,\dots,\pi_k$ be unitary automorphic subrepresentations of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$. Assume that $M$ and $n$ are such that Hypothesis ($\ast$) is satisfied, which is always the case if $n=2$. Then there exists an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$ such that $$\pi\cong{\widetilde{\otimes}}'_v\pi_v,$$ where each $\pi_v$ is the local metaplectic tensor product of Mezo. Moreover, if $\pi_1,\dots,\pi_k$ are cuspidal (resp. square-integrable modulo center), then $\pi$ is cuspidal (resp. square-integrable modulo center). In the above theorem, ${\widetilde{\otimes}}_v'$ indicates the metaplectic restricted tensor product, the meaning of which will be explained later in the paper. The existence and the local-global compatibility in the main theorem are proven in Theorem \[T:main\], and the cuspidality and square-integrability are proven in Theorem \[T:cuspidal\] and Theorem \[T:square\_integrable\], respectively. Let us note that by unitary, we mean that $\pi_i$ is equipped with a Hermitian structure invariant under the action of the group. Also we require $\pi_i$ be an automorphic subrepresentation, so that it is realized in a subspace of automorphic forms and hence each element in $\pi_i$ is indeed an automorphic form. (Note that usually an automorphic representation is a subquotient.) We need those two conditions for technical reasons, and they are satisfied if $\pi_i$ is in the discrete spectrum, namely either cuspidal or residual. Also we should emphasize that if $n>2$, we do not know if our construction works unless we impose a technical assumption as in Hypothesis ($\ast$). We will show in Appendix \[A:topology\] that this assumption is always satisfied if $n=2$, and if $n>2$ it is satisfied, for example, if $\gcd(n, r-1+2cr)=1$, where $c$ is the parameter to be explained. We hope that even for $n>2$ it is always satisfied, though at this moment we do not know how to prove it. As we will see, strictly speaking the metaplectic tensor product of $\pi_1,\dots,\pi_k$ might not be unique even up to equivalence but is dependent on a character $\omega$ on the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ of ${\widetilde{\operatorname{GL}}}_r$. Hence we write $$\pi_\omega:=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$$ for the metaplectic tensor product to emphasize the dependence on $\omega$.\ Also we will establish a couple of important properties of the metaplectic tensor product both locally and globally. The first one is that the metaplectic tensor product behaves in the expected way under the action of the Weyl group. Namely \ [**Theorem \[T:Weyl\_group\_local\] and \[T:Weyl\_group\_global\].**]{} [*Let $w\in W_M$ be a Weyl group element of ${\operatorname{GL}}_r$ that only permutes the ${\operatorname{GL}}_{r_i}$-factors of $M$. Namely for each $(g_1,\dots,g_k)\in{\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$, we have $w (g_1,\dots,g_k)w^{-1}=(g_{\sigma(1)},\dots,g_{\sigma(k)})$ for a permutation $\sigma\in S_k$ of $k$ letters. Then both locally and globally, we have $$^{w}(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega \cong(\pi_{\sigma(1)}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{\sigma(k)})_\omega,$$ where the left hand side is the twist of $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$ by $w$.* ]{}\ The second important property we establish is the compatibility of the metaplectic tensor product with parabolic inductions. Namely \ [**Theorem \[T:induction\_local\] and \[T:induction\_global\].**]{} [ *Both locally and globally, let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i, l_i}}$. For each $i$, we are given a representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i$, which is given as the metaplectic tensor product of the representations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}},\dots,{\widetilde{\operatorname{GL}}}_{r_{i, l_i}}$. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i}^{{\widetilde{\operatorname{GL}}}_{r_i}}\sigma_i$. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,l_k})_\omega,$$ where $Q$ is the standard parabolic subgroup of $M$ whose Levi part is $M_1\times\cdots\times M_k$.* ]{}\ In the above two theorems, it is implicitly assumed that if $n>2$ and $F$ is global, the metaplectic tensor products in the theorems exist in the sense that Hypothesis ($\ast$) is satisfied for the relevant Levi subgroups. Finally at the end, we will discuss the behavior of the global metaplectic tensor product when restricted to a smaller Levi. Namely for each automorphic form $\varphi\in(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_\omega$ in the metaplectic tensor product, we would like to know which space the restriction $\varphi|_{{\widetilde{M}}_2}$ belongs to, where $M_2=\{I_{r_1}\}\times{\operatorname{GL}}_{r_2}\times\cdots\times{\operatorname{GL}}_{r_k}\subset M$, viewed as a subgroup of $M$, is the Levi for the smaller group ${\operatorname{GL}}_{r-r_1}$. Somehow similarly to the non-metaplectic case, the restriction $\varphi|_{{\widetilde{M}}_2}$ belongs to the metaplectic tensor product of $\pi_2, \dots,\pi_k$. But the precise statement is a bit more subtle. Indeed, we will prove \ [**Theorem \[T:restriction\].**]{} [*Assume Hypothesis ($\ast\ast$) is satisfied, which is always the case if $n=2$ or $\gcd(n,r-1+2cr)=\gcd(n,r-r_1-1+2c(r-r_1))=1$. Then there exists a realization of the metaplectic tensor product $\pi_\omega=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$ such that, if we let $$\pi_\omega\|_{{\widetilde{M}}_2({\mathbb{A}})}=\{{\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}:{\widetilde{\varphi}}\in \pi_\omega\},$$ then $$\pi_\omega\|_{{\widetilde{M}}_2({\mathbb{A}})}\subseteq \bigoplus_\delta m_\delta(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta},$$ as a representation of ${\widetilde{M}}_2({\mathbb{A}})$, where $(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta}$ is the metaplectic tensor product of $\pi_2,\dots,\pi_k$, $\omega_\delta$ is a certain character twisted by $\delta$ which runs through a finite subset of ${\operatorname{GL}}_{r_1}(F)$ and $m_\delta\in{\mathbb{Z}}^{\geq 0}$ is a multiplicity.* ]{}\ The precise meanings of the notations will be explained in Section \[S:restriction\]. Even though the theory of metaplectic groups is an important subject in representation theory and automorphic forms and used in various important literatures such as [@Banks2; @F; @BBL; @BFH; @BH; @Suzuki] to name a few, and most importantly for the purpose of this paper, [@BG] which concerns the symmetric square $L$-function on ${\operatorname{GL}}(r)$ , it has an unfortunate history of numerous technical errors and as a result published literatures in this area are often marred by those errors which compromise their reliability. As is pointed out in [@BLS], this is probably due to the deep and subtle nature of the subject. At any rate, this has made people who work in the area particularly wary of inaccuracies in new works. For this reason, especially considering the foundational nature of this paper, we tried to provide detailed proofs for most of our assertions at the expense of the length of the paper. Furthermore, for large part, we rely only on the two fundamental works, namely the work on the metaplectic cocycle by Banks, Levy and Sepanski ([@BLS]) and the local metaplectic tensor product by Mezo ([@Mezo]), both of which are written carefully enough to be reliable. Finally, let us mention that the result of this paper will be used in our forthcoming [@Takeda2], which will improve the main result of [@Takeda1]. [**Notations**]{} Throughout the paper, $F$ is a local field of characteristic zero or a number field. If $F$ is a number field, we denote the ring of adeles by ${\mathbb{A}}$. As we did in the introduction we often use the notation $$R=\begin{cases} F\quad\text{if $F$ is local}\\ {\mathbb{A}}\quad\text{if $F$ is global}. \end{cases}$$ The symbol $R^\times$ has the usual meaning and we set $$R^{\times n}=\{a^n: a\in R^\times\}.$$ Both locally and globally, we denote by ${\mathcal{O}_F}$ the ring of integers of $F$. For each algebraic group $G$ over a global $F$, and $g\in G({\mathbb{A}})$, by $g_v$ we mean the $v^{{{\text{th}}}}$ component of $g$, and so $g_v\in G(F_v)$. For a positive integer $r$, we denote by $I_r$ the $r\times r$ identity matrix. Throughout we fix an integer $n\geq 2$, and we let $\mu_n$ be the group of $n^{\text{th}}$ roots of unity in the algebraic closure of the prime field. We always assume that $\mu_n\subseteq F$, where $F$ is either local or global. So in particular if $n\geq 3$, for archimedean $F$, we have $F={\mathbb C}$, and for global $F$, $F$ is totally complex. The symbol $(-,-)_F$ denotes the $n^{\text{th}}$ order Hilbert symbol of $F$ if $F$ is local, which is a bilinear map $$(-,-)_F:F^\times\times F^\times\rightarrow\mu_n.$$ If $F$ is global, we let $(-,-)_{\mathbb{A}}:=\prod_v(-,-)_{F_v}$, where the product is finite. We sometimes write simply $(-,-)$ for $(-,-)_R$ when there is no danger of confusion. Let us recall that both locally and globally the Hilbert symbol has the following properties: $$\begin{gathered} (a,b)^{-1}=(b,a)\\ (a^n,b)=(a, b^n)=1\\ (a, -a)=1\end{gathered}$$ for $a, b\in R^\times$. Also for the global Hilbert symbol, we have the product formula $(a,b)_{\mathbb{A}}=1$ for all $a, b\in F^\times$. We fix a partition $r_1+\cdots+r_k=r$ of $r$, and we let $$M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq{\operatorname{GL}}_r$$ and assume it is embedded diagonally as usual. We often denote each element $m\in M$ by $$m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \quad\text{or}\quad m={\operatorname{diag}}(g_1,\dots,g_k)$$ or sometimes simply $m=(g_1,\dots,g_k)$, where $g_i\in{\operatorname{GL}}_{r_i}$. For ${\operatorname{GL}}_r$, we let $B=TN_B$ be the Borel subgroup with the unipotent radical $N_B$ and the maximal torus $T$. If $\pi$ is a representation of a group $G$, we denote the space of $\pi$ by $V_{\pi}$, though we often confuse $\pi$ with $V_\pi$ when there is no danger of confusion. We say $\pi$ is unitary if $V_\pi$ is equipped with a Hermitian structure invariant under the action of $G$, but we do not necessarily assume that the space $V_\pi$ is complete. Now assume that the space $V_\pi$ is a space of functions or maps on the group $G$ and $\pi$ is the representation of $G$ on $V_\pi$ defined by right translation. (This is the case, for example, if $\pi$ is an automorphic subrepresentation.) Let $H\subseteq G$ be a subgroup. Then we define $\pi\|_H$ to be the representation of $H$ realized in the space $$V_{\pi\|_H}:=\{f|_H: f\in V_\pi\}$$ of restrictions of $f\in V_\pi$ to $H$, on which $H$ acts by right translation. Namely $\pi\|_H$ is the representation obtained by restricting the functions in $V_\pi$. Occasionally, we confuse $\pi\|_H$ with its space when there is no danger of confusion. Note that there is an $H$-intertwining surjection $\pi|_H\rightarrow\pi\|_H$, where $\pi|_H$ is the (usual) restriction of $\pi$ to $H$. For any group $G$ and elements $g, h\in G$, we define $^gh=ghg^{-1}$. For a subgroup $H\subseteq G$ and a representation $\pi$ of $H$, we define $^g\pi$ to be the representation of $gHg^{-1}$ defined by $^g\pi(h')=\pi(g^{-1}h'g)$ for $h'\in gHg^{-1}$. We let $W$ be the set of all $r\times r$ permutation matrices, so for each element $w\in W$ each row and each column has exactly one 1 and all the other entries are 0. The Weyl group of ${\operatorname{GL}}_r$ is identified with $W$. Also for our Levi $M$, we let $W_M$ be the subset of $W$ that only permutes the ${\operatorname{GL}}_{r_i}$-blocks of $M$. Namely $W_M$ is the collection of block matrices $$W_M:=\{(\delta_{\sigma(i),j}I_{r_j})\in W : \sigma\in S_k\},$$ where $S_k$ is the permutation group of $k$ letters. Though $W_M$ is not a group in general, it is in bijection with $S_k$. Note that if $w\in W_M$ corresponds to $\sigma\in S_k$, we have $$^w{\operatorname{diag}}(g_1,\dots,g_k)=w{\operatorname{diag}}(g_1,\dots,g_k)w^{-1} ={\operatorname{diag}}(g_{\sigma^{-1}(1)},\dots, g_{\sigma^{-1}(k)}).$$ In addition to $W$, in order to use various results from [@BLS], which gives a detailed description of the 2-cocycle $\sigma_r$ defining our metaplectic group ${\widetilde{\operatorname{GL}}}_r$, one sometimes needs to use another set of representatives of the Weyl group elements, which we as well as [@BLS] denote by ${\mathfrak{M}}$. The set ${\mathfrak{M}}$ is chosen to be such that for each element $\eta\in\mathfrak{M}$ we have $\det(\eta)=1$. To be more precise, each $\eta$ with length $l$ is written as $$\eta=w_{\alpha_1}\cdots w_{\alpha_l}$$ where $w_{\alpha_i}$ is a simple root refection corresponding to a simple root $\alpha_i$ and is the matrix of the form $$w_{\alpha_i}=\begin{pmatrix} \ddots&&&\\ &&-1&\\ &1&&\\ &&&\ddots \end{pmatrix}.$$ Though the set $\mathfrak{M}$ is not a group, it has the advantage that we can compute the cocycle $\sigma_r$ in a systematic way as one can see in [@BLS]. For each $w\in W$, we denote by $\eta_w$ the corresponding element in ${\mathfrak{M}}$. If $w\in W_M$, one can see that $\eta_w$ is of the form $(\varepsilon_j\delta_{\sigma(i),j}I_{r_j})$ for $\varepsilon_j\in\{\pm 1\}$. Namely $\eta_w$ is a $k\times k$ block matrix in which the non-zero entries are either $I_{r_j}$ or $-I_{r_j}$. [**Acknowledgements**]{} The author would like to thanks Paul Mezo for reading an early draf and giving him helpful comments, and Jeff Adams for sending him the preprint [@Adams] and explaining the construction of the metaplectic tensor product for the real case. The author is partially supported by NSF grant DMS-1215419. Also part of this research was done when he was visiting the I.H.E.S. in the summer of 2012 and he would like to thank their hospitality. **The metaplectic cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$** {#S:metaplectic_cover} ======================================================================================== In this section, we review the theory of the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r$ of ${\operatorname{GL}}_r$ for both local and global cases, which was originally constructed by Kazhdan and Patterson in [@KP]. **The local metaplectic cover ${\widetilde{\operatorname{GL}}}_r(F)$** ---------------------------------------------------------------------- Let $F$ be a (not necessarily non-archimedean) local field of characteristic $0$ which contains all the $n^\text{th}$ roots of unity. In this paper, by the metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(F)$ of ${\operatorname{GL}}_r(F)$ with a fixed parameter $c\in\{0,\dots,n-1\}$, we mean the central extension of ${\operatorname{GL}}_r(F)$ by $\mu_n$ as constructed by Kazhdan and Patterson in [@KP]. To be more specific, let us first recall that the $n$-fold cover ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ of ${\operatorname{SL}}_{r+1}(F)$ was constructed by Matsumoto in [@Matsumoto], and there is an embedding $$\label{E:embedding0} l_0:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad g\mapsto\begin{pmatrix}\det(g)^{-1}&\\ &g\end{pmatrix}.$$ Our metaplectic $n$-fold cover ${\widetilde{\operatorname{GL}}}_r(F)$ with $c=0$ is the preimage of $l_0({\operatorname{GL}}_r(F))$ via the canonical projection ${\widetilde{\operatorname{SL}}}_{r+1}(F)\rightarrow{\operatorname{SL}}_{r+1}(F)$. Then ${\widetilde{\operatorname{GL}}}_r(F)$ is defined by a 2-cocycle $$\sigma_r:{\operatorname{GL}}_r(F)\times{\operatorname{GL}}_r(F)\rightarrow\mu_n.$$ For arbitrary parameter $c\in\{0,\dots, n-1\}$, we define the twisted cocycle $\sigma_r^{(c)}$ by $$\sigma_r^{(c)}(g, g')=\sigma_r(g, g')(\det(g),\det(g'))^c_F$$ for $g, g'\in{\operatorname{GL}}_r(F)$, where recall from the notation section that $(-,-)_F$ is the $n^{\text{th}}$ order Hilbert symbol for $F$. The metaplectic cover with a parameter $c$ is defined by this cocycle. In [@KP], the metaplectic cover with parameter $c$ is denoted by ${\widetilde{\operatorname{GL}}}_r^{(c)}(F)$ but we avoid this notation. This is because later we will introduce the notation ${\widetilde{\operatorname{GL}}^{(n)}}_r(F)$, which has a completely different meaning. Also we suppress the superscript $(c)$ from the notation of the cocycle and always agree that the parameter $c$ is fixed throughout the paper. By carefully studying Matsumoto’s construction, Banks, Levy, and Sepanski ([@BLS]) gave an explicit description of the 2-cocycle $\sigma_r$ and shows that their 2-cocycle is “block-compatible” in the following sense: For the standard $(r_1,\dots,r_k)$-parabolic of ${\operatorname{GL}}_r$, so that its Levi $M$ is of the form ${\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ which is embedded diagonally into ${\operatorname{GL}}_r$, we have $$\begin{aligned} \label{E:compatibility} &\sigma_r(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix})\\ =&\prod_{i=1}^k\sigma_{r_i}(g_i,g_i') \prod_{1\leq i<j\leq k}(\det(g_i), \det(g_j'))_F \prod_{i\neq j}(\det(g_i), \det(g_j'))_F^c,\notag \end{aligned}$$ for all $g_i, g_i'\in{\operatorname{GL}}_{r_i}(F)$. (See [@BLS Theorem 11, §3]. Strictly speaking in [@BLS] only the case $c=0$ is considered but one can derive the above formula using the bilinearity of the Hilbert symbol.) This 2-cocycle generalizes the well-known cocycle given by Kubota [@Kubota] for the case $r=2$. Also we should note that if $r=1$, this cocycle is trivial. Note that ${\widetilde{\operatorname{GL}}}_r(F)$ is not the $F$-rational points of an algebraic group, but this notation seems to be standard. Let us list some other important properties of the cocycle $\sigma_r$, which we will use in this paper. \[P:BLS\] Let $B=TN_B$ be the Borel subgroup of ${\operatorname{GL}}_r$ where $T$ is the maximal torus and $N_B$ the unipotent radical. The cocycle $\sigma_r$ satisfies the following properties: (1) $\sigma_r(g,g')\sigma_r(gg', g'')=\sigma_r(g, g'g'')\sigma_r(g',g'')$ for $g,g',g''\in{\operatorname{GL}}_r$. (2) $\sigma_r(ng, g'n')=\sigma_r(g, g')$ for $g, g'\in{\operatorname{GL}}_r$ and $n, n'\in N_B$, and so in particular $\sigma_r(ng, n')=\sigma_r(n, g'n')=1$. (3) $\sigma_r(gn, g')=\sigma_r(g, ng')$ for $g, g'\in{\operatorname{GL}}_r$ and $n\in N_B$. (4) $\sigma(\eta, t)=\underset{\substack{\alpha=(i,j)\in\Phi^+\\ \eta\alpha<0}}{\prod}(-t_j, t_i)$ for $\eta\in{\mathfrak{M}}$ and $t={\operatorname{diag}}(t_1,\dots,t_r)\in T$, where $\Phi^+$ is the set of positive roots and each root $\alpha\in\Phi^+$ is identified with a pair of integers $(i,j)$ with $1\leq i<j\leq r$ as usual. (5) $\sigma_r(t,t')=\underset{i<j}{\prod}(t_i, t'_j)(\det(t),\det(t'))^c$ for $t={\operatorname{diag}}(t_1,\dots,t_r), t'={\operatorname{diag}}(t'_1,\dots,t'_r)\in T$. (6) $\sigma_r(t,\eta)=1$ for $t\in T$ and $\eta\in{\mathfrak{M}}$. The first one is simply the definition of 2-cocycle and all the others are some of the properties of $\sigma_r$ listed in [@BLS Theorem 7, p.153]. We need to recall how this cocycle is constructed. As mentioned earlier, Matsumoto constructed ${\widetilde{\operatorname{SL}}}_{r+1}(F)$. It is shown in [@BLS] that ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ is defined by a cocycle $\sigma_{{\operatorname{SL}}_{r+1}}$ which satisfies the block-compatibility in a much stronger sense as in [@BLS Theorem 7, §2, p145]. (Note that our ${\operatorname{SL}}_{r+1}$ corresponds to ${\mathbb{G}}^\flat$ of [@BLS].) Then the cocycle $\sigma_r$ is defined by $$\sigma_r(g, g')=\sigma_{{\operatorname{SL}}_{r+1}}(l(g), l(g'))(\det(g), \det(g'))_F (\det(g), \det(g'))^c_F,$$ where $l$ is the embedding defined by $$\label{E:embedding} l:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad g\mapsto\begin{pmatrix}g&\\ &\det(g)^{-1}\end{pmatrix}.$$ See [@BLS p.146]. (Note the difference between this embedding and the one in (\[E:embedding0\]). This is the reason we have the extra Hilbert symbol in the definition of $\sigma_r$.) Since we would like to emphasize the cocycle being used, we denote ${\widetilde{\operatorname{GL}}}_r(F)$ by ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)$ when the cocycle $\sigma$ is used. Namely ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)$ is the group whose underlying set is $${{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F) ={\operatorname{GL}}_r(F)\times\mu_n=\{(g,\xi):g\in{\operatorname{GL}}_r(F), \xi\in\mu_n\},$$ and the group law is defined by $$(g,\xi)\cdot (g',\xi')=(gg',\sigma_r(g, g')\xi\xi').$$ To use the block-compatible 2-cocycle of [@BLS] has obvious advantages. In particular, it has been explicitly computed and, of course, it is block-compatible. Indeed, when we consider purely local problems, we always assume that the cocycle $\sigma_r$ is used. However it does not allow us to construct the global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Namely one cannot define the adelic block-combatible 2-cocycle simply by taking the product of the local block-combatible 2-cocycles over all the places. Namely for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, the product $$\prod_v\sigma_{r,v}(g_v,g_v')$$ is not necessarily finite. This can be already observed for the case $r=2$. (See [@F p.125].) For this reason, we will use a different 2-cocycle $\tau_r$ which works nicely with the global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. To construct such $\tau_r$, first assume $F$ is non-archimedean. It is known that an open compact subgroup $K$ splits in ${\widetilde{\operatorname{GL}}}_r(F)$, and moreover if $|n|_F=1$, we have $K={\operatorname{GL}}_r({\mathcal{O}_F})$. (See [@KP Proposition 0.1.2].) Also for $k,k'\in K$, a property of the Hilbert symbol gives $(\det(k),\det(k'))_F=1$. Hence one has a continuous map $s_r:{\operatorname{GL}}_r(F)\rightarrow\mu_n$ such that $\sigma_r(k,k')s_r(k)s_r(k')=s_r(kk')$ for all $k,k'\in K$. Then define our 2-cocycle $\tau_r$ by $$\label{E:tau_sigma} \tau_r(g, g'):=\sigma_r(g, g')\cdot\frac{s_r(g)s_r(g')}{s_r(gg')}$$ for $g, g'\in{\operatorname{GL}}_r(F)$. If $F$ is archimedean, we set $\tau_r=\sigma_r$. The choice of $s_r$ and hence $\tau_r$ is not unique. However when $|n|_F=1$, there is a canonical choice with respect to the splitting of $K$ in the following sense: Assume that $F$ is such that $|n|_F=1$. Then the Hilbert symbol $(-,-)_F$ is trivial on ${\mathcal{O}_F}^\times\times{\mathcal{O}_F}^\times$, and hence, when restricted to ${\operatorname{GL}}_r({\mathcal{O}_F})\times{\operatorname{GL}}_r({\mathcal{O}_F})$, the cocycle $\sigma_r$ is the restriction of $\sigma_{{\operatorname{SL}}_{r+1}}$ to the image of the embedding $l$. Now it is known that the compact group ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$ also splits in ${\widetilde{\operatorname{SL}}}_{r+1}(F)$, and hence there is a map ${\mathfrak{s}}_r:{\operatorname{SL}}_{r+1}(F)\rightarrow\mu_n$ such that the section ${\operatorname{SL}}_{r+1}(F)\rightarrow{\widetilde{\operatorname{SL}}}_{r+1}(F)$ given by $(g,{\mathfrak{s}}_r(g))$ is a homomorphism on ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$. (Here we are assuming ${\widetilde{\operatorname{SL}}}_{r+1}(F)$ is realized as ${\operatorname{SL}}_{r+1}(F)\times\mu_n$ as a set and the group structure is defined by the cocycle $\sigma_{{\operatorname{SL}}_{r+1}}$.) Moreover ${\mathfrak{s}}_r|_{{\operatorname{SL}}_{r+1}({\mathcal{O}_F})}$ is determined up to twists by the elements in $H^1({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)={\operatorname{Hom}}({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)$. But ${\operatorname{Hom}}({\operatorname{SL}}_{r+1}({\mathcal{O}_F}), \mu_n)=1$ because ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})$ is a perfect group and $\mu_n$ is commutative. Hence ${\mathfrak{s}}_r|_{{\operatorname{SL}}_{r+1}({\mathcal{O}_F})}$ is unique. (See also [@KP p. 43] for this matter.) We choose $s_r$ so that $$\label{E:canonical_section} s_r|_{{\operatorname{GL}}_r({\mathcal{O}_F})}={\mathfrak{s}}_r|_{l({\operatorname{GL}}_r({\mathcal{O}_F}))}.$$ With this choice, we have the commutative diagram $$\label{E:canonical_diagram} \xymatrix{ {{^{\sigma}\widetilde{\operatorname{GL}}}}_r({\mathcal{O}_F})\ar[r]&{\widetilde{\operatorname{SL}}}_{r+1}({\mathcal{O}_F})\\ K\ar[r]\ar[u]^{k\mapsto (k,\;s_r(k))}&{\operatorname{SL}}_{r+1}({\mathcal{O}_F}),\ar[u]_{k\mapsto(k,\;{\mathfrak{s}}_r(k))} }$$ where the top arrow is $(g,\xi)\mapsto (\l(g),\xi)$, the bottom arrow is $l$, and all the arrows can be seen to be homomorphisms. This choice of $s_r$ will be crucial for constructing the metaplectic tensor product of automorphic representations. Also note that the left vertical arrow in the above diagram is what is called the canonical lift in [@KP] and denoted by $\kappa^\ast$ there. (Although we do not need this fact in this paper, if $r=2$ one can show that $\tau_r$ can be chosen to be block compatible, which is the cocycle used in [@F].) Using $\tau_r$, we realize ${\widetilde{\operatorname{GL}}}_r(F)$ as $${\widetilde{\operatorname{GL}}}_r(F)={\operatorname{GL}}_r(F)\times\mu_n,$$ as a set and the group law is given by $$(g,\xi)\cdot(g',\xi')=(gg', \tau_r(g,g')\xi\xi').$$ Note that we have the exact sequence $$\xymatrix{ 0\ar[r]&\mu_n\ar[r]&{\widetilde{\operatorname{GL}}}_r(F)\ar[r]^{p}&{\operatorname{GL}}_r(F)\ar[r]& 0 }$$ given by the obvious maps, where we call $p$ the canonical projection. We define a set theoretic section $$\kappa:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r(F),\; g\mapsto (g,1).$$ Note that $\kappa$ is not a homomorphism. But by our construction of the cocycle $\tau_r$, $\kappa|_K$ is a homomorphism if $F$ is non-archimedean and $K$ is a sufficiently small open compact subgroup, and moreover if $|n|_F=1$, one has $K={\operatorname{GL}}_r({\mathcal{O}_F})$. Also we define another set theoretic section $${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r(F),\; g\mapsto (g,s_r(g)^{-1})$$ where $s_r(g)$ is as above, and then we have the isomorphism $${\widetilde{\operatorname{GL}}}_r(F)\rightarrow{{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F),\quad (g,\xi)\mapsto (g,s_r(g)\xi),$$ which gives rise to the commutative diagram $$\xymatrix{ {\widetilde{\operatorname{GL}}}_r(F)\ar[rr]&&{{^{\sigma}\widetilde{\operatorname{GL}}}}_r(F)\\ &{\operatorname{GL}}_r(F)\ar[ul]^{{\mathbf{s}}_r}\ar[ur]_{g\mapsto (g,1)}& }$$ of set theoretic maps. Also note that the elements in the image ${\mathbf{s}}_r({\operatorname{GL}}_r(F))$ “multiply via $\sigma_r$” in the sense that for $g,g'\in{\operatorname{GL}}_r(F)$, we have $$\label{E:convenient} (g,s_r(g)^{-1}) (g',s_r(g')^{-1})=(gg', \sigma_r(g,g')s_r(gg')^{-1}).$$ Let us mention Assume $F$ is non-archimedean with $|n|_F=1$. We have $$\label{E:kappa_and_s} \kappa|_{T\cap K}={\mathbf{s}}_r|_{T\cap K},\quad \kappa|_{W}={\mathbf{s}}_r|_{W},\quad \kappa|_{N_B\cap K}={\mathbf{s}}_r|_{N_B\cap K},$$ where $W$ is the Weyl group and $K={\operatorname{GL}}_r({\mathcal{O}_F})$. In particular, this implies $s_r|_{T\cap K}=s_r|_{W}=s_r|_{N_B\cap K}=1$. See [@KP Proposition 0.I.3]. Though we do not need this fact in this paper, it should be noted that ${\mathbf{s}}_r$ splits the Weyl group $W$ if and only if $(-1,-1)_F=1$. So in particular it splits $W$ if $|n|_F=1$. See [@BLS §5]. If $P$ is a parabolic subgroup of ${\operatorname{GL}}_r$ whose Levi is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$, we often write $${\widetilde{M}}(F)={\widetilde{\operatorname{GL}}}_{r_1}(F){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_k}(F)$$ for the metaplectic preimage of $M(F)$. Next let $${\operatorname{GL}}_r^{(n)}(F)=\{g\in{\operatorname{GL}}_r(F):\det g\in F^{\times n}\},$$ and ${\widetilde{\operatorname{GL}}^{(n)}}_r(F)$ its metaplectic preimage. Also we define $$M^{(n)}(F)=\{(g_1,\dots,g_k)\in M(F): \det g_i\in F^{\times n}\}$$ and often denote its preimage by $${\widetilde{M}^{(n)}}(F)={\widetilde{\operatorname{GL}}^{(n)}}_{r_1}(F){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}(F).$$ The group ${\widetilde{M}^{(n)}}(F)$ is a normal subgroup of finite index. Indeed, we have the exact sequence $$\label{E:finite_quotient} 1\rightarrow{\widetilde{M}^{(n)}}(F)\rightarrow{\widetilde{M}}(F)\rightarrow \underbrace{F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times}_{\text{$k$ times}} \rightarrow 1,$$ where the third map is given by $({\operatorname{diag}}(g_1,\dots,g_k),\xi)\mapsto(\det(g_1),\dots,\det(g_k))$. We should mention the explicit isomorphism $F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times\rightarrow{\widetilde{M}^{(n)}}(F)\backslash{\widetilde{M}}(F)$ defined as follows: First for each $i\in\{1,\dots,k\}$, define a map $\iota_i:F^\times\rightarrow{\operatorname{GL}}_{r_i}$ by $$\label{E:iota} \iota_i(a)=\begin{pmatrix}a&\\ &I_{r_i-1}\end{pmatrix}.$$ Then the map given by $$(a_1,\dots,a_k)\mapsto(\begin{pmatrix}\iota_1(a_1)&&\\ &\ddots& \\ &&\iota_k(a_k)\end{pmatrix}, 1)$$ is a homomorphism. Clearly the map is well-defined and 1-1. Moreover this is surjective because each element $g_i\in{\operatorname{GL}}_{r_i}$ is written as $$g_i=g_i\iota_i(\det(g_i)^{n-1})\iota_i(\det(g_i)^{1-n})$$ and $g_i\iota_i(\det(g_i)^{n-1})\in{\operatorname{GL}}_{r_i}^{(n)}$. The following should be mentioned. \[L:closed\_subgroup\_local\] The groups $F^{\times n}, M^{(n)}(F)$ and ${\widetilde{M}^{(n)}}(F)$ are closed subgroups of $F^\times, M(F)$ and ${\widetilde{M}}(F)$, respectively. It is well-known that $F^{\times n}$ is closed and of finite index in $F^\times$. Hence the group $F^{\times n}\backslash F^\times\times\cdots\times F^{\times n}\backslash F^\times$ is discrete, in particular Hausdorff. But both ${\widetilde{M}^{(n)}}(F)\backslash{\widetilde{M}}(F)$ and $M^{(n)}(F)\backslash M(F)$ are, as topological groups, isomorphic to this Hausdorff space. This completes the proof. \[R:archimedean1\] If $F={\mathbb C}$, clearly ${\widetilde{M}^{(n)}}(F)={\widetilde{M}}(F)$. If $F={\mathbb{R}}$, then necessarily $n=2$ and ${\operatorname{GL}}_r^{(2)}({\mathbb{R}})$ consists of the elements of positive determinants, which is usually denote by ${\operatorname{GL}}_r^+({\mathbb{R}})$. Accordingly one may denote ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{R}})$ and ${\widetilde{M}^{(n)}}({\mathbb{R}})$ by ${\widetilde{\operatorname{GL}}}_r^+({\mathbb{R}})$ and ${{\widetilde{M}}}^+({\mathbb{R}})$ respectively. Both ${\widetilde{\operatorname{GL}}}_r^+({\mathbb{R}})$ and ${\widetilde{\operatorname{GL}}}_r({\mathbb{R}})$ share the identity component, and hence they have the same Lie algebra. The same applies to ${\widetilde{M}}^+({\mathbb{R}})$ and ${\widetilde{M}}({\mathbb{R}})$. Let us mention the following important fact. Let $Z_{{\operatorname{GL}}_r}(F)\subseteq{\operatorname{GL}}_r(F)$ be the center of ${\operatorname{GL}}_r(F)$. Then its metaplectic preimage $\widetilde{Z_{{\operatorname{GL}}_r}}(F)$ is not the center of ${\widetilde{\operatorname{GL}}}_r(F)$ in general. (It might not be even commutative for $n>2$.) The center, which we denote by $Z_{{\widetilde{\operatorname{GL}}}_r}(F)$, is $$\begin{aligned} \label{E:center_GLt} Z_{{\widetilde{\operatorname{GL}}}_r}(F)&=\{(aI_r, \xi):a^{r-1+2rc}\in F^{\times n}, \xi\in\mu_n\}\\ \notag&=\{(aI_r, \xi):a\in F^{\times \frac{n}{d}}, \xi\in\mu_n\},\end{aligned}$$ where $d=\gcd(r-1+2c, n)$. (The second equality is proven in [@GO Lemma 1].) Note that $Z_{{\widetilde{\operatorname{GL}}}_r}(F)$ is a closed subgroup. Let $\pi$ be an admissible representation of a subgroup $\widetilde{H}\subseteq {\widetilde{\operatorname{GL}}}_r(F)$, where ${\widetilde{H}}$ is the metaplectic preimage of a subgroup $H\subseteq{\operatorname{GL}}_r(F)$. We say $\pi$ is “genuine” if each element $(1,\xi)\in\widetilde{H}$ acts as multiplication by $\xi$, where we view $\xi$ as an element of ${\mathbb C}$ in the natural way. **The global metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$** {#S:group} ---------------------------------------------------------------------------------- In this subsection we consider the global metaplectic group. So we let $F$ be a number field which contains all the $n^\text{th}$ roots of unity and ${\mathbb{A}}$ the ring of adeles. Note that if $n>2$, then $F$ must be totally complex. We shall define the $n$-fold metaplectic cover ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ of ${\operatorname{GL}}_r({\mathbb{A}})$. (Just like the local case, we write ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ even though it is not the adelic points of an algebraic group.) The construction of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ has been done in various places such as [@KP; @FK]. First define the adelic 2-cocycle $\tau_r$ by $$\tau_r(g, g'):=\prod_v\tau_{r,v}({g}_v, g'_v),$$ for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, where $\tau_{r,v}$ is the local cocycle defined in the previous subsection. By definition of $\tau_{r,v}$, we have $\tau_{r,v}(g_v, g'_v)=1$ for almost all $v$, and hence the product is well-defined. We define ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ to be the group whose underlying set is ${\operatorname{GL}}_r({\mathbb{A}})\times\mu_n$ and the group structure is defined via $\tau_r$ as in the local case, [[*i.e.* ]{}]{}$$(g, \xi)\cdot(g', \xi')=(gg', \tau_r(g, g')\xi\xi'),$$ for $g, g'\in{\operatorname{GL}}_r({\mathbb{A}})$, and $\xi, \xi'\in\mu_n$. Just as the local case, we have $$\xymatrix{ 0\ar[r]&\{\pm1\}\ar[r]&{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\ar[r]^{p}&{\operatorname{GL}}_r({\mathbb{A}})\ar[r]&0, }$$ where we call $p$ the canonical projection. Define a set theoretic section $\kappa:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ by $g\mapsto(g,1)$. It is well-known that ${\operatorname{GL}}_r(F)$ splits in ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. However the splitting is not via $\kappa$. In what follows, we will see that the splitting is via the product of all the local ${\mathbf{s}}_r$. Let us start with the following “product formula” of $\sigma_r$. \[P:product\_formula\] For $g, g'\in{\operatorname{GL}}_r(F)$, we have $\sigma_{r,v}(g, g')=1$ for almost all $v$, and further $$\prod_v\sigma_{r,v}(g, g')=1.$$ From the explicit description of the cocycle $\sigma_{r,v}(g, g')$ given at the end of $\S 4$ of [@BLS], one can see that $\sigma_{r,v}(g, g')$ is written as a product of Hilbert symbols of the form $(t, t')_{F_v}$ for $t, t'\in F^\times$. This proves the first part of the proposition. The second part follows from the product formula for the Hilbert symbol. If $g\in{\operatorname{GL}}_r(F)$, then we have $s_{r,v}(g)=1$ for almost all $v$, where $s_{r,v}$ is the map $s_{r,v}:{\operatorname{GL}}(F_v)\rightarrow\mu_n$ defining the local section ${\mathbf{s}}_r:{\operatorname{GL}}(F_v)\rightarrow{\widetilde{\operatorname{GL}}}_r(F_v)$. By the Bruhat decomposition we have $g=bwb'$ for some $b, b'\in B(F)$ and $w\in W$. Then for each place $v$ $$\begin{aligned} s_{r,v}(g) &=s_{r,v}(bwb')\\ &=\sigma_{r,v}(b, wb')s_{r,v}(b)s_{r,v}(wb')/\tau_{r,v}(b,wb')\quad\text{by (\ref{E:tau_sigma})}\\ &=\sigma_{r,v}(b, wb')s_{r,v}(b) \sigma_{r,v}(w, b')s_{r,v}(w)s_{r,v}(b')/\tau_{r,v}(w,b')\tau_{r,v}(b,wb') \quad\text{again by (\ref{E:tau_sigma})}.\end{aligned}$$ By the previous proposition, $\sigma_{r,v}(b, wb')=\sigma_{r,v}(w, b')=1$ for almost all $v$. By (\[E:kappa\_and\_s\]) we know $s_{r,v}(b)=s_{r,v}(w)=s_{r,v}(b')=1$ for almost all $v$. Finally by definition of $\tau_{r,v}$, $\tau_{r,v}(w,b')=\tau_{r,v}(b,wb')=1$ for almost all $v$. This proposition implies that the expression $$s_r(g):=\prod_vs_{r,v}(g)$$ makes sense for all $g\in{\operatorname{GL}}_r(F)$, and one can define the map $${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),\quad g\mapsto (g, s_r(g)^{-1}).$$ Moreover, this is a homomorphism because of Proposition \[P:product\_formula\] and (\[E:convenient\]). Unfortunately, however, the expression $\prod_vs_{r,v}(g_v)$ does not make sense for every $g\in{\operatorname{GL}}_r({\mathbb{A}})$ because one does not know whether $s_{r,v}(g_v)=1$ for almsot all $v$. Yet, we have \[P:s\_split\] The expression $s_r(g)=\prod_vs_{r,v}(g_v)$ makes sense when $g$ is in ${\operatorname{GL}}_r(F)$ or $N_B({\mathbb{A}})$, so ${\mathbf{s}}_r$ is defined on ${\operatorname{GL}}_r(F)$ and $N_B({\mathbb{A}})$. Moreover, ${\mathbf{s}}_r$ is indeed a homomorphism on ${\operatorname{GL}}_r(F)$ and $N_B({\mathbb{A}})$. Also if $g\in{\operatorname{GL}}_r(F)$ and $n\in N_B({\mathbb{A}})$, both $s_r(gn)$ and $s_r(ng)$ make sense and further we have ${\mathbf{s}}_r(gn)={\mathbf{s}}_r(g){\mathbf{s}}_r(n)$ and ${\mathbf{s}}_r(ng)={\mathbf{s}}_r(n){\mathbf{s}}_r(g)$. We already know $s_r(g)$ is defined and ${\mathbf{s}}_r$ is a homomorphism on ${\operatorname{GL}}_r(F)$. Also $s_r(n)$ is defined thanks to (\[E:kappa\_and\_s\]) and ${\mathbf{s}}_r$ is a homomorphism on $N_B({\mathbb{A}})$ thanks to Proposition \[P:BLS\] (1). Moreover for all places $v$, we have $\sigma_{r,v}(g_v, n_v)=1$ again by Proposition \[P:BLS\] (1). Hence for all $v$, $s_{r,v}(gn_v)=s_{r,v}(g)s_{r,v}(n_v)/\tau_{r,v}(g,n_v)$. For almost all $v$, the right hand side is $1$. Hence the global $s_r(gn)$ is defined. Also this equality shows that ${\mathbf{s}}_r(gn)={\mathbf{s}}_r(g){\mathbf{s}}_r(n)$. The same argument works for $ng$. If $H\subseteq{\operatorname{GL}}_r({\mathbb{A}})$ is a subgroup on which ${\mathbf{s}}_r$ is not only defined but also a group homomorphism, we write $H^\ast:={\mathbf{s}}_r(H)$. In particular we have $$\label{E:star} {\operatorname{GL}}_r(F)^\ast:={\mathbf{s}}_r({\operatorname{GL}}_r(F))\quad\text{and}\quad N_B({\mathbb{A}})^\ast:={\mathbf{s}}_r(N_B({\mathbb{A}})).$$ We define the groups like ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})$, ${\widetilde{M}}({\mathbb{A}})$, ${\widetilde{M}^{(n)}}({\mathbb{A}})$, etc completely analogously to the local case. Let us mention \[L:closed\_subgroup\_global\] The groups ${\mathbb{A}}^{\times n}, M^{(n)}({\mathbb{A}})$ and ${\widetilde{M}^{(n)}}({\mathbb{A}})$ are closed subgroups of ${\mathbb{A}}^\times, M({\mathbb{A}})$ and ${\widetilde{M}}({\mathbb{A}})$, respectively. That ${\mathbb{A}}^{\times n}$ and $M^{(n)}({\mathbb{A}})$ are closed follows from the following lemma together with Lemma \[L:closed\_subgroup\_local\]. Once one knows $M^{(n)}({\mathbb{A}})$ is closed, one will know ${\widetilde{M}^{(n)}}({\mathbb{A}})$ is closed because it is the preimage of the closed $M^{(n)}({\mathbb{A}})$ under the canonical projection, which is continuous. \[L:closed\_subgroup\_local\_global\] Let $G$ be an algebraic group over $F$ and $G({\mathbb{A}})$ its adelic points. Let $H\subseteq G({\mathbb{A}})$ be a subgroup such that $H$ is written as $H=\prod'_vH_v$ (algebraically) where for each place $v$, $H_v:=H\cap G(F_v)$ is a closed subgroup of $G(F_v)$. Then $H$ is closed. Let $(x_i)_{i\in I}$ be a net in $H$ that converges in $G({\mathbb{A}})$, where $I$ is some index set. Let $g=\lim_{i\in I} x_i$. Assume $g\notin H$. Then there exists a place $w$ such that $g_w\notin H_w$. Since $H_w$ is closed, the set $U_w:=G(F_w)\backslash H_w$ is open. Then there exists an open neighborhood $U$ of $g$ of the form $U=\prod_v U_v$, where $U_v$ is some open neighborhood of $g_v$ and at $v=w$, $U_v=U_w$. But for any $i\in I$, $x_i\notin U$ because $x_{i, w}\notin U_w$, which contradicts the assumption that $g=\lim_{i\in I} x_i$. Hence $g\in H$, which shows $H$ is closed. Just like the local case, the preimage $\widetilde{Z_{{\operatorname{GL}}_r}}({\mathbb{A}})$ of the center $Z_{{\operatorname{GL}}_r}({\mathbb{A}})$ of ${\operatorname{GL}}_r({\mathbb{A}})$ is in general not the center of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ but the center, which we denote by $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$, is $$\begin{aligned} Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})&=\{(aI_r, \xi):a^{r-1+2rc}\in {\mathbb{A}}^{\times n}, \xi\in\mu_n\}\\ &=\{(aI_r, \xi):a\in {\mathbb{A}}^{\times \frac{n}{d}}, \xi\in\mu_n\},\end{aligned}$$ where $d=\gcd(r-1+2c, n)$. The center is a closed subgroup of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. We can also describe ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ as a quotient of a restricted direct product of the groups ${\widetilde{\operatorname{GL}}}_r(F_v)$ as follows. Consider the restricted direct product $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ with respect to the groups $\kappa(K_v)=\kappa({\operatorname{GL}}_r(\mathcal{O}_{F_v}))$ for all $v$ with $v\nmid n$ and $v\nmid\infty$. If we denote each element in this restricted direct product by $\Pi'_v(g_v,\xi_v)$ so that $g_v\in K_v$ and $\xi_v=1$ for almost all $v$, we have the surjection $$\label{E:surjection} \rho:{\prod_v}'{\widetilde{\operatorname{GL}}}_r(F_v)\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),\quad \Pi'_v(g_v,\xi_v)\mapsto (\Pi'_vg_v, \Pi_v\xi_v),$$ where the product $\Pi_v\xi_v$ is literary the product inside $\mu_n$. This is a group homomorphism because $\tau_r=\prod_v\tau_{r,v}$ and the groups ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ and ${\widetilde{\operatorname{GL}}}_r(F_v)$ are defined, respectively, by $\tau_r$ and $\tau_{r,v}$. We have $${\prod_v}'{\widetilde{\operatorname{GL}}}_r(F_v)/\ker\rho\cong {\widetilde{\operatorname{GL}}}_r({\mathbb{A}}),$$ where $\ker\rho$ consists of the elements of the form $(1,\xi)$ with $\xi\in\prod'_v\mu_n$ and $\Pi_v\xi_v=1$. Let $\pi$ be a representation of $\widetilde{H}\subseteq {\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ where ${\widetilde{H}}$ is the metaplectic preimage of a subgroup $H\subseteq{\operatorname{GL}}_r({\mathbb{A}})$. Just like the local case, we call $\pi$ genuine if $(1,\xi)\in\widetilde{H}({\mathbb{A}})$ acts as multiplication by $\xi$ for all $\xi\in\mu_n$. Also we have the notion of automorphic representation as well as automorphic form on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ or ${\widetilde{M}}({\mathbb{A}})$. In this paper, by an automorphic form, we mean a smooth automorphic form instead of a $K$-finite one, namely an automorphic form is $K_f$-finite, ${\mathcal{Z}}$-finite and of uniformly moderate growth. (See [@Cogdell p.17].) Hence if $\pi$ is an automorphic representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$), the full group ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$) acts on $\pi$. An automorphic form $f$ on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$) is said to be genuine if $f(g,\xi)=\xi f(g,1)$ for all $(g,\xi)\in{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ (or ${\widetilde{M}}({\mathbb{A}})$). In particular every automorphic form in the space of a genuine automorphic representation is genuine. Suppose we are given a collection of irreducible admissible representations $\pi_v$ of ${\widetilde{\operatorname{GL}}}_r(F_v)$ such that $\pi_v$ is $\kappa(K_v)$-spherical for almost all $v$. Then we can form an irreducible admissible representation of $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ by taking a restricted tensor product $\otimes_v'\pi_v$ as usual. Suppose further that $\ker\rho$ acts trivially on $\otimes_v'\pi_v$, which is always the case if each $\pi_v$ is genuine. Then it descends to an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, which we denote by ${\widetilde{\otimes}}'_v\pi_v$, and call it the “metaplectic restricted tensor product”. Let us emphasize that the space for ${\widetilde{\otimes}}'_v\pi_v$ is the same as that for $\otimes_v'\pi_v$. Conversely, if $\pi$ is an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, it is written as ${\widetilde{\otimes}}'_v\pi_v$ where $\pi_v$ is an irreducible admissible representation of ${\widetilde{\operatorname{GL}}}_r(F_v)$, and for almost all $v$, $\pi_v$ is $\kappa(K_v)$-spherical. (To see it, view $\pi$ as a representation of the restricted product $\prod_v'{\widetilde{\operatorname{GL}}}_r(F_v)$ by pulling it back by $\rho$ as in (\[E:surjection\]) and apply the usual tensor product theorem for the restricted direct product. This gives the restricted tensor product $\otimes_v'\pi_v$, where each $\pi_v$ is genuine, and hence it descends to ${\widetilde{\otimes}}_v'\pi_v$.) Finally in this section, let us mention that we define $$\label{L:Hasse} {\operatorname{GL}}_r^{(n)}(F):={\operatorname{GL}}_r(F)\cap{\operatorname{GL}}_r^{(n)}({\mathbb{A}}),$$ namely ${\operatorname{GL}}_r^{(n)}(F)=\{g\in{\operatorname{GL}}_r(F):\det g\in {\mathbb{A}}^{\times n}\}$. The author does not know if this is equal to $\{g\in{\operatorname{GL}}_r(F):\det g\in F^{\times n}\}$ unless $n=2$, in which case the Hasse-Minkowski theorem implies those two coincide. Similarly we define $$M^{(n)}(F)=M(F)\cap M^{(n)}({\mathbb{A}}).$$ **The metaplectic cover ${\widetilde{M}}$ of the Levi $M$** {#S:Levi} =========================================================== Both locally and globally, one cannot show the cocycle $\tau_r$ has the block-compatibility as in (\[E:compatibility\]) (except when $r=2$). Yet, in order to define the metaplectic tensor product, it seems to be necessary to have the block-compatibility of the cocycle. To get round it, we will introduce another cocycle $\tau_M$, but this time it is a cocycle only on the Levi $M$, and will show that $\tau_M$ is cohomologous to the restriction $\tau_r|_{M\times M}$ of $\tau_r$ to $M\times M$ both for the local and global cases. **The cocycle $\tau_M$** ------------------------ In this subsection, we assume that all the groups are over $F$ if $F$ is local and over ${\mathbb{A}}$ if $F$ is global, and suppress it from our notation. We define the cocycle $$\tau_M: M\times M\rightarrow\mu_n,$$ by $$\tau_M(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix}) =\prod_{i=1}^k\tau_{r_i}(g_i,g_i')\prod_{1\leq i<j\leq k}(\det(g_i), \det(g_j')) \prod_{i\neq j}(\det(g_i), \det(g_j'))^c,$$ where $(-,-)$ is the local or global Hilbert symbol. Note that the definition makes sense both locally and globally. Moreover the global $\tau_M$ is the product of the local ones. We define the group ${{^c\widetilde{M}}}$ to be $${{^c\widetilde{M}}}=M\times\mu_n$$ as a set and the group structure is given by $\tau_M$. The superscript $^c$ is for “compatible”. One advantage to work with ${{^c\widetilde{M}}}$ is that each ${\widetilde{\operatorname{GL}}}_{r_i}$ embeds into ${{^c\widetilde{M}}}$ via the natural map $$(g_i,\xi)\mapsto(\begin{pmatrix}I_{r_1+\cdots+r_{i-1}}&&\\ &g_i&\\ &&I_{r_{i+1}+\cdots+r_k}\end{pmatrix}, \xi).$$ Indeed, the cocycle $\tau_M$ is so chosen that we have this embedding. Also recall our notation $$M^{(n)}={\operatorname{GL}}_{r_1}^{(n)}\times\cdots\times{\operatorname{GL}}_{r_k}^{(n)},$$ and $${\widetilde{M}^{(n)}}={\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}.$$ We define ${{^c\widetilde{M}^{(n)}}}$ analogously to ${{^c\widetilde{M}}}$, namely the group structure of ${{^c\widetilde{M}^{(n)}}}$ is defined via the cocycle $\tau_M$. Of course, ${{^c\widetilde{M}^{(n)}}}$ is a subgroup of ${{^c\widetilde{M}}}$. Note that each ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ naturally embeds into ${{^c\widetilde{M}^{(n)}}}$ as above. The subgroups ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ and ${\widetilde{\operatorname{GL}}^{(n)}}_{r_j}$ in ${{^c\widetilde{M}^{(n)}}}$ commute pointwise for $i\neq j$. Locally or globally, it suffices to show $\tau_M(g_i,g_j)=\tau_M(g_j,g_i)$ for $g_i\in{\operatorname{GL}}^{(n)}_{r_i}$ and $g_j\in{\operatorname{GL}}^{(n)}_{r_j}$. But the block-compatibility of the 2-cocycle $\tau_M$, we have $\tau_M(g_i,g_j)=\tau_{r_i}(g_i, I_{r_j})\tau_{r_j}(I_{r_j}, g_j)=1$, and similarly have $\tau_M(g_j,g_i)=1$. There is a surjection $${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}\rightarrow\; {{^c\widetilde{M}^{(n)}}}$$ given by the map $$((g_1,\xi_1),\dots,(g_k,\xi_k))\mapsto (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \xi_1\cdots\xi_k),$$ whose kernel is $$\mathcal{K}_P:=\{((1,\xi_1),\dots,(1,\xi_k)):\xi_1\cdots\xi_k=1\},$$ so that ${{^c\widetilde{M}^{(n)}}}\cong {\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}/\mathcal{K}_P$. The block-compatibility of $\tau_M$ guarantees that the map is indeed a group homomorphism. The description of the kernel is immediate. **The relation between $\tau_M$ and $\tau_r$** ---------------------------------------------- Note that for the group ${\widetilde{M}}$ (instead of ${{^c\widetilde{M}}}$), the group structure is defined by the restriction of $\tau_r$ to $M\times M$, and hence each ${\widetilde{\operatorname{GL}}}_{r_i}$ might not embed into ${\widetilde{\operatorname{GL}}}_r$ in the natural way because of the possible failure of the block-compatibility of $\tau_r$ unless $r=2$. To make explicit the relation between ${{^c\widetilde{M}}}$ and ${\widetilde{M}}$, the discrepancy between $\tau_M$ and $\tau_r|_{M\times M}$ (which we denote simply by $\tau_r$) has to be clarified. [**Local case:**]{} Assume $F$ is local. Then we have $$\begin{aligned} &\tau_M(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix})\\ =&\sigma_r(\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}, \begin{pmatrix}g'_1&&\\ &\ddots&\\ &&g'_k\end{pmatrix}) \prod_{i=1}^k\frac{s_{r_i}(g_i)s_{r_i}(g_i')}{s_{r_i}(g_ig_i')},\end{aligned}$$ so $\tau_M$ and $\sigma_r|_{M\times M}$ are cohomologous via the function $\prod_{i=1}^ks_{r_i}$. Here recall from Section \[S:group\] that the map $s_{r_i}:{\operatorname{GL}}_{r_i}\rightarrow\mu_n$ relates $\tau_{r_i}$ with $\sigma_{r_i}$ by $$\sigma_{r_i}(g_i,g_i')=\tau_{r_i}(g_i,g_i')\cdot\frac{s_{r_i}(g_i,g_i')}{s_{r_i}(g_i)s_{r_i}(g_i')},$$ for $g_i,g_i'\in{\operatorname{GL}}_{r_i}$. Moreover if $|n|_F=1$, $s_{r_i}$ is chosen to be “canonical” in the sense that (\[E:canonical\_section\]) is satisfied. The block-compatibility of $\sigma_r$ implies $$\tau_r(m, m')\cdot\frac{s_r(mm')}{s_r(m)s_r(m')} =\sigma_r(m,m') =\tau_M(m,m')\cdot\prod_{i=1}^k\frac{s_{r_i}(g_ig_i')}{s_{r_i}(g_i)s_{r_i}(g_i')},$$ for $m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}$ and $m'=\begin{pmatrix}g_1'&&\\ &\ddots&\\ &&g_k'\end{pmatrix}$. Hence if we define $\hat{s}_M:M\rightarrow\mu_n$ by $$\label{E:s_hat_M} \hat{s}_M(m)=\frac{\prod_{i=1}^ks_{r_i}(g_i)}{s_r(m)},$$ we have $$\label{E:s_hat} \tau_M(m, m')=\tau_r(m, m')\cdot \frac{\hat{s}_M(m)\hat{s}_M(m')}{\hat{s}_M(mm')},$$ namely $\tau_r$ and $\tau_M$ are cohomologous via $\hat{s}_M$. Therefore we have the isomorphism $$\alpha_M:{{^c\widetilde{M}}}\rightarrow{\widetilde{M}},\quad (m,\xi)\mapsto (m, \hat{s}_M(m)\xi).$$ The following lemma will be crucial later for showing that the global $\tau_M$ is also cohomologous to $\tau_r|_{M({\mathbb{A}})\times M({\mathbb{A}})}$. \[L:s\_hat\_M\] Assume $F$ is such that $|n|_F=1$. Then for all $k\in M({\mathcal{O}_F})$, we have $\hat{s}_M(k)=1$. First note that if $k,k'\in M({\mathcal{O}_F})$, then $\tau_r(k,k')=\tau_M(k, k')=1$ and so by (\[E:s\_hat\]) we have $$\hat{s}_M(kk')=\hat{s}_M(k)\hat{s}_M(k'),$$ [[*i.e.* ]{}]{}$\hat{s}_M$ is a homomorphism on $M_M({\mathcal{O}_F})$. Hence it suffices to prove the lemma only for the elements $k\in M({\mathcal{O}_F})$ of the form $$k=\begin{pmatrix} I_{r_1+\cdots+r_{i-1}}&&\\ &k_i&\\ && I_{r_{i+1}+\cdots+r_k}\end{pmatrix}$$ where $k_i\in{\operatorname{GL}}_{r_i}$ is in the $i^{\text{th}}$ place on the diagonal. Namely we need to prove $$\frac{s_{r_i}(k_i)}{s_r(k)}=1.$$ In what follows, we will show that this follows from the “canonicality” of $s_r$ and $s_{r_i}$, and the fact that the cocycle for ${\operatorname{SL}}_{r+1}$ is block-compatible in a very strong sense as in [@BLS Lemma 5, Theorem 7 §2, p.145]. Recall from (\[E:canonical\_section\]) that $s_r$ has been chosen to satisfy $s_r|_{{\operatorname{GL}}_r({\mathcal{O}_F})}={{\mathfrak{s}}_r}|_{l({\operatorname{GL}}_r({\mathcal{O}_F}))}$, where ${\mathfrak{s}}_r$ is the map on ${\operatorname{SL}}_{r+1}(F)$ that makes the diagram (\[E:canonical\_diagram\]) commute. Similarly for $s_{r_i}$ with $r$ replaced by $r_i$. Let us write $$l_i:{\operatorname{GL}}_{r_i}(F)\rightarrow {\operatorname{SL}}_{r_i+1}(F),\quad g_i\mapsto\begin{pmatrix}g_i&\\ &\det(g_i)^{-1}\end{pmatrix}$$ for the embedding that is used to define the cocycle $\sigma_{r_i}$. Define the embedding $$F:{\operatorname{SL}}_{r_i+1}(F)\rightarrow{\operatorname{SL}}_{r+1}(F),\quad \begin{pmatrix} A&b\\ c&d\end{pmatrix}\mapsto \begin{pmatrix} I_{r_1+\cdots+r_{i-1}}&&&\\ & A&&b\\ &&I_{r_{i+1}+\cdots+r_{k}}&\\ &c&&d\end{pmatrix},$$ where $A$ is a $r_i\times r_i$-block and accordingly $b$ is $r_i\times 1$, $c$ is $1\times r_i$ and d is $1\times 1$. Note that this embedding is chosen so that we have $$\label{E:F_and_l} F(l_i(k_i))=l(k).$$ By the block compatibility of $\sigma_{{\operatorname{SL}}_{r+1}}$ we have $$\sigma_{{\operatorname{SL}}_{r+1}}|_{F({\operatorname{SL}}_{r_i+1})\times F({\operatorname{SL}}_{r_i+1})}=\sigma_{{\operatorname{SL}}_{r_i+1}}.$$ This is nothing but [@BLS Lemma 5, §2]. (The reader has to be careful in that the image $F({\operatorname{SL}}_{r_i+1})$ is not a standard subgroup in the sense defined in [@BLS p.143] if one chooses the set $\Delta$ of simple roots of ${\operatorname{SL}}_{r+1}$ in the usual way. One can, however, choose $\Delta$ differently so that $F({\operatorname{SL}}_{r_i+1})$ is indeed a standard subgroup. And all the results of [@BLS §2] are totally independent of the choice of $\Delta$.) This implies the map $(g_i, \xi)\mapsto (F(g_i),\xi)$ for $(g_i,\xi)\in{\widetilde{\operatorname{SL}}}_{r_i+1}$ is a homomorphism. Hence the canonical section ${\operatorname{SL}}_{r+1}({\mathcal{O}_F})\rightarrow {\widetilde{\operatorname{SL}}}_{r+1}(F)$, which is given by $g\mapsto (g, {\mathfrak{s}}_r(g))$, restricts to the canonical section ${\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})\rightarrow {\widetilde{\operatorname{SL}}}_{r_i+1}(F)$, which is given by $g_i\mapsto (g_i,{\mathfrak{s}}_{r_i}(g_i))$. Namely we have the commutative diagram $$\xymatrix{ {\widetilde{\operatorname{SL}}}_{r_i+1}({\mathcal{O}_F})\ar[rrr]^{(g,\;\xi)\rightarrow( F(g),\; \xi)}&&&{\widetilde{\operatorname{SL}}}_{r+1}({\mathcal{O}_F})\\ {\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})\ar[rrr]^F\ar[u]^{g_i\mapsto(g_i,\; {\mathfrak{s}}_{r_i}(g_i))}&&&{\operatorname{SL}}_{r+1}({\mathcal{O}_F})\ar[u]_{g\mapsto(g,\;{\mathfrak{s}}_r(g))}, }$$ where all the maps are homomorphisms. In particular, we have $$\label{E:sss} {\mathfrak{s}}_r(F(g_i))={\mathfrak{s}}_{r_i}(g_i),$$ for all $g_i\in{\operatorname{SL}}_{r_i+1}({\mathcal{O}_F})$. Thus $$\begin{aligned} s_r(k)&={\mathfrak{s}}_r(l(k))\quad\text{by (\ref{E:canonical_section})}\\ &={\mathfrak{s}}_r(F(l_i(k_i)))\quad\text{by (\ref{E:F_and_l})}\\ &={\mathfrak{s}}_{r_i}(l_i(k_i))\quad\text{by (\ref{E:sss})}\\ &=s_{r_i}(k_i)\quad\text{by (\ref{E:canonical_section}) with $r$ replaced by $r_i$}.\end{aligned}$$ The lemma has been proven. [**Global case:**]{} Assume $F$ is a number field. We define $\hat{s}_M: M({\mathbb{A}})\rightarrow\mu_n$ by $$\hat{s}_M(\prod_vm_v):=\prod_v{\hat{s}_{M_v}}(m_v)$$ for $\prod_vm_v\in M({\mathbb{A}})$. The product is finite thanks to Lemma \[L:s\_hat\_M\]. Since both of the cocycles $\tau_r$ and $\tau_M$ are the products of the corresponding local ones, one can see that the relation (\[E:s\_hat\]) holds globally as well. Thus analogously to the local case, we have the isomorphism $$\alpha_M:\; {{^c\widetilde{M}}}({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}}),\quad (m,\xi)\mapsto (m, \hat{s}_M(m)\xi).$$ \[L:splitting\_cMPt\] The splitting of $M(F)$ into ${{^c\widetilde{M}}}({\mathbb{A}})$ is given by $${\mathbf{s}}_M:M(F)\rightarrow\;{{^c\widetilde{M}}}({\mathbb{A}}),\quad \begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}\mapsto (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\; \prod_{i=1}^k s_i(g_i)^{-1}).$$ For each $i$ the splitting ${\mathbf{s}}_{r_i}:{\operatorname{GL}}_{r_i}(F)\rightarrow{\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ is given by $g_i\mapsto(g_i,\;s_{r_i}(g_i)^{-1})$, where ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ is defined via the cocycle $\tau_{r_i}$. The lemma follows by the block-compatibility of $\tau_M$ and the product formula for the Hilbert symbol. Just like the case of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, the section ${\mathbf{s}}_M$ as in this lemma cannot be defined on all of $M({\mathbb{A}})$ even set theoretically because the expression $\prod_is_{r_i}(g_i)$ does not make sense to all ${\operatorname{diag}}(g_1,\dots,g_k)\in M({\mathbb{A}})$. So we only have a partial set theoretic section $${\mathbf{s}}_M:M({\mathbb{A}})\rightarrow{{^c\widetilde{M}}}({\mathbb{A}}).$$ But analogously to Proposition \[P:s\_split\], we have \[P:s\_split\_M\] The partial section ${\mathbf{s}}_M$ is defined on both $M(F)$ and $N_M({\mathbb{A}})$, where $N_M({\mathbb{A}})$ is the unipotent radical of the Borel subgroup of $M$, and moreover it gives rise to a group homomorphism on each of these subgroups. Also for $m\in M(F)$ and $n\in N_M({\mathbb{A}})$, both ${\mathbf{s}}_M(mn)$ and ${\mathbf{s}}_M(nm)$ are defined and further ${\mathbf{s}}_M(mn)={\mathbf{s}}_M(m){\mathbf{s}}_M(n)$ and ${\mathbf{s}}_M(nm)={\mathbf{s}}_M(n){\mathbf{s}}_M(m)$. This follows from Proposition \[P:s\_split\] applied to each ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ together with the block-compatibility of the cocycle $\tau_M$. (Note that one also needs to use the fact that for all $g, g'$ in the subgroup generated by $M(F)$ and $N_M({\mathbb{A}})$, we have $(\det(g), \det(g'))_{\mathbb{A}}=1$.) This splitting is related to the splitting ${\mathbf{s}}_r:{\operatorname{GL}}_r(F)\rightarrow{\operatorname{GL}}_r({\mathbb{A}})$ by \[P:diagram\] We have the following commutative diagram: $$\xymatrix{{{^c\widetilde{M}}}({\mathbb{A}})\; \ar@{^{(}->}[r]^{\alpha_M}&{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\\ M(F)\;\ar@{^{(}->}[r]\; \ar[u]^{{\mathbf{s}}_M}&{\operatorname{GL}}_r(F)\ar[u]_{{\mathbf{s}}_r}. }$$ For $m=\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix} \in M(F)$, we have $$\alpha_M({\mathbf{s}}_M(m))=\alpha_M(m, \prod_{i=1}^ks_{r_i}(g_i)^{-1}) =(m, \hat{s}_M(m)\prod_{i=1}^ks_{r_i}(g_i)^{-1}) =(m, s_r(m)^{-1})={\mathbf{s}}_r(m),$$ where for the elements in $M(F)$, all of $s_{r_i}$ and $s_r$ are defined globally, and the second equality follows from the definition of $\hat{s}_M$ as in (\[E:s\_hat\_M\]). This proposition implies \[C:diagram\] Assume $\pi$ is an automorphic subrepresentation of ${{^c\widetilde{M}}}({\mathbb{A}})$. The representation of ${\widetilde{M}}({\mathbb{A}})$ defined by $\pi\circ\alpha_M^{-1}$ is also automorphic. If $\pi$ is realized in a space $V$ of automorphic forms on ${{^c\widetilde{M}}}({\mathbb{A}})$, then $\pi\circ\alpha_M^{-1}$ is realized in the space of functions of the form $f\circ\alpha_M^{-1}$ for $f\in V$. The automorphy follows from the commutativity of the diagram in the above lemma. The following remark should be kept in mind for the rest of the paper. The results of this subsection essentially show that we may identify ${{^c\widetilde{M}}}$ (locally or globally) with ${\widetilde{M}}$. We may even “pretend” that the cocycle $\tau_r$ has the block-compatibility property. We need to make the distinction between ${{^c\widetilde{M}}}$ and ${\widetilde{M}}$ only when we would like to view the group ${\widetilde{M}}$ as a subgroup of ${\widetilde{\operatorname{GL}}}_r$. For most part of this paper, however, we will not have to view ${\widetilde{M}}$ as a subgroup of ${\widetilde{\operatorname{GL}}}_r$. Hence we suppress the superscript $^c$ from the notation and always denote ${{^c\widetilde{M}}}$ simply by ${\widetilde{M}}$, when there is no danger of confusion. Accordingly, we denote the partial section ${\mathbf{s}}_M$ simply by ${\mathbf{s}}$. **The center $Z_{{\widetilde{M}}}$ of ${\widetilde{M}}$** --------------------------------------------------------- In this subsection $F$ is either local or global, and accordingly we let $R=F$ or ${\mathbb{A}}$ as in the notation section. And all the groups are over $R$. For any group $H$ (metaplectic or not), we denote its center by $Z_H$. In particular for each group $\widetilde{H}\subseteq{\widetilde{\operatorname{GL}}}_r$, we let $$Z_{\widetilde{H}}=\text{center of $\widetilde{H}$}.$$ For the Levi part $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_2}\subseteq{\operatorname{GL}}_r$, we of course have $$Z_{M}=\{\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_{k}I_{r_k}\end{pmatrix}:a_i\in R^\times\}.$$ But for the center $Z_{{\widetilde{M}}}$ of ${\widetilde{M}}$, we have $$Z_{{\widetilde{M}}}\subsetneq \widetilde{Z_{M}},$$ in general, and indeed $\widetilde{Z_{M}}$ might not be even commutative. In what follows, we will describe $Z_{{\widetilde{M}}}$ in detail. For this purpose, let us start with \[L:to\_compute\_center\] Assume $F$ is local. Then for each $g\in{\operatorname{GL}}_r(F)$ and $a\in F^\times$, we have $$\sigma_r(g, aI_r)\sigma_r(aI_r, g)^{-1}=(\det(g), a^{r-1+2cr}).$$ First let us note that if we write $\sigma_r=\sigma_r^{(c)}$ to emphasize the parameter $c$, then $$\sigma_r^{(c)}(g, aI_r)\sigma_r^{(c)}(aI_r, g)^{-1} =\sigma_r^{(0)}(g, aI_r)\sigma_r^{(0)}(aI_r, g)^{-1}(\det(g), a^r)^{2c}$$ because $(a^r,\det(g))^{-1}=(\det(g), a^r)$. Hence it suffices to show the lemma for the case $c=0$. But this can be done by using the recipe provided by [@BLS]. Namely let $g=nt\eta n'$ for $n, n'\in N_B$, $t\in T$ and $\eta\in{\mathfrak{M}}$. Then $$\begin{aligned} \sigma_r(g, aI_r)&=\sigma_r(nt\eta n', aI_r)\\ &=\sigma_r(t\eta, n' aI_r)\quad\text{by Proposition \ref{P:BLS} (1) and (2)}\\ &=\sigma_r(t\eta, aI_r)\quad\text{by $n'aI_r=aI_rn'$ and Proposition \ref{P:BLS} (1)}\\ &=\sigma_r(t, \eta aI_r)\sigma_r(\eta, aI_r)\sigma_r(t, \eta)^{-1}\quad\text{by Proposition \ref{P:BLS} (0)}\\ &=\sigma_r(t, aI_r\eta)\sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (5)}\\ &=\sigma_r(taI_r, \eta)\sigma_r(t, aI_r)\sigma_r(aI_r, \eta)^{-1} \sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (0)}\\ &=\sigma_r(t, aI_r)\sigma_r(\eta, aI_r)\quad\text{by Proposition \ref{P:BLS} (5)}.\end{aligned}$$ Now by Proposition \[P:BLS\] (3), $\sigma(\eta, aI_r)$ is a product of $(-a, a)$’s, which is $1$. Hence by using Proposition \[P:BLS\] (4), we have $$\sigma_r(g, aI_r)=\sigma_r(t, aI_r) =\prod_{i=1}^{r}(t_i, a)^{r-i}.$$ By an analogous computation, one can see $$\sigma_r(aI_r, g)=\sigma_r(aI_r, t)=\prod_{i=1}^{r}(a, t_i)^{i-1}.$$ Using $(a, t_i)^{-1}=(t_i, a)$, one can see $$\sigma_r(g, aI_r) \sigma_r(aI_r, g)^{-1}=\prod_{i=1}^r(t_i, a)^{r-1}.$$ But this is equal to $(\det(g), a^{r-1})$ because $\det(g)=\prod_{i=1}^rt_i$. Note that this lemma immediately implies that the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ of ${\widetilde{\operatorname{GL}}}_r$ is indeed as in (\[E:center\_GLt\]), though a different proof is provided in [@KP]. Also with this lemma, we can prove Both locally and globally, the center $Z_{{\widetilde{M}}}$ is described as $$Z_{{\widetilde{M}}}=\{\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_kI_{r_k}\end{pmatrix}: a_i^{r-1+2cr}\in R^{\times n}\;\text{ and }\;a_1\equiv\cdots\equiv a_r\mod{R^{\times n}}\}.$$ First assume $F$ is local. Let $m={\operatorname{diag}}(g_1,\dots,g_k)\in M$ and $a={\operatorname{diag}}(a_1I_{r_1},\dots,a_kI_{r_k})$. It suffices to show $\sigma_r(m,a)\sigma_r(a, m)^{-1}=1$ if and only if all $a_i$ are as in the proposition. But $$\begin{aligned} &\sigma_r(m,a) \sigma_r(a, m)^{-1}\\ =&\prod_{i=1}^r\sigma_{r_i}(g_i, a_iI_{r_i})\sigma_{r_i}(a_iI_{r_i}, g_i)^{-1}\prod_{1\leq i<j\leq r}(\det(g_i), a_j^{r_j})\prod_{i\neq j}(\det(g_i), a_j^{r_j})^c\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad \cdot\prod_{1\leq i<j\leq r}(a_i^{r_i},\det(g_j))^{-1}\prod_{i\neq j}(a_i^{r_i},\det(g_j))^{-c}\\ =&\prod_{i=1}^r\sigma_{r_i}(g_i, a_iI_{r_i})\sigma_{r_i}(a_iI_{r_i}, g_i)^{-1}\prod_{i\neq j}(\det(g_i), a_j^{r_j})^{1+2c}\\ =&\prod_{i=1}^r (\det(g_i), a_i^{r_i-1+2cr_i})\prod_{i\neq j}(\det(g_i), a_j^{r_j+2cr_j})\\ =&\prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}),\end{aligned}$$ where for the third equality we used the above lemma with $r$ replaced by $r_i$. Now assume $a$ is such that $(a,1)\in Z_{{\widetilde{M}}}$. Then the above product must be 1 for any $m$. In particular, choose $m$ so that $g_j=1$ for all $i\neq j$. Then we must have $(\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j})=1$ for all $g_i\in {\operatorname{GL}}_{r_i}$. This implies $$a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}\in F^{\times n}$$ for all $i$. Since this holds for all $i$, one can see $a_i^{-1}a_j\in F^{\times n}$ for all $i\neq j$, which implies $\;a_1\equiv\cdots\equiv a_r\mod{F^{\times n}}$. But if $\;a_1\equiv\cdots\equiv a_r\mod{F^{\times n}}$, then $$\begin{aligned} \prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j}) =&\prod_{i=1}^r(\det(g_i), a_i^{-1}\prod_{j=1}^ra_i^{r_j+2cr_j})\\ =&\prod_{i=1}^r(\det(g_i), a_i^{r-1+2cr}).\end{aligned}$$ This must be equal to 1 for any choice of $g_i$, which gives $a_i^{r-1+2cr}\in F^{\times n}$. Conversely if $a$ is of the form as in the proposition, one can see that $\sigma_r(m,a) \sigma_r(a, m)^{-1}=\prod_{i=1}^r (\det(g_i),\, a_i^{-1}\prod_{j=1}^ra_j^{r_j+2cr_j})=1$ for any $m$. The global case follows from the local one because locally by using (\[E:tau\_sigma\]) and $am=ma$, one can see $\sigma_r(m,a) \sigma_r(a, m)^{-1}=1$ if and only if $\tau_r(m,a) \tau_r(a, m)^{-1}=1$, and the global $\tau_r$ is the product of local ones. Lemma \[L:to\_compute\_center\] also implies \[L:center\_GLtt\] Both locally and globally, $\widetilde{Z_{{\operatorname{GL}}_r}}$ commutes with ${\widetilde{\operatorname{GL}}^{(n)}}_r$ pointwise. The local case is an immediate corollary of Lemma \[L:to\_compute\_center\] because if $g\in{\operatorname{GL}}_r^{(n)}$ the lemma implies $\sigma_r(g, aI_r)=\sigma_r(aI_r, g)$. Hence by (\[E:tau\_sigma\]), locally $\tau_r(g, aI_r)=\tau_r(aI_r, g)$ for all $g\in{\operatorname{GL}}_r^{(n)}$ and $a\in F^\times$. Since the global $\tau_r$ is the product of the local ones, the global case also follows. Let us mention that in particular, if $n=2$ and $r=\text{even}$, then $\widetilde{Z_{{\operatorname{GL}}_r}}\subseteq{\widetilde{\operatorname{GL}}^{(n)}}_r$ and $\widetilde{Z_{{\operatorname{GL}}_r}}$ is the center of ${\widetilde{\operatorname{GL}}^{(n)}}_r$. This fact is used crucially in [@Takeda1]. It should be mentioned that this description of the center $Z_{{\widetilde{M}}}$ easily implies $$\label{E:center=center} Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}=Z_{{\widetilde{M}}}{\widetilde{M}^{(n)}}.$$ Also we have \[P:Z\_M\_commute\_Mtn\] Both locally and globally, the groups $\widetilde{Z_M}$ and ${\widetilde{M}^{(n)}}$ commute pointwise, which gives $$\label{E:center_Mtn} Z_{{\widetilde{M}^{(n)}}}=\widetilde{Z_{M}}\cap {{\widetilde{M}^{(n)}}},$$ and hence $$\label{E:center_Mtn2} Z_{{\widetilde{\operatorname{GL}}}_r}Z_{{\widetilde{M}^{(n)}}}=Z_{{\widetilde{\operatorname{GL}}}_r}(\widetilde{Z_{M}}\cap {{\widetilde{M}^{(n)}}})=\widetilde{Z_{M}}\cap( Z_{{\widetilde{\operatorname{GL}}}_r} {\widetilde{M}^{(n)}}).$$ By the block compatibility of the cocycle $\tau_M$, one can see that an element of the form $(\begin{pmatrix}a_1I_{r_1}&&\\ &\ddots&\\ &&a_{k}I_{r_k}\end{pmatrix}, \xi)$ commutes with all the elements in ${\widetilde{M}^{(n)}}$ if and only if each $(a_iI_{r_i},\xi)$ commutes with all the elements in ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. But this is always the case by the above lemme (with $r$ replaced by $r_i$). This proves the proposition. If $F$ is global, we define $$Z_{{\widetilde{M}}}(F)=Z_{{\widetilde{M}}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)),$$ where recall that ${\mathbf{s}}:M(F)\rightarrow{\widetilde{M}}({\mathbb{A}})$ is the section that splits $M(F)$. Similarly we define groups like $Z_{{\widetilde{\operatorname{GL}}}_r}(F), {\widetilde{M}^{(n)}}(F)$, etc. Namely in general for any subgroup ${\widetilde{H}}\subseteq {\widetilde{M}}({\mathbb{A}})$, we define the “$F$-rational points” ${\widetilde{H}}(F)$ of ${\widetilde{H}}$ by $$\label{E:H(F)} {\widetilde{H}}(F):={\widetilde{H}}\cap{\mathbf{s}}(M(F)).$$ **The abelian subgroup $A_{{\widetilde{M}}}$** {#S:abelian} ---------------------------------------------- Again in this subsection, $F$ is local or global, and $R=F$ or ${\mathbb{A}}$. As we mentioned above, the preimage $\widetilde{Z_{M}}$ of the center $Z_M$ of the Levi $M$ might not be even commutative. For later purposes, we let $A_{{\widetilde{M}}}$ be a closed abelian subgroup of $\widetilde{Z_M}$ containing the center $Z_{{\widetilde{\operatorname{GL}}}_r}$. Namely $A_{{\widetilde{M}}}$ is a closed abelian subgroup such that $$Z_{{\widetilde{\operatorname{GL}}}_r}\subseteq A_{{\widetilde{M}}}\subseteq \widetilde{Z_{M}}.$$ We let $$A_M:=p(A_{{\widetilde{M}}}),$$ where $p$ is the canonical projection. If $F$ is global, we always assume $A_{{\widetilde{M}}}({\mathbb{A}})$ is chosen compatibly with the local $A_{{\widetilde{M}}}(F_v)$ in the sense that we have $$A_{M}({\mathbb{A}})={\prod_v}'A_{M}(F_v).$$ Note that if $A_{M}(F_v)$ (hence $A_{{\widetilde{M}}}(F_v)$) is closed, then $A_{M}({\mathbb{A}})$ (hence $A_{{\widetilde{M}}}({\mathbb{A}})$) is closed by Lemma \[L:closed\_subgroup\_local\_global\]. Of course there are many different choices for $A_{{\widetilde{M}}}$. But we would like to choose $A_{{\widetilde{M}}}$ so that the following hypothesis is satisfied: Assume $F$ is global. The image of $M(F)$ in the quotient $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete in the quotient topology. The author does not know if one can always find such $A_{{\widetilde{M}}}$ for general $n$. But at least we have \[P:hypothesis\] If $n=2$, the above hypothesis is satisfied for a suitable choice of $A_{{\widetilde{M}}}$. For $n>2$, if $d=\gcd(n, r-1+2cr)$ is such that $n$ divides $nr_i/d$ for all $i=1,\dots,k$, (which is the case, for example, if $d=1$,) then the above hypothesis is satisfied with $A_{{\widetilde{M}}}=Z_{{\widetilde{M}}}$. This is proven in Appendix \[A:topology\]. We believe that for any reasonable choice of $A_{{\widetilde{M}}}$ the above hypothesis is always satisfied, but the author does not know how to prove it at this moment. This is a bit unfortunate in that this subtle technical issue makes the main theorem of the paper conditional when $n>2$. However if $n=2$, our main results are complete, and this is the only case we need for our applications to symmetric square $L$-functions in [@Takeda1; @Takeda2], which is the main motivation for the present work. Let us mention that the group $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})$ (for any choice of $A_M$) is a normal subgroup of $M({\mathbb{A}})$, and hence the quotient $A_M({\mathbb{A}})M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is a group. Accordingly, if the hypothesis is satisfied, the image of $M(F)$ in the quotient is a discrete subgroup and hence closed. Also we have $$A_{{\widetilde{M}}}(F)=A_{{\widetilde{M}}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)).$$ following the convention as in (\[E:H(F)\]), and we set $$A_M(F)=p(A_{{\widetilde{M}}}(F)).$$ **On the local metaplectic tensor product** {#S:Mezo} =========================================== In this section we first review the local metaplectic tensor product of Mezo [@Mezo] and then extend his theory further, first by proving that the metaplectic tensor product behaves in the expected way under the Weyl group action, and second by establishing the compatibility of the metaplectic tensor product with parabolic inductions. Hence in this section, all the groups are over a local (not necessarily non-archimedean) field $F$ unless otherwise stated. Accordingly, we assume that our metaplectic group is defined by the block-compatible cocycle $\sigma_r$ of [@BLS], and hence by ${\widetilde{\operatorname{GL}}}_r$ we actually mean ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r$. **Mezo’s metaplectic tensor product** {#SS:Mezo} ------------------------------------- Let $\pi_1,\cdots,\pi_k$ be irreducible genuine representations of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$, respectively. The construction of the metaplectic tensor product takes several steps. First of all, for each $i$, fix an irreducible constituent ${\pi^{(n)}}_i$ of the restriction $\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}} $ of $\pi_i$ to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. Then we have $$\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}=\sum_{g}m_i\,^g({\pi^{(n)}}_i),$$ where $g$ runs through a finite subset of ${\widetilde{\operatorname{GL}}}_{r_i}$, $m_i$ is a positive multiplicity and $^g({\pi^{(n)}}_i)$ is the representation twisted by $g$. Then we construct the tenor product representation $${\pi^{(n)}}_1\otimes\cdots\otimes{\pi^{(n)}}_k$$ of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}$. Note that this group is merely the direct product of the groups ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$. The genuineness of the representations ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$ implies that this tensor product representation descends to a representation of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}$, [[*i.e.* ]{}]{}the representation factors through the natural surjection $${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}\twoheadrightarrow {\widetilde{\operatorname{GL}}^{(n)}}_{r_1}{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}^{(n)}}_{r_k}={\widetilde{M}^{(n)}}.$$ We denote this representation of ${\widetilde{M}^{(n)}}$ by $${\pi^{(n)}}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k,$$ and call it the metaplectic tensor product of ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$. Let us note that the space $V_{{\pi^{(n)}}}$ of ${\pi^{(n)}}$ is simply the tensor product $V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$ of the spaces of ${\pi^{(n)}}_i$. Let $\omega$ be a character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ such that for all $(aI_{r},\xi)\in Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$ where $a\in F^\times$ we have $$\omega(aI_r,\xi)={\pi^{(n)}}(aI_r,\xi)=\xi{\pi^{(n)}}_1(aI_{r_1},1)\cdots{\pi^{(n)}}_k(aI_{r_k},1).$$ Namely $\omega$ agrees with ${\pi^{(n)}}$ on the intersection $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$. We can extend ${\pi^{(n)}}$ to the representation $${\pi^{(n)}}_\omega:=\omega{\pi^{(n)}}$$ of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$ by letting $Z_{{\widetilde{\operatorname{GL}}}_r}$ act by $\omega$. Now extend the representation ${\pi^{(n)}}_\omega$ to a representation $\rho_\omega$ of a subgroup ${\widetilde{H}}$ of ${\widetilde{M}}$ so that $\rho_\omega$ satisfies Mackey’s irreducibility criterion and so the induced representation $$\label{E:Mezo_tensor} \pi_\omega:={\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}\rho_\omega$$ is irreducible. It is always possible to find such ${\widetilde{H}}$ and moreover ${\widetilde{H}}$ can be chosen to be normal. Mezo shows in [@Mezo] that $\pi_\omega$ is dependent only on $\omega$ and is independent of the other choices made throughout, namely the choices of ${\pi^{(n)}}_i$, ${\widetilde{H}}$ and $\rho_\omega$. We write $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ and call it the metaplectic tensor product of $\pi_1,\dots,\pi_k$ with the character $\omega$. Mezo also shows that the metaplectic tensor product $\pi_\omega$ is unique up to twist. Namely \[P:local\_uniqueness\] Let $\pi_1,\dots,\pi_k$ and $\pi'_1,\dots,\pi'_k$ be representations of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$. They give rise to isomorphic metaplectic tensor products with a character $\omega$, [[*i.e.* ]{}]{}$$(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega\cong (\pi'_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi'_k)_\omega,$$ if and only if for each $i$ there exists a character $\omega_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}$ such that $\pi_i\cong\omega_i\otimes\pi'_i$. This is [@Mezo Lemma 5.1]. \[R:dependence\_on\_omega\] Though the metaplectic tensor product generally depends on the choice of $\omega$, if the center $Z_{{\widetilde{\operatorname{GL}}}_r}$ is already contained in ${\widetilde{M}^{(n)}}$, we have ${\pi^{(n)}}_\omega={\pi^{(n)}}$ and hence there is no actual choice for $\omega$ and the metaplectic tensor product is canonical. This is the case, for example, when $n=2$ and $r$ is even, which is one of the important cases we consider in our applications in [@Takeda1; @Takeda2]. The equality (\[E:center=center\]) implies that extending a representation ${\pi^{(n)}}$ of ${\widetilde{M}^{(n)}}$ to ${\pi^{(n)}}_\omega$ multiplying the character $\omega$ on $Z_{{\widetilde{\operatorname{GL}}}_r}$ is the same as extending it by multiplying an appropriate character on $Z_{{\widetilde{M}}}$. Let us mention the following, which is not explicitly mentioned in [@Mezo]. \[L:always\_tensor\_product\] Let $\pi_\omega$ be an irreducible admissible representation of ${\widetilde{M}}$ where $\omega$ is the character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ defined by $\omega=\pi_\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}}$. Then there exist irreducible admissible representations $\pi_1,\dots\pi_k$ of ${\widetilde{\operatorname{GL}}}_{r_1},\dots,{\widetilde{\operatorname{GL}}}_{r_k}$, respectively, such that $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega.$$ Namely a representation of ${\widetilde{M}}$ is always a metaplectic tensor product. The restriction $\pi_\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ contains a representation of the form $\omega ({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ for some representations ${\pi^{(n)}}_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}$. Let $\pi_i$ be an irreducible constituent of ${\operatorname{Ind}}_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}^{{\widetilde{\operatorname{GL}}}_{r_i}}{\pi^{(n)}}_i$. Then one can see that $\pi_\omega$ is $(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$. From Mezo’s construction, one can tell that essentially the representation theory of the group ${\widetilde{M}}$ is determined by that of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Let us briefly explain why this is so. Let $\pi$ be an irreducible admissible representation of ${\widetilde{M}}$, and $\chi_\pi:{\widetilde{M}}\rightarrow{\mathbb C}$ be the distribution character. If $\pi$ is genuine, so is $\chi_\pi$. Namely $\chi_\pi((1,\xi){\tilde{m}})=\xi\chi_\pi({\tilde{m}})$ for all $\xi\in\mu_n$ and ${\tilde{m}}\in{\widetilde{M}}$. But if ${\tilde{m}}\in{\widetilde{M}}$ is a regular element but not in $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$, then one can find $\xi\in\mu_n$ with $\xi\neq 1$ such that $(1,\xi){\tilde{m}}$ is conjugate to ${\tilde{m}}$. This is proven in the same way as [@KP Proposition 0.1.4]. (The only modification one needs is to choose $A\subset M_r(F)$ in their proof so that $A\subset M_{r_1}(F)\times\cdots\times M_{r_k}(F)$.) Therefore for such ${\tilde{m}}$, one has $\chi_\pi({\tilde{m}})=0$. Namely, the support of $\chi_\pi$ is contained in $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. (Indeed, this argument by the distribution character is crucially used in [@Mezo Lemma 4.2]. ) This explains why $\pi$ is essentially determined by the restriction $\pi|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$. This idea can be observed in \[L:equivalent\_tensor\_product\] Let $\pi$ and $\pi'$ be irreducible admissible representations of ${\widetilde{M}}$. Then $\pi$ and $\pi'$ are equivalent if and only if $\pi|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ and $\pi'|_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}$ have an equivalent constituent. This follows from Proposition \[P:local\_uniqueness\] and Lemma \[L:always\_tensor\_product\]. Also let us mention \[P:Mezo\] we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega=m\pi_\omega$$ for some finite multiplicity $m$, so every constituent of ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$ is isomorphic to $\pi_\omega$. By inducting in stages, we have ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega ={\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{H}}}{\pi^{(n)}}_\omega$, where ${\widetilde{H}}$ is as in (\[E:Mezo\_tensor\]), and by [@Mezo Lemma 4.1] we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{H}}}{\pi^{(n)}}_\omega=\bigoplus_{\chi}\chi\otimes\rho_\omega$$ where $\chi$ runs over the finite set of characters of ${\widetilde{H}}$ that are trivial on $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Moreover it is shown in [@Mezo Lemma 4.1] that any extension of ${\pi^{(n)}}_\omega$ to ${\widetilde{H}}$ is of the form $\chi\otimes\rho_\omega$ and ${\operatorname{Ind}}^{{\widetilde{M}}}_{{\widetilde{H}}}\chi\otimes\rho_\omega=\pi_\omega$ for all $\chi$ by [@Mezo Lemma 4.2]. Hence we have $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega =\bigoplus_\chi {\operatorname{Ind}}_{{\widetilde{H}}}^{{\widetilde{M}}}\chi\otimes\rho_\omega =m\pi_\omega.$$ Let $\omega$ be as above and $A_{{\widetilde{M}}}$ as in Section \[S:abelian\]. The restriction ${\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}}$ gives a character on $A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}$ because ${A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}}$ is contained in the center of ${\widetilde{M}^{(n)}}$ by (\[E:center\_Mtn\]). The product $\omega ({\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}})$ of $\omega$ and ${\pi^{(n)}}|_{A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}} $ defines a character on $Z_{{\widetilde{\operatorname{GL}}}_r}(A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}})$ because the two characters agree on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap (A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}})$. Since the Pontryagin dual is an exact functor, one can extend it to a character on $A_{{\widetilde{M}}}$, which we denote again by $\omega$. Namely $\omega$ is a character on $A_{{\widetilde{M}}}$ extending $\omega$ such that $\omega(a)={\pi^{(n)}}(a)$ for all $a\in A_{{\widetilde{M}}}\cap{\widetilde{M}^{(n)}}$. With this said, we have \[C:local\_tensor\] Let $\omega$ be the character on $A_{{\widetilde{M}}}$ described above, and let ${\pi^{(n)}}_{\omega}:=\omega{\pi^{(n)}}$ be the representation of $A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}$ extending ${\pi^{(n)}}$ by letting $A_{{\widetilde{M}}}$ act as $\omega$. Then $${\operatorname{Ind}}_{A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}=m'\pi_\omega$$ where $m'$ is some finite multiplicity. This follows from the previous proposition because we have the inclusion ${\operatorname{Ind}}_{A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}\hookrightarrow {\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_{\omega}$. **The archimedean case** {#S:archimedean} ------------------------ Let us make some remarks when $F$ is archimedean. Strictly speaking, Mezo assumes that the field $F$ is non-archimedean. If $F={\mathbb C}$, then ${\widetilde{M}^{(n)}}={\widetilde{M}}$. Indeed, ${\widetilde{M}}({\mathbb C})=M({\mathbb C})\times\mu_n$ (direct product), and the metaplectic tensor product is obtained simply by taking the tensor product $\pi_1\otimes\cdots\otimes\pi_k$ and descending it to ${\widetilde{M}}({\mathbb C})$. Hence there is essentially no discrepancy between the metaplectic case and the non-metaplectic one. If $F={\mathbb{R}}$ (so necessarily $n=2$), one can trace the argument of Mezo and make sure the construction works for this case as well, with the proviso that equivalence has to be considered as infinitesimal equivalence. However, it has been communicated to the author by J. Adams that for this case, the induced representation ${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$ is always irreducible. (See [@Adams]). Hence one can simply define the metaplectic tensor product to be this induced representation. **Twists by Weyl group elements** {#S:Weyl_group_local} --------------------------------- As in the notation section, we let $W_M$ be the subset of the Weyl group $W_{{\operatorname{GL}}_r}$ consisting of only those elements which permute the ${\operatorname{GL}}_{r_i}$-factors of $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Though $W_M$ is not a group in general, it is identified with the group $S_k$ of permutations of $k$ letters. Assume $w\in W_M$ is such that $$M':=wMw^{-1}={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$$ for a permutation $\sigma\in S_k$, and so $w(g_1,\dots,g_k)w^{-1}=(g_{\sigma(1)},\dots,g_{\sigma(k)})$ for each $(g_1,\dots,g_k)\in M$. Namely $w$ corresponds to the permutation $\sigma^{-1}$. Then we have $${\widetilde{M'}}={\mathbf{s}}(w){\widetilde{M}}{\mathbf{s}}(w)^{-1}.$$ Let $\pi=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ be an irreducible admissible representation of ${\widetilde{M}}$. As in the notation section one can define the twist $^{{\mathbf{s}}(w)}\pi$ of $\pi$ by ${\mathbf{s}}(w)$ to be the representation of ${\widetilde{M'}}$ on the space $V_\pi$ given by $^{{\mathbf{s}}(w)}\pi({\tilde{m}}')=\pi({\mathbf{s}}(w)^{-1}{\tilde{m}}'{\mathbf{s}}(w))$ for ${\tilde{m}}'\in{\widetilde{M'}}$. To ease the notation we simply write $$^w\pi:=\,^{{\mathbf{s}}(w)}\pi.$$ Actually since $\mu_n\subseteq{\widetilde{M}}$ is in the center, for any preimage $\tilde{w}$ of $w$, we have $\,^{{\mathbf{s}}(w)}\pi=\,^{\tilde{w}}\pi$, and hence the notation $^w\pi$ is not ambiguous. The goal of this subsection is to show that the metaplectic tensor product behaves in the expected way under the Weyl group action. Namely, we will prove \[T:Weyl\_group\_local\] With the above notations, we have $$\label{E:Weyl_group_local} ^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega \cong(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega.$$ To prove this, we first need For each $(m,1)\in{\widetilde{M}^{(n)}}$ and $w\in W_M$, where $m\in M^{(n)}$, we have $${\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1}=(wmw^{-1},1),$$ namely ${\mathbf{s}}(w){\mathbf{s}}(m){\mathbf{s}}(w)^{-1}={\mathbf{s}}(wmw^{-1})$. Note that ${\mathbf{s}}(w)=(w,1)$ and ${\mathbf{s}}(w)^{-1}=(w^{-1},\sigma_r(w,w^{-1})^{-1})$ because we are using ${{^{\sigma}\widetilde{\operatorname{GL}}}}_r$, and hence $${\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1} =(wmw^{-1},\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}).$$ Let $$\varphi_w(m):=\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}.$$ We need to show $\varphi_w(m)=1$ for all $m\in M^{(n)}$. Let us first show that the map $m\mapsto\varphi_w(m)$ is a homomorphism on $M^{(n)}$. To see it, for $m, m'\in M^{(n)}$ we have $$\begin{aligned} {\mathbf{s}}(w)(m,1)(m',1){\mathbf{s}}(w)^{-1}&={\mathbf{s}}(w)(mm',\sigma_r(m,m')){\mathbf{s}}(w)^{-1}\\ &=(wmm'w^{-1},\sigma_r(m,m')\varphi_w(mm')).\end{aligned}$$ On the other hand, we have $$\begin{aligned} {\mathbf{s}}(w)(m,1)(m',1){\mathbf{s}}(w)^{-1}&={\mathbf{s}}(w)(m,1){\mathbf{s}}(w)^{-1}{\mathbf{s}}(w)(m',1){\mathbf{s}}(w)^{-1}\\ &=(wmw^{-1},\varphi_w(m))(wm'w^{-1},\varphi_w(m'))\\ &=(wmm'w^{-1}, \sigma_r(wmw^{-1},wm'w^{-1}) \varphi_w(m)\varphi_w(m'))\\ &=(wmm'w^{-1}, \sigma_r(m,m') \varphi_w(m)\varphi_w(m')),\end{aligned}$$ where the last equality follows because $\sigma_r(wmw^{-1},wm'w^{-1})=\sigma_r(m,m')$ by the block-compatibility of $\sigma_r$. Hence by comparing those two, one obtains $\varphi_w(mm')=\varphi_w(m)\varphi_w(m')$. Therefore to show $\varphi_w(m)=1$, it suffices to show it for the elmenets of the form $$\label{E:form_of_m} m={\operatorname{diag}}(I_{r_1},\dots,I_{r_{i-1}},g_i,I_{r_{i+1}},\dots, I_{r_k})$$ for $g_i\in{\operatorname{GL}}_{r_i}^{(n)}$. Then one can rewrite $\varphi_w(m)$ as follows: $$\begin{aligned} \varphi_w(m)&=\sigma_r(w, mw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(w, w^{-1}wmw^{-1})\sigma_r(m, w^{-1}) \sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(ww^{-1}, wmw^{-1})\sigma_r(w, w^{-1})\sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1})\sigma_r(w,w^{-1})^{-1}\\ &=\sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1}),\end{aligned}$$ where for the third equality we used Proposition \[P:BLS\] (0). So we only have to show $$\label{E:cocycle_computation1} \sigma_r(w^{-1}, wmw^{-1})^{-1}\sigma_r(m, w^{-1})=1.$$ This can be shown by using the algorithm computing the cocycle $\sigma_r$ given by [@BLS]. To use the results of [@BLS], it should be mentioned that one needs to use the set ${\mathfrak{M}}$ for a set of representatives of the Weyl group of ${\operatorname{GL}}_r$ as defined in the notation section. Also let us recall the following notation from [@BLS]: For each $g\in{\operatorname{GL}}_r$, the “torus part function” ${\mathbf{t}}:{\operatorname{GL}}_r\rightarrow T$ is the unique map such that $${\mathbf{t}}(nt\eta n')=t,$$ where $n, n'\in N_B$, $t\in T$ and $\eta\in\mathfrak{M}$ when ${\operatorname{GL}}_r$ is written as $${\operatorname{GL}}_r=\coprod_{\eta\in\mathfrak{M}}N_BT\eta N_B$$ by the Bruhat decomposition. Namely ${\mathbf{t}}(g)$ is the “torus part” of $g$. Using this language, each $w\in W_M$ is written as $$w={\mathbf{t}}(w)\eta_w$$ where $\eta_w\in\mathfrak{M}$, and ${\mathbf{t}}(w)\in{\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$ is of the form $${\mathbf{t}}(w)=(\varepsilon_{\sigma(1)}I_{\sigma(1)},\dots,\varepsilon_{\sigma(k)}I_{\sigma(k)}),$$ where $\varepsilon_i\in\{\pm 1\}$. We are now ready to carry out our cocycle computations for (\[E:cocycle\_computation1\]). Let us deal with $\sigma_r(m, w^{-1})$ first. Write $m=nt\eta n'$ by the Bruhat decomposition, so ${\mathbf{t}}(m)=t$. But recall that we are assuming $m$ is of the form as in (\[E:form\_of\_m\]), so the decomposition $nt\eta n'$ takes place essentially inside the ${\operatorname{GL}}_{r_i}$-block. In particular, we can write $$m={\operatorname{diag}}(I_{r_1},\dots,I_{r_{i-1}},n_it_i\eta_i n_i',I_{r_{i+1}}\dots,I_{r_k}),$$ where $t_i\in{\operatorname{GL}}^{(n)}_{r_i}$. (Note that $\det(t_i)\in F^{\times n}$.) Then one can compute $\sigma_r(m, w^{-1})$ as follows: $$\begin{aligned} \sigma_r(m, w^{-1})&=\sigma_r(nt\eta n', w^{-1})\\ &=\sigma_r(t\eta, n'w^{-1})\quad\text{by Proposition \ref{P:BLS} (1), (2)}\\ &=\sigma_r(t\eta, w^{-1}wn'w^{-1})\\ &=\sigma_r(t\eta, w^{-1})\quad\text{because $wn'w^{-1}\in N_B$ and by Proposition \ref{P:BLS} (1)}\\ &=\sigma_r(t\eta, {\mathbf{t}}(w^{-1})\eta_{w^{-1}}).\end{aligned}$$ Now since $\eta$ is essentially inside the ${\operatorname{GL}}_{r_i}$-factor of $M$ and $\eta_{w^{-1}}$ only permutes the ${\operatorname{GL}}_{r_j}$-factors of $M$, we have $l(\eta\eta_{w^{-1}})=l(\eta)+l(\eta_{w^{-1}})$, where $l$ is the length function. Hence by applying [@BLS Lemma 10, p.155], we have $$\label{E:cocycle_computation2} \sigma_r(t\eta, {\mathbf{t}}(w^{-1})\eta_{w^{-1}}) =\sigma_r(t,\eta{\mathbf{t}}(w^{-1})\eta^{-1})\sigma_r(\eta, {\mathbf{t}}(w^{-1})).$$ Here note that ${\mathbf{t}}(w^{-1})\in M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$ is of the form $(\varepsilon_1 I_{r_1},\dots, \varepsilon_k I_{r_k})$ and $\eta$ is in the ${\operatorname{GL}}_{r_i}$-block. Hence $\eta{\mathbf{t}}(w^{-1})\eta^{-1}={\mathbf{t}}(w^{-1})$. Thus by the block-compatibility of $\sigma_r$, (\[E:cocycle\_computation2\]) is written as $$\sigma_{r_i}(t_i, \varepsilon_i I_{r_i})\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i}).$$ Clearly, if $\varepsilon_i=1$, then both $\sigma_{r_i}(t_i, \varepsilon_i I_{r_i})$ and $\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i})$ are $1$. If $\varepsilon_i=-1$, then by Proposition \[P:BLS\] (3), one can see that $\sigma_{r_i}(\eta_i, \varepsilon_i I_{r_i})=1$. Hence in either case, one has $$\label{E:cocycle_computation3} \sigma_r(m,w^{-1}) =\sigma_{r_i}(t_i, \varepsilon_i I_{r_i}).$$ Next let us deal with $\sigma_r(w^{-1}, wmw^{-1})$ in (\[E:cocycle\_computation1\]). First by the analogous computation to what we did for $\sigma(m, w^{-1})$, one can write $$\label{E:cocycle_computation4} \sigma_r(w^{-1}, wmw^{-1}) =\sigma_r(w^{-1}, wt\eta w^{-1}) =\sigma_r({\mathbf{t}}(w^{-1})\eta_{w^{-1}}, wt\eta w^{-1}).$$ Since $w$ corresponds to the permutation $\sigma^{-1}$, if we let $$\tau_i:{\operatorname{GL}}_{r_i}\rightarrow wMw^{-1}={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$$ be the embedding of ${\operatorname{GL}}_{r_i}$ into the corresponding ${\operatorname{GL}}_{r_i}$-factor of $wMw^{-1}$, then (\[E:cocycle\_computation4\]) is written as $$\sigma_r({\mathbf{t}}(w^{-1})\eta_{w^{-1}}, \tau_i(t_i)\tau_i(\eta_i)).$$ Note that $\tau_i(\eta_i)\in\mathfrak{M}$ and $l(\eta_{w^{-1}}\tau_i(\eta_i))=l(\eta_{w^{-1}})+l(\tau_i(\eta_i))$. Hence by using [@BLS Lemma 10, p.155], this is written as $$\label{E:cocycle_computation5} \sigma_r({\mathbf{t}}(w^{-1}), \eta_{w^{-1}}\tau_i(t_i)\eta_{w^{-1}}^{-1}) \sigma_r(\eta_{w^{-1}}, \tau_i(t_i)).$$ By the block compatibility of $\sigma_r$, one can see $$\sigma_r({\mathbf{t}}(w^{-1}), \eta_{w^{-1}}\tau_i(t_i)\eta_{w^{-1}}^{-1}) =\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i).$$ Also to compute $\sigma_r(\eta_{w^{-1}}, \tau_i(t_i))$, one needs to use Proposition \[P:BLS\] (3). For this purpose, let us write $$t_i=\begin{pmatrix}a_1&&\\ &\ddots&\\ &&a_{r_i}\end{pmatrix}\in{\operatorname{GL}}_{r_i}$$ where $\det(t_i)=a_1\cdots a_{r_i}\in F^{\times n}$. By looking at the formula in Proposition \[P:BLS\] (3), one can see that $\sigma_r(\eta_{w^{-1}}, \tau_i(t_i))$ is a power of $$(-1, a_1)\cdots(-1,a_{r_i}),$$ which is equal to $(-1, a_1\cdots a_{r_i})=1$ because $\det(t_i)=a_1\cdots a_{r_i}\in F^{\times n}$. Hence (\[E:cocycle\_computation5\]), which is the same as (\[E:cocycle\_computation4\]), becomes $$\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i).$$ Hence the left hand side of (\[E:cocycle\_computation1\]) is written as $$\sigma_{r_i}(\varepsilon_iI_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i,\varepsilon_iI_{r_i}).$$ We need to show this is $1$. But clearly this is the case if $\varepsilon_i=1$. So let us assume $\varepsilon_i=-1$. Namely we will show $\sigma_{r_i}(-I_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i,-I_{r_i})=1$. But by Proposition \[P:BLS\] (4), one can compute $$\sigma_{r_i}(-I_{r_i}, t_i)=(-1, a_2)(-1, a_3)^2(-1,a_4)^3\cdots (-1, a_r)^{r-1+2c}$$ and $$\sigma_{r_i}(t_i, -I_{r_i})=(a_1, -1)^{r-1}(a_2,-1)^{r-2}(a_3,-1)^{r-3}\cdots (-1, a_{r-1}).$$ Noting that $(-1,a_i)^{-1}=(a_i, -1)$, we have $$\sigma_{r_i}(-I_{r_i}, t_i)^{-1}\sigma_{r_i}(t_i, -I_{r_i})=\prod_{i=1}^r(a_i, -1)^{r-1+2c}=(\prod_{i=1}^ra_i, -1)^{r-1+2c}=1,$$ where the last equality follows because $\det(t_i)=\prod_{i=1}^ra_i\in F^{\times n}$. This completes the proof. Now we are ready to prove Theorem \[T:Weyl\_group\_local\]. By restricting to ${\widetilde{M'}^{(n)}}$, one can see that the left hand side of (\[E:Weyl\_group\_local\]) contains the representation $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ and the right hand side of (\[E:Weyl\_group\_local\]) contains ${\pi^{(n)}}_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_{\sigma(k)}$, where $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ is the representation of ${\widetilde{M'}^{(n)}}={\mathbf{s}}(w){\widetilde{M}^{(n)}}{\mathbf{s}}(w)^{-1}$ whose space is the space of ${\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k$. Hence by Lemma \[L:equivalent\_tensor\_product\], it suffices to show that $$^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)\cong {\pi^{(n)}}_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_{\sigma(k)}.$$ But this can be seen from the commutative diagram $$\xymatrix{ &{\widetilde{\operatorname{GL}}^{(n)}}_{r_{\sigma(1)}}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_{\sigma(k)}}\ar[ddl]\ar[ddr]\ar[d]&\\ &{\widetilde{M'}^{(n)}}\ar[dl]\ar[dr]&\\ {\operatorname{Aut}}(V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k})\ar[rr]^{\sim}&& {\operatorname{Aut}}(V_{{\pi^{(n)}}_{\sigma(1)}}\otimes\cdots\otimes V_{{\pi^{(n)}}_{\sigma(k)}}), }$$ where the left most arrow is the representation of ${\widetilde{\operatorname{GL}}^{(n)}}_{r_\sigma(1)}\times\cdots\times{\widetilde{\operatorname{GL}}^{(n)}}_{r_\sigma(k)}$ (direct product) acting on the space of ${\pi^{(n)}}_1\otimes\cdots\otimes{\pi^{(n)}}_k$ by permuting each factor by $\sigma^{-1}$, which descends to the representation $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ of ${\widetilde{M}^{(n)}}$. To see this indeed descends to $^w({\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$, one needs the above lemma. **Compatibility with parabolic induction** {#S:parabolic_induction_local} ------------------------------------------ We will show the compatibility of the metaplectic tensor product with parabolic induction. Hence we consider the standard parabolic subgroup $P=MN\subseteq{\operatorname{GL}}_r$ where $M$ is the Levi part and $N$ the unipotent radical. First let us mention \[L:normalizer\_local\] The image $N^\ast$ of the unipotent radical $N$ via the section ${\mathbf{s}}:{\operatorname{GL}}_r\rightarrow{\widetilde{\operatorname{GL}}}_r$ is normalized by the metaplectic preimage ${\widetilde{M}}$ of the Levi part $M$. Though this seems to be well-known, we will give a proof here. Let ${\tilde{m}}\in{\widetilde{M}}$ and $(n,1)\in N^\ast$, where $n\in N$. (Note that since we are assuming the group ${\widetilde{\operatorname{GL}}}_r$ is defined by $\sigma_r$, each element in $N^\ast$ is written as $(n,1)$.) We may assume ${\tilde{m}}=(m,1)$ for $m\in M$. Noting that ${\tilde{m}}^{-1}=(m^{-1},\sigma_r(m,m^{-1})^{-1})$, we compute $$\begin{aligned} {\tilde{m}}(n,1){\tilde{m}}^{-1}&=(m,1)(n,1)(m^{-1}, \sigma_r(m,m^{-1})^{-1})\\ &=(mn, \sigma_r(m,n)) (m^{-1}, \sigma_r(m,m^{-1})^{-1})\\ &=(mnm^{-1},\sigma_r(mn, m^{-1})\sigma_r(m,n) \sigma_r(m,m^{-1})^{-1}).\end{aligned}$$ By Proposition \[P:BLS\] (1), $\sigma_r(m,n)=1$. Also since $mnm^{-1}\in N$, we have $\sigma_r(mn, m^{-1})=\sigma_r(mnm^{-1}m, m^{-1})=\sigma_r(m,m^{-1})$ again by Proposition \[P:BLS\] (1). Thus we have ${\tilde{m}}(n,1){\tilde{m}}^{-1}=(mnm^{-1},1)\in N^\ast$. By this lemma, we can write $${\widetilde{P}}={\widetilde{M}}N^\ast$$ where ${\widetilde{M}}$ normalizes $N^\ast$ and hence for a representation $\pi$ of ${\widetilde{M}}$ one can form the induced representation $${\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$$ by letting $N^\ast$ act trivially. \[T:induction\_local\] Let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i, l_i}}$. For each $i$, we are given a representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i$, which is given as the metaplectic tensor product of the representations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}},\dots,{\widetilde{\operatorname{GL}}}_{r_{i, l_i}}$. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i}^{{\widetilde{\operatorname{GL}}}_{r_i}}\sigma_i$. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,l_k})_\omega,$$ where $Q$ is the standard parabolic of $M$ whose Levi part is $M_1\times\cdots\times M_k$. First we need For a genuine representation $\pi$ of a Levi part ${\widetilde{M}}$, the map $${\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\rightarrow{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$$ given by the restriction $\varphi\mapsto \varphi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}$ for $\varphi\in {\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$ is an isomorphism, where $${(\widetilde{M})^{(n)}}={\widetilde{M}}\cap{\widetilde{\operatorname{GL}}^{(n)}}_r.$$ Hence in particular $$\left({\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\right)|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}\cong \left({\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi\right)\|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}\cong{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$$ as representations of ${\widetilde{\operatorname{GL}}^{(n)}}_r$. To show it is one-to-one, assume $\varphi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}=0$. We need to show $\varphi=0$. But for any $g\in{\operatorname{GL}}_r$, one can write $g=\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g$, where $\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}\in M$ and $\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g\in{\operatorname{GL}}_r^{(n)}$. Hence any ${\tilde{g}}\in{\widetilde{\operatorname{GL}}}_r$ is written as ${\tilde{g}}={\tilde{m}}{\tilde{g}}'$ for some ${\tilde{m}}\in{\widetilde{M}}$ and ${\tilde{g}}'\in{\widetilde{\operatorname{GL}}^{(n)}}_r$. Hence $\varphi({\tilde{g}})=\pi({\tilde{m}})\varphi({\tilde{g}}')$. But $\varphi({\tilde{g}}')=0$. Hence $\varphi({\tilde{g}})=0$. To show it is onto, let $\varphi\in{\operatorname{Ind}}_{{(\widetilde{M})^{(n)}}N^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_r}\pi|_{{(\widetilde{M})^{(n)}}}$. Define $\widetilde{\varphi}:{\widetilde{\operatorname{GL}}}_r\rightarrow\pi$ by $$\widetilde{\varphi}(g,\xi)=\xi\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)\varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),$$ where $\eta$ is chosen to be such that $(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)=(g,1)$. Namely $\eta$ is given by the cocycle as $$\eta=\sigma_r(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g)^{-1}.$$ That $\widetilde{\varphi}|_{{\widetilde{\operatorname{GL}}^{(n)}}_r}=\varphi$ follows because if $g\in{\operatorname{GL}}_r^{(n)}$ then $(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta)\in{(\widetilde{M})^{(n)}}$. Also one can check $\widetilde{\varphi}\in {\operatorname{Ind}}_{{\widetilde{M}}N^\ast}^{{\widetilde{\operatorname{GL}}}_r}\pi$ as follows: We need to check $\varphi({\tilde{m}}(g,\xi))=\pi({\tilde{m}})\varphi(g,\xi)$ for all ${\tilde{m}}\in{\widetilde{M}}$. But since $\pi$ (and hence $\varphi$) is genuine, we may assume ${\tilde{m}}$ is of the form $(m,1)$ for $m\in M$ and $\xi=1$. Then $$\begin{aligned} \notag\widetilde{\varphi}((m,1)(g,1))&=\widetilde{\varphi}(mg,\sigma_r(m,g))\\ \label{E:F_tilde}&=\sigma_r(m,g)\pi(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_1)\varphi(\begin{pmatrix}\det (mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg, 1),\end{aligned}$$ where $$\eta_1=\sigma_r(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det (mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg)^{-1}.$$ Now $$\begin{aligned} &(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}mg, 1)\\ =&(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_2) (\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),\end{aligned}$$ where $$\eta_2=\sigma_r(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g)^{-1}.$$ Since $(\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n-1}&\\ & I_{r-1}\end{pmatrix}, \eta_2)\in{(\widetilde{M})^{(n)}}$, the right hand side of (\[E:F\_tilde\]) becomes $$\begin{aligned} &\sigma_r(m,g)\pi\Big((\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_1) (\begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta_2)\Big)\\ &\qquad\qquad \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)\\ &=\sigma_r(m,g)\pi(m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix},\eta_1\eta_2\eta_3) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1)\\ &=\sigma_r(m,g)\pi(m,\eta_1\eta_2\eta_3\eta_4)\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, 1) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1),\end{aligned}$$ where $$\eta_3=\sigma_r(\begin{pmatrix}\det (mg)^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \begin{pmatrix}\det(mg)^{n-1}&\\ & I_{r-1}\end{pmatrix}m\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}),$$ and $$\eta_4=\sigma_r(m, \begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix})^{-1}.$$ Then one can compute $$\sigma_r(m,g)\eta_1\eta_2\eta_3\eta_4=\eta$$ by using Proposition \[P:BLS\] (0). Hence (\[E:F\_tilde\]) is written as $$\pi(m,1)\pi(\begin{pmatrix}\det g^{-n+1}&\\ & I_{r-1}\end{pmatrix}, \eta) \varphi(\begin{pmatrix}\det g^{n-1}&\\ & I_{r-1}\end{pmatrix}g, 1) =\pi(m,1)\widetilde{\varphi}(g,1).$$ This completes the proof. With this lemma, one can prove the theorem. Let $\pi_i^{(n)}$ be an irreducible constituent of the restriction $\pi_i|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}$. By the above lemma, it is an irreducible constituent of $${\operatorname{Ind}}_{{(\widetilde{M_i})^{(n)}}N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{(\widetilde{M_i})^{(n)}}}.$$ Noting that ${\widetilde{M}^{(n)}}_i\subseteq{(\widetilde{M_i})^{(n)}}$, we have the inclusion $${\operatorname{Ind}}_{{(\widetilde{M_i})^{(n)}}N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{(\widetilde{M_i})^{(n)}}} \hookrightarrow {\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_i N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\sigma_i|_{{\widetilde{M}^{(n)}}_i}.$$ But since $\sigma_i$ is a metaplectic tensor product of $\tau_{i,1},\dots,{\tau_{i,l_i}}$, the restriction $\sigma_i|_{{\widetilde{M}^{(n)}}_i}$ is a sum of representations of the form $$\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i}^{(n)}$$ where each $\tau_{i,t}^{(n)}$ is an irreducible constituent of the restriction $\tau_{i,t}|_{{\widetilde{\operatorname{GL}}^{(n)}}_{^ir_t}}$ of $\tau_{i,t}$ to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_{i,t}}$. Note that this is a metaplectic tensor product representation of ${\widetilde{M}^{(n)}}_i$. Hence the metaplectic tensor product $$\pi^{(n)}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k$$ is an irreducible constituent of $$\label{E:tensor_of_tensor} \widetilde{\bigotimes}_{i=1}^k{\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_i N_i^\ast}^{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}}\,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{i,l_i}^{(n)}}.$$ Note that the metaplectic tensor product for the group ${\widetilde{M}^{(n)}}$ can be defined for reducible representations, and hence $\widetilde{\bigotimes}_{i=1}^k$ is defined and the space of the representation is the same as the one for the usual tensor product. In particular, the space of the representation (\[E:tensor\_of\_tensor\]) is the usual tensor product. Then one can see that (\[E:tensor\_of\_tensor\]) is equivalent to $$\label{E:tensor_of_tensor2} {\operatorname{Ind}}_{{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{{\widetilde{M}^{(n)}}}{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{i,l_i}^{(n)}}.$$ (To see this one can define a map from (\[E:tensor\_of\_tensor\]) to (\[E:tensor\_of\_tensor2\]) by $\varphi_1\,\otimes\cdots\otimes\,\varphi_k\mapsto \varphi_1\cdots\varphi_k$ where $\varphi_1\cdots\varphi_k$ is the product of functions which can be naturally viewed as a function on ${\widetilde{M}^{(n)}}$.) Now let $\omega$ be a character on $Z_{{\widetilde{\operatorname{GL}}}_r}$ that agrees with ${\pi^{(n)}}$ on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$, so that the product $${\pi^{(n)}}_\omega:=\omega\cdot{\pi^{(n)}}_n$$ is a well-defined representation of $Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}$. Now all the constituents of the representation (\[E:tensor\_of\_tensor2\]) have the same central character, and hence $\omega$ agrees with (\[E:tensor\_of\_tensor2\]) on $Z_{{\widetilde{\operatorname{GL}}}_r}\cap{\widetilde{M}^{(n)}}$, and hence ${\pi^{(n)}}_\omega$ is a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}.$$ Recall that the metaplectic tensor product $\pi_\omega$ is a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\pi^{(n)}}_\omega$$ and hence a constituent of $${\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}},$$ which is $$\label{E:tensor_of_tensor3} {\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}{\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M}^{(n)}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}^{(n)}}_k (N_1\times\cdots\times N_k)^\ast}^{{\widetilde{Q}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}$$ by inducing in stages. Now one can see that the inner induced representation in (\[E:tensor\_of\_tensor3\]) is equal to $$\label{E:tensor_of_tensor4} {\operatorname{Ind}}_{Z_{{\widetilde{\operatorname{GL}}}_r}{\widetilde{M_Q}}^{(n)}}^{{\widetilde{M_Q}}}\omega\cdot{\widetilde{\otimes}}_{i=1}^k \,\tau_{i,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\tau_{i,l_i}^{(n)}}$$ where the unipotent group $(N_1\times\cdots\times N_k)^\ast$ acts trivially and ${\widetilde{M_Q}}$ is the Levi part of ${\widetilde{Q}}$, namely $${\widetilde{M_Q}}={\widetilde{M}}_1{\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{M}}_k.$$ By Proposition \[P:Mezo\] applied to the Levi ${\widetilde{M_Q}}$, the representation (\[E:tensor\_of\_tensor4\]) is a sum of the metaplectic tensor product $$(\tau_{1,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{1,l_1}^{(n)}}\,{\widetilde{\otimes}}\,\cdots{\widetilde{\otimes}}\, \tau_{k,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{k,l_k}^{(n)}})_\omega.$$ Hence $\pi_\omega$ is a constituent of ${\operatorname{Ind}}_{{\widetilde{Q}}}^{{\widetilde{M}}}(\tau_{1,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{1,l_1}^{(n)}}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k,1}^{(n)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, {\tau_{k,l_k}^{(n)}})_\omega$ as claimed. \[R:induction\_local\] In the statement of Theorem \[T:induction\_local\], one can replace “constituent” by “irreducible subrepresentation” or “irreducible quotient”, and the analogous statement is still true. Namely if each $\pi_i$ is an irreducible subrepresentation (resp. quotient) of the induced representation in the theorem, then the metaplectic tensor product $(\pi_1\,\otimes\cdots\otimes\,\pi_k)_\omega$ is also an irreducible subrepresentation (resp. quotient) of the corresponding induced representation. To prove it, one can simply replace all the occurrences of “constituent” by “irreducible subrepresentation” or “irreducible quotient” in the above proof. **The global metaplectic tensor product** ========================================= Starting from this section, we will show how to construct the metaplectic tensor product of unitary automorphic subrepresentations. Hence all the groups are over the ring of adeles unless otherwise stated, and it should be recalled here that as in the group ${\operatorname{GL}}_r(F)^\ast$ is the image of ${\operatorname{GL}}_r(F)$ under the partial map ${\mathbf{s}}:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, and we simply write ${\operatorname{GL}}_r(F)$ for ${\operatorname{GL}}_r(F)^\ast$, when there is no danger of confusion. Also throughout the section the group $A_{{\widetilde{M}}}({\mathbb{A}})$ is an abelian group that satisfies Hypothesis ($\ast$). **The construction** -------------------- The construction is similar to the local case in that first we consider the restriction to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$, though we need an extra care to ensure the automorphy. Let us start with the following. \[L:unitary\_restriction1\] Let $\pi$ be a genuine irreducible automorphic unitary subrepresentation of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Then the restriction $\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})}$ is completely reducible, namely $$\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r{{\mathbb{A}}}}=\bigoplus{\pi^{(n)}}_i$$ where $\pi_i$ is an irreducible unitary representation of ${\widetilde{\operatorname{GL}}^{(n)}}_r({\mathbb{A}})$. This follows from the admissibility and unitarity of $\pi|_{{\widetilde{\operatorname{GL}}^{(n)}}_r{{\mathbb{A}}}}$. The lemma implies that the restriction $\pi_i\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$ is also completely reducible. (See the notation section for the notation $\|$.) Hence each irreducible constituent of $\pi_i\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$ is a subrepresentation. Let $${\pi^{(n)}}_i\subseteq\pi_i$$ be an irreducible subrepresentation. Then each vector $f\in{\pi^{(n)}}_i$ is the restriction to ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ of an automorphic form on ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$. Hence one can naturally view each vector $f\in{\pi^{(n)}}_i$ as a function on the group $$H_i:={\operatorname{GL}}_{r_i}(F){\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}}).$$ Namely the representation ${\pi^{(n)}}_{i}$ is an irreducible representation of the group ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ realized in a space of “automorphic forms on $H_i$”. Note that $H_i$ is indeed a group and moreover it is closed in ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, which can be shown by using Lemma \[L:discrete\_sub\]. Also note that each element in $H_i$ is of the form $(h_i,\xi_i)$ for $h_i\in{\operatorname{GL}}_{r_i}(F){\operatorname{GL}}_{r_i}({\mathbb{A}})$ and $\xi_i\in\mu_n$. By the product formula for the Hilbert symbol and the block-compatibility of the cocycle $\tau_M$, we have the natural surjection $$\label{E:global_surjection} H_1\times\cdots\times H_k\rightarrow M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$$ given by the map $((h_1,\xi_1),\dots,(h_k,\xi_k))\mapsto (h_1\cdots h_k,\xi_1\cdots\xi_k)$ because $(\det(h_i),\det(h_j))_{\mathbb{A}}=1$ for all $i, j=1,\dots, k$. Now we can construct a metaplectic tensor product of ${\pi^{(n)}}_1,\dots,{\pi^{(n)}}_k$, which is an “automorphic representation” of ${\widetilde{M}^{(n)}}({\mathbb{A}})$ realized in a space of “automorphic forms on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$” as follows. \[P:tensor\_product\_Mtn\] Let $$V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$$ be the space of functions on the direct product $H_1\times\dots\times H_k$ which gives rise to an irreducible representation of ${\widetilde{\operatorname{GL}}^{(n)}}_{r_1}({\mathbb{A}})\times\dots\times {\widetilde{\operatorname{GL}}^{(n)}}_{r_i}(A)$ which acts by right translation. Then each function in this space can be viewed as a function on the group $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, namely it factors through the surjection as in (\[E:global\_surjection\]) and thus gives rise to an representation of ${\widetilde{M}^{(n)}}({\mathbb{A}})$, which we denote by $${\pi^{(n)}}:={\pi^{(n)}}_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k.$$ Moreover each function in $V_{{\pi^{(n)}}}=V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$ is “automorphic” in the sense that it is left invariant on $M(F)$. Since $\pi_i$ is genuine, for each $f_i\in V_{{\pi^{(n)}}_i}$ and $g\in H_i$, we have $f_i(g(1,\xi))=f_i((1,\xi)g)=\xi f_i(g)$ for all $\xi\in\mu_n$. Now the kernel of the map (\[E:global\_surjection\]) consists of the elements of the form $((I_{r_1},\xi_1),\dots,(I_{r_k},\xi_k))$ with $\xi_1\cdots\xi_k=1$. Hence each $f_1\otimes\cdots\otimes f_k\in V_{{\pi^{(n)}}_1}\otimes\cdots\otimes V_{{\pi^{(n)}}_k}$, viewed as a function on the direct product $H_1\times\cdots\times H_k$, factors through the map (\[E:global\_surjection\]), which we denote by $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$. Namely we can naturally define a function $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k $ on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$ by $$(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)( \begin{pmatrix}h_1&&\\ &\ddots&\\ &&h_k\end{pmatrix},\xi) =\xi f_1(h_1,1)\cdots f_k(h_k,1).$$ One can see each function $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$ is “automorphic” as follows: For $\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix}\in M(F)$ and $\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix}\in M(F)M^{(n)}({\mathbb{A}})$, we have $$\begin{aligned} &(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big({\mathbf{s}}\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix} (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\Big)\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big((\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix},\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}) (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\Big)\quad\text{by definition of ${\mathbf{s}}$}\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big(\begin{pmatrix}\gamma_1g_1&&\\ &\ddots&\\ &&\gamma_kg_k\end{pmatrix},\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1} \tau_M(\begin{pmatrix}\gamma_1&&\\ &\ddots&\\ &&\gamma_k\end{pmatrix}, \begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix})\Big)\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)\Big(\begin{pmatrix}\gamma_1g_1&&\\ &\ddots&\\ &&\gamma_kg_k\end{pmatrix},\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\Big) \quad\text{by block-compatibility of $\tau_M$}\\ =&\left(\xi\prod_{i=1}^ks_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\right) \left(\prod_{i=1}^k f_i(\gamma_ig_i,1)\right) \quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$}\\ =&\xi\prod_{i=1}^kf_i\Big(\gamma_ig_i,s_{r_i}(\gamma_i)^{-1}\tau_{r_i}(\gamma_i,g_i)\Big) \quad\text{because each $f_i$ is genuine}\\ =&\xi\prod_{i=1}^kf_i\Big((\gamma_i, s_{r_i}(\gamma_i)^{-1})(g_i,1)\Big) \quad\text{by definition of $\tau_{r_i}$}\\ =&\xi\prod_{i=1}^kf_i({\mathbf{s}}_{r_i}(\gamma_i)(g_i,1)) \quad\text{by definition of ${\mathbf{s}}_{r_i}$}\\ =&\xi\prod_{i=1}^kf_i(g_i,1) \quad\text{by automorphy of $f_i$}\\ =&(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k) (\begin{pmatrix}g_1&&\\ &\ddots&\\ &&g_k\end{pmatrix},\xi)\quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$} .\end{aligned}$$ Just like the local case, we would like to extend the representation ${\pi^{(n)}}$ to a representation of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ by letting $A_{{\widetilde{M}}}({\mathbb{A}})$ act as a character. This is certainly possible by choosing an appropriate character because $A_{{\widetilde{M}}}({\mathbb{A}})\cap{\widetilde{M}^{(n)}}({\mathbb{A}})$ is in the center of ${\widetilde{M}^{(n)}}({\mathbb{A}})$. To ensure the resulting representation is automorphic, however, one needs extra steps to do it. For this purpose, let us, first, define $$\label{E:F_points_of_AM} A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F):=A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\cap{\mathbf{s}}(M(F)).$$ Note that this is not necessarily the same as $A_{{\widetilde{M}}}(F){\widetilde{M}^{(n)}}(F)$. Also let $$H:=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F){\widetilde{M}^{(n)}}({\mathbb{A}}).$$ By our assumption on $A_{{\widetilde{M}}}$ (Hypothesis ($\ast$)), the image of ${\mathbf{s}}(M(F))$ (and hence $A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)$) in the quotient ${\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. Hence by Lemma \[L:discrete\_sub\], $H$ is a closed (and hence locally compact) subgroup of ${\widetilde{M}}({\mathbb{A}})$. Also note that the group $A_{{\widetilde{M}}}({\mathbb{A}})$ commutes pointwise with the group $H$ by Proposition \[P:Z\_M\_commute\_Mtn\] and hence $A_{{\widetilde{M}}}({\mathbb{A}})\cap H$ is in the center of $H$. We need the following subtle but important lemma. \[H:central\_character\_on\_H\] There exists a character $\chi$ on the center $Z_H$ of $H$ such that $f(ah)=\chi(a)f(h)$ for $a\in Z_H$, $h\in H$ and $f\in{\pi^{(n)}}$. (Note that each $f\in{\pi^{(n)}}$ is a function on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$ and hence can be viewed as a function on $H$.) Let ${\pi^{(n)}}_{H_i}$ be an irreducible subrepresentation of $\pi_i\|_{H_i}$ such that ${\pi^{(n)}}\subseteq{\pi^{(n)}}_{H_i}\|_{{\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})}$. Analogously to the construction of ${\pi^{(n)}}={\pi^{(n)}}_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_k$, one can construct the representation ${\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}$ of $M(F){\widetilde{M}}({\mathbb{A}})$. (The space of this representation is again a space of “automorphic forms on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$” but this time it is an irreducible representation of the group $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, rather than just ${\widetilde{M}^{(n)}}({\mathbb{A}})$. The construction is completely the same as ${\pi^{(n)}}$ and one can just modify the proof of Proposition \[P:tensor\_product\_Mtn\].) Then one can see $$V_{{\pi^{(n)}}}\subseteq V_{{\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}},$$ and $$({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})\|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}\cong ({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}.$$ Let ${\pi^{(n)}}_H$ be an irreducible subrepresentation of $({\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k})|_H$ such that $$V_{{\pi^{(n)}}}\subseteq V_{{\pi^{(n)}}_H},$$ where both sides are spaces of functions on $M(F){\widetilde{M}^{(n)}}({\mathbb{A}})$. Such ${\pi^{(n)}}_H$ certainly exists, since each $\pi_i$ is unitary and the unitary structure descends to ${\pi^{(n)}}_{H_1}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{H_k}$ making it unitary. Now since ${\pi^{(n)}}_H$ is unitary and $H$ is locally compact, ${\pi^{(n)}}_H$ admits a central character $\chi$. Thus for each $f\in V_{{\pi^{(n)}}_H}$ and [*a fortiori*]{} each $f\in V_{{\pi^{(n)}}}$, we have $f(ah)=\chi(a)f(h)$ for $a\in Z_H$ and $h\in H$. In the above lemma, if $a\in Z_H\cap{\mathbf{s}}(M(F))$, we have $\chi (a)=1$ by the automorphy of $f$, namely $\chi$ is a “Hecke character on $Z_H$”. Now define a character $\omega$ on $A_{{\widetilde{M}}}({\mathbb{A}})$ such that $\omega$ is trivial on $A_{{\widetilde{M}}}(F)$ and $$\omega|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}=\chi|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}.$$ Such $\omega$ certainly exists because $\chi|_{A_{{\widetilde{M}}}({\mathbb{A}})\cap H}$ is viewed as a character on the group ${\mathbf{s}}(M(F))\cap( A_{{\widetilde{M}}}({\mathbb{A}})\cap H)\backslash A_{{\widetilde{M}}}({\mathbb{A}})\cap H$, which is a locally compact abelian group naturally viewed as a closed subgroup of the locally compact abelian group $A_{{\widetilde{M}}}(F)\backslash A_{{\widetilde{M}}}({\mathbb{A}})$, and thus it can be extended to $A_{{\widetilde{M}}}(F)\backslash A_{{\widetilde{M}}}({\mathbb{A}})$. For each $f\in {{\pi^{(n)}}}$, viewed as a function on $H=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F){\widetilde{M}^{(n)}}({\mathbb{A}})$, we extend it to a function $f_\omega:A_{{\widetilde{M}}}({\mathbb{A}})H\rightarrow{\mathbb C}$ by $$f_\omega(ah)=\omega(a)f(h),\quad\text{for all $a\in A_{{\widetilde{M}}}({\mathbb{A}})$ and $h\in H$}.$$ This is well-defined because of our choice of $\omega$, and \[L:automorphy\_of\_f\_omega\] The function $f_\omega$ is a function on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ such that $$f_\omega(\gamma m)=f_\omega(m)$$ for all $\gamma\in A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)$ and $m\in{\widetilde{M}^{(n)}}({\mathbb{A}})$. Namely $f_\omega$ is an “automorphic form on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$”. The lemma follows from the definition of $f_\omega$ and the obvious equality $A_{{\widetilde{M}}}({\mathbb{A}})H=A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. The group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}) $ acts on the space of functions of the form $f_\omega$, giving rise to an “automorphic representation” ${\pi^{(n)}}_\omega$ of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}) $, namely $$V_{{\pi^{(n)}}_\omega}:=\{f_\omega:f\in{\pi^{(n)}}\}$$ and $A_{{\widetilde{M}}}({\mathbb{A}})$ acts as the character $\omega$. As abstract representations, we have $$\label{E:pin_omega} {\pi^{(n)}}_\omega\cong\omega\cdot{\pi^{(n)}}$$ where by $\omega\cdot{\pi^{(n)}}$ is the representation of the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ extended from ${\pi^{(n)}}$ by letting $A_{{\widetilde{M}}}({\mathbb{A}})$ act via the character $\omega$. We need to make sure the relation between ${\pi^{(n)}}_\omega$ and its local analogue we constructed in the previous section. For this, let us start with \[L:local\_global\] Let $\pi\cong{\widetilde{\otimes}}'_v\pi_v$ be a genuine admissible representation of ${\widetilde{M}}({\mathbb{A}})$. Let ${\pi^{(n)}}$ be an irreducible quotient of the restriction $\pi|_{{\widetilde{M}^{(n)}}({\mathbb{A}})}$. If we write $${\pi^{(n)}}\cong\underset{v}{{\widetilde{\otimes}}}'{\pi^{(n)}}_v,$$ then each ${\pi^{(n)}}_v$ is an irreducible constituent of the restriction $\pi_v|_{{\widetilde{M}^{(n)}}_r(F_v)}$. Since ${\pi^{(n)}}$ is an irreducible quotient, there is a surjective ${\widetilde{M}^{(n)}}({\mathbb{A}})$ map $$T:{\underset{v}{{\widetilde{\otimes}}'}\pi_v}\rightarrow {\underset{v}{{\widetilde{\otimes}}}'{\pi^{(n)}}_v}.$$ Fix a place $v_0$. Since $T\neq0$, there exists a pure tensor $\otimes w_v\in {{{\widetilde{\otimes}}}'\pi_v}$ such that $T(\otimes w_v)\neq 0$. (Note that, as we have seen, the space of ${{\widetilde{\otimes}}'\pi_v}$ is the space of the usual restricted tensor product $\otimes'_v\pi_v$.) Define $$i:{\pi_{v_0}}\rightarrow {{\widetilde{\otimes}}' \pi_v}$$ by $$i(w)=w\otimes(\otimes_{v\neq v_0} w_v)$$ for $w\in V_{\pi_{v_0}}$. Then the composite $T\circ i: {\pi_{v_0}}\rightarrow {\otimes'_v{\pi^{(n)}}_v}$ is a non-zero ${\widetilde{M}^{(n)}}(F_{v_0})$ intertwining. Let $w\in {\pi_{v_0}}$ be such that $T\circ i(w)\neq 0$. Then $T\circ i(w)$ is a finite linear combination of pure tensors, and indeed it is written as $$T\circ i(w)=x_1\otimes y_1+\cdots+x_t\otimes y_t,$$ where $x_i\in {{\pi^{(n)}}_{v_0}}$ and $y_i\in\otimes'_{v\neq v_0} {{\pi^{(n)}}_v}$. Here one can assume that $y_1,\dots,y_t$ are linearly independent. Let $\lambda: \otimes_{v\neq v_0} {{\pi^{(n)}}_v}\rightarrow{\mathbb C}$ be a linear functional such that $\lambda (y_1)\neq 0$ and $\lambda (y_2)=\cdots=\lambda (y_t)=0$. (Such $\lambda$ certainly exits because $y_1,\dots,y_t$ are linearly independent.) Consider the map $$U:{{\widetilde{\otimes}}'{\pi^{(n)}}_v}\rightarrow {{\pi^{(n)}}_{v_0}}$$ defined on pure tensors by $$U(\otimes x_v)=\lambda (\otimes_{v\neq v_0} x)x_{v_0}.$$ This is a non-zero ${\widetilde{M}^{(n)}}(F_v)$ intertwining map. Moreover the composite $U\circ T\circ i$ gives a non-zero ${\widetilde{M}^{(n)}}(F_v)$ intertwining map from ${\pi_{v_0}}$ to ${{\pi^{(n)}}_{v_0}}$. Hence ${\pi^{(n)}}_{v_0}$ is an irreducible constituent of the restriction $\pi_{v_0}|_{{\widetilde{M}^{(n)}}(F_{v_0})}$. By taking $k=1$ in the above lemma, one can see that if one writes $${\pi^{(n)}}_i\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_{i,v}$$ then each local component ${\pi^{(n)}}_{i,v}$ is an irreducible constituent of $\pi_{i,v}|_{{\widetilde{\operatorname{GL}}}_{r_i}(F_v)}$ where $\pi_{i,v}$ is the $v$-component of $\pi_i\cong\underset{v}{{\widetilde{\otimes}}'}\pi_{i,v}$. Then one can see that for ${\pi^{(n)}}={\pi^{(n)}}_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_k$, if we write ${\pi^{(n)}}\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_v$, we have $${\pi^{(n)}}_v\cong{\pi^{(n)}}_{1,v}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{k,v}$$ where the right hand side is the local metaplectic tensor product representation of ${\widetilde{M}^{(n)}}(F_v)$. Also one can see that the character $\omega$ decomposes as $\omega=\underset{v}{{\widetilde{\otimes}}'}\omega_v$ where $\omega_v$ is a character on $A_{{\widetilde{M}}(F_v)}$. Hence by (\[E:pin\_omega\]) we have \[P:global\_local\_pin\_omega\] As abstract representations of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$, we have $${\pi^{(n)}}_\omega\cong\underset{v}{{\widetilde{\otimes}}'}{\pi^{(n)}}_{\omega_v},$$ where $${\pi^{(n)}}_{\omega_v}=\omega_v\cdot {\pi^{(n)}}_{1,v}{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}{\pi^{(n)}}_{k,v}$$ is the representation of $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ as defined in the previous section. Now that we have constructed the representation ${\pi^{(n)}}_\omega$ of ${\widetilde{M}^{(n)}}({\mathbb{A}})$, we can construct an automorphic representation of ${\widetilde{M}}({\mathbb{A}})$ analogously to the local case by inducing it to ${\widetilde{M}}({\mathbb{A}})$, though we need extra care for the global case. First consider the compactly induced representation $${\operatorname{c-Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega =\{\varphi:{\widetilde{M}}({\mathbb{A}})\rightarrow{\pi^{(n)}}_\omega\}$$ where $\varphi$ is such that $\varphi(hm)={\pi^{(n)}}_\omega(h)\varphi(m)$ for all $ h\in A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$ and $m\in{\widetilde{M}}({\mathbb{A}})$, and the map $m\mapsto \varphi(m;1)$ is a smooth function on ${\widetilde{M}}({\mathbb{A}})$ whose support is compact modulo $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. (Note here that for each $\varphi\in{\operatorname{Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega$ and $m\in{\widetilde{M}}({\mathbb{A}})$, $\varphi(m)\in V_{{\pi^{(n)}}_\omega}$ is a function on $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. For $m'\in A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$, we use the notation $\varphi(m;m')$ for the value of $\varphi(m)$ at $m'$ instead of writing $\varphi(m)(m')$.) Also consider the metaplectic restricted tensor product $$\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v},$$ where for almost all $v$ at which all the data defining ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ are unramified, we choose the spherical vector $\varphi_v^\circ\in {\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ to be the one defined by $$\varphi_v^\circ(m)=\begin{cases}{\pi^{(n)}}_{\omega_v}(h)f_v^\circ&\text{if $m=h(k,1)$ for $h\in A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ and $(k,1)\in{\widetilde{M}}({\mathcal{O}_{F_v}})$};\\ 0&\text{otherwise}, \end{cases}$$ where $f_v^\circ\in{\pi^{(n)}}_{\omega_v}$ is the spherical vector defining the restricted metaplectic tensor product ${\pi^{(n)}}_\omega={\widetilde{\otimes}}_v'{\pi^{(n)}}_{\omega_v}$. (Let us mention that we do not know if the dimension of the spherical vectors in ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ is one or not.) One has the injection $$T:\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v} {\ensuremath{\lhook\joinrel\relbar\joinrel\rightarrow}}{\operatorname{c-Ind}}_{A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}{\pi^{(n)}}_\omega$$ given by $T(\otimes_v\varphi_v)(m)=\otimes_v \varphi_v(m_v)\in{\widetilde{\otimes}}_v'{\pi^{(n)}}_{\omega_v}$. The reason the image of $T$ lies in the compactly induced space is because for almost all $v$, the support of $\varphi^\circ$ is $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v){\widetilde{M}}({\mathcal{O}_{F_v}})$ and for all $v$ the index of $A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)$ in ${\widetilde{M}}(F_v)$ is finite by (\[E:finite\_quotient\]). (Indeed, the support property and the finiteness of this index imply that $T$ is actually onto as well, though we do not use this fact.) Let $$V({\pi^{(n)}}_\omega)= T\left(\underset{v}{{\widetilde{\otimes}}'}{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}\right),$$ namely $V({\pi^{(n)}}_\omega) $ is the image of $T$. For each $\varphi\in V({\pi^{(n)}}_\omega)$, define ${\widetilde{\varphi}}:{\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ by $$\label{E:definition} {\widetilde{\varphi}}(m)=\sum_{\gamma\in A_{M}M^{(n)}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1).$$ Let us note that by $A_{M}M^{(n)}(F)$ we mean $p(A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F))$, which is not necessarily the same as $A_{M}(F){M^{(n)}}(F)$, and $${\mathbf{s}}(A_{M}M^{(n)}(F))=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)\subseteq A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}}).$$ By the automorphy of ${\pi^{(n)}}_\omega$, $\varphi$ is left invariant on ${\mathbf{s}}(A_{M}M^{(n)}(F))$ and hence the sum is well-defined. Also note that for each fixed $m\in{\widetilde{M}}({\mathbb{A}})$ the map $m'\mapsto\varphi(m'm;1)$ is compactly supported modulo $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})$. Now by our assumption on $A_{{\widetilde{M}}}$ (Hypothesis ($\ast$)), the image of $M(F)$ is discrete in $A_M({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$, and hence the group $A_{M}M^{(n)}(F)\backslash M(F)$ naturally viewed as a subgroup of $A_M({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete. Now a discrete subgroup is always closed by [@Deitmar Lemma 9.1.3 (b)]. Thus the above sum is a finite sum, and in particular the sum is convergent. Moreover one can find $\varphi$ with the property that the support of the map $m'\mapsto\varphi(m';1)$ is small enough so that if $\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)$, then $\varphi(\gamma;1)\neq 0$ only at $\gamma=1$. Thus the map $\varphi\mapsto{\widetilde{\varphi}}$ is not identically zero. \[R:hypothesis\] It should be mentioned here that Hypothesis ($\ast$) is needed to make sure that the sum in (\[E:definition\]) is convergent and not identically zero. The author suspects that either one can always find $A_{{\widetilde{M}}}$ so that Hypothesis ($\ast$) is satisfied (which is the case if $n=2$), or even without Hypothesis ($\ast$) one can show that the sum in (\[E:definition\]) is convergent and not identically zero. But the thrust of this paper is our application to symmetric square $L$-functions ([@Takeda1; @Takeda2]) for which we only need the case for $n=2$. One can verify that ${\widetilde{\varphi}}$ is a smooth automorphic form on ${\widetilde{M}}({\mathbb{A}})$: The automorphy is clear. The smootheness and $K_f$-finiteness follows from the fact that at each non-archimedean $v$, the induced representation ${\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ is smooth and admissible. That ${\widetilde{\varphi}}$ is ${\mathcal{Z}}$-finiteness and of uniform moderate growth follows from the analogous property of $\varphi({\mathbf{s}}(\gamma)m)$, because the Lie algebra of ${\widetilde{M}}(F_v)$ at archimedean $v$ is the same as that of ${\widetilde{M}^{(n)}}(F_v)$. As we already mentioned, the sum in (\[E:definition\]) is finite. But [*a priori*]{} which $\gamma\in A_{M}M^{(n)}(F)\backslash M(F)$ contributes to the above sum depends on $m$. Yet, one can show that as $m$ runs through ${\widetilde{M}}({\mathbb{A}})$, only finitely many $\gamma$’s contribute. To show it, we need the following lemma. \[L:strong\_approximation\] Let $S$ be a finite set of places including all the infinite places and let $\mathcal{O}_S=\prod_{v\notin S}{{\mathcal{O}_F}}_v$. Then the group $$F^\times{\mathbb{A}}^{\times n}\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times$$ is finite. It is well-known that the strong approximation theorem implies $${\mathbb{A}}^\times=F^\times \prod_{v\in S}F_v^\times\mathcal{O}_S^\times,$$ and the index of $(F_v^\times)^n$ in $F_v^\times$ is finite. This proves the lemma. Occasionally, $F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times$ can be shown to be the trivial group. This is the case for example if $n=2$ and $F={\mathbb{Q}}$. But there are cases where it is not trivial even when $n=2$. An interested reader might want to look at [@Kable Appendix]. \[L:finite\_sum\] For each $\varphi\in V({\pi^{(n)}}_\omega)$, there exist finitely many $\gamma_1,\dots,\gamma_N\in A_{M}M^{(n)}(F)\backslash M(F)$, depending only on $\varphi$, such that $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\varphi({\mathbf{s}}(\gamma_i) m;1).$$ We may assume that $\varphi$ is of the form $T(\otimes_v\varphi_v)$ for a simple tensor $\otimes_v\varphi_v$. Then there exists a finite set $S$ of places such that for all $k\in\kappa(K_S)$, we have $k\cdot\varphi=\varphi$, where $K_S=\prod_{v\notin S}M({{\mathcal{O}_F}}_v)\subset M({\mathbb{A}})$ and $\kappa: M({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}})$ is the section $m\mapsto(m,1)$. Namely the stabilizer of $\varphi$ contains $\kappa(K_S)$. Then one can see that $${\operatorname{supp}}(\varphi)={\operatorname{supp}}(m\cdot\varphi)$$ for all $m\in {\widetilde{M}^{(n)}}({\mathbb{A}})\kappa(K_S)$. Also we have ${\widetilde{\varphi}}({\mathbf{s}}(\gamma)m)={\widetilde{\varphi}}(m)$. Hence, noting that ${\widetilde{\varphi}}(m)=\widetilde{m\cdot\varphi}(1)$, we see that the $\gamma$’s that contribute to the sum in (\[E:definition\]) depend only on the class in $${\mathbf{s}}(M(F)){\widetilde{M}^{(n)}}({\mathbb{A}})\kappa(K_S)\backslash{\widetilde{M}}({\mathbb{A}}).$$ But one can see that this set can be identified with the product of $k$ copies of $$F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times,$$ where $\mathcal{O}_S=\prod_{v\notin S}{{\mathcal{O}_F}}_v$, and this set is finite by Lemma \[L:strong\_approximation\]. This implies that there are only finitely many $\gamma_1,\dots,\gamma_N\in A_{M}(F)M^{(n)}(F)\backslash M(F)$ such that $\varphi({\mathbf{s}}(\gamma_i)m;1)\neq 0$ for at least some $m\in{\widetilde{M}}({\mathbb{A}})$. This completes the proof. \[T:main\] Let $${\widetilde{V}}({\pi^{(n)}}_\omega)=\{{\widetilde{\varphi}}:\varphi\in V({\pi^{(n)}}_\omega)\}$$ and $\pi_\omega$ an irreducible constituent of ${\widetilde{V}}({\pi^{(n)}}_\omega)$. Then it is an irreducible automorphic representation of ${\widetilde{M}}({\mathbb{A}})$ and $$\pi_\omega\cong\underset{v}{{\widetilde{\otimes}}}'\pi_{\omega_v},$$ where $\pi_{\omega_v} $ is the local metaplectic tensor product of Mezo. Also the isomorphism class of $\pi_\omega$ depends only on the choice of the character $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})}$. Since the map $\varphi\mapsto{\widetilde{\varphi}}$ is ${\widetilde{M}}({\mathbb{A}})$-intertwining, the space ${\widetilde{V}}({\pi^{(n)}}_\omega)$ provides a space of (possibly reducible) automorphic representation of ${\widetilde{M}}({\mathbb{A}})$. Hence $\pi_\omega$ is an automorphic representation of ${\widetilde{M}}({\mathbb{A}})$. Since each $\pi_i$ is unitary, so is each ${\pi^{(n)}}_i$, from which one can see that ${\pi^{(n)}}_\omega$ is unitary. Since $V({\pi^{(n)}}_\omega)$ is a subrepresentation of the compactly induced representation induced from the unitary ${\pi^{(n)}}_\omega$, $V({\pi^{(n)}}_\omega)$ is unitary. Hence $\pi_\omega$, which is a subquotient of $V({\pi^{(n)}}_\omega)\cong\underset{v}{{\widetilde{\otimes}}}'{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$, is actually a quotient of $\underset{v}{{\widetilde{\otimes}}}'{\operatorname{Ind}}_{A_{{\widetilde{M}}}(F_v){\widetilde{M}^{(n)}}(F_v)}^{{\widetilde{M}}(F_v)}{\pi^{(n)}}_{\omega_v}$ by admissibility. With this said, one can derive the isomorphism $\pi_\omega\cong\underset{v}{{\widetilde{\otimes}}}'\pi_{\omega_v}$ from Lemma \[L:local\_global\]. Since the local $\pi_{\omega_v}$ depends only on the choice of $\omega_v|_{Z_{{\widetilde{\operatorname{GL}}}_r}(F_v)}$, the global $\pi_\omega$ depends only on $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})}$ up to equivalence. We call the above constructed $\pi_\omega$ the global metaplectic tensor product of $\pi_1,\dots,\pi_k$ (with respect to $\omega$) and write $$\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega.$$ We do not know if the multiplicity one theorem holds for the group ${\widetilde{M}}({\mathbb{A}})$, and hence do not know if the space ${\widetilde{V}}({\pi^{(n)}}_\omega)$ has only one irreducible constituent. In this sense, the definition of $\pi_\omega$ depends on the choice of the irreducible constituent. For this reason, the metaplectic tensor product should be construed as an equivalence class of automorphic representations, although we know more or less explicit way of expressing automorphic forms in $\pi_\omega$. **The uniqueness** ------------------ Just like the local case, the metaplectic tensor product of automorphic representations is unique up to twist. \[P:global\_uniqueness\] Let $\pi_1,\dots,\pi_k$ and $\pi'_1,\dots,\pi'_k$ be unitary automorphic subrepresentations of ${\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}})$. They give rise to isomorphic metaplectic tensor products with a character $\omega$, [[*i.e.* ]{}]{}$$(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega\cong (\pi'_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi'_k)_\omega,$$ if and only if for each $i$ there exists an automorphic character $\omega_i$ of ${\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}({\mathbb{A}})$ such that $\pi_i\cong\omega_i\otimes\pi'_i$. By Theorem \[T:main\], the global metaplectic tensor product is written as the metaplectic restricted tensor product of the local metaplectic tensor products of Mezo. Hence by Proposition \[P:local\_uniqueness\] for each $i$ and each place $v$, there is a character $\omega_{i,v}$ on ${\widetilde{\operatorname{GL}}}_{r_i}(F_v)$ trivial on ${\widetilde{\operatorname{GL}}^{(n)}}_{r_i}(F_v)$ such that $\pi_{i, v}\cong\omega_{i, v}\otimes\pi'_{i, v}$. Let $\omega_i={\widetilde{\otimes}}_v'\omega_{i,v}$. Then $\pi_i\cong\omega_i\otimes\pi'_i$. The automorphy of $\omega$ follows from that of $\pi_i$ and $\pi'_i$. This proves the only if part. The if part follows similarly. **Cuspidality and square-integrability** {#S:cuspidality} ---------------------------------------- In this subsection, we will show that the cuspidality and square-integrability are preserved for the metaplectic tensor product. \[T:cuspidal\] Assume $\pi_1,\dots,\pi_k$ are all cuspidal. Then the metaplectic tensor product $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ is cuspidal. Assume $\pi_1,\dots,\pi_k$ are all cuspidal. It suffices to show that for each $\varphi\in V({\pi^{(n)}}_\omega)$ $$\int_{U(F)\backslash U({\mathbb{A}})}{\widetilde{\varphi}}({\mathbf{s}}(u))\,du=0$$ for all unipotent radical $U$ of the standard proper parabolic subgroup of $M$, where recall from Proposition \[P:s\_split\_M\] that the partial set theoretic section ${\mathbf{s}}:M({\mathbb{A}})\rightarrow{\widetilde{M}}({\mathbb{A}})$ is defined (and a group homomorphism) on the groups $M(F)$ and $U({\mathbb{A}})$. We know that ${\widetilde{\varphi}}({\mathbf{s}}(u))$ is a finite sum of $\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)$ for $\gamma\in A_{M}M^{(n)}(F)\backslash M(F)$. Hence it suffices to show $\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)\,du=0$. Note that for each $\gamma={\operatorname{diag}}(\gamma_1,\dots,\gamma_k)$ with $\gamma_i\in{\operatorname{GL}}_{r_i}(F)$, we have $$\gamma_i=\gamma_i\begin{pmatrix}\det(\gamma_i)^{n-1}&\\ &I_{r_i-1}\end{pmatrix}\begin{pmatrix}\det(\gamma_i)^{-n+1}&\\ &I_{r_i-1}\end{pmatrix},$$ where $\gamma_i\begin{pmatrix}\det(\gamma_i)^{n-1}&\\ &I_{r_i-1}\end{pmatrix}\in {\operatorname{GL}}_r^{(n)}(F)$. Hence we may assume $\gamma\in M(F)$ is a diagonal matrix, and so $\gamma u\gamma^{-1}\in U({\mathbb{A}})$. Then we have $$\begin{aligned} \allowdisplaybreaks &\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma){\mathbf{s}}(u);1)\,du\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma) {\mathbf{s}}(u){\mathbf{s}}(\gamma^{-1}){\mathbf{s}}(\gamma);1)\,du\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( u){\mathbf{s}}(\gamma^{-1}))\,du\quad\text{because ${\mathbf{s}}(\gamma){\mathbf{s}}( u){\mathbf{s}}(\gamma^{-1})\in\widetilde{U}({\mathbb{A}})$}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( u\gamma^{-1}))\,du\quad\text{by Proposition \ref{P:s_split_M}}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( \gamma^{-1}u)\,du\quad\text{by change of variables $\gamma u\gamma^{-1}\mapsto u$}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(\gamma){\mathbf{s}}( \gamma^{-1}){\mathbf{s}}(u)\,du\quad\text{by Proposition \ref{P:s_split_M}}\\ =&\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))\,du \quad\text{by Proposition \ref{P:s_split_M}}.\end{aligned}$$ We would like to show this is equal to zero. For this purpose, recall that for each $\gamma$, $\varphi({\mathbf{s}}(\gamma))$ is in the space $V_{{\pi^{(n)}}_\omega}$ and hence is (a finite sum of functions) of the form $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$ with $f_i\in V_{\pi_i}$ and each $f_i$ is a cusp form. We may assume $\varphi({\mathbf{s}}(\gamma))$ is a simple tensor $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$. Now we can write $U=U_1\times\cdots\times U_k$, where each $U_i$ is a unipotent subgroup of ${\operatorname{GL}}_{r_i}$ with at least one of $U_i$ non-trivial, and accordingly we denote each element $u\in U$ by $u={\operatorname{diag}}(u_1,\dots,u_k)$. Then by definition of ${\mathbf{s}}$, we have $${\mathbf{s}}(u)=(u, \prod_is_{r_i}(u_i)^{-1}),$$ and $$\begin{aligned} \varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))&=(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)({\mathbf{s}}(u))\\ &=(f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k)(u, \prod_is_{r_i}(u_i)^{-1})\\ &=\left(\prod_is_{r_i}(u_i)^{-1}\right)f_1(u_1,1)\cdots f_k(u_k,1)\quad\text{by definition of $f_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}f_k$}\\ &=f_1(u_1,s_{r_i}(u_i)^{-1})\cdots f_k(u_k,s_{r_k}(u_k)^{-1})\quad \text{because each $f_i$ is genuine}\\ &=f_1({\mathbf{s}}_{r_1}(u_1))\cdots f_k({\mathbf{s}}_{r_k}(u_k))\quad \text{by definition of ${\mathbf{s}}_{r_i}$}.\end{aligned}$$ Hence $$\int_{U(F)\backslash U({\mathbb{A}})}\varphi({\mathbf{s}}(\gamma);{\mathbf{s}}(u))\,du =\prod_{i=1}^k\int_{U_i(F)\backslash U_i({\mathbb{A}})}f_i({\mathbf{s}}_{r_i}(u_i))\,du_i.$$ This is equal to zero because each $f_i$ is cuspidal and at least one of $U_i$ is non-trivial. Next let us take care of the square-integrability. \[T:square\_integrable\] Assume $\pi_1,\dots,\pi_k$ are all square-integrable modulo center. Then the metaplectic tensor product $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$ is square-integrable modulo center. We need a few lemmas for the proof of this theorem. \[L:quotient\_measure\] Let $G$ be a locally compact group and $H, N\subset G$ be closed subgroups such that $NH$ is a closed subgroup. Further assume that the quotient measures for $N\backslash G, H\backslash NH$ and $NH\backslash G$ all exist. (Recall that in general the quotient measure for $N\backslash G$ exists if the modular characters of $G$ and $N$ agree on $N$.) Then $$\begin{aligned} \int_{N\backslash G}f(g)\;dg&=\int_{NH\backslash G}\int_{N\backslash NH}\;f(hg)\;dh\;dg\\ &=\int_{NH\backslash G}\int_{N\cap H\backslash H}\;f(hg)\;dh\;dg.\end{aligned}$$ for all $f\in L^1(N\backslash G)$. The first equality is [@Bourbaki Cor. 1 VII 47], and the second equality follows from the natural identification $N\backslash NH\cong N\cap H\backslash H$. Now let $f:{\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ be any function. Then the absolute value $|f|$ is non-genuine in the sense that it factors through $M({\mathbb{A}})$. Also we let $$Z^{(n)}_M({\mathbb{A}}):= \{\begin{pmatrix}a_1^nI_{r_1}&&\\ &\ddots&\\ &&a_k^nI_{r_k}\end{pmatrix}:a_i\in{\mathbb{A}}^\times\}.$$ This is a closed subgroup by Lemma \[L:closed\_subgroup\_local\] and \[L:closed\_subgroup\_local\_global\]. Note the inclusions $$Z_M^{(n)}({\mathbb{A}})\subseteq p(Z_{{\widetilde{M}}}({\mathbb{A}}))\subseteq Z_M({\mathbb{A}}),$$ where all the groups are closed subgroups of $M({\mathbb{A}})$. Then we have \[L:M\_square\_integrable\] Let $f:M(F)\backslash {\widetilde{M}}({\mathbb{A}})\rightarrow{\mathbb C}$ be an automorphic form with a unitary central character. Then $f$ is square-integrable modulo the center $Z_{{\widetilde{M}}}({\mathbb{A}})$ if and only if $|f|\in L^2( Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}}))$ where $|f|$ is viewed as a function on $M({\mathbb{A}})$ as noted above. Let $f$ be an automorphic form on ${\widetilde{M}}({\mathbb{A}})$ with a unitary central character. Since $|f|$ is non-genuine, we have $$\int_{Z_{{\widetilde{M}}}({\mathbb{A}}) M(F)\backslash{\widetilde{M}}({\mathbb{A}})}|f(\tilde{m})|^2\;d\tilde{m} =\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm,$$ where recall that $p:{\widetilde{M}}({\mathbb{A}})\rightarrow M({\mathbb{A}})$ is the canonical projection. Note that the quotient measure on the right hand side exists because the group $p(Z_{{\widetilde{M}}}({\mathbb{A}}))M(F)$ is closed by [@MW Lemma I.1.5, p.8] and is unimodular because $p(Z_{{\widetilde{M}}}({\mathbb{A}}))$ is unimodular and $M(F)$ is discrete and countable. By Lemma \[L:quotient\_measure\], we have $$\int_{Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm =\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} \int_{Z_M^{(n)}({\mathbb{A}}) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}}))}|f(\kappa(zm))|^2\;dz\;dm.$$ Since for each fixed $m\in M({\mathbb{A}})$, the function $z\mapsto f(\kappa(zm))$ is a smooth function on $p(Z_{{\widetilde{M}}}({\mathbb{A}}))$, there exists a finite set $S$ of places such that for all $z'\in p(Z_{{\widetilde{M}}}(\mathcal{O}_S))=Z_M(\mathcal{O}_S)\cap p(Z_{{\widetilde{M}}}({\mathbb{A}}))$ we have $f(\kappa(z'zm))=f(\kappa(zm))$. Hence the inner integral of the above integral is written as $$\label{E:square_integrable} \int_{Z^{(n)}_M({\mathbb{A}})p(Z_{{\widetilde{M}}}(\mathcal{O}_S)) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}}))}|f(\kappa(zm))|^2\;dz.$$ Note that we have the inclusion $$Z^{(n)}_M({\mathbb{A}})p(Z_{{\widetilde{M}}}(\mathcal{O}_S)) p(Z_{{\widetilde{M}}}(F))\backslash p(Z_{{\widetilde{M}}}({\mathbb{A}})) \subseteq Z^{(n)}_M({\mathbb{A}})Z_M(\mathcal{O}_S)Z_M(F)\backslash Z_M({\mathbb{A}}),$$ because $p(Z_{{\widetilde{M}}}(\mathcal{O}_S))\cap p(Z_{{\widetilde{M}}}(F))=Z_M(\mathcal{O}_S)\cap Z_M(F)=1$, and note that $Z^{(n)}_M({\mathbb{A}})Z_M(\mathcal{O}_S)Z_M(F)\backslash Z_M({\mathbb{A}})$ can be identified with the product of $k$ copies of $$F^\times{\mathbb{A}}^{\times n}\;\mathcal{O}_S^\times\backslash{\mathbb{A}}^\times.$$ By Lemma \[L:strong\_approximation\], we know that this is a finite group, and hence the integral in (\[E:square\_integrable\]) is just a finite sum. Thus for some finite $z_1,\dots,z_N\in p(Z_{{\widetilde{M}}}({\mathbb{A}}))$, we have $$\begin{aligned} \int_{Z^{(n)}_{M}({\mathbb{A}}) M(F)\backslash M({\mathbb{A}})}|f(\kappa(m))|^2\;dm &=\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} \sum_{i=1}^N |f(\kappa(z_im))|^2\;dm\\ &=\sum_{i=1}^N \int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} |f(\kappa(m))|^2\;dm\\ &=N\int_{p(Z_{{\widetilde{M}}}({\mathbb{A}})) M(F)\backslash M({\mathbb{A}})} |f(\kappa(m))|^2\;dm,\end{aligned}$$ where for the second equality we used $$|f(\kappa(z_im))|=|f((\kappa(z_i)\kappa(m)))| =|\omega(\kappa(z_1))||f(\kappa(m))|=|f(\kappa(m))|$$ where $\omega$ is the central character of $f$ which is assumed to be unitary. The lemma follows from this. \[L:pin\_square\_integrable\] Assume $\pi_1,\dots,\pi_k$ are as in Theorem \[T:square\_integrable\]. Let $\varphi_i\in{\pi^{(n)}}_i$ for $i=1,\dots,k$ and $\varphi=\varphi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\varphi_k\in{\pi^{(n)}}$, which is a function on ${\widetilde{M}^{(n)}}({\mathbb{A}})$. Then $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|\varphi(\kappa(m))|^2\;dm<\infty.$$ Write each element $m\in M({\mathbb{A}})$ as $m={\operatorname{diag}}(g_1,\dots,g_k)$ where $g_i\in {\operatorname{GL}}_{r_i}({\mathbb{A}})$. Then ${\operatorname{diag}}(g_1,\dots,g_k)\in M^{(n)}({\mathbb{A}})$ if and only if $g_i\in {\operatorname{GL}}_{r_i}^{(n)}({\mathbb{A}})$ for all $i$. Hence the integral in the lemma is the product of integrals $$\int_{Z_{{\operatorname{GL}}_{r_i}}^{(n)}({\mathbb{A}}){\operatorname{GL}}_{r_i}^{(n)}(F)\backslash {\operatorname{GL}}_{r_i}^{(n)}({\mathbb{A}})}|\varphi_i(\kappa(g_i))|^2\;dg_i,$$ where $Z_{{\operatorname{GL}}_{r_i}}^{(n)}({\mathbb{A}})$ consists of the elements of the form $a_iI_{r_i}$ with $a_i\in{\mathbb{A}}^{\times n}$. So we have to show that this integral converges. But with Lemma \[L:M\_square\_integrable\] applied to $M={\operatorname{GL}}_{r_i}$, we know $$\int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}_{r_i}(F)\backslash{\operatorname{GL}}_{r_i}({\mathbb{A}})}|\varphi_i(\kappa(g_i))|^2\;dg_i<\infty,$$ because each $\varphi_i$ is square-integrable modulo center. By Lemmas \[L:quotient\_measure\] and \[L:discrete\_sub\], this is written as $$\int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}^{(n)}_{r_i}({\mathbb{A}}){\operatorname{GL}}_{r_i}(F)\backslash{\operatorname{GL}}_{r_i}({\mathbb{A}})} \int_{Z^{(n)}_{{\operatorname{GL}}_{r_i}}({\mathbb{A}}){\operatorname{GL}}^{(n)}_{r_i}(F)\backslash{\operatorname{GL}}^{(n)}_{r_i}({\mathbb{A}})} |\varphi_i(\kappa(m_i'm_i))|^2\;dm_i'\;dm_i<\infty.$$ In particular the inner integral converges, which proves the lemma. Now we are ready to prove Theorem \[T:square\_integrable\]. By Lemma \[L:M\_square\_integrable\], we have only to show $$\int_{Z_M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m))|^2\;dm<\infty.$$ By Lemma \[L:quotient\_measure\], we have $$\begin{aligned} \label{E:outer_integral} \notag&\int_{Z_M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m))|^2\;dm\\ =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m'm))|^2\;dm'\;dm\\ \notag =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}\left|\sum_{\gamma}\varphi(\kappa(\gamma m'm);1)\right|^2\;dm'\;dm\\ \notag =&\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}\left|\sum_{\gamma}\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2\;dm'\;dm.\end{aligned}$$ Note that by Lemma \[L:finite\_sum\], the sum of the integrand is finite. Also the map $m'\mapsto \left|\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2$ is invariant under $Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)$ on the left. Hence to show the inner integral converges, it suffices to show the integral $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})} \left|\varphi(\kappa(\gamma m);\kappa(\gamma m'\gamma^{-1}))\right|^2\;dm'$$ converges. But this follows from Lemma \[L:pin\_square\_integrable\]. To show the outer integral converges, note that the map $m\mapsto |{\widetilde{\varphi}}(\kappa(m'm))|^2$ is smooth and hence there exists a finite set of places $S$ so that ${\widetilde{\varphi}}(\kappa(m'm k))={\widetilde{\varphi}}(\kappa(m'm k))$ for all $k\in M(\mathcal{O}_S)$. Thus the integral in is (a scalar multiple of) $$\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)}\int_{Z_M^{(n)}({\mathbb{A}})M^{(n)}(F)\backslash M^{(n)}({\mathbb{A}})}|{\widetilde{\varphi}}(\kappa(m'm))|^2\;dm'\;dm.$$ Now the set theoretic map $$F^\times{\mathbb{A}}^{\times n}\underbrace{\mathcal{O}^\times_S\backslash{\mathbb{A}}^\times \times\cdots\times F^\times}_{\text{$k$ copies}}{\mathbb{A}}^{\times n}\mathcal{O}^\times_S \backslash{\mathbb{A}}^\times \rightarrow Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)$$ given by $$(a_1,\dots,a_k)\mapsto\begin{pmatrix}\iota_1(a_1)&&\\ &\ddots&\\ &&\iota_k(a_k)\end{pmatrix}$$ where $\iota_i$ is as in (\[E:iota\]) is a well-defined surjection. Hence Lemma \[L:strong\_approximation\] implies that the set $$Z_M^{(n)}({\mathbb{A}})M^{(n)}({\mathbb{A}})M(F)\backslash M({\mathbb{A}})/M(\mathcal{O}_S)$$ is a finite set. Therefore the outer integral of the above integral is a finite sum and hence converges. This completes the proof. **Twists by Weyl group elements** {#S:Weyl_group_global} --------------------------------- Just as we saw in Section \[S:Weyl\_group\_global\] for the local case, the global metaplectic tensor product behaves in the expected way under the action of the Weyl group elements in $W_M$. Namely \[T:Weyl\_group\_global\] Let $w\in W_M$ be such that $^w({\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}) ={\operatorname{GL}}_{r_{\sigma(1)}}\times\cdots\times{\operatorname{GL}}_{r_{\sigma(k)}}$. Then we have $$^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega \cong(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega,$$ where $w$ is viewed as an element in ${\operatorname{GL}}_r(F)$. Note that each ${\mathbf{s}}(w)\in{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ is written as $\prod_v(w, s_{r,v}(w))$, where we view $(w, s_{r,v}(w))\in{\widetilde{\operatorname{GL}}}_r(F_v)$ as an element of ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ view the natural embedding ${\widetilde{\operatorname{GL}}}_r(F_v)\hookrightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$, and the product $\prod_v$ is literally the product inside ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. Then one can see that $$^w(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega={\widetilde{\otimes}}'_v\, ^w(\pi_{1,v}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{k,v})_{\omega_v}.$$ Hence the theorem follows from the local counter part (Theorem \[T:Weyl\_group\_local\]). The following is immediate: Let $\pi_\omega=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$. For $w$ as in the theorem and each automorphic form ${\widetilde{\varphi}}\in\pi_\omega$, define $^w{\widetilde{\varphi}}:\;^w{\widetilde{M}}(A)\rightarrow{\mathbb C}$ by $$^w{\widetilde{\varphi}}(m)={\widetilde{\varphi}}({\mathbf{s}}(w)^{-1}m{\mathbf{s}}(w))$$ for $m\in\;^w{\widetilde{M}}({\mathbb{A}})$. Then the representation $^w\pi_\omega$ is realized in the space $$\{^w{\widetilde{\varphi}}:{\widetilde{\varphi}}\in V_{\pi_\omega}\}.$$ Let us mention the following subtle point. Here we have (at least) two different realizations of $^w\pi_\omega$ in a space of automorphic forms on $^w{\widetilde{M}}({\mathbb{A}})$, the one is in the space $\{^w{\widetilde{\varphi}}:{\widetilde{\varphi}}\in V_{\pi_\omega}\}$ as in the proposition and the other as in the definition of the metaplectic tensor product $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ by choosing an appropriate $A_{^w{\widetilde{M}}}$ that satisfies Hypothesis ($\ast$) with respect to the Levi $^w{\widetilde{M}}$ (if possible at all). Without the multiplicity one property for the group $^w{\widetilde{M}}$, we do not know if they coincide. But one can see that if $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}$, then the group $^wA_{{\widetilde{M}}}:=wA_{{\widetilde{M}}}w^{-1}$ satisfies Hypothesis ($\ast$) with respect to $^w{\widetilde{M}}$. Then if we define $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ by choosing $A_{^w{\widetilde{M}}}=\,^wA_{{\widetilde{M}}}$, one can see from the construction of our metaplectic tensor product that the space of $(\pi_{\sigma(1)}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_{\sigma(k)})_\omega$ is indeed a space of automorphic forms of the form $^w{\widetilde{\varphi}}$ for ${\widetilde{\varphi}}\in V_{\pi_\omega}$. **Compatibility with parabolic induction** {#S:parabolic_induction_global} ------------------------------------------ Just as the local case, we have the compatibility with parabolic inductions. But before stating the theorem, let us mention Let $P=MN$ be the standard parabolic subgroup of ${\operatorname{GL}}_r$. Then ${\widetilde{M}}({\mathbb{A}})$ normalizes $N({\mathbb{A}})^\ast$, where $N({\mathbb{A}})^\ast$ is the image of $N({\mathbb{A}})$ under the partial section ${\mathbf{s}}:{\operatorname{GL}}_r({\mathbb{A}})\rightarrow{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$. \[L:normalizer\_global\] One can prove it by using the local analogue (Lemma \[L:normalizer\_local\]). Namely let ${\tilde{m}}=(m,1)\in{\widetilde{M}}({\mathbb{A}})$, so ${\tilde{m}}^{-1}=(m^{-1},\tau_r(m,m^{-1})^{-1})$. Also let $n^\ast=(n,s_r(n)^{-1})\in N({\mathbb{A}})^\ast$. Then $$\begin{aligned} {\tilde{m}}n^\ast{\tilde{m}}^{-1}&=(m,1)(n,s_r(n)^{-1})(m^{-1},\tau_r(m,m^{-1})^{-1})\\ &=(mnm^{-1}, s_r(n)^{-1}\tau_r(m,n)\tau(m,m^{-1})^{-1}\tau_r(mn,m^{-1})).\end{aligned}$$ Then one needs to show $$s_r(n)^{-1}\tau_r(m,n)\tau_r(m,m^{-1})^{-1}\tau_r(mn,m^{-1}) =s(mnm^{-1})^{-1},$$ so that ${\tilde{m}}n^\ast{\tilde{m}}^{-1}=(mnm^{-1})^\ast\in N({\mathbb{A}})^\ast$. But one can show it by arguing “semi-locally”. Namely for a sufficiently large finite set $S$ of places, we have $$\begin{aligned} &s_r(n)^{-1}\tau_r(m,n)\tau_r(m,m^{-1})^{-1}\tau_r(mn,m^{-1})\\ =&\prod_{v\in S}s_r(n_v)^{-1}\tau_r(m_v,n_v)\tau_r(m_v,m_v^{-1})^{-1}\tau_r(m_vn_v,m_v^{-1})\\ =&\prod_{v\in S}s_r(n_v)^{-1}\sigma_r(m_v,n_v)\frac{s_r(m_v)s_r(n_v)}{s_r(m_vn_v)}\\ &\qquad\qquad\cdot\sigma_r(m_v,m_v^{-1})^{-1}\frac{s_r(m_vm_v^{-1})}{s_r(m_v)s_r(m_v^{-1})} \sigma_r(m_vn_v,m_v^{-1})\frac{s_r(m_vn_v)s_r(m_v^{-1})}{s_r(m_vn_vm_v^{-1})}\\ =&\prod_{v\in S}s_r(m_vn_vm_v^{-1})^{-1}\\ =&s_r(mnm^{-1})^{-1},\end{aligned}$$ where for the second equality we used (\[E:tau\_sigma\]), for the third equality we used the same cocycle computation as in the proof of Lemma \[L:normalizer\_local\] and finally for the last equality we used $s_r(m_vn_vm_v^{-1})=1$ for all $v\notin S$. Let us mention that for the case at hand one can prove this lemma as we did here. However this lemma holds not just for our ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ but for covering groups in general . (See [@MW I.1.3(4), p.4].) At any rate, this lemma allows one to form the global induced representation $${\operatorname{Ind}}_{{\widetilde{M}}({\mathbb{A}})N({\mathbb{A}})^\ast}^{{\widetilde{\operatorname{GL}}}_r({\mathbb{A}})}\pi$$ for an automorphic representation $\pi$ of ${\widetilde{M}}({\mathbb{A}})$, and hence one can form Eisenstein series on ${\widetilde{\operatorname{GL}}}_r({\mathbb{A}})$ just like the non-metaplectic case. With this said, we have \[T:induction\_global\] Let $P=MN\subseteq{\operatorname{GL}}_r$ be the standard parabolic subgroup whose Levi part is $M={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_k}$. Further for each $i=1,\dots,k$ let $P_i=M_iN_i\subseteq{\operatorname{GL}}_{r_i}$ be the standard parabolic of ${\operatorname{GL}}_{r_i}$ whose Levi part is $M_i={\operatorname{GL}}_{r_{i,1}}\times\cdots\times{\operatorname{GL}}_{r_{i,l_i}}$. For each $i$, assume we can find $A_{{\widetilde{M}}_i}$ that satisfies Hypothesis ($\ast$) with respect to $M_i$ (which is always the case if $n=2$), and we are given an automorphic representation $$\sigma_i:=(\tau_{i,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{i,l_i})_{\omega_i}$$ of ${\widetilde{M}}_i({\mathbb{A}})$, which is given as the metaplectic tensor product of the unitary automorphic subrepresentations $\tau_{i,1},\dots,\tau_{i,l_i}$ of ${\widetilde{\operatorname{GL}}}_{r_{i,1}}({\mathbb{A}}),\dots,{\widetilde{\operatorname{GL}}}_{r_{i,l_i}}({\mathbb{A}})$, respectively. Assume that $\pi_i$ is an irreducible constituent of the induced representation ${\operatorname{Ind}}_{{\widetilde{P}}_i({\mathbb{A}})}^{{\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}})}\sigma_i$ and is realized as an automorphic subrepresentation. Then the metaplectic tensor product $$\pi_\omega:=(\pi_1\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_\omega$$ is an irreducible constituent of the induced representation $${\operatorname{Ind}}_{{\widetilde{Q}}({\mathbb{A}})}^{{\widetilde{M}}({\mathbb{A}})}(\tau_{1,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{1, l_1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\, \tau_{k,1}\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\tau_{k, l_k})_\omega,$$ where $Q$ is the standard parabolic of $M$ whose Levi part is $M_1\times\cdots\times M_k$, where $M_i\subseteq{\operatorname{GL}}_{r_i}$ for each $i$. This follows from its local analogue (Theorem \[T:induction\_local\]) and the local-global compatibility of the metaplectic tensor product $\pi_\omega\cong{\widetilde{\otimes}}'\pi_{\omega_v}$. Just as we mentioned in Remark \[R:induction\_local\] for the local case, in the above theorem one may replace “constituent” by “irreducible subrepresentation” or “irreducible quotient”, and the analogous statement still holds. **Restriction to a smaller Levi** {#S:restriction} --------------------------------- As the last thing in this paper, let us mention an important property of the metaplectic tensor product which one needs when one computes constant terms of metaplectic Eisenstein series. (See [@Takeda2].) Both locally and globally, let $$M_2={\operatorname{GL}}_{r_2}\times\cdots\times{\operatorname{GL}}_{r_k} =\{\begin{pmatrix}I_{r_1}&&&\\ &g_2&&\\ &&\ddots&\\ &&&g_k\end{pmatrix}\in M: g_i\in{\operatorname{GL}}_{r_i}\}$$ be viewed as a subgroup of $M$ in the obvious way. We view ${\operatorname{GL}}_{r-r_1}$ as a subgroup of ${\operatorname{GL}}_r$ embedded in the right lower corner, and so $M_2$ can be also viewed as a Levi subgroup of ${\operatorname{GL}}_{r-r_1}$ embedded in this way. Both locally and globally, we let $$\tau_{M_2}:M_2\times M_2\rightarrow\mu_n$$ be the block-compatible 2-cocycle on $M_2$ defined analogously to $\tau_M$. One can see that the block-compatibility of $\tau_M$ and $\tau_{M_2}$ implies $$\label{E:tau_M2} \tau_{M_2}={\tau_M}|_{M_2\times M_2},$$ which gives the embeddings $$\begin{aligned} {\widetilde{M}}_2\subseteq{\widetilde{M}}\hookrightarrow{\widetilde{\operatorname{GL}}}_r.\end{aligned}$$ (Note that the last map is not the natural inclusion because here ${\widetilde{M}}$ is actually ${{^c\widetilde{M}}}$, and that is why we use $\hookrightarrow$ instead of $\subseteq$.) For each automorphic form ${\widetilde{\varphi}}\in V_{\pi_\omega}$ in the space of the metaplectic tensor product, one would like to know which space the restriction ${\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}$ belongs to. Just like the non-metaplectic case, it would be nice if this restriction is simply in the space of the metaplectic tensor product of $\pi_2,\dots,\pi_k$ with respect to the character $\omega$ restricted to, say, $A_{{\widetilde{M}}}\cap{\widetilde{M}}_2$. But as we will see, this is not necessarily the case. The metaplectic tensor product is more subtle. Let us first introduce the subgroup $A_{{\widetilde{M}}_2}$ of ${\widetilde{M}}_2$ which plays the role analogous to that of $A_{{\widetilde{M}}}$: $$A_{{\widetilde{M}}_2}(R):=\{(\begin{pmatrix}I_{r_1}&\\ &A_2\end{pmatrix},\xi): (\begin{pmatrix}a_1I_{r_1}&\\ &A_2\end{pmatrix},\xi)\in A_{{\widetilde{M}}}(R) \text{ for some $a_1\in R^{\times n}$}\}.$$ Note that $A_{{\widetilde{M}}}(R)\cap{\widetilde{M}}_2(R) \subseteq A_{{\widetilde{M}}_2}(R)$, but the equality might not hold in general. The following lemma implies that $A_{{\widetilde{M}}_2}$ is abelian. Let $(\begin{pmatrix}I_{r_1}&\\ &A_2\end{pmatrix}, \xi), (\begin{pmatrix}I_{r_1}&\\ &A'_2\end{pmatrix}, \xi')\in A_{{\widetilde{M}}_2}(R)$. Then $$\tau_{M_2}(A_2,A'_2)=\tau_{M_2}(A'_2,A_2).$$ This follows by the block-compatibility of $\tau_M$ and the fact that $A_{{\widetilde{M}}}(R)$ is abelian. Also one can see that the image of $A_{{\widetilde{M}}_2}(R)$ under the canonical projection is closed, and hence $A_{{\widetilde{M}}_2}(R)$ is closed. Another property to be mentioned is \[L:M\_2\] For $R={\mathbb{A}}$ or $F_v$, we have $$A_{{\widetilde{M}}_2}(R){\widetilde{M}^{(n)}}_2(R)=A_{{\widetilde{M}}}(R){\widetilde{M}^{(n)}}(R)\cap {\widetilde{M}}_2(R).$$ Also for global $F$ we have $$A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F)=A_{{\widetilde{M}}}{\widetilde{M}^{(n)}}(F)\cap {\mathbf{s}}(M_2(F)),$$ where by definition $$A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F):=A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})\cap{\mathbf{s}}(M(F)),$$ which is not necessarily the same as $A_{{\widetilde{M}}_2}(F){\widetilde{M}^{(n)}}_2(F)$. The lemma can be verified by direct computations. Note that for both cases, the inclusion $\subseteq$ is immediate. To show the reverse inclusion, we need that if $a\in A_{{\widetilde{M}}}(R)$ and $m\in {\widetilde{M}^{(n)}}(R)$ are such that $am\in A_{{\widetilde{M}}}(R){\widetilde{M}^{(n)}}(R)\cap {\widetilde{M}}_2(R)$, one can always write $a=a_2a_1$ with $a_2\in A_{{\widetilde{M}}_2}(R)$ such that $a_1m\in{\widetilde{M}^{(n)}}_2(R)$, and hence $am=a_2(a_1m)\in A_{{\widetilde{M}}_2}(R){\widetilde{M}^{(n)}}_2(R)\subseteq A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F)$. Now assume that our group $A_{{\widetilde{M}}}$ satisfies the following: The group $A_{{\widetilde{M}}}$ satisfies: 1. $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) 2. $A_{{\widetilde{M}}_2}$ as defined above contains the center $Z_{{\widetilde{\operatorname{GL}}}_{r-r_1}}$. 3. $A_{{\widetilde{M}}_2}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}_2$. As an example of $A_{{\widetilde{M}}}$ satisfying the above hypothesis, we have If $n=2$, the choice of $A_{{\widetilde{M}}}$ as in Proposition \[P:A\_M\_for\_n=2\] satisfies this hypothesis. Moreover, one has $$A_{{\widetilde{M}}_2}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_2$$ both locally and globally. This can be merely checked case-by-case. Next for each $\delta\in{\operatorname{GL}}_{r_1}(F)$, define $\omega_{\delta}:A_{{\widetilde{M}}_2}(F)\backslash A_{{\widetilde{M}}_2}({\mathbb{A}})\rightarrow{\mathbb C}^1$ by $$\omega_{\delta}(a)=\omega({\mathbf{s}}(\delta) a {\mathbf{s}}(\delta^{-1})).$$ Since ${\mathbf{s}}(\delta) A_{{\widetilde{M}}_2}({\mathbb{A}}){\mathbf{s}}(\delta^{-1})=A_{{\widetilde{M}}_2}({\mathbb{A}})$ and $A_{{\widetilde{M}}_2}({\mathbb{A}})\subseteq A_{{\widetilde{M}}}({\mathbb{A}})$, this is well-defined, and since ${\mathbf{s}}$ is a homomorphism on $M(F)$, $\omega_\delta$ is a character. Indeed, one can compute $$\label{E:omega_delta} \omega_{\delta}(a)=(\det\delta,\det a)^{1+2c}\omega(a)$$ because one can see $${\mathbf{s}}(\delta) a {\mathbf{s}}(\delta^{-1})=(1,(\det\delta,\det a)^{1+2c})a$$ and $\omega$ is genuine. Hence for each $a\in A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})$ we have $\omega_\delta(a)=\omega(a)$ because $(\det\delta,\det a)=1$, namely $$\omega_\delta|_{A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})} =\omega|_{A_{{\widetilde{M}}_2}({\mathbb{A}})\cap A_{{\widetilde{M}}_2}{\widetilde{M}^{(n)}}_2(F){\widetilde{M}^{(n)}}_2({\mathbb{A}})}.$$ Therefore using $\pi_2,\dots,\pi_k$ and $\omega_\delta$, one can construct the metaplectic tensor product representation of ${\widetilde{M}}_2({\mathbb{A}})$ with respect to $A_{{\widetilde{M}}_2}$, namely $$\label{E:pi_restricted_to_M_2} \pi_{\omega_\delta}:=(\pi_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,\pi_k)_{\omega_\delta}.$$ Then we have \[P:restriction\] Assume $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast\ast$). For each ${\widetilde{\varphi}}\in\pi_\omega=(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$, $${\widetilde{\varphi}}|_{{\widetilde{M}}_2({\mathbb{A}})}\in\bigoplus_\delta m_\delta\pi_{\omega_\delta},$$ where $\pi_{\omega_\delta}=(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta}$ as in (\[E:pi\_restricted\_to\_M\_2\]) and $\delta$ runs through a finite subset of ${\operatorname{GL}}_{r_1}(F)$, and $m_\delta\in{\mathbb{Z}}^{>0}$ is a multiplicity. (Note that which $\delta$ appears in the sum could depend on $\varphi$.) Recall that $${\widetilde{\varphi}}(m)=\sum_{\gamma\in A_{M}M^{(n)}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1),$$ where the sum is finite by Lemma \[L:finite\_sum\]. Note that $A_M{M^{(n)}}(F)$ is a normal subgroup of $M(F)$, and hence $A_{M}{M^{(n)}}(F)\backslash M(F)$ is a group, which is actually an abelian group because it is a subgroup of the abelian group $A_{M}({\mathbb{A}}){M^{(n)}}({\mathbb{A}})\backslash M({\mathbb{A}})$. By Lemma \[L:M\_2\] we have the inclusion $$A_{M_2}{M^{(n)}}_2(F)\backslash M_2(F) \hookrightarrow A_{M}{M^{(n)}}(F)\backslash M(F).$$ Hence we have $$\begin{aligned} {\widetilde{\varphi}}(m)&=\sum_{\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)}\varphi({\mathbf{s}}(\gamma) m;1)\\ &=\sum_{\gamma\in M_2(F)A_{M}{M^{(n)}}(F)\backslash M(F)}\; \sum_{\mu\in A_{M_2}{M^{(n)}}_2(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\mu){\mathbf{s}}(\gamma) m;1).\end{aligned}$$ By using Lemma \[L:M\_2\], one can see that the map on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $m_2\mapsto \varphi(m_2{\mathbf{s}}(\gamma)m)$ is in the induced space ${\operatorname{c-Ind}}_{A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})}^{{\widetilde{M}}_2({\mathbb{A}})}{\pi^{(n)}}_{\omega, 2}$, where ${\pi^{(n)}}_{\omega,2}:=\omega({\pi^{(n)}}_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$ and $\omega$ is actually the restriction of $\omega$ to $A_{{\widetilde{M}}_2}({\mathbb{A}})$. Now since we are assuming that $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast\ast$), the inner sum is finite. Since the sum over $\gamma\in A_{M}{M^{(n)}}(F)\backslash M(F)$ is finite, the outer sum is also finite. Thus there are only a finitely many $\gamma_1,\dots,\gamma_N$ that contribute to the sum over $\gamma\in M_2(F)A_{M}{M^{(n)}}(F)\backslash M(F)$ and so we have $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\mu\gamma_i) m;1).$$ Note that one can choose $\gamma_i$ to be in ${\operatorname{GL}}_{r_1}(F)$, so ${\mathbf{s}}(\mu\gamma_i)={\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu)$. So we have $${\widetilde{\varphi}}(m)=\sum_{i=1}^N\sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu) m;1).$$ One can see by using Lemma \[L:M\_2\] that the map on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $$m_2\mapsto \varphi({\mathbf{s}}(\gamma_i)m;1)$$ is in the induced space ${\operatorname{c-Ind}}_{A_{{\widetilde{M}}_2}({\mathbb{A}}){\widetilde{M}^{(n)}}_2({\mathbb{A}})}^{{\widetilde{M}}_2({\mathbb{A}})}{\pi^{(n)}}_{\omega_{\gamma_i}}$, where ${\pi^{(n)}}_{\omega_{\gamma_i}}=\omega_{\gamma_i}({\pi^{(n)}}_2\,{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\,{\pi^{(n)}}_k)$. Hence the function on ${\widetilde{M}}_2({\mathbb{A}})$ defined by $$m_2\mapsto \sum_{\mu\in A_{M_2}M_2^{(n)}(F)\backslash M_2(F)}\varphi({\mathbf{s}}(\gamma_i){\mathbf{s}}(\mu) m_2;1)$$ belongs to $\pi_{\omega_{\gamma_i}}$. Since we do not know the multiplicity one property for the group ${\widetilde{M}}_2$, we might have a possible multiplicity $m_\delta$. This completes the proof. \[T:restriction\] Assume that the metaplectic tensor product $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega} $ is realized with the group $A_{{\widetilde{M}}}$ which satisfies Hypothesis ($\ast\ast$). Then we have $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_2({\mathbb{A}})}\subseteq \bigoplus_{\delta\in{\operatorname{GL}}_{r_1}(F)} m_\delta(\pi_2{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega_\delta},$$ where $m_\delta\in {\mathbb{Z}}^{\geq 0}$ This is immediate from the above proposition. Now we can restrict the metaplectic tensor product “from the bottom”, and get the same result. Let $$M_{k-1}={\operatorname{GL}}_{r_1}\times{\operatorname{GL}}_{r_{k-1}}= \{\begin{pmatrix}g_1&&&\\ &\ddots&&\\ &&g_{k-1}&\\ &&&I_{r_k}\end{pmatrix}\in M: g_i\in{\operatorname{GL}}_{r_i}\},$$ and embed $M_{k-1}$ in ${\operatorname{GL}}_r$ in the upper left corner. Then define $A_{{\widetilde{M}}_{k-1}}$ and the character $\omega_\delta$ analogously. Also consider the analogue of Hypothesis ($\ast\ast$), namely The group $A_{{\widetilde{M}}}$ satisfies: 1. $A_{{\widetilde{M}}}$ satisfies Hypothesis ($\ast$) 2. $A_{{\widetilde{M}}_{k-1}}$ as defined above contains the center $Z_{{\widetilde{\operatorname{GL}}}_{r-r_k}}$. 3. $A_{{\widetilde{M}}_{k-1}}$ satisfies Hypothesis ($\ast$) with respect to ${\widetilde{M}}_{k-1}$. Then we have \[T:restriction\] Assume that the metaplectic tensor product $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega} $ is realized with the group $A_{{\widetilde{M}}}$ which satisfies Hypothesis ($\ast\ast\ast$). Then we have $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_{k-1}({\mathbb{A}})}\subseteq \bigoplus_{\delta\in{\operatorname{GL}}_{r_k}(F)} m_\delta(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{k-1})_{\omega_\delta},$$ where $m_\delta\in {\mathbb{Z}}^{>0}$ The proof is essentially the same as the case for the restriction to ${\widetilde{M}}_2$. We will leave the verification to the reader. Also for the case $n=2$, we can do even better. \[T:restriction\_n=2\] Assume $n=2$. (a) Choose $A_{{\widetilde{M}}}$ to be as in Proposition \[P:A\_M\_for\_n=2\]. For $j=2,\dots,k$, let $M_j={\operatorname{GL}}_{r_j}\times\cdots\times{\operatorname{GL}}_{r_k}\subseteq M$ embedded into the right lower corner. Then $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_j({\mathbb{A}})}\subseteq \bigoplus_{\omega'} m_{\omega'}(\pi_j{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega'},$$ where $\omega'$ runs through a countable number of characters on $A_{{\widetilde{M}}_j}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_j$. (b) Choose $A_{{\widetilde{M}}}$ to be as in Proposition \[P:A\_M\_for\_n=2\_2\]. For $j=1,\dots,k-1$, let $M_{k-j}={\operatorname{GL}}_{r_1}\times\cdots\times{\operatorname{GL}}_{r_{k-j}}\subseteq M$ embedded into the left upper corner. Then $$(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}\|_{{\widetilde{M}}_{k-j}({\mathbb{A}})}\subseteq \bigoplus_{\omega'} m_{\omega'}(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_{k-j})_{\omega'},$$ where $\omega'$ runs through a countable number of characters on $A_{{\widetilde{M}}_{k-j}}=A_{{\widetilde{M}}}\cap{\widetilde{M}}_{k-j}$. For (a), one can inductively show that $A_{{\widetilde{M}}_j}=A_{{\widetilde{M}}_{j-1}}\cap{\widetilde{M}}_{j-1}$ satisfies both Hypotheses ($\ast$) and ($\ast\ast$) for the Levi $M_j$. Thus one can successively apply the above theorem for $j=2,\dots,k$, which proves the theorem. The case (b) can be treated similarly. In the above theorem, we choose different $A_{{\widetilde{M}}}$ for the two cases to define $(\pi_1{\widetilde{\otimes}}\cdots{\widetilde{\otimes}}\pi_k)_{\omega}$. They are, however, equivalent, because, though the character $\omega$ is a character on $A_{{\widetilde{M}}}$, the metaplectic tensor product is dependent only on the restriction $\omega|_{Z_{{\widetilde{\operatorname{GL}}}_r}}$ to the center. **On the discreteness of the group $A_{M} M^{(n)}(F)\backslash M (F)$** {#A:topology} ======================================================================= In this appendix, we will discuss the issue of when $A_{{\widetilde{M}}}$ can be chosen so that the group $A_{M} M^{(n)}(F)\backslash M (F)$ is a discrete subgroup of $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$, and hence the metaplectic tensor product can be defined. In particular, we will show that if $n=2$, one can always choose such $A_{{\widetilde{M}}}$, and hence all the global results hold without any condition. If $n>2$, the author does not know if it is always possible to chose such nice $A_{{\widetilde{M}}}$, though he suspects that this is always the case. Throughout this appendix the field $F$ is a number field. Also for topological groups $H\subseteq G$, we always assume $H\backslash G$ is equipped with the quotient topology. The crucial fact is \[P:F\_is\_discrete\] For any positive integer $m$, the image of $F^\times$ in ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^\times$ is discrete in the quotient topology. Let $K=\prod_{v}K_v\subseteq{\mathbb{A}}^\times$ be the open neighborhood of the identity defined by $K_v=\mathcal{O}_{F_v}^\times$ for all finite $v$ and $K_v=F_v^\times$ for all infinite $v$. To show the discreteness of the image of $F^\times$, it suffices to show that the set ${\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times$ has only finitely many points modulo $A^{\times m}$. This is because the image of $F^\times$ in ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^\times$ will then have an open neighborhood of the identity in the subspace topology for ${\mathbb{A}}^{\times m}\backslash {\mathbb{A}}^{\times m}F^\times$ containing finitely many points, and the quotient ${\mathbb{A}}^{\times m}\backslash{\mathbb{A}}^\times$ is Hausdorf since ${\mathbb{A}}^{\times m}$ is closed. Now let $a^m\in{\mathbb{A}}^{\times m}$ and $u\in F^\times$ be such that $a^mu\in {\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times$. Then $u\in{\mathbb{A}}^{\times m}K$, and so for each finite $v$, we have $u_v\in F_v^{\times m}K_v$, which implies the fractional ideal $(u)$ generated by $u$ is $m^{\text{th}}$ power in the group $I_F$ of fractional ideals of $F$. Namely $(u)\in P_F\cap I_F^m$, where $P_F$ the group of principal fractional ideals. On the other hand for any $(u)\in P_F\cap I_F^m$, one can see that $u\in{\mathbb{A}}^{\times m}K$. Accordingly, if we define $$G:=\{u\in F^\times: (u)\in P_F\cap I_F^m\},$$ we have the surjection $$F^{\times m}\backslash G\rightarrow {\mathbb{A}}^{\times m}\backslash( {\mathbb{A}}^{\times m} K\cap {\mathbb{A}}^{\times m}F^\times),$$ given by $u\mapsto{\mathbb{A}}^{\times m}u$. So we have only to show that the group $F^{\times m}\backslash G$ is finite. But note that the map $u\mapsto (u)$ gives rise to the short exact sequence $$0\rightarrow U_F^{m}\backslash U_F\rightarrow F^{\times m}\backslash G\rightarrow P_F^m\backslash P_F\cap I_F^m\rightarrow 0,$$ where $U_K$ is the group of units for $F$. Now the group $U_F^{m}\backslash U_F$ is finite by Dirichlet’s unit theorem. The group $P_F^m\backslash P_F\cap I_F^m$ is isomorphic to the group of $m$-torsions in the class group of $F$ via the map $$P_F^m\backslash P_F\cap I_F^m\rightarrow P_F\backslash I_F,\quad \mathfrak{A}^m\mapsto\mathfrak{A}$$ for each fractional ideal $\mathfrak{A}^m \in I_F^m$, and hence finite. Therefore $F^{\times m}\backslash G$ is finite. As a first consequence of this, we have \[P:M(F)\_is\_discrete\] The image of $M(F)$ in $M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})$ is discrete. Let $${\operatorname{Det}}_M:M({\mathbb{A}})\rightarrow \underbrace{{\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times}_{k-\text{times}}$$ be the map defined by ${\operatorname{Det}}_M({\operatorname{diag}}(g_1,\dots,g_k))=(\det(g_1),\dots,\det(g_k))$. Then $\ker({\operatorname{Det}}_M)=M^{(n)}({\mathbb{A}})$. Moreover the map ${\operatorname{Det}}_M$ is continuous. Hence we have a continuous group isomorphism $$M^{(n)}({\mathbb{A}})\backslash M({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times.$$ Moreover, one can construct the continuous inverse by sending each $a_i\in {\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times$ to the first entry of the $i^{\text{th}}$ block ${\operatorname{GL}}_{r_i}({\mathbb{A}})$. But the image of $M(F)$ in ${\mathbb{A}}^{\times n}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times n}\backslash {\mathbb{A}}^\times$ under ${\operatorname{Det}}_M$ is discrete by the above proposition. The proposition follows. As a corollary, \[C:GCD\] If the center $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ is contained in ${\widetilde{M}^{(n)}}({\mathbb{A}})$, which is the case if $n$ divides $nr_i/d$ for all $i=1,\dots,k$ where $d=\gcd(n, r-1+2cr)$, then Hypothesis ($\ast$) is satisfied and the metaplectic tensor product can be defined. If the center is already in ${\widetilde{M}^{(n)}}({\mathbb{A}})$, one can choose $A_{{\widetilde{M}}}({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ and then $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})={\widetilde{M}^{(n)}}({\mathbb{A}})$, and so $A_MM^{(n)}(F)=M^{(n)}(F)$. Then by the above proposition, $A_MM^{(n)}(F)\backslash M(F)$ is discrete in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(n)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$. Proposition \[P:M(F)\_is\_discrete\] also implies The group $M(F)M^{(n)}({\mathbb{A}})$ (resp. $M(F)^\ast{\widetilde{M}^{(n)}}({\mathbb{A}})$) is a closed subgroup of $M({\mathbb{A}})$ (resp. ${\widetilde{M}}({\mathbb{A}})$). It suffices to show it for $M(F)M^{(n)}({\mathbb{A}})$ because the canonical projection is continuous. But for this, one can apply the following lemma with $G=M({\mathbb{A}}), Y=M^{(n)}({\mathbb{A}})$ and $\Gamma=M(F)$, which will complete the proof. \[L:discrete\_sub\] Let $G$ be a Hausdorf topological group. If $\Gamma\subset G$ is a discrete subgroup and $Y\subset G$ a closed normal subgroup such that the image of $\Gamma$ in $G\slash Y$ is discrete in the quotient topology, then the group $\Gamma Y$ is closed in $G$. Let $p:G\rightarrow G/Y$ be the canonical projection. By our assumption, the image $p(\Gamma)$ of $\Gamma$ is discrete in the quotient topology. Now since $Y$ is closed, the quotient $G/Y$ is a Hausdorf topological group. Hence $p(\Gamma)$ is closed by [@Deitmar Lemma 9.1.3 (b)]. To show $\Gamma Y$ is closed, it suffices to show every net $\{\gamma_iy_i\}_{i\in I}$ that converges in $G$, where $\gamma_i\in\Gamma$ and $y_i\in Y$, converges in $\Gamma Y$. But since $p$ is continuous, the net $\{p(\gamma_iy_i)\}$ converges in $G/Y$. But $p(\gamma_iy_i)=p(\gamma_i)$ and $p(\gamma_i)\in p(\Gamma)$. Since $p(\Gamma)$ is closed and discrete, in order for the net $\{p(\gamma_i)\}$ to converge, there exists $\gamma\in\Gamma$ such that $p(\gamma_i)=p(\gamma)$ for all sufficiently large $i\in I$, namely, the net $\{p(\gamma_i)\}$ is eventually constant. Hence for sufficiently large $i$, we have $\gamma_iy_i=\gamma y_i'$ for some $y_i'\in Y$. This means that the net $\{\gamma_iy_i\}$ is eventually in the set $\gamma Y$. But since $Y$ is closed, so is $\gamma Y$, which implies that the net $\{\gamma_iy_i\}$ converges in $\gamma Y\subset \Gamma Y$. Finally in this appendix, we will show that if $n=2$, one can always choose $A_{{\widetilde{M}}}$ so that the group $A_MM^{(n)}(F)\backslash M(F)$ is discrete and hence the metaplectic tensor product is defined, and moreover the metaplectic tensor product can be realized in such a way that it behaves nicely with the restriction to the smaller rank groups. First let us note that for any $r$, the center $Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})$ is given by $$Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}})=\{(aI_r, \xi): a\in {\mathbb{A}}^{\times \varepsilon}\},\quad \varepsilon= \begin{cases} 1\quad\text{if $r$ is odd};\\ 2\quad\text{if $r$ is even}. \end{cases}\\$$ Accordingly, one can see $$Z_{{\widetilde{\operatorname{GL}}}_r}({\mathbb{A}}){\widetilde{\operatorname{GL}}^{(2)}}_r({\mathbb{A}})= \begin{cases} {\widetilde{\operatorname{GL}}}_r({\mathbb{A}})\quad\text{if $r$ is odd};\\ {\widetilde{\operatorname{GL}}^{(2)}}_r({\mathbb{A}})\quad\text{if $r$ is even}. \end{cases}$$ With this said, one can see \[P:A\_M\_for\_n=2\] Assume $n=2$. Let $${\widetilde{Z}}_i({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}}({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_i}({\mathbb{A}}){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_k}({\mathbb{A}}) \subseteq{\widetilde{M}}({\mathbb{A}}),$$ and $$A_{{\widetilde{M}}}({\mathbb{A}})={\widetilde{Z}}_1({\mathbb{A}}){\widetilde{Z}}_2({\mathbb{A}})\cdots{\widetilde{Z}}_k({\mathbb{A}}).$$ Then $A_{{\widetilde{M}}}({\mathbb{A}})$ is a closed abelian subgroup of $\widetilde{Z_M}({\mathbb{A}})$ and further the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed and the image of $M(F)$ in $A_M({\mathbb{A}})M^{(2)}({\mathbb{A}})\backslash M({\mathbb{A}})$ as well as in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. It is clear that $A_{{\widetilde{M}}}({\mathbb{A}})$ is abelian since for each $i=1,\dots, k$, ${\widetilde{Z}}_i$ is the center of ${\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}({\mathbb{A}})$, and hence commutes pointwise with ${\widetilde{Z}}_j({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_i+\cdots+r_k}({\mathbb{A}})$ for all $j\geq i$. To show $A_{{\widetilde{M}}}({\mathbb{A}})$ is closed, it suffices to show $A_{M}({\mathbb{A}}):=p(A_{{\widetilde{M}}}({\mathbb{A}}))$ is closed. Now one can write $A_{M}({\mathbb{A}})=\prod_v'A_M(F_v)$, where $A_M(F_v)$ is defined analogously to the global case. Then one can see that $Z^{(2)}_M(F_v)\subseteq A_M(F_v)\subseteq Z_M(F_v)$, and since $Z^{(2)}_M(F_v)$ is closed and of finite index in $Z_M(F_v)$, so is $A_M(F_v)$. But $Z_M(F_v)$ is closed in $M(F_v)$ and so $A_M(F_v)$ is closed in $M(F_v)$. Then one can show that $A_M({\mathbb{A}})$ is closed in $M({\mathbb{A}})$ by Lemma \[L:closed\_subgroup\_local\_global\]. Now one can show by induction on $k$ that the group $A_{M}({\mathbb{A}})M^{(2)}({\mathbb{A}})$ is the kernel of the map $${\operatorname{Det}}_M:M({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times \varepsilon_1}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times \varepsilon_k}\backslash {\mathbb{A}}^\times,$$ where $\varepsilon_i$ is either $1$ or $2$. Hence one has a continuous group isomorphism $$A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})\rightarrow {\mathbb{A}}^{\times \varepsilon_1}\backslash{\mathbb{A}}^\times\times\cdots\times{\mathbb{A}}^{\times \varepsilon_k}\backslash {\mathbb{A}}^\times,$$ where the space on the right is Hausdorff. Hence the space on the left is Hausdorf as well, which shows $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed. Also one can show that the image of $M(F)$ is discrete as we did for Proposition \[P:M(F)\_is\_discrete\]. \[P:A\_M\_for\_n=2\_2\] Assume $n=2$. Let $${\widetilde{Z}}_j({\mathbb{A}})=Z_{{\widetilde{\operatorname{GL}}}_{r_1+\cdots+r_{k-j}}}({\mathbb{A}})\subseteq {\widetilde{\operatorname{GL}}}_{r_1}({\mathbb{A}}){\widetilde{\times}}\cdots{\widetilde{\times}}{\widetilde{\operatorname{GL}}}_{r_{k-j}}({\mathbb{A}}) \subseteq{\widetilde{M}}({\mathbb{A}}),$$ and $$A_{{\widetilde{M}}}({\mathbb{A}})={\widetilde{Z}}_1({\mathbb{A}}){\widetilde{Z}}_2({\mathbb{A}})\cdots{\widetilde{Z}}_k({\mathbb{A}}).$$ Then $A_{{\widetilde{M}}}({\mathbb{A}})$ is a closed abelian subgroup of $\widetilde{Z_M}({\mathbb{A}})$ and further the group $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})$ is closed and the image of $M(F)$ in $A_M({\mathbb{A}})M^{(2)}({\mathbb{A}})\backslash M({\mathbb{A}})$ as well as in $A_{{\widetilde{M}}}({\mathbb{A}}){\widetilde{M}^{(2)}}({\mathbb{A}})\backslash{\widetilde{M}}({\mathbb{A}})$ is discrete. Identical to the previous proposition. Let us make the following final remark. The above proposition and Corollary \[C:GCD\] imply Proposition \[P:hypothesis\]. Also for $n>2$, if $n$ and $r=r_1+\cdots+r_k$ are such that $n$ divides $nr_i/d$ for all $i=1\cdots k$ where $d=\gcd(n, r-1+2cr)$ and $n$ divides $ nr_i/d_2$ for all $i=2\cdots k$ where $d_2=\gcd(n, r-r_1-1+2c(r-r_2)$, then $A_{{\widetilde{M}}}=Z_{{\widetilde{\operatorname{GL}}}_r}$ satisfies Hypothesis ($\ast\ast$), and hence one has the restriction property to the smaller rank group. Moreover this is always the case, for example, if $\gcd(n, r-1+2cr)=\gcd(n, r-r_1-1+2c(r-r_1))=1$. Similarly one can satisfy Hypothesis ($\ast\ast\ast$) if $n$ divides $nr_i/d$ for all $i=1\cdots k$ and divides $ nr_i/d_{k-1}$ for all $i=1\cdots k-1$ where $d_{k-1}=\gcd(n, r-r_{k-1}-1+2c(r-r_{k-1}))$. Those conditions are indeed often satisfied especially when $n$ is a prime. [999999]{} J. Adams, [*Non-linear real groups*]{}, (preprint). W. Banks, [*Twisted symmetric-square $L$-functions and the nonexistence of Siegel zeros on ${{\operatorname{GL}}}(3)$*]{}, Duke Math. J. 87 (1997), 343–353. W. Banks, D. Bump and D. Lieman, [*Whittaker-Fourier coefficients of metaplectic Eisenstein series*]{}, Compositio Math. 135 (2003), 153–178. W. Banks, J. Levy, M. Sepanski, [*Block-compatible metaplectic cocycles*]{}, J. Reine Angew. Math. 507 (1999), 131–163. N. Bourbaki, Integration. II, Chapters 7–9, translated from the 1963 and 1969 French originals by Sterling K. Berberian. Elements of Mathematics, Springer-Verlag, Berlin, 2004. D. Bump, S. Friedberg and J. Hoffstein, [*p-adic Whittaker functions on the metaplectic group*]{}, Duke Math. J. 63 (1991), 379–397. D. Bump and D. Ginzburg, [*Symmetric square $L$-functions on ${\rm GL}(r)$*]{}, Ann. of Math. 136 (1992), 137–205. D. Bump and J. Hoffstein, [*On Shimura’s correspondence*]{}, Duke Math. J. 55 (1987), 661–691. J. Cogdell, [*Lectures on $L$-functions, converse theorems, and functoriality of ${\operatorname{GL}}(n)$*]{}, in Lectures on Automorphic $L$-functions, Fields Institute Monographs, AMS (2004), 5–100. A. Deitmar and S. Echterhoff, Principles of harmonic analysis, Springer, New York, 2009. Y. Flicker, [*Automorphic forms on covering groups of ${{\operatorname{GL}}}(2)$*]{}, Invent. Math. 57 (1980), 119–182. Y. Flicker and D. Kazhdan, [*Metaplectic correspondence*]{}, Inst. Hautes Etudes Sci. Publ. Math. No. 64 (1986), 53–110 G. Chinta and O. Offen, [*A metaplectic Casselman-Shalika formula for ${\operatorname{GL}}_r$*]{}, to appear in Amer. J. Math. A. Kable, [*Exceptional representations of the metaplectic double cover of the general linear group*]{}, PH.D thesis, Oklahoma State University (1997). A. Kable, [*The tensor product of exceptional representations on the general linear group*]{}, Ann. Sci. École Norm. Sup. (4) 34 (2001), 741–769. D. A. Kazhdan and S. J. Patterson, [*Metaplectic forms*]{}, Inst. Hautes Etudes Sci. Publ. Math. No. 59 (1984), 35–142. T. Kubota, [*On automorphic functions and the reciprocity law in a number field*]{}, Lectures in Mathematics, Department of Mathematics, Kyoto University, No. 2 Kinokuniya Book-Store Co., Ltd., Tokyo 1969 iii+65 pp. H. Matsumoto, [*Sur les sous-groupes arithmétiques des groupes semi-simples déployés*]{}, Ann. Sci. École Norm. Sup. 2 (1969), 1–62. P. Mezo, [*Metaplectic tensor products for irreducible representations*]{}, Pacific J. Math. 215 (2004), 85–96. C. Moeglin and J.-L. Waldspurger, [*Spectral decomposition and Eisenstein series*]{}, Cambridge Tracts in Mathematics, 113. Cambridge University Press, Cambridge, 1995. T. Suzuki, [*Metaplectic Eisenstein series and the Bump-Hoffstein conjecture*]{}, Duke Math. J. 90 (1997), 577–630. S. Takeda, [*The twisted symmetric square $L$-function of ${\operatorname{GL}}(r)$*]{}, Duke Math. J. (to appear). S. Takeda, [*On a certain metaplectic Eisenstein series and the twisted symmetric square $L$-function*]{}, preprint.

ArXiv

--- abstract: 'We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.' address: ' Universidade do Minho, Centro de Matemática, Escola de Ciências, Campus de Gualtar, 4710-057 Braga, Portugal' author: - 'A. Yu. Karlovich' title: ALGEBRAS OF SINGULAR INTEGRAL OPERATORS ON NAKANO SPACES WITH KHVEDELIDZE WEIGHTS OVER CARLESON CURVES WITH LOGARITHMIC WHIRL POINTS --- Introduction ============ Fredholm theory of one-dimensional singular integral operators (SIOs) with piecewise continuous ($PC$) coefficients on weighted Lebesgue spaces was constructed by Gohberg and Krupnik [@GK92] and [@GK70; @GK71] in the beginning of 70s in the case of Khvedelidze weights and piecewise Lyapunov curves (see also the monographs [@CG81; @KS01; @LS87; @MP86]). Simonenko and Chin Ngok Min [@SCNM86] suggested another approach to the study of the Banach algebra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces with Khvedelidze weights over piecewise Lyapunov curves. This approach is based on Simonenko’s local principle [@Simonenko65]. In 1992 Spitkovsky [@Spitkovsky92] made a next significant step: he proved a Fredholm criterion for an individual SIO with $PC$ coefficients on Lebesgue spaces with Muckenhoupt weights over Lyapunov curves. Finally, Böttcher and Yu. Karlovich extended Spitkovsky’s result to the case of arbitrary Carleson curves and Banach algebras of SIOs with $PC$ coefficients. After their work the Fredholm theory of SIOs with $PC$ coefficients is available in the maximal generality (that, is, when the Cauchy singular integral operator $S$ is bounded on weighted Lebesgue spaces). We recommend the nice paper [@BK01] for a first reading about this topic and the book [@BK97] for a complete and self-contained exposition. It is quite natural to consider the same problems in other, more general, spaces of measurable functions on which the operator $S$ is bounded. Good candidates for this role are rearrangement-invariant spaces (that is, spaces with the property that norms of equimeasurable functions are equal). These spaces have nice interpolation properties and boundedness results can be extracted from known results for Lebesgue spaces applying interpolation theorems. The author extended (some parts of) the Böttcher-Yu. Karlovich Fredholm theory of SIOs with $PC$ coefficients to the case of rearrangement-invariant spaces with Muckenhoupt weights [@K98; @K02]. Notice that necessary conditions for the Fredholmness of an individual singular integral operator with $PC$ coefficients are obtained in [@K03] for weighted reflexive Banach function spaces on which the operator $S$ is bounded. Nakano spaces $L^{p(\cdot)}$ (generalized Lebesgue spaces with variable exponent) are a nontrivial example of Banach function spaces which are not rearrangement-invariant, in general. Many results about the behavior of some classical operators on these spaces have important applications to fluid dynamics (see [@DR03] and the references therein). Kokilashvili and Samko [@KS-GMJ] proved that the operator $S$ is bounded on weighted Nakano spaces for the case of nice curves, nice weights, and nice (but variable!) exponents. They also extended the Gohberg-Krupnik Fredholm criterion for an individual SIO with $PC$ coefficients to this situation [@KS-Proc] (see also [@Samko05]). The author [@K05] has found a Fredholm criterion and a formula for the index of an arbitrary operator in the Banach algebra of SIOs with $PC$ coefficients on Nakano spaces with Khvedelidze weights over either Lyapunov curves or Radon curves without cusps. Very recently Kokilashvili and Samko [@KS-Memoirs] (see also [@Kokilashvili05 Theorem 7.1]) have proved a boundedness criterion for the Cauchy singular integral operator $S$ on Nakano spaces with Khvedelidze weights over arbitrary Carleson curves. Combining this boundedness result with the machinery developed in [@K03], we are able to prove a Fredholm criterion for an individual SIO on a Nakano space with a Khvedelidze weight over a Carleson curve satisfying a “logarithmic whirl condition" (see [@BK95], [@BK97 Ch. 1]) at each point. Further, we extend this result to the case of Banach algebras of SIOs with $PC$ coefficients, using the approach developed in [@BK95; @BK97; @K03; @K05]. The paper is organized as follows. In Section \[sect:preliminaries\] we define weighted Nakano spaces and discuss the boundedness of the operator $S$ on these spaces. Section \[sect:individual\] contains a Fredholm criterion for an individual SIO with $PC$ coefficients on weighted Nakano spaces. The proof of this result is based on the local principle of Simonenko type and factorization technique. In Section \[sect:tools\] we formulate the Allan-Douglas local principle and the two projections theorem. The results of Section \[sect:tools\] are the main tools allowing us to construct a symbol calculus for the Banach algebra of SIOs with $PC$ coefficients acting on a Nakano space with a Khvedelidze weight over a Carleson curve with logarithmic whirl points in Section \[sect:symbol\]. Preliminaries {#sect:preliminaries} ============= Weighted Nakano spaces $L_w^{p(\cdot)}$ --------------------------------------- Function spaces $L^{p(\cdot)}$ of Lebesgue type with variable exponent $p$ were studied for the first time by Orlicz [@Orlicz31] in 1931, but notice that another kind of Banach spaces is called after him. Inspired by the successful theory of Orlicz spaces, Nakano defined in the late forties [@Nakano50; @Nakano51] so-called *modular spaces*. He considered the space $L^{p(\cdot)}$ as an example of modular spaces. In 1959, Musielak and Orlicz [@MO59] extended the definition of modular spaces by Nakano. Actually, that paper was the starting point for the theory of Musielak-Orlicz spaces (generalized Orlicz spaces generated by Young functions with a parameter), see [@Musielak83]. Let $\Gamma$ be a Jordan (i.e., homeomorphic to a circle) rectifiable curve. We equip $\Gamma$ with the Lebesgue length measure $|d\tau|$ and the counter-clockwise orientation. Let $p:\Gamma\to(1,\infty)$ be a measurable function. Consider the convex modular (see [@Musielak83 Ch. 1] for definitions and properties) $$m(f,p):=\int_\Gamma|f(\tau)|^{p(\tau)}|d\tau|.$$ Denote by $L^{p(\cdot)}$ the set of all measurable complex-valued functions $f$ on $\Gamma$ such that $m(\lambda f,p)<\infty$ for some $\lambda=\lambda(f)>0$. This set becomes a Banach space when equipped with the *Luxemburg-Nakano norm* $$\|f\|_{L^{p(\cdot)}}:=\inf\big\{\lambda>0: \ m(f/\lambda,p)\le 1\big\}$$ (see, e.g., [@Musielak83 Ch. 2]). Thus, the spaces $L^{p(\cdot)}$ are a special case of Musielak-Orlicz spaces. Sometimes the spaces $L^{p(\cdot)}$ are referred to as Nakano spaces. We will follow this tradition. Clearly, if $p(\cdot)=p$ is constant, then the Nakano space $L^{p(\cdot)}$ is isometrically isomorphic to the Lebesgue space $L^p$. Therefore, sometimes the spaces $L^{p(\cdot)}$ are called generalized Lebesgue spaces with variable exponent or, simply, variable $L^p$ spaces. We shall assume that $$\label{eq:reflexivity} 1<{\rm ess}\inf_{\!\!\!\!\!\!\!\!t\in\Gamma} p(t), \quad {\rm ess}\sup_{\!\!\!\!\!\!\!\!\!t\in\Gamma} p(t)<\infty.$$ In this case the conjugate exponent $$q(t):=\frac{p(t)}{p(t)-1} \quad (t\in\Gamma)$$ has the same property. A nonnegative measurable function $w$ on the curve $\Gamma$ is referred to as a [*weight*]{} if $0<w(t)<\infty$ almost everywhere on $\Gamma$. The [*weighted Nakano space*]{} is defined by $$L_w^{p(\cdot)}= \big\{f\mbox{ is measurable on }\Gamma\mbox{ and }fw\in L^{p(\cdot)}\big\}.$$ The norm in $L_w^{p(\cdot)}$ is defined by $\|f\|_{L_w^{p(\cdot)}}=\|fw\|_{L^{p(\cdot)}}$. Carleson curves --------------- A rectifiable Jordan curve $\Gamma$ is said to be a [*Carleson*]{} (or [*Ahlfors-David regular*]{}) [*curve*]{} if $$\sup_{t\in\Gamma}\sup_{R>0}\frac{|\Gamma(t,R)|}{R}<\infty,$$ where $\Gamma(t,R):=\{\tau\in\Gamma:|\tau-t|<R\}$ for $R>0$ and $|\Omega|$ denotes the measure of a measurable set $\Omega\subset\Gamma$. We can write $$\tau-t=|\tau-t|e^{i\arg(\tau-t)} \quad\mbox{for}\quad\tau\in\Gamma\setminus\{t\},$$ and the argument can be chosen so that it is continuous on $\Gamma\setminus\{t\}$. Seifullaev [@Seif80] showed that for an arbitrary Carleson curve the estimate $\arg(\tau-t)=O(-\log|\tau-t|)$ as $\tau\to t$ holds for every $t\in\Gamma$. A simpler proof of this result can be found in [@BK97 Theorem 1.10]. One says that a Carleson curve $\Gamma$ satisfies the *logarithmic whirl condition* at $t\in\Gamma$ if $$\label{eq:spiralic} \arg(\tau-t)=-\delta(t)\log|\tau-t|+O(1)\quad (\tau\to t)$$ with some $\delta(t)\in{\mathbb{R}}$. Notice that all piecewise smooth curves satisfy this condition at each point and, moreover, $\delta(t)\equiv 0$. For more information along these lines, see [@BK95], [@BK97 Ch. 1], [@BK01]. The Cauchy singular integral operator ------------------------------------- The *Cauchy singular integral* of $f\in L^1$ is defined by $$(Sf)(t):=\lim_{R\to 0}\frac{1}{\pi i}\int_{\Gamma\setminus\Gamma(t,R)} \frac{f(\tau)}{\tau-t}d\tau \quad (t\in\Gamma).$$ Not so much is known about the boundedness of the Cauchy singular integral operator $S$ on weighted Nakano spaces $L_w^{p(\cdot)}$ for general curves, general weights, and general exponents $p(\cdot)$. [From]{} [@K03 Theorem 6.1] we immediately get the following. Let $\Gamma$ be a rectifiable Jordan curve, let $w:\Gamma\to[0,\infty]$ be a weight, and let $p:\Gamma\to(1,\infty)$ be a measurable function satisfying . If the Cauchy singular integral generates a bounded operator $S$ on the weighted Nakano space $L_w^{p(\cdot)}$, then $$\label{eq:Ap} \sup_{t\in\Gamma}\sup_{R>0}\frac{1}{R} \|w\chi_{\Gamma(t,R)}\|_{L^{p(\cdot)}} \|\chi_{\Gamma(t,R)}/w\|_{L^{q(\cdot)}}<\infty.$$ [From]{} the Hölder inequality for Nakano spaces (see, e.g., [@Musielak83] or [@KR91]) and we deduce that if $S$ is bounded on $L_w^{p(\cdot)}$, then $\Gamma$ is necessarily a Carleson curve. If the exponent $p(\cdot)=p\in(1,\infty)$ is constant, then is simply the famous Muckenhoupt condition $A_p$. It is well known that for classical Lebesgue spaces $L^p$ this condition is not only necessary, but also sufficient for the boundedness of the Cauchy singular integral operator $S$. A detailed proof of this result can be found in [@BK97 Theorem 4.15]. Let $N\in{\mathbb{N}}$. Consider now a power weight $$\label{eq:power} \varrho(t):=\prod_{k=1}^N|t-\tau_k|^{\lambda_k}, \quad \tau_k\in\Gamma, \quad k\in\{1,\dots,N\},$$ where all $\lambda_k$ are real numbers. Introduce the class ${\mathcal{P}}$ of exponents $p:\Gamma\to(1,\infty)$ satisfying and $$\label{eq:Dini-Lipschitz} |p(\tau)-p(t)|\le\frac{A}{-\log |\tau-t|}$$ for some $A\in(0,\infty)$ and all $\tau,t\in\Gamma$ such that $|\tau-t|<1/2$. The following criterion for the boundedness of the Cauchy singular integral operator on Nakano spaces with power weights has been recently proved by Kokilashvili and Samko [@KS-Memoirs] (see also [@Kokilashvili05 Theorem 7.1]). \[th:KS\] Let $\Gamma$ be a Carleson Jordan curve, let $\varrho$ be a power weight of the form , and let $p\in{\mathcal{P}}$. The Cauchy singular integral operator $S$ is bounded on the weighted Nakano space $L_\varrho^{p(\cdot)}$ if and only if $$\label{eq:Khvedelidze} 0<\frac{1}{p(\tau_k)}+\lambda_k<1 \quad\mbox{for all}\quad k\in\{1,\dots,N\}.$$ For weighted Lebesgue spaces over Lyapunov curves the above theorem was proved by Khvedelidze [@Khvedelidze56] (see also the proof in [@GK92; @Khvedelidze75; @MP86]). Therefore the weights of the form are often called *Khvedelidze weights*. We shall follow this tradition. Notice that if $p$ is constant and $\Gamma$ is a Carleson curve, then is equivalent to the fact that $\varrho$ is a Muckenhoupt weight (see, e.g., [@BK97 Chapter 2]). Analogously one can prove that if the exponent $p$ belong to the class ${\mathcal{P}}$ and the curve $\Gamma$ is Carleson, then the power weight satisfies the condition if and only if is fulfilled. The proof of this fact is based on certain estimates for the norms of power functions in Nakano spaces with exponents in the class ${\mathcal{P}}$ (see also [@K03 Lemmas 5.7 and 5.8] and [@KS-GMJ]). Singular integral operators with $PC$ coefficients {#sect:individual} ================================================== The local principle of Simonenko type ------------------------------------- Let $I$ be the identity operator on $L_\varrho^{p(\cdot)}$. Under the conditions of Theorem \[th:KS\], the operators $$P:=(I+S)/2, \quad Q:=(I-S)/2$$ are bounded projections on $L_\varrho^{p(\cdot)}$ (see [@K03 Lemma 6.4]). Let $L^\infty$ denote the space of all measurable essentially bounded functions on $\Gamma$. The operators of the form $aP+Q$ with $a \in L^\infty$ are called [*singular integral operators*]{} (SIOs). Two functions $a,b\in L^\infty$ are said to be locally equivalent at a point $t\in\Gamma$ if $$\inf\big\{\|(a-b)c\|_\infty\ :\ c\in C,\ c(t)=1\big\}=0.$$ \[th:local\_principle\] Suppose the conditions of Theorem [\[th:KS\]]{} are satisfied and $a\in L^\infty$. Suppose for each $t\in\Gamma$ there is a function $a_t\in L^\infty$ which is locally equivalent to $a$ at $t$. If the operators $a_tP+Q$ are Fredholm on $L_\varrho^{p(\cdot)}$ for all $t\in\Gamma$, then $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$. For weighted Lebesgue spaces this theorem is known as Simonenko’s local principle [@Simonenko65]. It follows from [@K03 Theorem 6.13]. Simonenko’s factorization theorem --------------------------------- The curve $\Gamma$ divides the complex plane $\mathbb{C}$ into the bounded simply connected domain $D^+$ and the unbounded domain $D^-$. Without loss of generality we assume that $0\in D^+$. We say that a function $a\in L^\infty$ admits a *Wiener-Hopf factorization on* $L_\varrho^{p(\cdot)}$ if $1/a\in L^\infty$ and $a$ can be written in the form $$\label{eq:WH} a(t)=a_-(t)t^\kappa a_+(t) \quad\mbox{a.e. on}\ \Gamma,$$ where $\kappa\in{\mathbb{Z}}$, and the factors $a_\pm$ enjoy the following properties: 1. $a_-\in QL_\varrho^{p(\cdot)}\stackrel{\cdot}{+}\mathbb{C}, \quad 1/a_-\in QL_{1/\varrho}^{q(\cdot)}\stackrel{\cdot}{+}\mathbb{C}, \quad a_+\in PL_{1/\varrho}^{q(\cdot)},\quad 1/a_+\in PL_\varrho^{p(\cdot)}$, 2. the operator $(1/a_+)Sa_+I$ is bounded on $L_\varrho^{p(\cdot)}$. One can prove that the number $\kappa$ is uniquely determined. \[th:factorization\] Suppose the conditions of Theorem [\[th:KS\]]{} are satisfied. A function $a\in L^\infty$ admits a Wiener-Hopf factorization [(\[eq:WH\])]{} on $L_\varrho^{p(\cdot)}$ if and only if the operator $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$. If $aP+Q$ is Fredholm, then its index is equal to $-\kappa$. This theorem goes back to Simonenko [@Simonenko64; @Simonenko68]. For more about this topic we refer to [@BK97 Section 6.12], [@BS90 Section 5.5], [@GK92 Section 8.3] and also to [@CG81; @LS87] in the case of weighted Lebesgue spaces. Theorem \[th:factorization\] follows from [@K03 Theorem 6.14]. Fredholm criterion for singular integral operators with $PC$ coefficients ------------------------------------------------------------------------- We denote by $PC$ the Banach algebra of all piecewise continuous functions on $\Gamma$: a function $a\in L^\infty$ belongs to $PC$ if and only if the finite one-sided limits $$a(t\pm 0):=\lim_{\tau\to t\pm 0}a(\tau)$$ exist for every $t\in\Gamma$. \[th:criterion\] Let $\Gamma$ be a Carleson Jordan curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$. Suppose $p\in{\mathcal{P}}$ and $\varrho$ is a power weight of the form which satisfies . The operator $aP+Q$, where $a\in PC$, is Fredholm on the weighted Nakano space $L^{p(\cdot)}_\varrho$ if and only if $a(t\pm 0)\ne 0$ and $$-\frac{1}{2\pi}\arg\frac{a(t-0)}{a(t+0)} + \frac{\delta(t)}{2\pi}\log\left|\frac{a(t-0)}{a(t+0)}\right| + \frac{1}{p(t)}+\lambda(t)\notin{\mathbb{Z}}\label{eq:Fredholm}$$ for all $t\in\Gamma$, where $$\lambda(t):=\left\{ \begin{array}{lcl} \lambda_k, &\mbox{if} & t=\tau_k, \quad k\in\{1,\dots,N\},\\ 0, &\mbox{if} & t\notin\Gamma\setminus\{\tau_1,\dots,\tau_N\}. \end{array} \right.$$ The [*necessity*]{} part follows from [@K03 Theorem 8.1] because the Böttcher-Yu. Karlovich indicator functions $\alpha_t$ and $\beta_t$ in that theorem for a Khvedelidze weight $\varrho$ and a Carleson curve $\Gamma$ satisfying the logarithmic whirl condition at $t\in\Gamma$ are calculated by $$\alpha_t(x)=\beta_t(x)=\lambda(t)+\delta(t)x\quad \mbox{for}\quad x\in{\mathbb{R}}$$ (see [@BK97 Ch. 3] or [@K02 Lemma 3.9]). [*Sufficiency.*]{} If $aP+Q$ is Fredholm, then, by [@K03 Theorem 6.11], $a(t\pm 0)\ne 0$ for all $t\in\Gamma$. Fix $t\in\Gamma$. For the function $a$ we construct a “canonical” function $g_{t,\gamma}$ which is locally equivalent to $a$ at the point $t\in\Gamma$. The interior and the exterior of the unit circle can be conformally mapped onto $D^+$ and $D^-$ of $\Gamma$, respectively, so that the point $1$ is mapped to $t$, and the points $0\in D^+$ and $\infty\in D^-$ remain fixed. Let $\Lambda_0$ and $\Lambda_\infty$ denote the images of $[0,1]$ and $[1,\infty)\cup\{\infty\}$ under this map. The curve $\Lambda_0\cup\Lambda_\infty$ joins $0$ to $\infty$ and meets $\Gamma$ at exactly one point, namely $t$. Let $\arg z$ be a continuous branch of argument in $\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. For $\gamma\in\mathbb{C}$, define the function $z^\gamma:=|z|^\gamma e^{i\gamma\arg z}$, where $z\in\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. Clearly, $z^\gamma$ is an analytic function in $\mathbb{C}\setminus(\Lambda_0\cup\Lambda_\infty)$. The restriction of $z^\gamma$ to $\Gamma\setminus\{t\}$ will be denoted by $g_{t,\gamma}$. Obviously, $g_{t,\gamma}$ is continuous and nonzero on $\Gamma\setminus\{t\}$. Since $a(t\pm 0)\ne 0$, we can define $\gamma_t=\gamma\in\mathbb{C}$ by the formulas $$\operatorname{Re}\gamma_t:=\frac{1}{2\pi}\arg\frac{a(t-0)}{a(t+0)}, \quad \operatorname{Im}\gamma_t:=-\frac{1}{2\pi}\log\left|\frac{a(t-0)}{a(t+0)}\right|,$$ where we can take any value of $\arg(a(t-0)/a(t+0))$, which implies that any two choices of $\operatorname{Re}\gamma_t$ differ by an integer only. Clearly, there is a constant $c_t\in\mathbb{C}\setminus\{0\}$ such that $a(t\pm 0)=c_tg_{t,\gamma_t}(t\pm 0)$, which means that $a$ is locally equivalent to $c_tg_{t,\gamma_t}$ at the point $t\in\Gamma$. [From]{} it follows that there exists an $m_t\in{\mathbb{Z}}$ such that $$0<m_t-\operatorname{Re}\gamma_t- \delta(t)\operatorname{Im}\gamma_t+\frac{1}{p(t)}+\lambda(t)<1.$$ By Theorem \[th:KS\], the operator $S$ is bounded on $L_{\widetilde{\varrho}}^{p(\cdot)}$, where $$\widetilde{\varrho}(\tau)= |\tau-t|^{m_t-\operatorname{Re} \gamma_t-\delta(t)\operatorname{Im}\gamma_t}\varrho(\tau)$$ for $\tau\in\Gamma$. In view of the logarithmic whirl condition we have $$\begin{aligned} |(\tau-t)^{m_t-\gamma_t}| &=& |\tau-t|^{m_t-\operatorname{Re}\gamma_t} e^{\operatorname{Im}\gamma_t\arg(\tau-t)} \\ &=& |\tau-t|^{m_t-\operatorname{Re}\gamma_t} e^{-\operatorname{Im}\gamma_t(\delta(t)\log|\tau-t|+O(1))} \\ &=& |\tau-t|^{m_t-\operatorname{Re}\gamma_t-\delta(t)\operatorname{Im}\gamma_t} e^{-\operatorname{Im}\gamma_t O(1)}\end{aligned}$$ as $\tau\to t$. Therefore the operator $\varphi_{t,m_t-\gamma_t}S\varphi_{t,\gamma_t-m_t}I$, where $$\varphi_{t,m_t-\gamma_t}(\tau)=|(\tau-t)^{m_t-\gamma_t}|,$$ is bounded on $L_\varrho^{p(\cdot)}$. Then, by [@K03 Lemma 7.1], the function $g_{t,\gamma_t}$ admits a Wiener-Hopf factorization on $L_\varrho^{p(\cdot)}$. Due to Theorem \[th:factorization\], the operator $g_{t,\gamma_t}P+Q$ is Fredholm. Then the operator $c_tg_{t,\gamma_t}P+Q$ is Fredholm, too. Since the function $c_tg_{t,\gamma_t}$ is locally equivalent to the function $a$ at every point $t\in\Gamma$, in view of Theorem \[th:local\_principle\], the operator $aP+Q$ is Fredholm on $L_\varrho^{p(\cdot)}$. Double logarithmic spirals -------------------------- Given $z_1,z_2\in\mathbb{C}$, $\delta\in{\mathbb{R}}$, and $r\in(0,1)$, put $$\mathcal{S}(z_1,z_2;\delta,r) := \{z_1,z_2\} \cup \Big\{ z\in\mathbb{C}\setminus\{z_1,z_2\}: \arg\frac{z-z_1}{z-z_2}- \delta\log\left|\frac{z-z_1}{z-z_2}\right|\in2\pi (r+{\mathbb{Z}}) \Big\}.$$ The set $\mathcal{S}(z_1,z_2;\delta,r)$ is a double logarithmic spiral whirling about the points $z_1$ and $z_2$. It degenerates to a familiar Widom-Gohberg-Krupnik circular arc whenever $\delta=0$ (see [@BK97; @GK92]). Fix $t\in\Gamma$ and consider a function $\chi_t\in PC$ which is continuous on $\Gamma\setminus\{t\}$ and satisfies $\chi_t(t-0)=0$ and $\chi_t(t+0)=1$. [From]{} Theorem \[th:criterion\] we get the following. \[co:important\] Let $\Gamma$ be a Carleson Jordan curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$. Suppose $p\in{\mathcal{P}}$ and $\varrho$ is a power weight of the form which satisfies . Then $$\big\{\lambda\in\mathbb{C}:(\chi_t-\lambda)P+Q \mbox{ is not Fredholm on }L_\varrho^{p(\cdot)}\big\} =\mathcal{S}\big(0,1;\delta(t),1/p(t)+\lambda(t)\big).$$ Tools for the construction of the symbol calculus {#sect:tools} ================================================= The Allan-Douglas local principle --------------------------------- Let $B$ be a Banach algebra with identity. A subalgebra $Z$ of $B$ is said to be a central subalgebra if $zb=bz$ for all $z\in Z$ and all $b\in B$. \[th:AllanDouglas\] [(see [@BS90 Theorem 1.34(a)]).]{} Let $B$ be a Banach algebra with unit $e$ and let $Z$ be closed central subalgebra of $B$ containing $e$. Let $M(Z)$ be the maximal ideal space of $Z$, and for $\omega\in M(Z)$, let $J_\omega$ refer to the smallest closed two-sided ideal of $B$ containing the ideal $\omega$. Then an element $b$ is invertible in $B$ if and only if $b+J_\omega$ is invertible in the quotient algebra $B/J_\omega$ for all $\omega\in M(Z)$. The two projections theorem --------------------------- The following two projections theorem was obtained by Finck, Roch, Silbermann [@FRS93] and Gohberg, Krupnik [@GK93]. \[th:2proj\] Let $F$ be a Banach algebra with identity $e$, let ${\mathcal{C}}$ be a Banach subalgebra of $F$ which contains $e$ and is isomorphic to ${\mathbb{C}}^{n \times n}$, and let $p$ and $q$ be two projections in $F$ such that $cp=pc$ and $cq=qc$ for all $c \in {\mathcal{C}}$. Let $W={\mathrm{alg}}({\mathcal{C}},p,q)$ be the smallest closed subalgebra of $F$ containing ${\mathcal{C}},p,q$. Put $$x=pqp+(e-p)(e-q)(e-p),$$ denote by $\mathrm{sp}\,x$ the spectrum of $x$ in $F$, and suppose the points $0$ and $1$ are not isolated points of $\mathrm{sp}\,x$. Then 1. for each $\mu \in \mathrm{sp}\,x$ the map $\sigma_{\mu}$ of ${\mathcal{C}}\cup \{p,q\}$ into the algebra ${\mathbb{C}}^{2n\times 2n}$ of all complex $2n\times 2n$ matrices defined by $$\label{eq:2proj1} \sigma_{\mu}c=\left( \begin{array}{cc} c & 0\\ 0 & c \end{array} \right), \quad \sigma_{\mu}p=\left( \begin{array}{cc} E & 0\\ 0 & 0 \end{array} \right),$$ $$\label{eq:2proj2} \sigma_{\mu}q=\left( \begin{array}{cc} \mu E & \sqrt{\mu(1-\mu)}E \\ \sqrt{\mu(1-\mu)}E & (1-\mu)E \end{array} \right),$$ where $c\in {\mathcal{C}}, E$ denotes the $n \times n$ unit matrix and $\sqrt{\mu(1-\mu)}$ denotes any complex number whose square is $\mu(1-\mu)$, extends to a Banach algebra homomorphism $$\sigma_{\mu}: W \to {\mathbb{C}}^{2n \times 2n};$$ 2. every element $a$ of the algebra $W$ is invertible in the algebra $F$ if and only if $$\det \sigma_{\mu} a \neq 0 \quad\mbox{for all}\quad \mu \in \mathrm{sp}\,x;$$ 3. the algebra $W$ is inverse closed in $F$ if and only if the spectrum of $x$ in $W$ coincides with the spectrum of $x$ in $F$. A further generalization of the above result to the case of $N$ projections is contained in [@BK97]. Algebra of singular integral operators with $PC$ coefficients {#sect:symbol} ============================================================= The ideal of compact operators ------------------------------ In this section we will suppose that $\Gamma$ is a Carleson curve satisfying with $\delta(t)\in{\mathbb{R}}$ for every $t\in\Gamma$, $p\in{\mathcal{P}}$, and $\varrho$ is a Khvedelidze weight of the form which satisfies . Let $X_n:=[L_\varrho^{p(\cdot)}]_n$ be the direct sum of $n$ copies of weighted Nakano spaces $X:=L_\varrho^{p(\cdot)}$, let ${\mathcal{B}}:={\mathcal{B}}(X_n)$ be the Banach algebra of all bounded linear operators on $X_n$, and let ${\mathcal{K}}:={\mathcal{K}}(X_n)$ be the closed two-sided ideal of all compact operators on $X_n$. We denote by $C^{n\times n}$ (resp. $PC^{n\times n}$) the collection of all continuous (resp. piecewise continuous) $n\times n$ matrix functions, that is, matrix-valued functions with entries in $C$ (resp. $PC$). Put $I^{(n)}:={\mathrm{diag}}\{I,\dots, I\}$ and $S^{(n)}:={\mathrm{diag}}\{S,\dots,S\}$. Our aim is to get a Fredholm criterion for an operator $$A\in{\mathcal{U}}:={\mathrm{alg}}(PC^{n\times n},S^{(n)}),$$ the smallest Banach subalgebra of ${\mathcal{B}}$ which contains all operators of multiplication by matrix-valued functions in $PC^{n\times n}$ and the operator $S^{(n)}$. \[le:compact\] The ideal ${\mathcal{K}}$ is contained in the algebra ${\mathrm{alg}}(C^{n\times n},S^{(n)})$, the smallest closed subalgebra of ${\mathcal{B}}$ which contains the operators of multiplication by continuous matrix-valued functions and the operator $S^{(n)}$. The proof of this statement is standard and can be developed as in [@K96 Lemma 9.1] or [@K05 Lemma 5.1]. Operators of local type ----------------------- We shall denote by ${\mathcal{B}}^\pi$ the Calkin algebra ${\mathcal{B}}/{\mathcal{K}}$ and by $A^\pi$ the coset $A+{\mathcal{K}}$ for any operator $A\in{\mathcal{B}}$. An operator $A\in{\mathcal{B}}$ is said to be of [*local type*]{} if $AcI^{(n)}-cA$ is compact for all $c\in C$, where $cI^{(n)}$ denotes the operator of multiplication by the diagonal matrix-valued function ${\mathrm{diag}}\{c,\dots,c\}$. This notion goes back to Simonenko [@Simonenko65] (see also the presentation of Simonenko’s local theory in his joint monograph with Chin Ngok Min [@SCNM86]). It easy to see that the set ${\mathcal{L}}$ of all operators of local type is a closed subalgebra of ${\mathcal{B}}$. \[pr:OLT\] 1. We have ${\mathcal{K}}\subset{\mathcal{U}}\subset{\mathcal{L}}$. 2. An operator $A\in{\mathcal{L}}$ is Fredholm if and only if the coset $A^\pi$ is invertible in the quotient algebra ${\mathcal{L}}^\pi:={\mathcal{L}}/{\mathcal{K}}$. \(a) The embedding ${\mathcal{K}}\subset{\mathcal{U}}$ follows from Lemma \[le:compact\], the embedding ${\mathcal{U}}\subset{\mathcal{L}}$ follows from the fact that $cS-ScI$ is a compact operator on $L_\varrho^{p(\cdot)}$ for $c\in C$ (see, e.g., [@K03 Lemma 6.5]). \(b) The proof of this fact is straightforward. Localization ------------ [From]{} Proposition \[pr:OLT\](a) we deduce that the quotient algebras ${\mathcal{U}}^\pi:={\mathcal{U}}/{\mathcal{K}}$ and ${\mathcal{L}}^\pi:={\mathcal{L}}/{\mathcal{K}}$ are well defined. We shall study the invertibility of an element $A^\pi$ of ${\mathcal{U}}^\pi$ in the larger algebra ${\mathcal{L}}^\pi$ by using a localization techniques (more precisely, Theorem \[th:AllanDouglas\]). To this end, consider $${\mathcal{Z}}^\pi:=\big\{(cI^{(n)})^\pi:c\in C\big\}.$$ [From]{} the definition of ${\mathcal{L}}$ it follows that ${\mathcal{Z}}^\pi$ is a central subalgebra of ${\mathcal{L}}^\pi$. The maximal ideal space $M({\mathcal{Z}}^\pi)$ of ${\mathcal{Z}}^\pi$ may be identified with the curve $\Gamma$ via the Gelfand map ${\mathcal{G}}$ given by $${\mathcal{G}}:{\mathcal{Z}}^\pi\to C, \quad \big({\mathcal{G}}(cI^{(n)})^\pi\big)(t)=c(t) \quad (t\in\Gamma).$$ In accordance with Theorem \[th:AllanDouglas\], for every $t\in\Gamma$ we define ${\mathcal{J}}_t\subset{\mathcal{L}}^\pi$ as the smallest closed two-sided ideal of ${\mathcal{L}}^\pi$ containing the set $$\big\{(cI^{(n)})^\pi\ :\ c\in C,\ c(t)=0\big\}.$$ Consider a function $\chi_t\in PC$ which is continuous on $\Gamma\setminus\{t\}$ and satisfies $\chi_t(t-0)=0$ and $\chi_t(t+0)=1$. For $a\in PC^{n\times n}$ define the function $a_t\in PC^{n\times n}$ by $$\label{eq:at} a_t:=a(t-0)(1-\chi_t)+a(t+0)\chi_t.$$ Clearly $(aI^{(n)})^\pi-(a_tI^{(n)})^\pi\in{\mathcal{J}}_t$. Hence, for any operator $A\in{\mathcal{U}}$, the coset $A^\pi+{\mathcal{J}}_t$ belongs to the smallest closed subalgebra ${\mathcal{W}}_t$ of ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ containing the cosets $$\label{eq:projections} p:=\big((I^{(n)}+S^{(n)})/2\big)^\pi+{\mathcal{J}}_t, \ q:=(\chi_tI^{(n)})^\pi+{\mathcal{J}}_t,$$ where $\chi_tI^{(n)}$ denotes the operator of multiplication by the diagonal matrix-valued function ${\mathrm{diag}}\{\chi_t,\dots,\chi_t\}$ and the algebra $$\label{eq:algebra} {\mathcal{C}}:=\big\{(cI^{(n)})^\pi+{\mathcal{J}}_t\ : \ c\in\mathbb{C}^{n\times n}\big\}.$$ The latter algebra is obviously isomorphic to $\mathbb{C}^{n\times n}$, so ${\mathcal{C}}$ and $\mathbb{C}^{n\times n}$ can be identified with each other. The spectrum of $pqp+(e-p)(e-q)(e-p)$ ------------------------------------- Since $P^2=P$ on $L_\varrho^{p(\cdot)}$ (see, e.g., [@K03 Lemma 6.4]) and $\chi_t^2-\chi_t\in C$, $(\chi_t^2-\chi_t)(t)=0$, it is easy to see that $$\label{eq:2proj-conditions} p^2=p, \quad q^2=q, \quad pc=cp, \quad qc=cq$$ for every $c\in{\mathcal{C}}$, where $p,q$ and ${\mathcal{C}}$ are given by and . To apply Theorem \[th:2proj\] to the algebras $F={\mathcal{L}}^\pi/{\mathcal{J}}_t$ and $W={\mathcal{W}}_t={\mathrm{alg}}({\mathcal{C}},p,q)$, we have to identify the spectrum of $$pqp+(e-p)(e-q)(e-p) =\big(P^{(n)}\chi_tP^{(n)}+Q^{(n)}(1-\chi_t)Q^{(n)}\big)^\pi+{\mathcal{J}}_t \label{eq:element}$$ in the algebra $F={\mathcal{L}}^\pi/{\mathcal{J}}_\tau$; here $P^{(n)}:=(I^{(n)}+S^{(n)})/2$ and $Q^{(n)}:=(I^{(n)}-S^{(n)})/2$. \[le:spectrum\] Let $\chi_t\in PC$ be a continuous function on $\Gamma\setminus\{t\}$ such that $$\chi_t(t-0)=0, \quad \chi_t(\tau+0)=1,$$ and $$\chi_t(\Gamma\setminus\{t\})\cap\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))=\emptyset.$$ Then the spectrum of in ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ coincides with $\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$. Once we have Corollary \[co:important\] at hand, the proof of this lemma can be developed by a literal repetition of the proof of [@K96 Lemma 9.4]. Symbol calculus --------------- Now we are in a position to prove the main result of this paper. \[th:symbol\] Define the “double logarithmic spirals bundle” $${\mathcal{M}}:= \bigcup\limits_{t\in\Gamma} \Big(\{t\} \times \mathcal{S}\big(0,1;\delta(t),1/p(t)+\lambda(t)\big) \Big).$$ 1. For each point $(t,\mu)\in{\mathcal{M}}$, the map $$\sigma_{t,\mu} \: : \: \{S^{(n)}\}\cup\{aI^{(n)}\: :\: a\in PC^{n\times n}\} \to \mathbb{C}^{2n\times 2n}$$ given by $$\sigma_{t,\mu}(S^{(n)}) = \left( \begin{array}{ll} E & O\\ O & -E \end{array} \right), \ \sigma_{t,\mu}(aI^{(n)}) = \left( \begin{array}{ll} a_{11}(t,\mu) & a_{12}(t,\mu)\\ a_{21}(t,\mu) & a_{22}(t,\mu) \end{array} \right),$$ where $$\begin{aligned} a_{11}(t,\mu) &:=& a(t+0)\mu + a(t-0)(1-\mu),\\ a_{12}(t,\mu) &=& a_{21}(t,\mu) := (a(t+0)-a(t-0)) \sqrt{\mu(1-\mu)}, \\ a_{22}(t,\mu) &:=& a(t+0)(1-\mu) + a(t-0)\mu,\end{aligned}$$ and $O$ and $E$ are the zero and identity $n\times n$ matrices, respectively, extends to a Banach algebra homomorphism $$\sigma_{t,\mu} :{\mathcal{U}}\to\mathbb{C}^{2n\times 2n}$$ with the property that $\sigma_{t,\mu}(K)$ is the zero matrix for every compact operator $K$ on $X_n$; 2. an operator $A\in{\mathcal{U}}$ is Fredholm on $X_n$ if and only if $$\det\sigma_{t,\mu} (A)\neq 0 \quad\mbox{for all}\quad (t,\mu)\in{\mathcal{M}};$$ 3. the quotient algebra ${\mathcal{U}}^\pi$ is inverse closed in the Calkin algebra ${\mathcal{B}}^\pi$, that is, if a coset $A^\pi\in{\mathcal{U}}^\pi$ is invertible in ${\mathcal{B}}^\pi$, then $(A^\pi)^{-1}\in{\mathcal{U}}^\pi$. The idea of the proof of this theorem is borrowed from [@BK97] and is based on the Allan-Douglas local principle (Theorem \[th:AllanDouglas\]) and the two projections theorem (Theorem \[th:2proj\]). Fix $t\in\Gamma$ and choose a function $\chi_t\in PC$ such that $\chi_t$ is continuous on $\Gamma\setminus\{t\}$, $\chi_t(t-0)=0$, $\chi_t(t+0)=1$, and $\chi_t(\Gamma\setminus\{t\})\cap\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))=\emptyset$. [From]{} and Lemma \[le:spectrum\] we deduce that the algebras ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ and ${\mathcal{W}}_t={\mathrm{alg}}({\mathcal{C}},p,q)$, where $p,q$ and ${\mathcal{C}}$ are given by and , respectively, satisfy all the conditions of the two projections theorem (Theorem \[th:2proj\]). \(a) In view of Theorem \[th:2proj\](a), for every $\mu\in\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$, the map $\sigma_\mu:\mathbb{C}^{n\times n}\cup\{p,q\}\to\mathbb{C}^{2n\times 2n}$ given by – extends to a Banach algebra homomorphism $\sigma_\mu:{\mathcal{W}}_t\to\mathbb{C}^{2n\times 2n}$. Then the map $$\sigma_{t,\mu}=\sigma_\mu\circ\pi_t:{\mathcal{U}}\to\mathbb{C}^{2n\times 2n},$$ where $\pi_t:{\mathcal{U}}\to{\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ is acting by the rule $A\mapsto A^\pi+{\mathcal{J}}_t$, is a well defined Banach algebra homomorphism and $$\sigma_{t,\mu}(S^{(n)})=2\sigma_\mu p-\sigma_\mu e= \left(\begin{array}{cc}E & O \\O & -E\end{array}\right).$$ If $a\in PC^{n\times n}$, then in view of and $(aI^{(n)})^\pi-(a_tI^{(n)})^\pi\in{\mathcal{J}}_t$ it follows that $$\begin{aligned} \sigma_{t,\mu}(aI^{(n)}) &=& \sigma_{t,\mu}(a_tI^{(n)}) = \sigma_\mu(a(t-0))\sigma_\mu(e-q)+\sigma_\mu(a(t+0))\sigma_\mu q \\ &=& \left(\begin{array}{cc} a_{11}(t,\mu) & a_{12}(t,\mu)\\ a_{21}(t,\mu) & a_{22}(t,\mu)\end{array}\right).\end{aligned}$$ [From]{} Proposition \[pr:OLT\](a) it follows that $\pi_t(K)=K^\pi+{\mathcal{J}}_t={\mathcal{J}}_t$ for every $K\in{\mathcal{K}}$ and every $t\in\Gamma$. Hence, $$\sigma_{t,\mu}(K)=\sigma_\mu(0)= \left(\begin{array}{cc} O & O\\ O & O\end{array}\right).$$ Part (a) is proved. \(b) [From]{} Proposition \[pr:OLT\] it follows that the Fredholmness of $A\in{\mathcal{U}}$ is equivalent to the invertibility of $A^\pi\in{\mathcal{L}}^\pi$. By Theorem \[th:AllanDouglas\], the former is equivalent to the invertibility of $\pi_t(A)=A^\pi+{\mathcal{J}}_t$ in ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ for every $t\in\Gamma$. By Theorem \[th:2proj\](b), this is equivalent to $$\begin{aligned} \label{eq:symbol1} \det\sigma_{t,\mu}(A)=\det\sigma_\mu\pi_t(A)\ne 0 \mbox{ for all }(t,\mu)\in{\mathcal{M}}. &&\end{aligned}$$ Part (b) is proved. \(c) Since $\mathcal{S}(0,1;\delta(t),1/p(t)+\lambda(t))$ does not separate the complex plane $\mathbb{C}$, it follows that the spectra of in the algebras ${\mathcal{L}}^\pi/{\mathcal{J}}_t$ and ${\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ coincide, so we can apply Theorem \[th:2proj\](c). If $A^\pi$, where $A\in{\mathcal{U}}$, is invertible in ${\mathcal{B}}^\pi$, then holds. Consequently, by Theorem \[th:2proj\](b), (c), $\pi_t(A)=A^\pi+{\mathcal{J}}_t$ is invertible in ${\mathcal{W}}_t={\mathcal{U}}^\pi/{\mathcal{J}}_t$ for every $t\in\Gamma$. Applying Theorem \[th:AllanDouglas\] to ${\mathcal{U}}^\pi$, its central subalgebra ${\mathcal{Z}}^\pi$, and the ideals ${\mathcal{J}}_t$, we obtain that $A^\pi$ is invertible in ${\mathcal{U}}^\pi$, that is, ${\mathcal{U}}^\pi$ is inverse closed in the Calkin algebra ${\mathcal{B}}^\pi$. Note that the approach to the study of Banach algebras of SIOs based on the Allan-Douglas local principle and the two projections theorem is nowadays standard. It was successfully applied in many situations (see, e.g., [@BK95; @BK97; @FRS93; @K96; @K98; @K02; @K05]). However, it does not allow to get formulas for the index of an arbitrary operator in the Banach algebra of SIOs with $PC$ coefficients. 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ArXiv

--- abstract: 'We present new mid-infrared $N$-band spectroscopy and $Q$-band photometry of the local luminous infrared galaxy NGC 1614, one of the most extreme nearby starbursts. We analyze the mid-IR properties of the nucleus (central 150pc) and four regions of the bright circumnuclear (diameter$\sim 600$pc) star-forming (SF) ring of this object. The nucleus differs from the circumnuclear SF ring by having a [strong 8–12 continuum]{} (low 11.3 PAH equivalent width). These characteristics, together with the nuclear X-ray and sub-mm properties, can be explained by an X-ray weak active galactic nucleus (AGN), or by peculiar SF with a short molecular gas depletion time and producing an enhanced radiation field density. In either case, the nuclear luminosity ($L_{\rm IR}<$6$\times$10$^{43}$ergs$^{-1}$) is only $<$5% of the total bolometric luminosity of NGC 1614. So this possible AGN does not dominate the energy output in this object. We also compare three star-formation rate (SFR) tracers (Pa$\alpha$, 11.3 PAH, and 24 emissions) at 150pc scales [in the circumnuclear ring]{}. In general, we find that the SFR is underestimated (overestimated) by a factor of 2–4 (2–3) using the 11.3 PAH (24) emission with respect to the extinction corrected Pa$\alpha$ SFR. The former can be explained because we do not include diffuse PAH emission in our measurements, while the latter might indicate that the dust temperature is particularly warmer in the central regions of NGC 1614.' author: - | \ $^{1}$Centro de Astrobiología (CSIC/INTA), Ctra de Torrejón a Ajalvir, km 4, 28850, Torrejón de Ardoz, Madrid, Spain\ $^{2}$ASTRO-UAM, UAM, Unidad Asociada CSIC\ $^{3}$Instituto de Física de Cantabria, CSIC-Universidad de Cantabria, 39005 Santander, Spain\ $^{4}$Observatorio Astronómico Nacional (OAN-IGN)-Observatorio de Madrid, Alfonso XII, 3, 28014, Madrid, Spain\ $^{5}$Núcleo de Astronomía de la Facultad de Ingeniería, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile\ $^{6}$Instituto de Astrofísica de Andalucía, Glorieta de las Astronomía, s/n, 18008 Granada, Spain\ $^{7}$Centro de Radioastronomía y Astrofísica (CRyA-UNAM), 3-72 (Xangari), 8701, Morelia, Mexico\ $^{8}$Subaru Telescope, 650 North A’ohoku Place, Hilo, Hawaii, 96720, U.S.A.\ $^{9}$Gemini Observatory, Casilla 603, La Serena, Chile\ $^{10}$Centro de Estudios de la Física del Cosmos de Aragón, 44001 Teruel, Spain\ $^{11}$Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain title: 'Sub-arcsec mid-IR observations of NGC 1614: Nuclear star-formation or an intrinsically X-ray weak AGN?' --- \[firstpage\] galaxies: active – galaxies: nuclei – galaxies: starburst – galaxies: individual: NGC 1614 – infrared: galaxies Introduction {#s:intro} ============ Ultra-luminous and luminous infrared galaxies (U/LIRGs) are objects with infrared (IR) luminosities ($ L_{\rm IR}$) between 10$^{11}$ and 10$^{12}$ (LIRGs) and $>$10$^{12}$  (ULIRGs). Locally, objects with such high IR luminosities are unusual. However, between $z\sim 1$ and 2, galaxies in the LIRG and ULIRG luminosity ranges dominate the star-formation rate (SFR) density of the Universe [@PerezGonzalez2005; @LeFloch2005; @Caputi2007; @Magnelli2011]. Therefore, the study at high-angular resolution of local LIRGs provides a unique insight into extreme SF environments similar to those of high-$z$ galaxies near the SFR density peak of the Universe [@Madau2014]. NGC 1614 (Mrk 617) is the second most luminous galaxy within 75Mpc ($\log L_{\rm IR}=11.6$; @SandersRBGS) and according to optical spectroscopy its nuclear activity is classified as composite [@Yuan2010]. It is an advanced minor merger (3:1–5:1 mass ratio; @Vaisanen2012) located at 64Mpc (310pcarcsec$^{-1}$) with long tidal tails. Its bolometric luminosity is dominated by a strong starburst in the central kpc [@AAH01; @Imanishi2010], and, so far, there is no clear evidence of an active galactic nucleus (AGN) in NGC 1614 [@Herrero-Illana2014]. The central kpc of NGC 1614 contains a compact nucleus (45-80pc), which dominates the near-IR continuum emission, and a bright circumnuclear SF ring (diameter$\sim600$pc), which is predominant in Pa$\alpha$ [@AAH01] and other SF indicators like the polycyclic aromatic hydrocarbon (PAH) emission [@DiazSantos2008; @Vaisanen2012], cold molecular gas [@Konig2013; @Sliwa2014; @Xu2015], and radio continuum [@Olsson2010; @Herrero-Illana2014]. In addition, @GarciaBurillo2015 found a massive cold molecular gas outflow (3$\times$10$^7$$M_\odot$; $\dot{M}_{\rm out}\sim$40$M_{\odot}$yr$^{-1}$) which can be powered by the SF in the ring. A bright obscured AGN is discarded by X-ray observations [@Pereira2011; @Herrero-Illana2014]. However, previous mid-IR $N$-band imaging of NGC 1614 showed that the compact nucleus has a relatively high surface brightness [@Soifer2001; @DiazSantos2008; @Siebenmorgen2008]. Therefore, these observations suggest an enhanced mid-IR luminosity to SFR (as inferred from the observed Pa$\alpha$ luminosity) ratio in the nucleus [@DiazSantos2008], which might indicate the presence of an active nucleus. However, without high angular resolution spectroscopy no detailed studies were possible. {width="\textwidth"} In this paper we present the first high-angular resolution ($\sim$05 ) $N$-band (7.5–13) spectroscopy of the nucleus and surrounding star-forming ring of NGC 1614, as well as $Q$-band 24.5 imaging using CanariCam on the 10.4m Gran Telescopio CANARIAS (GTC). First, we describe the new observations in Section \[s:data\]. The extraction of the spectra and photometry, and a simple two component modeling are presented in Section \[s:analysis\]. We explore the AGN or SF nature of the nucleus in Section \[ss:agn\_vs\_sf\], and, in Section \[ss:sfr\_tracers\], the reliability of several SFR tracers at 150pc scales is discussed. The main conclusions are presented in Section \[s:conclusions\]. Throughout this paper we assume the following cosmology $H_{\rm 0} = 70$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$, and $\Omega_{\rm \Lambda}=0.7$ and the @Kroupa2001 IMF. Observations and Data Reduction {#s:data} =============================== Mid-IR Imaging -------------- We obtained $Q$-band diffraction limited (05) images of NGC 1614 using the Q8 filter ($\lambda_{\rm c}=24.5$, width at 50% cut-on/off of $\Delta\lambda = 0.8$) of CanariCam (CC; @Telesco2003CC) on the 10.4m GTC during December 2nd 2014. These observations are part of the ESO/GTC large program 182.B-2005 (PI Alonso-Herrero). The plate scale of CC is 008/pixel and its field of view is 26$\times$19, so it covers the central 6kpc of NGC 1614. Three exposures were taken with an on-source integration of 400s each. To reduce the data we used the <span style="font-variant:small-caps;">redcan</span> pipeline [@GonzalezMartin2013RedCan]. It performs the flat-fielding, stacking, and flux calibration of the individual exposures. The three reduced images were then combined after correcting the different background levels (right panel of Figure \[fig:maps\]). For the flux calibration the standard star HD 28749 was observed. It is relatively weak at 24.5 (1.2Jy; @Cohen1999) so the absolute calibration error of our $Q$-band observations is $\sim$20%. To check the flux calibration we also compared the integrated flux of NGC 1614 in our 24.5 image (6.0$\pm$0.9Jy) with the [*Spitzer*]{}/MIPS 24 flux (5.7$\pm$0.3Jy; @Pereira2015not). Both values are in good agreement. In addition, $N$-band imaging of this galaxy was previously obtained using Gemini/T-ReCS in the Si2 filter ($\lambda_{\rm c}=8.7$, $\Delta\lambda = 0.8$). This image was published by @DiazSantos2008 and it is shown in the middle panel of Figure \[fig:maps\]. The angular resolution of this observation estimated from the calibration star image is 04. Mid-IR Spectroscopy ------------------- We obtained $N$-band spectroscopy (7.5–13) of NGC 1614 with GTC/CanariCam on September 8th 2013 and January 5th 2014. The low spectral resolution ($R\sim175$) grating was used. These observations are also part of the ESO/GTC large program 182.B-2005. The nucleus of NGC 1614 was observed with a slit of 052 width using two perpendicular orientations (position angles 0 and 90[$^\circ$]{}). The approximate location of the slits is overplotted in the middle panel of Figure \[fig:maps\]. The on-source integration time for each of the slit orientations was 1200s. The standard star HD 28749 was observed in spectroscopy mode to provide the absolute flux calibration and telluric correction. From the two-dimensional spectrum of the standard star we derive that the angular resolution, $\sim$05, is approximately constant with the wavelength both nights. That is, the spectroscopy was not obtained in diffraction limited conditions. The data were reduced using the <span style="font-variant:small-caps;">redcan</span> pipeline. Flat-fielding, stacking, wavelength calibration, and flux calibration of the exposures are performed by this software. The spectra were extracted using a custom procedure (see Section \[ss:image\_modeling\]) instead of the default <span style="font-variant:small-caps;">redcan</span> extraction. Analysis and Results {#s:analysis} ==================== Image modeling and spectral extraction {#ss:image_modeling} -------------------------------------- The *HST*/NICMOS Pa$\alpha$ image of NGC 1614 (@AAH01, see Fig 1) revealed that the angular separation between the nucleus and the star-forming ring is 05–07, which is comparable to the angular resolution of the CC Q-band imaging and N-band spectroscopy. Therefore, to disentangle the emission produced by the different regions we modeled the mid-IR image with the highest resolution (i.e., the 8.7 T-ReCS image) with <span style="font-variant:small-caps;">galfit</span> [@Peng2010GALFIT]. @Imanishi2011 published $Q$-band imaging of NGC 1614 at 17.7. In this image, the emissions from the SF ring and the nucleus are not as clearly separated as in the CC 24.5 image, probably due to the slightly worse angular resolution (07; @Asmus2014). We used six Gaussian spatial components (nucleus, north, south, east, west, and diffuse) convolved with the PSF, to reproduce the 8.7 image (see Figure \[fig:galfit\]). These components are motivated by the Pa$\alpha$ morphology (Figure \[fig:maps\]) and is the minimum number of components needed to reproduce the mid-IR images. The position and full width half maximum (FWHM) of these components are listed in Table \[tab:components\]. According to this decomposition, the ring is located $\sim$06 away from the nucleus (190pc) and has a FWHM of $\sim$05–07 (160–220pc). The residuals of the model are less than 20% (Figure \[fig:galfit\]). To extract the fluxes from the CC Q-band image, we used <span style="font-variant:small-caps;">galfit</span> fixing the relative positions and widths of these components, but allowing their intensities to vary (see Figure \[fig:galfit\] and Table \[tab:fluxes\]). ![<span style="font-variant:small-caps;">galfit</span> models of the T-ReCS 8.7 (top panels), and CanariCam 24.5 (bottom panels) observations of the nuclear regions of NGC 1614. The observed image, the best model and the residuals are shown in the left, middle, and right panels, respectively. The color scale is in Jyarcsec$^{-2}$ units. \[fig:galfit\]](NGC1614_galfit.pdf){width="50.00000%"} --------- ------------------ ---------- -- -- -- -- -- -- -- -- Region $d^a$ FWHM$^b$ (arcsec) (arcsec) Nucleus [  $\cdots$  ]{} 0.21 Diffuse [  $\cdots$  ]{} 2.4 N 0.52 0.73 S 0.59 0.72 E 0.60 0.54 W 0.61 0.45 --------- ------------------ ---------- -- -- -- -- -- -- -- -- : Spatial decomposition nuclear region and circumnuclear ring of star formation of NGC 1614\[tab:components\] **Notes:** $^{(a)}$ Angular distance between the nucleus and the component. $^{(b)}$ Deconvolved FWHM. Similarly, we used this information to extract the CC N-band spectra. For each wavelength we generated a synthetic image taking into account the CC N-band PSF, and then we simulated the two slit orientations (P.A. 0 and 90[$^\circ$]{}) to obtain the one-dimensional spatial profiles. We varied the intensities of the different regions to reproduce simultaneously the observed N-S and E-W profiles at each wavelength. The resulting spectra are plotted in Figure \[fig:cc\_spec\]. The fluxes at 10 and 12, and the 11.3 PAH flux and equivalent width (EW) are listed in Table \[tab:fluxes\]. ![Mid-IR CanariCam spectra of the nucleus and different regions in the star-forming ring. The nuclear (orange), north (green), south (dark blue), and east (purple) spectra are shifted by 1.3, 0.8, 0.55, and 0.3Jy, respectively. The vertical lines mark the wavelength of the 7.7, 8.6, and 11.3 PAH features (dashed line). The shaded gray regions mark low-atmospheric transmission spectral ranges. \[fig:cc\_spec\]](fig_spec_cc.pdf){width="46.00000%"} --------- ----------------- ----------------- ------------------- ---------------------------------- ---------------------------- ------------------------- -- -- -- -- Region f$_\nu$(10)$^a$ f$_\nu$(12)$^a$ f$_\nu$(24.1)$^b$ 11.3 PAH$^c$ ${EW_{\rm 11.3\micron}}^d$ f$_\nu$(Pa$\alpha$)$^e$ (mJy) (mJy) (Jy) (10$^{-13}$ergcm$^{-2}$s$^{-1}$) (10$^{-3}$) (mJy) Nucleus 120 $\pm$ 8 210 $\pm$ 20 $<$0.5 3.2 $\pm$ 0.6 79 $\pm$ 8 2.2 N 91 $\pm$ 10 390 $\pm$ 30 1.7 $\pm$ 0.6 12.7 $\pm$ 0.2 220 $\pm$ 5 5.2 S 81 $\pm$ 10 390 $\pm$ 50 1.6 $\pm$ 0.4 9.3 $\pm$ 0.5 170 $\pm$ 3 5.0 E 61 $\pm$ 6 130 $\pm$ 10 2.0 $\pm$ 0.7 13.2 $\pm$ 0.3 500 $\pm$ 20 5.5 W 73 $\pm$ 20 280 $\pm$ 30 0.6 $\pm$ 0.3 14.2 $\pm$ 0.4 350 $\pm$ 10 4.6 --------- ----------------- ----------------- ------------------- ---------------------------------- ---------------------------- ------------------------- -- -- -- -- **Notes:** 3$\sigma$ upper limits are indicated for non-detections. The uncertainties do not include the $\sim$10–15% absolute calibration error. All the wavelengths are rest-frame. $^{(a)}$ The monochromatic 10 and 12 fluxes are measured in the CC spectra of each region (see Section \[ss:image\_modeling\]). $^{(b)}$ 24.1 fluxes derived from the CC $Q$-band imaging (see Section \[ss:image\_modeling\]). $^{(c)}$ Flux of the 11.3 PAH feature. $^{(d)}$ EW of the 11.3 PAH. $^{(e)}$ Pa$\alpha$ flux measured in the continuum subtracted F190N NICMOS images [@AAH06s]. To convert to flux units, these values should be multiplied by 1.56$\times$10$^{-14}$ ergcm$^{-2}$s$^{-1}$mJy$^{-1}$. Spectral Modeling {#ss:modeling} ----------------- For the five selected regions, we decomposed the $N$-band spectra together with the 24 photometry using a two component model consisting of a modified black-body with $\beta=2$ and a PAH emission template. The latter is derived from the [*Spitzer*]{}/IRS starburst template presented by @Smith07 after removing the dust continuum emission (see @Pereira2015not for details). We excluded the low-atmospheric transmission spectral ranges marked in Figure \[fig:cc\_spec\] for the fitting. The black-body temperature and the intensities of the black-body and the PAH template are free parameters of the model. In addition, we let the relative strength of the 11.3 PAH feature free during the fit since the strength of the different PAH features varies both in starbursts [@Smith07] and Seyfert galaxies [@Diamond2010]. We calculated the warm dust mass using the following relation $$M_{\rm dust} = \frac{D^2 f_{\nu}}{\kappa_{\nu} B_{\nu} (T_{\rm dust})}$$ where $D$ is the distance, $f_{\nu}$ the observed flux, $\kappa_{\nu}$ the absorption opacity coefficient, and $B_{\nu} (T_{\rm dust})$ the Planck’s blackbody law, all of them evaluated at 10. We assumed $\kappa_{10\mu m}=1920$cm$^{2}$g$^{-1}$ [@Li2001]. The results of the fits are shown in Table \[tab:models\] and Figure \[fig:models\]. The mid-IR emission of the SF ring regions are well fitted by a combination of a PAH component, which dominates the emission below 9, and a warm ($T\sim$110K) dust continuum component which dominates the emission at longer wavelengths. By contrast, the nuclear 8–13 spectrum is completely dominated by a warmer ($T\sim$160K) dust component. ![Best-fit of the modified blackbody+PAH template models to the CC spectroscopy (solid black line) and 24.5 photometry (white circle) of the different regions of NGC 1614. The 1$\sigma$ range of the best-fit model is indicated by the red shaded area. The solid green line and the dashed blue line represent the PAH template and the modified blackbody continuum, respectively. \[fig:models\]](fig_modelfit.pdf){width="46.00000%"} --------- -------------------------------------------------------- ------------------------------- ------------------------------- -- -- -- -- -- -- -- Region $\frac{\rm 7.7\micron\ PAH}{\rm 11.3\micron\ PAH}$$^a$ $T^{\rm warm}_{\rm dust}$$^b$ $M^{\rm warm}_{\rm dust}$$^c$ (K) (10$^3$) Nucleus 1.7 $\pm$ 0.2 160 $\pm$ 8 0.2$^{+0.3}_{-0.05}$ N 3.6 $\pm$ 0.5 110 $\pm$ 3 7.6$^{+6.7}_{-3.4}$ S 3.1 $\pm$ 0.4 109 $\pm$ 4 8.9$^{+3.2}_{-2.3}$ E 2.9 $\pm$ 0.5 108 $\pm$ 6 6.3$^{+1.8}_{-4.3}$ W 3.3 $\pm$ 0.6 115 $\pm$ 3 3.9$^{+2.2}_{-1.3}$ --------- -------------------------------------------------------- ------------------------------- ------------------------------- -- -- -- -- -- -- -- : Results from the modeling of the CC data\[tab:models\] **Notes:** $^{(a)}$ Ratio between the intensities of the modeled 7.7 and 11.3 PAH features (Section \[ss:modeling\]). $^{(b)}$ Temperature of the warm dust component detected in the mid-IR. $^{(c)}$ Mass of the warm dust (see Section \[ss:modeling\] for details). The AGN or SF Nature of the Nucleus {#ss:agn_vs_sf} =================================== The nature of the nucleus (central 150pc) of NGC 1614 is not well established. In part, this is because it is surrounded by a circumnuclear ring with strong star-formation (ring SFR$\sim$40yr$^{-1}$; @AAH01), which masks the relatively weak nuclear emission when observed at lower angular resolutions. In the high-angular resolution (011) *HST*/NICMOS images, the nucleus is slightly resolved and shows near-IR colors compatible with stellar emission, although the CO index is inconsistent with an old stellar population [@AAH01]. To explain this, @AAH01 suggested that the nuclear SF is more evolved than that of the star-forming ring. Based on *ASCA* X-ray observations, @Risaliti2000 suggested that NGC 1614 may host a Compton-thick AGN. However, the FeK 6.4keV line, which usually has a high EW in Compton-thick AGNs (although it depends on the obscuring matter geometry; see e.g., @Fabian2002), is not detected in more sensitive *XMM-Newton* observations [@Pereira2011]. More recently, the non-detection of CO(6–5) emission and 435 continuum in the nucleus in high-resolution ALMA observations implies that the amount of dust and molecular gas is much lower than that expected for a Compton-thick AGN [@Xu2015]. Similarly, interferometric radio continuum observations reveal that the nuclear emission is mostly thermal and relatively weak, which also supports the non-AGN nuclear activity [@Herrero-Illana2014]. Consequently, SF, as traced by the nuclear Pa$\alpha$ emission, would be the dominant energy source of the nucleus of NGC 1614. Our new mid-IR data challenge these previous results. The nuclear spectrum shows a strong 12 mid-IR (and a low EW of the 11.3 PAH feature), and the nuclear 24 continuum is weak in comparison with the SF ring emission. Therefore, the nucleus of NGC 1614 presents some characteristics (weak X-ray and far-IR emissions, lacking molecular gas, strong 12 mid-IR continuum, and Pa$\alpha$ emission) that cannot be explained in a standard AGN or SF context. In the following, we discuss possible modifications to the AGN and SF scenarios to explain the observations available so far. X-ray weak AGN? --------------- ![Comparison of the nuclear NGC 1614 mid-IR spectrum (black) and the average spectra of Type 1 (red) and Type 2 (blue) Seyfert galaxies from @AAH2014 normalized at 12. The shaded regions represent the 1$\sigma$ dispersion of the averaged spectra. \[fig:agn\_comparison\]](compara_agn.png){width="45.00000%"} ### Mid-IR AGN evidences The CC spectrum of the nucleus is remarkably different from the spectra of the star-forming regions in the ring of SF. It shows a strong mid-IR continuum relative to the PAH emission, a dust temperature higher than in the SF regions of the ring (Table \[tab:models\]), and a continuum peak at around $\sim$20 (Figure \[fig:models\]). Differences in the dust continuum emission are also evident if we consider the 24 to 10 flux ratio which is $\sim$10-30 in the ring and $<$5 in the nucleus (see Table \[tab:fluxes\] and Figure \[fig:maps\]). Low 24/10 ratios are predicted by AGN torus models because of the high dust temperatures reached in the torus (150–1500K; @Nenkova2008). Moreover, mid-IR [*Spitzer*]{}/IRS spectroscopy of active galaxies shows that for $\sim$30% of them (including Type 1 and 2 Seyfert objects) the mid-IR spectra peaks at $\sim$20 indicating that a warm dust component ($T\sim150-170$K) dominates the mid-IR emission [@Buchanan2006; @Wu2009]. This trend is also observed in ground-based sub-arcsecond mid-IR spectroscopic surveys of Seyfert galaxies (e.g., @AAH2011torus [@RamosAlmeida2009; @RamosAlmeida2011]). In Figure \[fig:agn\_comparison\] we compare the average Type 1 and Type 2 AGN mid-IR spectra obtained by @AAH2014 for nearby Seyfert galaxies. It shows that they all have similar continuum slopes. This suggests that the warm dust conditions in the nucleus of NGC 1614 are similar to those found in Seyfert galaxies. [Although, the emission of hotter dust (at $\sim$8) is weaker in NGC 1614.]{} The minimum 11.3 PAH EW is located at the nucleus (79 $\pm$ 8)$\times10^{-3}$ (Table \[tab:fluxes\]). This behavior is also observed in local Seyfert galaxies, and it is explained in these objects by the increased AGN continuum contribution in the nucleus [@AAH2014; @Esquej2014; @RamosAlmeida2014; @GarciaBernete2015]. In addition, in the nucleus of NGC 1614, the 11.3 PAH feature is enhanced by a factor of $\sim$2 with respect to the 7.7 PAH feature (Table \[tab:models\]). Similar enhancements of the 11.3 PAH feature are observed in active galaxies although on kpc scales [@Diamond2010]. ### Weak X-ray Emission {#ss:xrayweak} A correlation between the 12 and the 2–10keV luminosities is observed for Seyfert galaxies [@Horst2008; @Levenson2009; @Gandhi2009; @Asmus2011]. For the nuclear 12 luminosity measured from the spectrum of NGC 1614 ($\nu L_{\nu}=$2.6$\times$10$^{43}$ergs$^{-1}$) the expected hard X-ray luminosity would be 1.6$\times$10$^{43}$ergs$^{-1}$ according to the @Gandhi2009 relation. Threfore, both the nuclear 12 and expected 2–10keV luminosities are comparable to that of an average local Seyfert galaxy (see Figure 1 of @Gandhi2009). However, the observed integrated hard X-ray luminosity of this galaxy is just 1.4$\times$10$^{41}$ergs$^{-1}$, almost a factor of 200 lower than expected for an AGN, and most of it can be explained by the hard X-ray emission from star-formation (i.e., high-mass X-ray binaries; @Pereira2011). Similarly, the soft X-ray emission is also better explained by star-formation [@Pereira2011; @Herrero-Illana2014]. If an AGN is present in the nucleus of NGC 1614, three possibilities may explain the weakness of the X-ray emission: it may be a strongly variable source observed during its low state; it may be a Compton-thick AGN so the 2–10keV emission is absorbed; or it may be an intrinsically X-ray weak AGN. There are three hard X-ray observations of NGC 1614 during 18yr (Table \[tab:xray\]) which show that the variability is less than a factor of 2. So it is not likely that X-ray variability is the reason for the X-ray weakness. The Compton-thick AGN possibility was rejected by @Xu2015 based on the low amount of molecular gas and cold dust in the nucleus. Moreover, NGC 1614 is not detected in the 14–195keV [*Swift*]{}/BAT 70-Month Hard X-ray Survey [@Baumgartner2013]. If NGC 1614 would be a Compton-thick AGN with an intrinsic 2–10keV luminosity of 1.6$\times$10$^{43}$ergs$^{-1}$ (see above), its 14–195keV flux would be[^1] 6$\times$10$^{-11}$ergcm$^{-2}$s$^{-1}$, which is $\sim$4 times the 5$\sigma$ sensitivity of the [*Swift*]{}/BAT survey. Finally, it is also possible that the X-ray emission of the NGC 1614 AGN is intrinsically weak. The ultra-luminous IR galaxy Mrk 231 [@Teng2014], as well as several quasars [@Leighly2007; @Miniutti2012; @Luo2014], have X-ray luminosities 30–100 times weaker than those predicted by the $\alpha_{\rm OX}$[^2] vs. $L_{\rm 2500A})$ correlation, probably due to a distortion of the accretion disk corona [@Miniutti2012; @Luo2013]. In the case of NGC 1614, the nuclear UV emission is completely obscured (see @Petty2014), so a direct comparison with the results for these X-ray weak AGNs is not possible. However, using the 12 emission we obtain that the [observed]{} 2–10keV emission is more than two orders of magnitude lower than the expected value, similar to the X-ray weakness observed on those objects. ------------ ------------------ ---------------------------------- ------ -- -- -- -- -- -- -- Date Telescope Flux Ref. (10$^{-13}$ergcm$^{-2}$s$^{-1}$) 1994-02-16 [*ASCA*]{} 5.6 1 2003-02-13 [*XMM-Newton*]{} 2.7$\pm$0.4 2 2012-04-10 [*Swift*]{} 2.5$\pm$0.4 3 ------------ ------------------ ---------------------------------- ------ -- -- -- -- -- -- -- : 2–10keV X-ray observations of NGC 1614\[tab:xray\] **References:** (1) @Risaliti2000; (2) @Pereira2011; (3) @Evans2014. or Nuclear Star-formation? -------------------------- Alternatively, it is possible to explain the nuclear observations assuming only star-formation (SF). However, the nuclear SF and the SF taking place in the ring surrounding the nucleus must have very different characteristics. In particular, the nuclear mid-IR spectrum shows a strong 8–12 continuum that is not present in the ring spectra (Figure \[fig:models\]), and the nucleus remains undetected in the 435 far-IR continuum and CO(3–2) maps [@Xu2015; @Usero2015] while the ring is clearly detected. In our nuclear mid-IR spectrum, we detect the 11.3 PAH feature which is usually associated with SF (mostly B stars, see @Peeters2004). Using the $L_{\rm 11.3\mu m\,PAH}$ SFR calibration of @Diamond-Stanic2012, we estimate a nuclear SFR of $\sim$0.9yr$^{-1}$ (Table \[tab:sfr\] and see Section \[ss:sfr\_tracers\]). We also used the nuclear Pa$\alpha$ flux [@DiazSantos2008] to derive a SFR $\sim$1.5yr$^{-1}$ (assuming $A_{\rm k}=$0.3mag; @AAH01), so both SFR tracers are in agreement within a factor of 2. Finally, we used the IR continuum upper limits at 24 and 432 to derive an upper limit for the nuclear IR (4-1000) luminosity of $<$6$\times$10$^{43}$ergs$^{-1}$. This upper limit is compatible with the expected IR luminosity for a SFR$\sim$1.5yr$^{-1}$ ($\sim$4$\times$10$^{43}$ergs$^{-1}$; @Kennicutt2012). Therefore, all these IR SFR tracers are compatible and they indicate that the nuclear SFR is $\leq$1.5yr$^{-1}$, that is, less than $<$2% of the total SFR of NGC 1614 ($\sim$100yr$^{-1}$; @Pereira2015not). However, the nuclear and the integrated IR (8–500) spectral energy distributions are very different. The ring is detected at 435 [@Xu2015] and 24 (Figure \[fig:maps\]),but the nucleus is not. Therefore, this implies that the dust temperature is much higher in the nucleus, as already suggested by our mid-IR data. This higher nuclear dust temperature (Table \[tab:models\]) can be explained by the enhanced radiation field density, which is expected to increase the dust temperature (see @Draine07), due to an increased density of young stars in the nucleus (or an AGN, see Section \[ss:xrayweak\]). Molecular gas is not detected in the nucleus of NGC 1614. From the 05 resolution CO(3-2) ALMA observations of NGC 1614, @Usero2015 estimate an upper limit to the nuclear molecular gas mass of 3$\times$10$^{6}$[^3]. This low molecular gas mass puts the nucleus of NGC 1614 well above the Kennicutt-Schmidt relation (see Figure 8 of @Xu2015). Consequently, the molecular gas depletion time is $<$3Myr, much lower than in normal galaxies at 100pc scales (1–3Gyr; e.g., @Leroy2013), and also lower than in local ULIRGs (70–100Myr; e.g., @Combes2013). A short depletion time might indicate that the ignition of the nuclear SF occurred earlier than in the ring (see @AAH01). [Therefore, the nuclear starburst would have consumed a larger fraction of the original cold molecular gas than the younger starburst of the ring. Actually, the evolutionary state of the SF regions is commonly used to explain the dispersion of individual SF regions in the Kennicutt-Schmidt relation (e.g., @Onodera2010 [@Schruba2010; @Kruijssen2014]).]{} However, the integrated (including nucleus and SF ring) dense molecular gas depletion time in NGC 1614 is also shorter ($\sim$10Myr) than in other LIRGs ($\sim$50Myr; @GarciaBurillo2012), so it is not obvious to associate the particularly short nuclear depletion time with older SF. Alternatively, a massive molecular outflow, produced by an AGN or SN explosions (see @GarciaBurillo2015), could have swept most the molecular gas away from the nucleus. On the other hand, the hard X-ray luminosity of this object is also compatible with a SF origin [@Pereira2011], although most of the emission would be produced in the ring. Unfortunately, the angular resolution of the *Chandra* X-ray data is not sufficient to separate the nucleus and the ring [@Herrero-Illana2014]. Note that, in principle, a combination of SF and a normal AGN would be also possible. However, this assumption suffers the same problems explaining the observations than the SF and AGN individually. For these reason, we do not discuss this AGN$+$SF composite possibility. ------------------- ------------------- ------------------ ---------------- ---------- -- -- -- -- -- -- \[-1.5ex\] \[-2.5ex\] Region $A_{\rm k}$$^{a}$ Pa$\alpha$$^{b}$ 11.3 PAH$^{c}$ 24$^{d}$ Nucleus$^\star$ 0.3 1.5 0.9 $<$6 N 0.7 9.3 4.1 22 S 0.8 11.2 3.1 21 E 0.6 7.8 4.1 26 W 1.0 16.4 5.2 7 ------------------- ------------------- ------------------ ---------------- ---------- -- -- -- -- -- -- : SFR from different IR tracers\[tab:sfr\] **Notes:** $^{(a)}$ $K$-band extinction in magnitudes derived from the stellar colors [@AAH01]. $^{(b)}$ Extinction corrected Pa$\alpha$ SFR using the @Kennicutt2012 calibration assuming H$\alpha$/Pa$\alpha$ = 8.51. $^{(c)}$ SFR obtained from the 11.3 PAH luminosities (Table \[tab:fluxes\]) based on the @Diamond-Stanic2012 calibration. We multiplied by 2 our 11.3 PAH luminosities to account for the different method used to measure the PAH features (local continuum vs. full decomposition, see @Smith07). $^{(d)}$ SFR derived from the monochromatic 24 luminosities (Table \[tab:fluxes\]) using the @Rieke2009 calibration. $^{(\star)}$ Nuclear SFR derived assuming that all the nuclear emission is produced by SFR (i.e., no AGN). SFR Tracers at $\sim$150 Scales {#ss:sfr_tracers} =============================== Using the new CC mid-IR data (11.3 PAH and 24 continuum) in combination with the NICMOS Pa$\alpha$ image, we can test several SFR calibrations at 150pc scales in this galaxy. In Table \[tab:sfr\] we show a summary of the SFR derived using these tracers for the five regions we defined in NGC 1614. We used the calibrations of @Kennicutt2012, @Diamond-Stanic2012, and @Rieke2009 for the Pa$\alpha$, 11.3 PAH, and 24 tracers, respectively. The Pa$\alpha$ emission was corrected for extinction using the near-IR continuum colors (see @AAH01). Since the extinction corrected Pa$\alpha$ calibration is a direct measurement of the number of ionizing photons produced by young stars, we consider it as the reference SFR tracer. The 24 luminosity gives the highest SFR values (2–3 and 5–7 times higher than those derived from the Pa$\alpha$ and 11.3 PAH luminosities, respectively), except in the W region of the ring. The modeling of the radio emission of the W region indicates the presence of supernovae (SN; @Herrero-Illana2014), so it could be more evolved than the rest of the ring. Therefore, a lower amount of young stars would be dust embedded in this region reducing the warm dust emission. The disagreement between the extinction corrected Pa$\alpha$ and the 24 SFR values is $\sim$0.4dex, which is higher than the calibration uncertainty (0.2dex). Although, in principle, both tracers should produce similar SFR estimates (see Equations 5 and 8 of @Rieke2009). There are two possibilities to explain this. First, it is possible that even the extinction corrected Pa$\alpha$ emission underestimates the SFR. In extremely obscured regions (e.g., $A_{\rm v}>$15–20mag), dust might absorb the Pa$\alpha$ emission completely, as well as part of the ionizing photons, and therefore, rendering any extinction correction ineffective. Alternatively, an increase of the dust temperature at high SFR densities, like in the SF ring of NGC 1614, can produce enhanced 24 emission that might not be taken into account by the 24 SFR calibration which is valid for integrated emission of galaxies (e.g., @Calzetti2010). The stellar $A_{\rm k}$ measured in the SF ring of NGC 1614 is 0.6–1.0mag ($A_{\rm v}=$5–10mag; @AAH01), so the obscuration level is not as extreme as observed in some ULIRGs ($A_{\rm v}=$8–80mag; @Armus07). In addition, the 9.7 silicate absorption in the SF ring spectra is not very deep (Figure \[fig:models\]). Therefore, this favors the second possibility. That is, an increased 24 emission in the SF ring of NGC 1614 due to a warmer dust emission. According to Table \[tab:sfr\], the SFR derived from the 11.3 PAH luminosity is 2–4 times lower than that derived from Pa$\alpha$. The 11.3 PAH SFR calibration is based on $\sim$kpc integrated measurements [@Diamond-Stanic2012]. However, it is known that the PAH emission, and in particular the 11.3 PAH emission, is more extended than the warm dust continuum and other ionized gas tracers (e.g., \[\]12.81; @DiazSantos2011). Actually, $\sim$30–40% of the total PAH emission is not related to recent SF [@Crocker2013]. Therefore, this SFR calibration possibly includes a considerable amount of PAH emission not produced by young stars. [In addition, using templates of SF galaxies, @Rieke2009 showed that the 11.3 PAH contribution to the total IR luminosity drops by a factor of $\sim$2.5 for galaxies with $L_{\rm IR}>10^{11}$. A similar result was found by @AAH2013 for a sample of local LIRGs.]{} [A combination of these reasons might]{} explain why we obtain these relatively low SFR estimates from the 11.3 PAH luminosities for the $\sim$150pc SF regions in the ring of NGC 1614. Conclusions {#s:conclusions} =========== We analyzed new GTC/CC high-angular resolution ($\sim$05) mid-IR observations of the local LIRG NGC 1614. The new $N$-band spectroscopy and $Q$-band imaging are combined with existing *HST*/NICMOS Pa$\alpha$ and T-ReCS 8.7 images to study the properties of the bright circumnuclear SF ring and the nucleus of this object. The main results are the following: 1. We extracted mid-IR spectra from four different regions in the circumnuclear SF ring and from the nuclear region (central 05$\sim$150pc). The spectra from the SF ring are typical of a SF region with strong PAH emission and a shallow 9.7 silicate absorption. By contrast, the nuclear spectrum has a strong mid-IR continuum, which dominates its mid-IR emission, and weak PAH emission (EW$_{\rm 11.3\micron}$=80$\times$10$^{-3}$). Similarly, the SF ring is clearly detected in the 24.5 image, as expected for a SF region, while the nucleus is [weaker]{} at this wavelength. 2. A two component model, consisting of a modified black-body with $\beta=2$ and a PAH emission template, reproduces the observed $N$ spectra and $Q$ photometry well. The main differences between the nuclear and the SF ring observations are: the higher dust temperature in the nucleus (160K in the nucleus vs. $\sim$110K in the ring); the lower PAH EW; and the lower nuclear 7.7/11.3 PAH ratio. 3. The above results based on the mid-IR data, suggest that an AGN might be present in the nucleus. However, this is at odds with the low X-ray luminosity of NGC 1614 ($\sim$200 times lower than that expected for an AGN with the observed 12 continuum luminosity). Since the hard (2–10keV) X-ray emission shows no variability, and likely it is not a Compton-thick AGN, if an AGN is present in NGC 1614, it must be an intrinsically X-ray weak AGN. We also calculated an upper limit to the IR luminosity of the nucleus, $<$6$\times$10$^{43}$ergs$^{-1}$. 4. Alternatively, SF can explain the observations of the nucleus too. However, we need to invoke extremely short molecular gas depletion times ($<$3Myr [for a nuclear SFR of $\sim$1–1.5yr$^{-1}$]{}), and an increased radiation field density to explain the observed hot dust in the nucleus. 5. Finally, we compared three SFR tracers at 150pc scales [in the circumnuclear ring]{}: extinction corrected Pa$\alpha$, 11.3 PAH, and 24 continuum. Since the extinction is not extremely high ($A_{\rm v}<10$mag), we take as reference the Pa$\alpha$ derived SFR. In general, the 24 SFR overestimates the SFR by a factor of 2–3, while the 11.3 PAH underestimates the SFR by a factor of 2–4. The former might be explained if the dust temperature is higher in the SF regions of NGC 1614, while the latter [could be because we do not include diffuse PAH emission in our measurements as well as because the PAH contribution to the total IR luminosity might be reduced in LIRGs.]{} 6. In the West region of the ring, the 24 emission is $\sim$5 times weaker than expected based on the observed Pa$\alpha$/24 ratio in this galaxy. We propose that this is because this is a more evolved SF region (SN are present; @Herrero-Illana2014) where a larger fraction of the young stars are not dust embedded. In summary, our mid-IR data suggest that an intrinsically X-ray weak AGN ($L^{\rm AGN}_{\rm bol}\sim$10$^{43}$ergs$^{-1}$, $<$5% of the NGC 1614 bolometric luminosity) might be present in the nucleus of NGC 1614. However, SF with a short molecular gas depletion time and increased dust temperatures can explain the observations as well. In order to further investigate the nature of the nucleus of this galaxy, IR and sub-mm high-angular resolution observations are needed. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for useful comments and suggestions. We thank the GTC staff for their continued support on the CanariCam observations. We acknowledge support from the Spanish Plan Nacional de Astronomía y Astrofísica through grants AYA2010-21161-C02-01, and AYA2012-32295. AAH and AA acknowledges funding from the Spanish Ministry of Economy and Competitiveness under grants AYA2012-31447 and AYA2012-38491-CO2-02, which are party funded by the FEDER program. MAPT acknowledges support from the Spanish MICINN through grant AYA2012-38491-C02-02. CRA acknowledges support from a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (PIEF-GA-2012-327934). Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma. Partially based on observations obtained at the Gemini Observatory (program GS-2006B-Q-9), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). 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SPIE, 4841, 913 , S. H., [Brandt]{}, W. N., [Harrison]{}, F. A., [et al.]{} 2014, , 785, 19 , A. 2015 in prep. , P., [Rajpaul]{}, V., [Zijlstra]{}, A. A., [Reunanen]{}, J., & [Kotilainen]{}, J. 2012, , 420, 2209 , Y., [Charmandaris]{}, V., [Huang]{}, J., [Spinoglio]{}, L., & [Tommasin]{}, S. 2009, , 701, 658 , C. K., [Cao]{}, C., [Lu]{}, N., [et al.]{} 2015, , 799, 11 , T.-T., [Kewley]{}, L. J., & [Sanders]{}, D. B. 2010, , 709, 884 \[lastpage\] [^1]: Using <span style="font-variant:small-caps;">XSPEC</span> [@Arnaud1996] and assuming a power-law spectrum with $\Gamma=1.9$ [@Marconi2004] and $N_{\rm H}=$3$\times$10$^{24}$cm$^{-2}$. Increasing the $N_{\rm H}$ up to 10$^{26}$cm$^{-2}$ the flux would be reduced by factor of two. [^2]: $\alpha_{\rm OX}=-0.384\log (L(2{\rm \,keV}) \slash L(2500\AA) $ [^3]: Assuming a CO(3-2) to CO(1-0) ratio of $\sim$1 and the Galactic CO-to-H$_2$ conversion factor [@Bolatto2013]. Using the conversion factor for ULIRGs it would be a factor of $\sim$4 lower.

ArXiv

--- abstract: 'We study various aspects of extracting spectral information from time correlation functions of lattice QCD by means of Bayesian inference with an entropic prior, the maximum entropy method (MEM). Correlator functions of a heavy-light meson-meson system serve as a repository for lattice data with diverse statistical quality. Attention is given to spectral mass density functions, inferred from the data, and their dependence on the parameters of the MEM. We propose to employ simulated annealing, or cooling, to solve the Bayesian inference problem, and discuss practical issues of the approach.' author: - H Rudolf Fiebig title: Spectral density analysis of time correlation functions in lattice QCD using the maximum entropy method --- \[sec:intro\]Introduction ========================= Numerical simulations of quantum chromodynamics (QCD) on a Euclidean space-time lattice provides access to mass spectra of hadronic systems through the analysis of time correlation functions. In theory the latter are linear combinations of exponential functions $$C(t,t_0)=Z_1e^{-E_1(t-t_0)}+Z_2e^{-E_2(t-t_0)}+\ldots\,, \label{exp2}$$ where the $E_n$ are the excitation energies of the system and the strength coefficients $$Z_n=|\langle n|\hat{\Phi}(t_0)|0\rangle|^2 \label{Zp}$$ are matrix elements of some vacuum-subtracted operator $\hat{\Phi}(t_0)=\Phi(t_0)-\langle 0|\Phi(t_0)| 0\rangle$ between the vacuum $|0\rangle$ and a ground or excited state $|n\rangle, n>0$. In practice the exponential model (\[exp2\]) is fitted to noisy numerical simulation ‘data’. The statistical quality of simulation data rarely is good enough for the two-exponential fit (\[exp2\]) to succeed. It is common practice to look at the large-$t$ behavior of the correlation function $C(t,t_0)$ in a $t$-interval where it is dominated by only one exponential, with the lowest energy, and then make a one-parameter fit to a plateau of the effective-mass function $\mu_{\rm eff}(t,t_0)=-{\partial}\ln C(t,t_0)/{\partial t}$. Possible discretizations are $$\begin{aligned} \mu_{\rm eff,0}(t,t_0)&=&-\ln\left(\frac{C(t+1,t_0)}{C(t,t_0)}\right) \simeq m_{\rm eff,0}\label{eff0}\\ \mu_{\rm eff,1}(t,t_0)&=&\frac{C(t+1,t_0)}{C(t,t_0)}\simeq e^{-m_{\rm eff,1}}\label{eff1}\\ \mu_{\rm eff,2}(t,t_0)&=&\frac{C(t+1,t_0)-C(t-1,t_0)}{2C(t,t_0)}\label{eff2}\\ & &\simeq -\sinh(m_{\rm eff,2})\nonumber\\ \mu_{\rm eff,3}^2(t,t_0)&=&\frac{C(t+1,t_0)+C(t-1,t_0)-2C(t,t_0)}{C(t,t_0)}\nonumber\\ & &\simeq 2(\cosh(m_{\rm eff,3})-1)\label{eff3}\,.\end{aligned}$$ The expressions after the $\simeq$ are the values of $\mu_{\rm eff}$ for a pure plateau of mass $m_{\rm eff}$. The procedure implies the selection of consecutive time slices $t=t_1\ldots t_2$ for which $\mu_{\rm eff}={\rm const}$, within errors, and an appropriate fit. The selection of this, so-called, plateau is a matter of judgment. A condition for reliable results is that the correlation function (\[exp2\]) is dominated by just one exponential term, usually the ground state. The latter can be enhanced by the use of smeared operators [@Alexandrou:1994ti] and fuzzy link variables [@Alb87a]. This analysis procedure discourages consideration of excited states. In fact it will only produce reliably results if those are suppressed. Workarounds involve diagonalization of a correlation matrix of several operators or variational techniques [@Morningstar:1999rf]. Those however, still rely on plateau selection without utilizing the information contained in the entire available time-slice range of a correlation function. As lattice simulations of QCD now aim at excited hadron states, $N^\ast$’s for example [@Lee:1998cx; @Gockeler:2001db; @Sasaki:2001nf], this situation is unsatisfactory. Alternative methods employing Bayesian inference [@Jar96] are a viable option. The maximum entropy method (MEM), which involves a particular choice of the Bayesian prior probability, falls in this class. Bayesian statistics [@Box73] is a classic subject with a vast range of applications. However, application within the context of lattice QCD is relatively new [@Nakahara:1999vy; @Nakahara:1999bm; @Asakawa:2000pv; @Asakawa:2000tr; @Lepage:2001ym]. In this work we report on our experience using the MEM for extracting spectral mass density functions $\rho(\omega)$ from lattice-generated time correlators $$C(t,t_0)=\int d\omega\,\rho(\omega) e^{-\omega(t-t_0)}\,, \label{Crho}$$ where a discrete set of time slices $t$ is understood. Discretization of the $\omega$-integral with reasonably fine resolution leads to an ambiguous problem where the number of parameters values $\rho(\omega)$ is (typically much) larger than the number of lattice data $C(t,t_0)$. In the MEM an entropy term involving the spectral density is used as a Bayesian prior to infer $\rho(\omega)$ from the data. We here apply MEM analysis to sets of lattice correlation functions of a meson-meson system. Those particular simulations are aimed at learning about mechanisms of hadronic interaction. This will be discussed separately [@Fie02c]. The lattice data generated within that project involve local and nonlocal operators. They exhibit a wide range of statistical quality from ‘very good’ to ‘marginally acceptable’. Our focus here is to utilize those data as a testing ground for Bayesian MEM analysis. In contrast to other works we employ simulated annealing to the solution of the Bayesian inference problem. The main aim of this work is to explore the feasibility of this approach for extracting masses from a lattice simulation using realistic lattice data, including excitations. For the most part this translates into studying the sensitivity of the method to to its native parameters. \[sec:BayesCF\]Bayesian Inference for Curve Fitting =================================================== From a Bayesian point of view the spectral density function $\rho$ in (\[Crho\]) is a random variable subject to a certain probability distribution functional ${\cal P}[\rho]$. Solution of the curve fitting problem consists in finding the function $\rho$ which maximizes the conditional probability ${\cal P}[\rho\leftarrow C]$, the [*posterior probability*]{}, given a ‘measured’ data set $C$. Computation of $\rho$ is then based on Bayes’ theorem [@Jar96] $${\cal P}[\rho\leftarrow C]\, {\cal P}[C] ={\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]\,, \label{BayesT}$$ also known as ‘detailed balance’ in a different context. The functional ${\cal P}[C]$, the [*evidence*]{}, gives the probability of measuring a data set $C$. The conditional probability ${\cal P}[C\leftarrow \rho]$, the [*likelihood function*]{}, determines the probability of measuring $C$ given a spectral function $\rho$. Finally ${\cal P}[\rho]$, the Bayesian [*prior*]{}, defines a constraint on the spectral density function $\rho$. Its choice is a matter of judgment. Ideally, the prior should reflect the physics known about the system, for example an upper limit on the hadronic mass scale. The posterior probability is the product of the likelihood function and the prior ${\cal P}[\rho\leftarrow C] = {\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]/{\cal P}[C]$, where the [*evidence*]{} merely plays the role of a normalization constant [@Jar96]. Indeed, the normalization condition $\int[d\rho]{\cal P}[\rho\leftarrow C]=1$ applied to (\[BayesT\]) gives ${\cal P}[C] = \int[d\rho]{\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]$. Thus, for a fixed $C$, we have $${\cal P}[\rho\leftarrow C]\propto{\cal P}[C\leftarrow \rho]\, {\cal P}[\rho]\,. \label{BayesT3}$$ The curve fitting problem requires the product of the [*likelihood function*]{} and the [*prior*]{} function. \[sec:spectralD\]Spectral density --------------------------------- Our lattice data come from correlation functions built from heavy-light meson-meson operators $$\Phi_v=v_1\Phi_1+v_2\Phi_2\,, \label{Phiv}$$ where $\Phi_1$ and $\Phi_2$ involve local and non-local meson-meson fields, respectively, at relative distance $r$, and $v$ are some coefficients [@Fiebig:2001mr; @Fiebig:2001nn]. On a finite lattice the corresponding correlator $C_v(t,t_0)=\langle\hat{\Phi}^\dagger_v(t)\hat{\Phi}_v(t_0)\rangle$, where $\hat{\Phi}=\Phi-\langle\Phi\rangle$, has a purely discrete spectrum $$C_v(t,t_0)=\sum_{n\neq 0} |\langle n|\Phi_v(t_0)|0\rangle|^2 e^{-\omega_n(t-t_0)}\,. \label{Cvn}$$ Here $|n\rangle$ denotes a complete set of states with energies $\omega_n$, some of which may be negative due to periodic lattice boundary conditions and operator structure. Our normalization conventions for forward and backward going propagators are determined by defining $$\exp_T(\omega,t)=\Theta(\omega)e^{-\omega t} +\Theta(-\omega)e^{+\omega(T-t)}\,, \label{expT}$$ where $0\leq t< T$, and $\Theta$ denotes the step function. We then expect the lattice data to fit the following model $$F(\rho_T|t,t_0)=\int_{-\infty}^{+\infty}d\omega\/ \rho_T(\omega)\exp_T(\omega,t-t_0)\,, \label{Fc}$$ where $\rho_T(\omega)$ is a spectral density function, defined for positive (forward) and negative (backward) frequencies. The requirement that the model be exact, $F(\rho_T|t,t_0)=C_v(t,t_0)$, leads to $$\begin{aligned} \rho_T(\omega)&=&\sum_{n\neq 0}\delta(\omega-\omega_n)\, |\langle n|\Phi_v(t_0)|0\rangle|^2\times\nonumber\\ & &[\Theta(\omega_n)+\Theta(-\omega_n)e^{-\omega_n T}]\,.\label{rhoT}\end{aligned}$$ Thus a discrete sum over $\delta$-peaks is the theoretical form of the spectral function. Our objective is to compute $\rho_T(\omega)$ from lattice data using Bayesian inference. \[sec:likely\]Likelihood function --------------------------------- Toward this end we proceed to construct the likelihood function. The lattice data come in the form of an average over $N_U$ gauge configurations $$C_v(t,t_0)=\frac{1}{N_U}\sum_{n=1}^{N_U}C_v(U_n|t,t_0)\,, \label{CNU}$$ where $C_v(U_n|t,t_0)$ is the value of an operator, in this case $\hat{\Phi}^\dagger_v(t)\hat{\Phi}_v(t_0)$, in one gauge field configuration $U_n$. Correlation function data on different time slices are stochastically dependent. Their errors are described by the covariance matrix $$\begin{aligned} \Gamma_v(t_1,t_2)=\frac{1}{N_U}\sum_{n=1}^{N_U} &&\left(\rule{0mm}{4mm}C_v(t_1,t_0)-C_v(U_n|t_1,t_0)\right)\times\nonumber\\ &&\left(\rule{0mm}{4mm}C_v(t_2,t_0)-C_v(U_n|t_2,t_0)\right).\label{Ecov}\end{aligned}$$ The $\chi^2$-distance of the spectral model (\[Fc\]) from the lattice data then is $$\begin{aligned} \chi^2=\sum_{t_1,t_2} &&\left(\rule{0mm}{4mm}C_v(t_1,t_0)-F(\rho_T|t_1,t_0)\right)\Gamma_v^{-1}(t_1,t_2) \times\nonumber\\ &&\left(\rule{0mm}{4mm}C_v(t_2,t_0)-F(\rho_T|t_2,t_0)\right).\label{Chi2}\end{aligned}$$ For numerical work a discretization scheme of the $\omega$-integral in (\[Fc\]) is required. Our choice is $$F(\rho_T|t,t_0)\simeq\sum_{k=K_-}^{K_+} \rho_k\,\exp_T(\omega_k,t-t_0) \label{Fd}$$ where $\omega_k=\Delta\omega k$, $\Delta\omega$ is an appropriate (small) interval, $\rho_k=\Delta\omega \rho_T(\omega_k)$, and $K_- < 0 < K_+$. The likelihood function ${\cal P}[C\leftarrow \rho]$ describes the probability distribution of the data $C$ given a certain parameter set $\rho$. If we imagine that the data are obtained by a large number of measurements, at fixed $\rho$, then the probability distribution for $C$ is Gaussian by virtue of the central limit theorem, $${\cal P}[C\leftarrow \rho]\propto e^{-\chi^2/2}\,. \label{Pchi}$$ This is the standard argument for employing the above form of the likelihood function in the context of Bayesian inference [@Jar96; @Bra76]. \[sec:prior\]Entropic prior --------------------------- In case some information is available about the physics of the system it can be used to constrain the parameter space of the model. This is the role of the Bayesian prior. In the standard approach plateau methods are a severe form of imposing restrictions. A two-exponential fit (\[exp2\]), if feasible, is less constraining. In a Bayesian context it is possible to gradually increase the number of exponentials until convergence is reached. This is a strategy advocated in [@Lepage:2001ym], see also [@Morningstar:2001je]. There, the model for the correlation function is $\sum_n A_n e^{-E_nt}$, initially with small number of terms, which is then constrained by the Bayesian prior $e^{-\sum_n[(A_n-\bar{A}_n)^2/2\bar{\sigma}_{A_n}^2 +(E_n-\bar{E}_n)^2/2\bar{\sigma}_{E_n}^2]}$. The quantities $\bar{A}_n,\bar{\sigma}_{A_n},\bar{E}_n,\bar{\sigma}_{E_n}$ are input. Their choice is inspired by prior knowledge about the physics of the system. On the other hand, there is usually no [*a priory*]{} information about the location and the strengths of the peaks in the mass spectrum. The view that only [*minimal information*]{} is available about the spectral density function can also be implemented in the Bayesian prior. The information content, in the sense of [@Sha49; @Jay57a; @Jay57b], is measured by the entropy $S=-\sum_k \rho_k\ln(\rho_k/m)$, on some scale $m$. Rather, a commonly used variant is the Shannon-Jaynes entropy [@Jar96] $${\cal S}[\rho]=\sum_{k=K_-}^{K_+}\left(\rho_k-m_k-\rho_k\ln\frac{\rho_k}{m_k}\right)\,. \label{Smem}$$ Note that $\rho_k\ge 0$, according to (\[rhoT\]). The configuration $m=\{m_k : K_- \leq k \leq K_+\}$ is called the default model. We have ${\cal S}\leq 0$, $\forall\rho$, while ${\cal S}=0 \iff \rho=m$. The default model is a unique absolute maximum of ${\cal S}$. Choosing the prior probability as $${\cal P}[\rho]\propto e^{\alpha{\cal S}} \label{Pent}$$ entails that ${\cal P}[\rho]$ is maximal in the absence of information about $\rho$. An argument for (\[Pent\]) can be found in [@Jar96]. The entropy strength $\alpha$ and the default model $m$ are parameters. \[sec:compute\]Computing the spectral density --------------------------------------------- With (\[Pchi\]) and (\[Pent\]) the posterior probability (\[BayesT3\]) becomes $${\cal P}[\rho\leftarrow C]\propto e^{-(\chi^2/2-\alpha{\cal S})}\,. \label{Ppost}$$ We wish to maximize ${\cal P}[\rho\leftarrow C]$ with respect to $\rho$, at fixed $C$. It can be shown that both $\chi^2[\rho]$ and $-{\cal S}[\rho]$ are convex functions of $\rho=\{\rho_k : K_- \leq k \leq K_+\}$. Thus $$W[\rho]=\chi^2/2-\alpha S \label{Wrho}$$ has a unique absolute minimum. The functional $W[\rho\/]$ is nonlinear and maximally nonlocal since all degrees of freedom $\rho_k$ are coupled via the covariance matrix (\[Ecov\]) in (\[Chi2\]). To find the minimum of $W[\rho]$ one option is to use singular value decomposition (SVD), see [@Asakawa:2000tr]. In keeping with the Bayesian probabilistic interpretation of $\rho$ an attractive alternative is to employ stochastic methods to solve the optimization problem $W[\rho\/]=\min$. In this work we employ simulated annealing [@Kir84], equivalently known as cooling. The algorithm is based on the partition function $$Z_W=\int [d\rho\/] e^{-\beta_W W[\rho\/]}\,. \label{Zmem}$$ It involves the generation of equilibrium configurations $\rho$ while gradually increasing $\beta_W$ from an initially small value, following some annealing schedule. The latter is subject to experimentation. We have used the power law $$\beta_W(n)=(\beta_1-\beta_0)\left(n/N\right)^\gamma+\beta_0 \label{powerlaw}$$ with annealing steps $n=0\ldots N$ between an initial $\beta_0$ and a final $\beta_1$. A standard Metropolis algorithm was used to generate configurations $\rho$ with the distribution in (\[Zmem\]). In consecutive sweeps local updates were done by multiplying the spectral parameters with positive random numbers, $\rho_k\rightarrow x\rho_k$. Some experimenting showed that $\Gamma$-distributed random deviates of order two, $p_a(x)=x^{a-1}e^{-x}/\Gamma(a), a=2$, work quite efficiently at an acceptance rate centered at about 50%. \[sec:results\]Results ====================== All simulations were done on an $L^3\times T=10^3\times 30$ lattice. The gauge field and fermion actions are both anisotropic, with bare aspect ratio of $a_s/a_t=3$, and tadpole improved. The gauge field action is that of [@Morningstar:1999rf] with $\beta=2.4$, leading to a spatial lattice constant of $a_s\simeq 0.25{\rm fm}, a_s^{-1}\simeq 800{\rm MeV}$. For the light fermions we use a clover improved Wilson action. The hopping parameter $\kappa=0.0679$ results in a mass ratio $m_\pi/m_\rho \simeq 0.75$. Following [@Morningstar:1999rf] only spatial directions are improved with spatial tadpole renormalization factors $u_s=\langle\, \framebox(5,5)[t]{}\,\rangle^{1/4}$, while $u_t=1$ in the time direction. Clover terms involving time directions are omitted. Some guidance for a reasonable $\omega$-discretization (\[Fd\]), of (\[Fc\]), may be derived from the physical value of the lattice constant $a_t$, and the time extent $Ta_t$ of the lattice. Admissible lattice energies thus lie approximately between $\pi/a_t\approx 7.5{\rm GeV}$ and $\pi/Ta_t\approx 250{\rm MeV}$, or $\approx 3$ and $\approx 0.1$ in units of $a_t^{-1}$. In practice these are somewhat extreme bounds. Typical hadronic excitation energies are much less than $\pi/a_t\approx 7.5{\rm GeV}$. The lower bound, on the other hand, may well be ignored as a criterion for choosing the discretization interval $\Delta\omega$, because the theoretical form of $\rho$ is a superposition of $\delta$-peaks. Thus the resolution $\Delta\omega$ should be small, in fact much smaller than $\approx 0.1$. A reasonable lower bound is the likely statistical error on spectral masses. For most of the results presented here $\Delta\omega=0.04$, and $ K_-=-40, K_+=+80$, leading to $-1.6\leq\omega\leq +3.2$, were used with (\[Fd\]). With the annealing schedule (\[powerlaw\]), we have used $N=2048$ cooling steps, at 128 sweeps per temperature, starting at $\beta_0=1.0\times 10^{-5}\beta$ and ending at $\beta_1=1.0\times 10^{+5}\beta$, with a geometric average of $\beta=1.0\times 10^{+3}$. These choices are an outcome of experimentation. With $\gamma\simeq 16.61$ in (\[powerlaw\]) about half of the cooling steps operate in the regions $\beta_W(n)<\beta$ and $\beta_W(n)>\beta$, respectively. The average value $\beta$ is such that $\beta_W W[\rho\/]$ fluctuates about one at around $N/2$ cooling steps. With the final annealing temperature kept constant, $\beta_W=\beta_1$, an additional 1024 steps were done keeping 16 configurations $\rho$ in order to measure cooling fluctuations. Results are robust within reasonable changes of the annealing schedule parameters, they were used throughout this work. \[sec:alpha\]Entropy weight dependence -------------------------------------- The extent to which the spectral density $\rho$ depends on the value of the entropy weight parameter $\alpha$, in (\[Wrho\]), is a primary concern. We are interested in testing the $\alpha$ dependence for a case where both ground and excited states are prominently present in a time correlation function. For this reason we have constructed a mock correlator $C_{\rm X}(t,t_0)$. Its building blocks were the eigenvalues of the $2\times 2$ correlation matrix $C_{ij}(t,t_0) = \langle\hat{\Phi}_i^\dagger(t)\hat{\Phi}_j(t_0)\rangle$ using the above mentioned local and non-local meson fields. Pieces of those were arbitrarily matched and enhanced in order to exhibit a multi-exponential correlation function. While $C_{\rm X}(t,t_0)$ bears no physical significance, its rich structure provides a useful laboratory for testing the $\alpha$ dependence of the spectral density function. In Fig. \[fig1\] we show a sequence of six pairs of Bayesian fits to the mock correlator $C_{\rm X}(t,t_0)$ and the corresponding spectral densities $\rho$ for a wide range of entropy weights $\alpha$. The stability of the global structure of $\rho$ while $\alpha$ changes from $1.4\times 10^{-2}$ to $1.4\times 10^{+7}$ is most notable[^1]. As $\alpha$ becomes larger entire peaks vanish starting with the smallest one. The reason is that the annealing action (\[Wrho\]) gradually loses memory of the data, contained in $\chi^2$, in favor of the entropy. The fit at $\alpha=1.4\times 10^{+7}$ exhibits the onset of a smoothing of the micro structure, starting with the largest peak. This is the signature of emerging entropy dominance over the data. In practice this situation should be avoided. In our case entropy strengths in the region $\alpha < 10^{+6}$ over eight orders of magnitude give stable consistent results. It has been proposed that spectral functions be integrated over $\alpha$ to avoid the parameter dependence [@Jar96]. Inspection of our results clearly indicates that averaging over $\alpha$ would be without consequence to the gross structure of $\rho$, only the micro structure would be affected. Even the region $\alpha>10^{+6}$ could be included, since the magnitude of $\rho$ quickly becomes insignificant. {width="84mm"}{width="84mm"}\ {width="84mm"}{width="84mm"}\ {width="84mm"}{width="84mm"} In order to decide on a tuning criterion for $\alpha$ it is useful to monitor quantities like $$\begin{aligned} Y_{S/W}&=&\frac{\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}} {\langle W\rangle_{\beta_W\rightarrow\infty}}\label{SW}\\ Y_{S/\chi^2}&=&\frac{\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}} {\langle \chi^2/2\rangle_{\beta_W\rightarrow\infty}}\label{SC}\,,\end{aligned}$$ where $\langle\ldots\rangle_{\beta_W\rightarrow\infty}$ refers to the annealing average measured at the final cooling temperature, $\beta_1$. We will refer to the above quantities as entropy loads. Those are shown in Fig. \[fig2\]. It turns out that $\log(Y)$ depends linearly on $\log(\alpha)$ in the regions $\log(\alpha)<+1$ and $\log(\alpha)<+4$, for $Y_{S/W}$ and $Y_{S/\chi^2}$, respectively. (In fact $Y\approx 6.2\times10^{-4}\alpha$.) Beyond the linear region too much entropy is loaded into the annealing action $W$, leading to a smoothing of peaks, as seen in Fig. \[fig1\]. Empirically, the criterion emerging from this observation is to tune the entropy weight such that $\log(Y)\approx -2\pm 1$ within the linear region. The precise value of $\log(Y)$ is not important, also $Y=Y_{S/W}$ and $Y=Y_{S/\chi^2}$ work equally well. As is evident from Fig. \[fig1\] results are extremely robust against varying $\alpha$. ![\[fig2\]Empirical dependence of the entropy loads $Y_{S/W}$ and $Y_{S/\chi^2}$ on the entropy weight parameter $\alpha$, see (\[SW\], \[SC\]). These results are for the mock correlator $C_X(t,t_0)$. The lines indicate the extent of linear relationships.](specdens-fig2a.eps "fig:"){width="42mm"} ![\[fig2\]Empirical dependence of the entropy loads $Y_{S/W}$ and $Y_{S/\chi^2}$ on the entropy weight parameter $\alpha$, see (\[SW\], \[SC\]). These results are for the mock correlator $C_X(t,t_0)$. The lines indicate the extent of linear relationships.](specdens-fig2b.eps "fig:"){width="42mm"} \[sec:single\]Single-meson spectrum ----------------------------------- The correlation function $c(t,t_0)=\langle\hat{\phi}^\dagger(t)\hat{\phi}(t_0)\rangle$ of a single pseudoscalar heavy-light meson operator $\phi(t)=\sum_{\vec{x}}\overline{Q}_A(\vec{x}t)\gamma_5 q_A(\vec{x}t)$ delivers high quality data in this simulation. We use these to compare with plateau methods and make some observations relevant to the present stochastic approach to the MEM. In Fig. \[fig3\] plots of the mass function discretizations (\[eff0\]–\[eff3\]), built from $c(t,t_0)$, and the corresponding plateau fits are displayed. Plateau fits were made directly to $\mu_{\rm eff,i=0\ldots 3}$. The resulting masses, other than $m_{\rm eff,0}$, are from solving (\[eff1\]–\[eff3\]). Table \[tab1\] shows that those are consistent within statistical (jackknife) errors. ![\[fig3\]Effective mass functions (\[eff0\]–\[eff3\]) for a single heavy-light meson. The horizontal lines are plateau fits in the time slice range $6\leq t\leq 18$.](specdens-fig3.eps){width="84mm"} ----------------- ----------------- ----------------- ----------------- ----------- ------------ $m_{\rm eff,0}$ $m_{\rm eff,1}$ $m_{\rm eff,2}$ $m_{\rm eff,3}$ $E_1$ $\Delta_1$ 0.468(8) 0.468(7) 0.468(3) 0.47(2) 0.471(15) 0.017(6) ----------------- ----------------- ----------------- ----------------- ----------- ------------ : \[tab1\]Plateau masses derived from (\[eff0\]–\[eff3\]) on the time slice range $6\leq t\leq 18$. The entry $E_1$ is the Bayesian result with $\Delta_1$ being the peak width (standard deviation) computed from the spectral density function $\rho$. Statistical errors are derived from a gauge configuration jackknife analysis. Figure \[fig4\] gives a sense of the annealing dynamics. Beside (\[SW\]) and (\[SC\]) also shown are $$\begin{aligned} Y_{S}&=&\langle -\alpha S\rangle_{\beta_W\rightarrow\infty}\label{YS}\\ Y_{\chi^2}&=&\langle \chi^2/2\rangle_{\beta_W\rightarrow\infty}\,.\label{YC}\end{aligned}$$ In [@Fiebig:2001nn] and [@Fiebig:2001mr] the use of $Y_{S/W}$ was advocated as a tuning criterion. In view of Fig. \[fig4\] $Y_{S/\chi^2}$ appears to be a better choice given its monotonic nature. A target entropy load of $Y_{S/\chi^2}\approx 10^{-1\pm 1}$ is a safe tuning criterion, provided the cooling algorithm runs in the (upper) linear region, see Fig. \[fig2\]. ![\[fig4\]Annealing dynamics in terms of the tuning functions $Y_{S/W}$, $Y_{S/\chi^2}$, and $Y_{S}$, $Y_{\chi^2}$, versus the cooling parameter $\beta_W$. The graphs are labeled with reference to the entropy loads (\[SW\], \[SC\]), and (\[YS\], \[YC\]). This example is for the single-meson correlator, with entropy strength $\alpha=5.0\times 10^{-5}$ and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig4.eps){width="56mm"} The Bayesian analysis of the time correlation function $c(t,t_0)$ is shown in Fig. \[fig5ab\]. The solid line in Fig. \[fig5ab\](a) derives from the computed spectral density $\rho$, via (\[Fd\]). With the exception of $t_0=0$ all available time slices were used. Parameters are $\alpha=5.0\times 10^{-5}$, for the entropy strength, a constant default model $m=1.0\times 10^{-12}$, and a random annealing start about $m$. The graph of $\rho$ in Fig. \[fig5ab\](b) exhibits a global structure consisting of distinct peaks, some broad, and a micro structure of fluctuations on the scale of $\Delta\omega$. The micro structure depends on details of the annealing process, particularly the start configuration. Clearly, it makes no sense to infer the micro structure from the data. The reason is that only $T-1=29$ data points do not contain enough information to determine $K_{+}-K_{-}+1=K=121$ spectral parameters (with any sizable probability). ![\[fig5ab\]Time correlation function for a single heavy-light meson together with a Bayesian fit (a), and the corresponding spectral density function (b). This result stems from a single random start, with entropy weight $\alpha=5.0\times 10^{-5}$, and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig5a.eps "fig:"){width="42mm"} ![\[fig5ab\]Time correlation function for a single heavy-light meson together with a Bayesian fit (a), and the corresponding spectral density function (b). This result stems from a single random start, with entropy weight $\alpha=5.0\times 10^{-5}$, and a constant default model $m=1.0\times 10^{-12}$.](specdens-fig5b.eps "fig:"){width="42mm"} On the other hand the global structure is a stable feature. In the region $\omega>0$ three peaks can be distinguished in Fig. \[fig5ab\](b). By way of inspection we loosely define $$\delta_n=\{\omega:\omega\in{\rm peak}\ \#n\}\quad n=1,2\ldots\,. \label{deltan}$$ Then, for each peak $n$, we may calculate the volume $Z_n$, the mass $E_n$, and the width $\Delta_n$, according to $$\begin{aligned} Z_n&=&\int_{\delta_n}d\omega\/\rho_T(\omega)\label{Zn}\\ E_n&=&Z_n^{-1}\int_{\delta_n}d\omega\/\rho_T(\omega)\omega\label{En}\\ \Delta_n^2&=&Z_n^{-1}\int_{\delta_n}d\omega\/\rho_T(\omega)\left(\omega-E_n\right)^2\,. \label{Dn}\end{aligned}$$ These integrated, low moment, quantities are evidently insensitive to the micro structure. They constitute the information that reasonably can be expected to flow from the Bayesian analysis. The spectral density of Fig. \[fig5ab\](b) is replotted in Fig. \[fig5cd\] on linear scales. The tall narrow peak in Fig. \[fig5cd\](c) corresponds to the plateau masses of Fig. \[fig3\], as listed in Tab. \[tab1\]. There, the entries $E_1$ and $\Delta_1$ are the Bayesian results. Their statistical errors are derived from a jackknife analysis selecting four subsets of gauge configurations. (Note that the uncertainties in Fig. \[fig5cd\] are standard deviations from eight annealing starts.) Cold starts from the default model $m$ were used to suppress the dependence on the annealing start configuration. The peak width $\Delta_1$ is comparable to the gauge configuration statistical error. This is the exception. With correlation function data of lesser quality (like with the two-meson operators below) the size of the peak width is typically larger than the statistical error. It appears that the peak width $\Delta_n$ is related to the size $\Theta_n$ of the corresponding effective mass function plateau, like in Fig. \[fig3\], or the size of the $\log$-linear stretch in a plot like in Fig. \[fig5ab\](a). As a very coarse description $\Delta_n \Theta_n\approx {\rm const}$ comes to mind. Using $\Theta_1=12$ and $\Delta_1=0.017$ we have ${\rm const}\approx 0.2$. The peaks $n=2$ and $n=3$ seen in Fig. \[fig5cd\](d) would thus appear to originate from $\Theta_n\approx 0.2/\Delta_n$, or 1.3 and 0.8 time slices, respectively. (By inspection of Fig. \[fig3\] as many as 5 time slices appear involved, however.) The physical relevance of, at least, peak $n=3$ is therefore questionable. On the other hand it is remarkable that the maximum entropy method is sensitive to the slightest details in the correlation function data. ![\[fig5cd\]Spectral density $\rho$ for a single heavy-light meson, same as in Fig. \[fig5ab\](b), but on linear scales, emphasizing the ground and the excites states (c) and (d), respectively. The uncertainties of $Z_n$, $E_n$, and $\Delta_n$ are standard deviations from eight annealing runs.](specdens-fig5c.eps "fig:"){width="41.2mm"} ![\[fig5cd\]Spectral density $\rho$ for a single heavy-light meson, same as in Fig. \[fig5ab\](b), but on linear scales, emphasizing the ground and the excites states (c) and (d), respectively. The uncertainties of $Z_n$, $E_n$, and $\Delta_n$ are standard deviations from eight annealing runs.](specdens-fig5d.eps "fig:"){width="42.8mm"} \[sec:defaultm\]Default model dependence ---------------------------------------- The Shannon-Jaynes entropy (\[Smem\]) implies the possible dependence of the computed spectral density $\rho$ on the default model $m=\{m_k : K_- \leq k \leq K_+\}$. We explore the $m$ dependence using as an example the time correlation function $C_v$ with $v_1=1$ $v_2=0$, in the notation of (\[Phiv\]), at relative distance $r=4$. Figure \[fig6d7d\] shows the time correlation function data together with the Bayesian fit, and the corresponding spectral density $\rho$. The latter is the average over eight random annealing start configurations. This has the effect of smoothing out the micro structure of $\rho$. We have used a constant default model $m_k=1.0\times 10^{-12}$, all $k$. ![\[fig6d7d\]Correlation function $C_{11}=\langle\hat{\Phi}^\dagger_1(t)\hat{\Phi}_1(t_0)\rangle$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=2\times 10^{-6}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations. The average entropy load is $Y_{S/\chi^2}=0.477$ for these runs.](specdens-fig6d.eps "fig:"){width="42.2mm"} ![\[fig6d7d\]Correlation function $C_{11}=\langle\hat{\Phi}^\dagger_1(t)\hat{\Phi}_1(t_0)\rangle$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=2\times 10^{-6}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations. The average entropy load is $Y_{S/\chi^2}=0.477$ for these runs.](specdens-fig7d.eps "fig:"){width="41.8mm"} The stability of this result is tested by varying the default model through 15 orders of magnitude, $m=10^{-12}\ldots 10^{+3}$, as shown in Fig. \[fig18\]. To keep effects of the annealing start configuration small cold starts from $\rho=m$, using the same random seed, were employed for all values of $m$. In each case the entropy strength parameter $\alpha$ was tuned such that the entropy load $Y_{S/\chi^2}$ remained constant. Aside from the familiar micro structure fluctuations, the global (physical) features are stable within the range of, a remarkable, fifteen orders of magnitude. Numerical experiments with non-constant $m$ do not change this assessment. In Tab. \[tab2\] are listed the three integral quantities (\[Zn\])–(\[Dn\]) averaged over the six default models together with the corresponding standard deviations. Their smallness (0.3–3%) attests to the default model independence of the Bayesian fits. Given the huge variation of the default model the stability of $\rho$ is remarkable. ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18A.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18B.eps "fig:"){width="42mm"}\ ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18C.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18D.eps "fig:"){width="42mm"}\ ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18E.eps "fig:"){width="42mm"} ![\[fig18\]A sequence of spectral densities $\rho$ obtained from a wide range of constant default models $m$, see inserts. The entropy strength parameter $\alpha$ was tuned to keep the entropy load constant, $Y_{S/\chi^2}\approx 0.045$. The operator is the same as in Fig. \[fig6d7d\].](specdens-fig18F.eps "fig:"){width="42mm"} ------------ ---------- ------------ ----------------- $Z_1$ $E_1$ $\Delta_1$ $m_{\rm eff,0}$ 3923.(18.) 0.972(3) 0.100(3) 0.94(1) ------------ ---------- ------------ ----------------- : \[tab2\]Averages of volume, energy, and width of the dominant peak seen in Fig. \[fig18\] over the six default model choices $m=10^{-12}\ldots 10^{+3}$ at fixed entropy load $Y_{S/\chi^2}\approx 0.045$. The uncertainties are the corresponding standard deviations. The entry $m_{\rm eff,0}$ is the plateau mass (\[eff0\]) from Fig.\[fig3Rab\] with the statistical (jackknife) error, see Sect. \[sec:plateau\]. \[sec:anneal\]Annealing start dependence ---------------------------------------- The annealing algorithm starts with some initial spectral configuration $\rho_{\rm ini}$. Depending on the purpose we have used cold starts from the default model, $\rho_{\rm ini}=m$, or random starts from the default model, $\rho_{{\rm ini},k}=x_km_k$, where the $x_k$ are drawn from a gamma distribution of order two, $p_a(x)=x^{a-1}e^{-x}/\Gamma(a), a=2$. The global features of the final spectral density are of course independent of the start configuration, but the micro structure of $\rho$ is not. The reason is that in practice the annealing process is neither infinitely slow nor is the final cooling temperature $\beta_1^{-1}$ exactly zero. Therefore the annealing result for $\rho$ settles close to the global minimum, say $\rho_{\rm min}$, of $W[\rho]$. Considering annealing (thermal) fluctuations only, we expect the deviation $|\rho-\rho_{\rm min}|$ to be large in directions (of $\rho$ space) where the minimum is shallow. Thermal fluctuations are easily controlled, however. Those were kept negligible in the present study. More importantly, there may be local minima close to $\rho_{\rm min}$ which are only slightly larger than $W[\rho_{\rm min}]$. This situation invites computing a set of spectral densities from different, say random, initial configurations. The averages and standard deviations of the $\rho_k$ then gives some insight into the structure of the peak and the nature of the minimum of $W$ and its neighborhood. To present an example we have selected an excited state time correlation function $C_2(t,t_0)$ of the meson-meson system at relative distance $r=4$. $C_2(t,t_0)$ is the smaller of the eigenvalues of the $2\times 2$ correlation matrix $C_{ij}(t,t_0) = \langle\hat{\Phi}_i^\dagger(t)\hat{\Phi}_j(t_0)\rangle$, on each time slice. The reason for selecting this operator is to see how the MEM responds to a data set that is marginally acceptable, at best. Figure \[fig15d16d\] shows the correlator and the corresponding spectral density obtained from an average over eight Bayesian fits based on different random annealing start configurations. The same spectral density is displayed in the first frame of Fig. \[fig19\] on a linear scale. The dotted lines represent the limits within one standard deviation. The remaining three frames of Fig. \[fig19\] show spectral functions from selected single start configurations. They illustrate the micro structure fluctuations. ![\[fig15d16d\]Excited state correlation function $C_2$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=5\times 10^{-7}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations.](specdens-fig15d.eps "fig:"){width="42.2mm"} ![\[fig15d16d\]Excited state correlation function $C_2$ of a heavy-light meson-meson operator at relative distance $r=4$. The Bayesian fit (solid line) is from the spectral density $\rho$ shown on the right. At $\alpha=5\times 10^{-7}$ and constant default model $m=1.0\times 10^{-12}$ the spectral density $\rho$ is obtained from an average over eight random annealing start configurations.](specdens-fig16d.eps "fig:"){width="41.8mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19.eps "fig:"){width="42mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19c.eps "fig:"){width="42mm"}\ ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19e.eps "fig:"){width="42mm"} ![\[fig19\]Spectral densities $\rho$ of the excited state correlation function of Fig. \[fig15d16d\]. The sequence of four frames shows the average (ave) over a sample of eight random annealing start configurations including the bounds (dotted lines) of one standard deviation ($\pm$sig), and three selected examples of spectral functions making up that sample.](specdens-fig19h.eps "fig:"){width="42mm"} We argue that the micro structure, on a fine discretization scale $\Delta\omega$, is extraneous information. On the basis that the number of measured data points, as supplied by the time correlation function $C_v(t,t_0)$, is much smaller than the number of inferred parameters $\rho_k$, exact knowledge of $\rho$ would actually constitute an information gain not supported by the data. Rather, only averages of suitable observables based on the inferred spectral density, like (\[Zn\])–(\[Dn\]) for example, are relevant information that can be extracted from the Bayesian analysis. Whether or not the $\rho$ average of a certain observable is relevant information supported by the data may possibly be decided by the criterion that the standard deviation with respect to different annealing starts be small. From Tab. \[tab3\] we see that the standard deviations for the small-moment averages (\[Zn\],\[En\],\[Dn\]) are comparable to typical gauge configuration statistical errors, for example those in Tab. \[tab1\]. This should be an acceptable test, certainly high resolution operators would fail it. ------------ ---------- ------------ ----------------- $Z_1$ $E_1$ $\Delta_1$ $m_{\rm eff,0}$ 2156.(11.) 2.012(7) 0.214(11) 1.92(3) ------------ ---------- ------------ ----------------- : \[tab3\]Averages of volume, energy, and width of the dominant peak seen in Figs. \[fig19\] over eight random annealing start configurations, at fixed $\alpha=5.0\times 10^{-7}$ and constant default model $m=10^{-12}$. The entry $m_{\rm eff,0}$ is the plateau mass (\[eff0\]) from Fig.\[fig3Rab\] with the statistical (jackknife) error, see Sect. \[sec:plateau\]. \[sec:plateau\]Relation to plateau methods ------------------------------------------ Aside from the obvious differences in algorithm and philosophy it is important to understand that the traditional plateau method and the celebrated Bayesian approach also are distinctly different in the way they utilize the lattice correlator data. First, the former uses data on only a (subjectively) truncated contiguous set of time slices while completely ignoring the rest, whereas the latter utilizes the data on all available time slices without bias. Second, in the plateau method the stochastic dependence of the data between the plateau time slices is often ignored[^2] whereas in the Bayesian approach the dependence is fully accounted for through the covariance matrix (\[Ecov\]). Hence, the traditional plateau method and the Bayesian inference approach cannot be compared on an equal footing. In particular, their systematic errors are in principle different. A comparison of those methods is thus reduced to observing their responses to the same data sets. If the numerical quality of data is very good both methods (in fact any two methods) will of course give the same answers. An example is the single-meson case discussed above, see Tab. \[tab1\]. In case of imperfect numerical data, however, the two methods should be expected to give different results. We illustrate this point by showing in Fig. \[fig3Rab\] the effective mass functions (\[eff0\]) of the correlators $C_{11}$ and $C_2$ displayed in Figs. \[fig6d7d\] and \[fig15d16d\], respectively. While the $C_{11}$ data are somewhat level within 9 time slices, the $C_2$ data are extreme in the sense that only 2 data points are available to the plateau method. Bayesian inference, as illustrated by Fig. \[fig15d16d\] and also Fig. \[fig19\], has no problem responding with a distinct peak. The reason, of course, is that the entire set of correlator data including their correlations is available to the Bayesian approach. ![\[fig3Rab\]Effective mass functions $\mu_{\rm eff,0}$, see (\[eff0\]), of the correlator examples $C_{11}$ and $C_2$ shown in Figs.\[fig6d7d\] and \[fig15d16d\]. The plateaus are shown as horizontal lines extending over 9 and 2 time slices, respectively.](specdens-fig3Ra.eps "fig:"){width="42.0mm"} ![\[fig3Rab\]Effective mass functions $\mu_{\rm eff,0}$, see (\[eff0\]), of the correlator examples $C_{11}$ and $C_2$ shown in Figs.\[fig6d7d\] and \[fig15d16d\]. The plateaus are shown as horizontal lines extending over 9 and 2 time slices, respectively.](specdens-fig3Rb.eps "fig:"){width="42.0mm"} In Tabs. \[tab2\] and \[tab3\] we compare the plateau masses $m_{\rm eff,0}$ obtained from (\[eff0\]) to the Bayesian results $E_1$. The numbers differ by about 3–5%. Note that the statistical (jackknife) errors on the plateau masses are much smaller. Because of the data truncation the method has no way of ‘knowing’ about the poor quality of the correlator data, particularly in the $C_2$ case of the exited state correlator. The Bayesian method, on the other hand, is fully ‘aware’ of this fact and conveys this information by responding with a sizable peak width $\Delta_1$, which easily encompasses the plateau masses. This raises the question whether Bayesian peak widths or plateau mass statistical errors are a better measure for the uncertainty of masses extracted from lattice simulations. The answer is beyond the scope of this work. \[sec:conclusion\]Summary and conclusion ======================================== We have reported on our experience using Bayesian inference with an entropic prior, the maximum entropy method, to extract spectral information from lattice generated time correlation functions. The latter were taken from a simulation aimed at studying hadronic interaction, but used here only as a repository of simulation data of diverse quality. In contrast to other works the method of choice for extracting spectral densities was simulated annealing. Between the maximum entropy method and simulated annealing there were three major concerns about the parameter and algorithm dependence of the results: Dependence on (i) the entropy weight, (ii) the default model, and (iii) the annealing start configuration. Besides suggesting strategies for parameter tuning, independence of the Bayesian inferred spectral density $\rho$ on (i) the entropy weight, and (ii) the default model could be demonstrated within a range of eight and fifteen orders of magnitude of the parameters, respectively. Concerning the annealing start configuration dependence (iii) we argued that only spectral density averages of certain operators are acceptable. From an information theory point of view [@Sha49], those should be operators insensitive to the micro structure of the inferred spectral density. In particular, keeping in mind that the theoretical structure of the lattice spectral function is a superposition of distinct peaks, those operators include the spectral peak volume $Z_n$, or normalization, the peak energy $E_n$, or mass, and the peak width $\Delta_n$, or standard deviation. Bayesian inference has too long been ignored by the lattice community as an analysis tool. It has an advantage over conventional plateau methods for extracting hadron masses from lattice simulations because the entire information contained in the correlator function, or matrix, is utilized. This aspect is particularly important where excited state masses are desired, since the noise contamination of their signal can be significant. The maximum entropy method is very robust with respect to changing its parameters. Simulated annealing is practical for obtaining spectral density functions. The method should be given serious consideration as an alternative for conventional ways. This material is based upon work supported by the National Science Foundation under Grant No. 0073362. Resources made available through the Lattice Hadron Physics Collaboration (LHPC) were used in this project. [24]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , , , ****, (), . , ****, (). , ****, (), . , ****, (), . () (), . , , , ****, (), . , ****, (). , ** (, , ). , , , ****, (), . , , , ****, (), . , , , ****, (). , , , ****, (), . , , , , , , , ****, (), . , **, . (), ****, (), . (), ****, (), . , ** (, , ). (), . , ** (, , ). , ****, (). , ****, (). , ****, (). , ****, (), . , ****, (), . [^1]: In [@Fiebig:2001nn] and [@Fiebig:2001mr] a different normalization of the covariance matrix (\[Ecov\]) was used. This can be accounted for by a rescaling of the entropy weight $\alpha=(N_U-1)\bar{\alpha}$, where $\bar{\alpha}$ refers to the above references and $N_U=708$. [^2]: Uncorrelated fits to a mass function may be justified if the number of gauge configurations $N$ is large compared to the number of plateau times slices $D$, see [@Michael:1994yj; @Michael:1995sz]. There the condition $N>\max(D^2,10(D+1))$ applied to the situation of Fig. \[fig3\] gives $708>169$.

ArXiv

--- abstract: | Following [@Visintin], we exploit the fractional perimeter of a set to give a definition of fractal dimension for its measure theoretic boundary. We calculate the fractal dimension of sets which can be defined in a recursive way and we give some examples of this kind of sets, explaining how to construct them starting from well known self-similar fractals.\ In particular, we show that in the case of the von Koch snowflake $S\subset{\mathbb R}^2$ this fractal dimension coincides with the Minkowski dimension, namely $$P_s(S)<\infty\qquad\Longleftrightarrow\qquad s\in\Big(0,2-\frac{\log4}{\log3}\Big).$$ We also study the asymptotics as $s\to1^-$ of the fractional perimeter of a set having finite (classical) perimeter. author: - Luca Lombardini title: Fractional perimeter from a fractal perspective --- [Introduction and main results]{} It is well known (see e.g. [@Gamma] and [@cafenr]) that sets with a regular boundary have finite $s$-fractional perimeter for every $s\in(0,1)$. In this paper we show that also sets with an irregular, “fractal”, boundary can have finite $s$-perimeter for every $s$ below some threshold $\sigma<1$.\ Actually, the $s$-perimeter can be used to define a “fractal dimension” for the measure theoretic boundary $$\partial^-E:=\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n\textrm{ for every }r>0\},$$ of a set $E\subset{\mathbb R}^n$. Indeed, in [@Visintin] the author suggested using the index $s$ of the seminorm $[\chi_E]_{W^{s,1}}$ as a way to measure the codimension of $\partial^-E$ and he proved that the fractal dimension obtained in this way is less or equal than the (upper) Minkowski dimension. We give an example of a set, the von Koch snowflake, for which these two dimensions coincide. Moreover, exploiting the roto-translation invariance and the scaling property of the $s$-perimeter, we calculate the dimension of sets which can be defined in a recursive way similar to that of the von Koch snowflake. On the other hand, as remarked above, sets with a regular boundary have finite $s$-perimeter for every $s$ and actually their $s$-perimeter converges, as $s$ tends to 1, to the classical perimeter, both in the classical sense (see [@cafenr]) and in the $\Gamma$-convergence sense (see [@Gamma]).\ As a simple byproduct of the computations developed in this paper, we exploit Theorem 1 of [@Davila] to prove this asymptotic property for a set $E$ having finite classical perimeter in a bounded open set with Lipschitz boundary.\ This last result is probably well known to the expert, though not explicitly stated in the literature (as far as we know).\ In particular, we remark that this lowers the regularity requested in [@cafenr], where the authors asked the boundary $\partial E$ to be $C^{1,\alpha}$.\ We begin by recalling the definition of $s$-perimeter. Let $s\in(0,1)$ and let $\Omega\subset\mathbb R^n$ be an open set. The $s$-fractional perimeter of a set $E\subset\mathbb R^n$ in $\Omega$ is defined as $$P_s(E,\Omega):=\mathcal L_s(E\cap\Omega,{\mathcal C}E\cap\Omega)+ \mathcal L_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+ \mathcal L_s(E\setminus\Omega,{\mathcal C}E\cap\Omega),$$ where $$\mathcal L_s(A,B):=\int_A\int_B\frac{1}{|x-y|^{n+s}}\,dx\,dy, $$ for every couple of disjoint sets $A,\,B\subset\mathbb R^n$. We simply write $P_s(E)$ for $P_s(E,{\mathbb R}^n)$.\ We can also write the fractional perimeter as the sum $$P_s(E,\Omega)=P_s^L(E,\Omega)+P_s^{NL}(E,\Omega),$$ where $$\begin{split} &P_s^L(E,\Omega):=\mathcal L_s(E\cap\Omega,{\mathcal C}E\cap\Omega)=\frac{1}{2}[\chi_E]_{W^{s,1}(\Omega)},\\ & P_s^{NL}(E,\Omega):={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega). \end{split}$$ We can think of $P^L_s(E,\Omega)$ as the local part of the fractional perimeter, in the sense that if $|(E\Delta F)\cap\Omega|=0$, then $P^L_s(F,\Omega)=P^L_s(E,\Omega)$. We say that a set $E$ has locally finite $s$-perimeter if it has finite $s$-perimeter in every bounded open set $\Omega\subset{\mathbb R}^n$.\ Now we give precise statements of the results obtained, starting with the fractional analysis of fractal dimensions. [Fractal boundaries]{} First of all, we prove in Section 3.1 that in some sense the measure theoretic boundary $\partial^-E$ is the “right definition” of boundary for working with the $s$-perimeter. To be more precise, we show that $$\partial^-E=\{x\in{\mathbb R}^n\,|\,P_s^L(E,B_r(x))>0,\,\forall\,r>0\},$$ and that if $\Omega$ is a connected open set, then $$P_s^L(E,\Omega)>0\quad\Longleftrightarrow\quad \partial^-E\cap\Omega\not=\emptyset.$$ This can be thought of as an analogue in the fractional framework of the fact that for a Caccioppoli set $E$ we have $\partial^-E=$ supp $|D\chi_E|$. Now the idea of the definition of the fractal dimension consists in using the index $s$ of $P_s^L(E,\Omega)$ to measure the codimension of $\partial^- E\cap\Omega$, $${\textrm{Dim}}_F(\partial^-E,\Omega):=n-\sup\{s\in(0,1)\,|\,P^L_s(E,\Omega)<\infty\}.$$ As shown in [@Visintin] (Proposition 11 and Proposition 13), the fractal dimension $\textrm{Dim}_F$ defined in this way is related to the (upper) Minkowski dimension by $$\label{intro_dim_ineq} {\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal M(\partial^-E,\Omega),$$ (for the convenience of the reader we provide a proof in Proposition $\ref{vis_prop}$). If $\Omega$ is a bounded open set with Lipschitz boundary, this means that $$\label{intro_dim_ineq2} P_s(E,\Omega)<\infty\qquad\textrm{for every }s\in\big(0,n-\overline{{\textrm{Dim}}}_\mathcal M(\partial^-E,\Omega)\big),$$ since the nonlocal part of the $s$-perimeter of any set $E\subset{\mathbb R}^n$ is $$P_s^{NL}(E,\Omega)\leq2P_s(\Omega)<\infty,\qquad\textrm{for every }s\in(0,1).$$ We show that for the von Koch snowflake $(\ref{intro_dim_ineq})$ is actually an equality. ![[*The first three steps of the construction of the von Koch snowflake*]{}](fiocco){width="100mm"} Namely, we prove the following \[von\_koch\_snow\] Let $S\subset{\mathbb R}^2$ be the von Koch snowflake. Then $$\label{koch1} P_s(S)<\infty,\qquad\forall\,s\in\Big(0,2-\frac{\log4}{\log3}\Big),$$ and $$\label{koch2} P_s(S)=\infty,\qquad\forall\,s\in\Big[2-\frac{\log4}{\log3},1\Big).$$ Therefore $${\textrm{Dim}}_F(\partial S)={\textrm{Dim}}_\mathcal{M}(\partial S)=\frac{\log4}{\log3}.$$ Actually, exploiting the self-similarity of the von Koch curve, we have $${\textrm{Dim}}_F(\partial S,\Omega)=\frac{\log4}{\log3},$$ for every $\Omega$ s.t. $\partial S\cap\Omega\not=\emptyset$. In particular, this is true for every $\Omega=B_r(p)$ with $p\in S$ and $r>0$ as small as we want.\ We remark that this represents a deep difference between the classical and the fractional perimeter.\ Indeed, if a set $E$ has (locally) finite perimeter, then by De Giorgi’s structure Theorem we know that its reduced boundary $\partial^*E$ is locally $(n-1)$-rectifiable. Moreover $\overline{\partial^*E}=\partial^-E$, so the reduced boundary is, in some sense, a “big” portion of the measure theoretic boundary. On the other hand, there are (open) sets, like the von Koch snowflake, which have a “nowhere rectifiable” boundary (meaning that $\partial^-E\cap B_r(p)$ is not $(n-1)$-rectifiable for every $p\in\partial^-E$ and $r>0$) and still have finite $s$-perimeter for every $s\in(0,\sigma_0)$.\ Moreover our argument for the von Koch snowflake is quite general and can be adapted to calculate the dimension ${\textrm{Dim}}_F$ of all sets which can be constructed in a similar recursive way (see Section 3.4).\ Roughly speaking, these sets are defined by adding scaled copies of a fixed “building block” $T_0$, that is $$T:=\bigcup_{k=1}^\infty \bigcup_{i=1}^{ab^{k-1}}T_k^i,$$ where $T_k^i:=F_k^i(T_0)$ is a roto-traslation of the scaled set $\lambda^{-k}T_0$ (see Figure 2 below for an example). We also assume that $\frac{\log b}{\log\lambda}\in(n-1,n)$. Theorem $\ref{fractal_bdary_selfsim_dim}$ shows that if such a set $T$ satisfies an additional assumption, namely that “near” each set $T_k^i$ we can find a set $S_k^i=F^i_k(S_0)$ contained in ${\mathcal C}T$, then the fractal dimension of its measure theoretic boundary is $${\textrm{Dim}}_F(\partial^-T)=\frac{\log b}{\log\lambda}.$$ Many well known self-simlar fractals can be written either as (the boundary of) a set $T$ defined as above, like the von Koch snowflake, or as the difference $E=T_0\setminus T$, like the Sierpinski triangle and the Menger sponge. However sets of this second kind are often s.t. $|T\Delta T_0|=0$.\ Since the $s$-perimeters of two sets which differ only in a set of measure zero are equal, in this case the $s$-perimeter can not detect the “fractal nature” of $T$. Consider for example the Sierpinski triangle, which is defined as $E=T_0\setminus T$ with $T_0$ an equilateral triangle.\ Then $\partial^-T=\partial T_0$ and $P_s(T,\Omega)=P_s(T_0,\Omega)<\infty$ for every $s\in(0,1)$. Roughly speaking, the reason of this situation is that the fractal object is the topological boundary of $T$, while its measure theoretic boundary is regular and has finite (classical) perimeter.\ Still, we show how to modify such self-similar sets, without altering their “structure”, to obtain new sets which satisfy the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$. However, the measure theoretic boundary of such a new set will look quite different from the original fractal (topological) boundary and in general it will be a mix of smooth parts and unrectifiable parts. ![[*Example of a “fractal” set constructed exploiting the structure of the Sierpinski triangle (seen at the fourth iterative step), which satisfies the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$*]{}](Star31211color1){width="60mm"} The most interesting examples of this kind of sets are probably represented by bounded sets, like the one in Figure 2, because in this case the measure theoretic boundary does indeed have, in some sense, a “fractal nature”.\ Indeed, if $T$ is bounded, then its boundary $\partial^-T$ is compact. Nevertheless, it has infinite (classical) perimeter and actually $\partial^-T$ has Minkowski dimension strictly greater than $n-1$, thanks to $(\ref{intro_dim_ineq})$.\ However, even unbounded sets can have an interesting behavior. Indeed we obtain the following \[expl\_farc\_prop1\] Let $n\geq2$. For every $\sigma\in(0,1)$ there exists a Caccioppoli set $E\subset{\mathbb R}^n$s.t. $$P_s(E)<\infty\qquad\forall\,s\in(0,\sigma)\quad\textrm{and}\quad P_s(E)=\infty\qquad\forall\,s\in[\sigma,1).$$ Roughly speaking, the interesting thing about this Proposition is the following. Since $E$ has locally finite perimeter, $\chi_E\in BV_{loc}({\mathbb R}^n)$, it also has locally finite $s$-perimeter for every $s\in(0,1)$, but the global perimeter $P_s(E)$ is finite if and only if $s<\sigma<1$. [Asymptotics as $s\to1^-$]{} We have shown that sets with an irregular, eventually fractal, boundary can have finite $s$-perimeter. On the other hand, if the set $E$ is “regular”, then it has finite $s$-perimeter for every $s\in(0,1)$.\ Indeed, if $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary (or $\Omega={\mathbb R}^n$), then $BV(\Omega)\hookrightarrow W^{s,1}(\Omega)$. As a consequence of this embedding, we obtain $$P(E,\Omega)<\infty\qquad\Longrightarrow\qquad P_s(E,\Omega)<\infty\quad\textrm{for every }s\in(0,1).$$ Actually we can be more precise and obtain a sort of converse, using only the local part of the $s$-perimeter and adding the condition $$\liminf_{s\to1^-}(1-s)P^L_s(E,\Omega)<\infty.$$ Indeed one has the following result, which is just a combination of Theorem 3’ of [@BBM] and Theorem 1 of [@Davila], restricted to characteristic functions, \[Davila\_conv\_local\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then $E\subset{\mathbb R}^n$ has finite perimeter in $\Omega$ if and only if $P_s^L(E,\Omega)<\infty$ for every $s\in(0,1)$, and $$\label{asymptotics_fin_cond} \liminf_{s\to1}(1-s)P_s^L(E,\Omega)<\infty.$$ In this case we have $$\label{asymptotics_local_part} \lim_{s\to1}(1-s)P_s^L(E,\Omega)=\frac{n\omega_n}{2}K_{1,n}P(E,\Omega).$$ We briefly show how to get this result (and in particular why the constant looks like that) from the two Theorems cited above. We compute the constant $K_{1,n}$ in an elementary way, showing that $$\frac{n\omega_n}{2}K_{1,n}=\omega_{n-1}.$$ Moreover we show the following Condition $(\ref{asymptotics_fin_cond})$ is necessary. Indeed, there exist bounded sets (see the following Example) having finite $s$-perimeter for every $s\in(0,1)$ which do not have finite perimeter.\ This also shows that in general the inclusion $BV(\Omega)\subset\bigcap_{s\in(0,1)}W^{s,1}(\Omega)$ is strict. \[inclusion\_counterexample\] Let $0<a<1$ and consider the open intervals $I_k:=(a^{k+1},a^k)$ for every $k\in\mathbb{N}$. Define $E:=\bigcup_{k\in\mathbb{N}}I_{2k}$, which is a bounded (open) set.\ Due to the infinite number of jumps $\chi_E\not\in BV(\mathbb{R})$. However it can be proved that $E$ has finite $s$-perimeter for every $s\in(0,1)$. We postpone the proof to Appendix A. The main result of Section 2 is the following Theorem, which extends the asymptotic convergence of $(\ref{asymptotics_local_part})$ to the whole $s$-perimeter, at least when the boundary $\partial E$ intersects the boundary of $\Omega$ “transversally”. \[asymptotics\_teo\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Suppose that $E$ has finite perimeter in $\Omega_\beta$, for some $\beta\in(0,r_0)$, with $r_0>0$ small enough. Then $$\label{asymptotics_nonlocal_estimate} \limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega) \leq2\omega_{n-1}\lim_{\rho\to0^+}P(E,N_\rho(\partial\Omega)).$$ In particular, if $P(E,\partial\Omega)=0$, then $$\lim_{s\to1}(1-s)P_s(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Moreover, there exists a set $S\subset(-r_0,\beta)$, at most countable, s.t. $$\label{asymptotics_ae_convergence} \lim_{s\to1}(1-s)P_s(E,\Omega_\delta)=\omega_{n-1}P(E,\Omega_\delta),$$ for every $\delta\in(-r_0,\beta)\setminus S$. Roughly speaking, the second part of this Theorem says that even if we do not have the asymptotic convergence of the $s$-perimeter in $\Omega$, we can slighltly enlarge or restrict $\Omega$ to obtain it. Actually, since $S$ has null measure, we can restrict or enlarge $\Omega$ as little as we want.\ In [@cafenr] the authors obtained a similar result for $\Omega=B_R$ a ball, but asking $C^{1,\alpha}$ regularity of $\partial E$ in $B_R$. They proved the convergence in every ball $B_r$ with $r\in(0,R)\setminus S$, with $S$ at most countable, exploiting uniform estimates.\ On the other hand, asking $E$ to have finite perimeter in a neighborhood (as small as we want) of the open set $\Omega$ is optimal.\ In [@Gamma] the authors studied the asymptotics as $s\longrightarrow1^-$ in the $\Gamma$-convergence sense. In particular, for the proof of a $\Gamma$-limsup inequality, which is typically constructive and by density, they show that if $\Pi$ is a polyhedron, then $$\limsup_{s\to1}(1-s)P_s(\Pi,\Omega) \leq\Gamma_n^*P(\Pi,\Omega)+2\Gamma_n^*\lim_{\rho\to0^+}P(\Pi,N_\rho(\partial\Omega)),$$ which is $(\ref{asymptotics_nonlocal_estimate})$, once we sum the local part of the perimeter. Their proof relies on the fact that $\Pi$ is a polyhedron to obtain the convergence of the local part of the perimeter, which is then used also in the estimate of the nonlocal part. Moreover they need an approximation result to prove that the constant is $\Gamma_n^*=\omega_{n-1}$. They also prove, in particular $$\Gamma-\liminf_{s\to1}(1-s)P_s^L(E,\Omega)\geq\omega_{n-1}P(E,\Omega),$$ which is a stronger result than the first part of Theorem $\ref{Davila_conv_local}$. [Notation and assumptions]{} - All sets and functions considered are assumed to be Lebesgue measurable. - We write $A\subset\subset B$ to mean that the closure of $A$ is compact and $\overline{A}\subset B$. - In ${\mathbb R}^n$ we will usually write $|E|=\mathcal{L}^n(E)$ for the $n$-dimensional Lebesgue measure of a set $E\subset{\mathbb R}^n$. - We write ${\mathcal H}^d$ for the $d$-dimensional Hausdorff measure, for any $d\geq0$. - We define the dimensional constants $$\omega_d:=\frac{\pi^\frac{d}{2}}{\Gamma\big(\frac{d}{2}+1\big)},\qquad d\geq0.$$ In particular, we remark that $\omega_k=\mathcal{L}^k(B_1)$ is the volume of the $k$-dimensional unit ball $B_1\subset{\mathbb R}^k$ and $k\,\omega_k={\mathcal H}^{k-1}(\mathbb{S}^{k-1})$ is the surface area of the $(k-1)$-dimensional sphere $$\mathbb{S}^{k-1}=\partial B_1=\{x\in{\mathbb R}^k\,|\,|x|=1\}.$$ - Since $$|E\Delta F|=0\quad\Longrightarrow\quad P(E,\Omega)=P(F,\Omega)\quad\textrm{and}\quad P_s(E,\Omega)=P_s(F,\Omega),$$ in Section 2 we implicitly identify sets up to sets of negligible Lebesgue measure.\ Moreover, whenever needed we can choose a particular representative for the class of $\chi_E$ in $L^1_{loc}({\mathbb R}^n)$, as in the Remark below.\ We will not make this assumption in Section 3, since the Minkowski content can be affected even by changes in sets of measure zero, that is, in general $$|\Gamma_1\Delta\Gamma_2|=0\quad\not\Rightarrow\quad \overline{\mathcal{M}}^r(\Gamma_1,\Omega)=\overline{\mathcal{M}}^r(\Gamma_2,\Omega)$$ (see Section 3 for a more detailed discussion). - We consider the open tubular $\rho$-neighborhood of $\partial\Omega$, $$N_\rho(\partial\Omega):=\{x\in{\mathbb R}^n\,|\,d(x,\partial\Omega)<\rho\}=\{|\bar{d}_\Omega|<\rho\}=\Omega_\rho\setminus\overline{\Omega_{-\rho}}$$ (see Appendix B). \[gmt\_assumption\] Let $E\subset{\mathbb R}^n$. Up to modifying $E$ on a set of measure zero, we can assume (see Appendix C) that $$\label{gmt_assumption_eq} \begin{split} &E_1\subset E,\qquad E\cap E_0=\emptyset\\ \textrm{and}\quad\partial E=\partial^-E&=\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n,\,\forall\,r>0\}. \end{split}$$ [Asymptotics as $s\to1^-$]{} We say that an open set $\Omega\subset{\mathbb R}^n$ is an extension domain if $\exists C=C(n,s,\Omega)>0$ s.t. for every $u\in W^{s,1}(\Omega)$ there exists $\tilde{u}\in W^{s,1}({\mathbb R}^n)$ with $\tilde{u}_{|\Omega}=u$ and $$\|\tilde{u}\|_{W^{s,1}({\mathbb R}^n)}\leq C\|u\|_{W^{s,1}(\Omega)}.$$ Every open set with bounded Lipschitz boundary is an extension domain (see [@HitGuide] for a proof). For simplicity we consider ${\mathbb R}^n$ itself as an extension domain. We begin with the following embedding. \[embedding\_prop\] Let $\Omega\subset{\mathbb R}^n$ be an extension domain. Then $\exists C(n,s,\Omega)\geq 1$ s.t. for every $u:\Omega\longrightarrow{\mathbb R}$ $$\label{embedding_ineq} \|u\|_{W^{s,1}(\Omega)}\leq C\|u\|_{BV(\Omega)}.$$ In particular we have the continuous embedding $$BV(\Omega)\hookrightarrow W^{s,1}(\Omega).$$ The claim is trivially satisfied if the right hand side of $(\ref{embedding_ineq})$ is infinite, so let $u\in BV(\Omega)$. Let $\{u_k\}\subset C^\infty(\Omega)\cap BV(\Omega)$ be an approximating sequence as in Theorem 1.17 of [@Giusti], that is $$\|u-u_k\|_{L^1(\Omega)}\longrightarrow0\qquad\textrm{and}\qquad\lim_{k\to\infty}\int_\Omega|\nabla u_k|\,dx=|Du|(\Omega).$$ We only need to check that the $W^{s,1}$-seminorm of $u$ is bounded by its $BV$-norm.\ Since $\Omega$ is an extension domain, we know (see Proposition 2.2 of [@HitGuide]) that $\exists C(n,s)\geq1$ s.t. $$\|v\|_{W^{s,1}(\Omega)}\leq C\|v\|_{W^{1,1}(\Omega)}.$$ Then $$[u_k]_{W^{s,1}(\Omega)}\leq\|u_k\|_{W^{s,1}(\Omega)}\leq C\|u_k\|_{W^{1,1}(\Omega)} =C\|u_k\|_{BV(\Omega)},$$ and hence, using Fatou’s Lemma, $$\begin{split} [u]_{W^{s,1}(\Omega)}&\leq\liminf_{k\to\infty}[u_k]_{W^{s,1}(\Omega)} \leq C\liminf_{k\to\infty}\|u_k\|_{BV(\Omega)}=C\lim_{k\to\infty}\|u_k\|_{BV(\Omega)}\\ & =C\|u\|_{BV(\Omega)}, \end{split}$$ proving $(\ref{embedding_ineq})$. \[embedding\_fin\_per\_coroll\] $(i)\quad$ If $E\subset{\mathbb R}^n$ has finite perimeter, i.e. $\chi_E\in BV({\mathbb R}^n)$, then $E$ has also finite $s$-perimeter for every $s\in(0,1)$.\ $(ii)\quad$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then there exists $r_0>0$ s.t. $$\label{unif_bound_lip_frac_per} \sup_{|r|<r_0}P_s(\Omega_r)<\infty.$$ $(iii)\quad$ If $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary, then $$P_s^{NL}(E,\Omega)\leq 2P_s(\Omega)<\infty$$ for every $E\subset{\mathbb R}^n$.\ $(iv)\quad$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then $$P(E,\Omega)<\infty\qquad\Longrightarrow\qquad P_s(E,\Omega)<\infty\quad\textrm{for every }s\in(0,1).$$ $(i)$ follows from $$P_s(E)=\frac{1}{2}[\chi_E]_{W^{s,1}({\mathbb R}^n)}$$ and previous Proposition with $\Omega={\mathbb R}^n$. $(ii)$ Let $r_0$ be as in Proposition $\ref{bound_perimeter_unif}$ and notice that $$P(\Omega_r)={\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big),$$ so that $$\|\chi_{\Omega_r}\|_{BV({\mathbb R}^n)}=|\Omega_r|+{\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big).$$ Thus $$\sup_{|r|<r_0}P_s(\Omega_r)\leq C\Big(|\Omega_{r_0}|+\sup_{|r|<r_0}{\mathcal H}^{n-1}\big(\{\bar{d}_\Omega=r\}\big)\Big)<\infty.$$ $(iii)$ Notice that $$\begin{split} &{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)\leq {\mathcal L}_s(\Omega,{\mathcal C}\Omega)=P_s(\Omega),\\ & {\mathcal L}_s({\mathcal C}E\cap\Omega,E\setminus\Omega)\leq {\mathcal L}_s(\Omega,{\mathcal C}\Omega)=P_s(\Omega), \end{split}$$ and use $(\ref{unif_bound_lip_frac_per})$ (just with $\Omega_0=\Omega$). $(iv)$ The nonlocal part of the $s$-perimeter is finite thanks to $(iii)$. As for the local part, remind that $$P(E,\Omega)=|D\chi_E|(\Omega)\qquad\textrm{and}\qquad P_s^L(E,\Omega)=\frac{1}{2}[\chi_E]_{W^{s,1}(\Omega)},$$ then use previous Proposition. [Theorem $\ref{Davila_conv_local}$, asymptotics of the local part of the $s$-perimeter]{} \[bb\] Let $\Omega\subset{\mathbb R}^n$ be a smooth bounded domain. Let $u\in L^1(\Omega)$. Then $u\in BV(\Omega)$ if and only if $$\liminf_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy<\infty,$$ and then $$\label{rough} \begin{split} C_1|Du|(\Omega)&\leq\liminf_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy\\ & \leq\limsup_{n\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_n(x-y)\,dxdy\leq C_2|Du|(\Omega), \end{split}$$ for some constants $C_1$, $C_2$ depending only on $\Omega$. This result was refined by Davila Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Let $u\in BV(\Omega)$. Then $$\label{correct} \lim_{k\to\infty}\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_k(x-y)\,dxdy=K_{1,n}|Du|(\Omega),$$ where $$K_{1,n}=\frac{1}{n\omega_n}\int_{\mathbb{S}^{n-1}}|v\cdot e|\,d\sigma(v),$$ with $e\in{\mathbb R}^n$ any unit vector. In the above Theorems $\rho_k$ is any sequence of radial mollifiers i.e. of functions satisfying $$\label{rule1} \rho_k(x)\geq0,\quad\rho_k(x)=\rho_k(|x|),\quad\int_{{\mathbb R}^n}\rho_k(x)\,dx=1$$ and $$\label{rule2} \lim_{k\to\infty}\int_\delta^\infty\rho_k(r)r^{n-1}dr=0\quad\textrm{for all }\delta>0.$$ In particular, for $R$ big enough, $R>$ diam$(\Omega)$, we can consider $$\rho(x):=\chi_{[0,R]}(|x|)\frac{1}{|x|^{n-1}}$$ and define for any sequence $\{s_k\}\subset(0,1),\,s_k\nearrow1$, $$\rho_k(x):=(1-s_k)\rho(x)c_{s_k}\frac{1}{|x|^{s_k}},$$ where the $c_{s_k}$ are normalizing constants. Then $$\begin{split} \int_{{\mathbb R}^n}\rho_k(x)\,dx&=(1-s_k)c_{s_k}n\omega_n\int_0^R\frac{1}{r^{n-1+s_k}}r^{n-1}\,dr\\ & =(1-s_k)c_{s_k}n\omega_n\int_0^R\frac{1}{r^{s_k}}\,dr=c_{s_k}n\omega_nR^{1-s_k}, \end{split}$$ and hence taking $c_{s_k}:=\frac{1}{n\omega_n}R^{s_k-1}$ gives $(\ref{rule1})$; notice that $c_{s_k}\to\frac{1}{n\omega_n}$.\ Also $$\begin{split} \lim_{k\to\infty}\int_\delta^\infty\rho_k(r)r^{n-1}\,dr&= \lim_{k\to\infty}(1-s_k)c_{s_k}\int_\delta^R\frac{1}{r^{s_k}}\,dr\\ & =\lim_{k\to\infty}c_{s_k}(R^{1-s_k}-\delta^{1-s_k})=0, \end{split}$$ giving $(\ref{rule2})$.\ With this choice we get $$\int_\Omega\int_\Omega\frac{|u(x)-u(y)|}{|x-y|}\rho_k(x-y)\,dxdy=c_{s_k}(1-s_k)[u]_{W^{s_k,1}(\Omega)}.$$ Then, if $u\in BV(\Omega)$, Davila’s Theorem gives $$\label{limitperimeter}\begin{split} \lim_{s\to1}(1-s)[u]_{W^{s,1}(\Omega)}&=\lim_{s\to1}\frac{1}{c_s}(c_s(1-s)[u]_{W^{s,1}(\Omega)})\\ & =n\omega_nK_{1,n}|Du|(\Omega). \end{split}$$ [Proof of Theorem $\ref{asymptotics_teo}$]{} [The constant $\omega_{n-1}$]{} We need to compute the constant $K_{1,n}$. Notice that we can choose $e$ in such a way that $v\cdot e=v_n$.\ Then using spheric coordinates for ${\mathbb S}^{n-1}$ we obtain $|v\cdot e|=|\cos\theta_{n-1}|$ and $$d\sigma=\sin\theta_2(\sin\theta_3)^2\ldots(\sin\theta_{n-1})^{n-2}d\theta_1\ldots d\theta_{n-1},$$ with $\theta_1\in[0,2\pi)$ and $\theta_j\in[0,\pi)$ for $j=2,\ldots,n-1$. Notice that $$\begin{split} {\mathcal H}^k({\mathbb S}^k)&=\int_0^{2\pi}\,d\theta_1\int_0^\pi\sin\theta_2\,d\theta_2\ldots \int_0^\pi(\sin\theta_{k-1})^{k-2}\,d\theta_{k-1}\\ & ={\mathcal H}^{k-1}({\mathbb S}^{k-1})\int_0^\pi(\sin t)^{k-2}\,dt. \end{split}$$ Then we get $$\begin{split} \int_{{\mathbb S}^{n-1}}|v\cdot e|&\,d\sigma(v)={\mathcal H}^{n-2}({\mathbb S}^{n-2})\int_0^\pi(\sin t)^{n-2}|\cos t|\,dt\\ & ={\mathcal H}^{n-2}({\mathbb S}^{n-2})\Big(\int_0^\frac{\pi}{2}(\sin t)^{n-2}\cos t\,dt-\int_\frac{\pi}{2}^\pi(\sin t)^{n-2}\cos t\,dt\Big)\\ & =\frac{{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}\Big(\int_0^\frac{\pi}{2}\frac{d}{dt}(\sin t)^{n-1}\,dt-\int_\frac{\pi}{2}^\pi\frac{d}{dt}(\sin t)^{n-1}\,dt\Big)\\ & =\frac{2{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}. \end{split}$$ Therefore $$n\omega_nK_{1,n}=2\frac{{\mathcal H}^{n-2}({\mathbb S}^{n-2})}{n-1}=2{\mathcal L}^{n-1}(B_1(0))=2\omega_{n-1},$$ and hence $(\ref{limitperimeter})$ becomes $$\lim_{s\to1}(1-s)[u]_{W^{s,1}(\Omega)}=2\omega_{n-1}|Du|(\Omega),$$ for any $u\in BV(\Omega)$. [Estimating the nonlocal part of the $s$-perimeter]{} We prove something slightly more general than $(\ref{asymptotics_nonlocal_estimate})$. Namely, that to estimate the nonlocal part of the $s$-perimeter we do not necessarily need to use the sets $\Omega_\rho$: any “regular” approximation of $\Omega$ would do. Let $A_k,\, D_k\subset{\mathbb R}^n$ be two sequences of bounded open sets with Lipschitz boundary strictly approximating $\Omega$ respectively from the inside and from the outside, that is $(i)\quad A_k\subset A_{k+1}\subset\subset\Omega$ and $A_k\nearrow\Omega$, i.e. $\bigcup_k A_k=\Omega$, $(ii)\quad \Omega\subset\subset D_{k+1}\subset D_k$ and $D_k\searrow\overline{\Omega}$, i.e. $\bigcap_k D_k=\overline{\Omega}$.\ We define for every $k$ $$\begin{split} &\Omega_k^+:=D_k\setminus\overline{\Omega},\qquad\Omega_k^-:=\Omega\setminus\overline{A_k} \qquad T_k:=\Omega_k^+\cup\partial\Omega\cup\Omega_k^-,\\ &\qquad\qquad d_k:=\min\{d(A_k,\partial\Omega),\,d(D_k,\partial\Omega)\}>0. \end{split}$$ In particular we can consider $\Omega_\rho$ with $\rho<0$ in place of $A_k$ and with $\rho>0$ in place of $D_k$. Then $T_k$ would be $N_\rho(\partial\Omega)$ and $d_k=\rho$. Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary and let $E\subset{\mathbb R}^n$ be a set having finite perimeter in $D_1$. Then $$\limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega)\leq 2\omega_{n-1}\lim_{k\to\infty}P(E,T_k).$$ In particular, if $P(E,\partial\Omega)=0$, then $$\lim_{s\to1}(1-s)P_s(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Since $\Omega$ is regular and $P(E,\Omega)<\infty$, we already know that $$\lim_{s\to1}(1-s)P_s^L(E,\Omega)=\omega_{n-1}P(E,\Omega).$$ Notice that, since $|D\chi_E|$ is a finite Radon measure on $D_1$ and $T_k\searrow\partial\Omega$ as $k\nearrow\infty$, we have $$\exists\lim_{k\to\infty}P(E,T_k)=P(E,\partial\Omega).$$ Consider the nonlocal part of the fractional perimeter, $$P_s^{NL}(E,\Omega)={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)+{\mathcal L}_s({\mathcal C}E\cap\Omega,E\setminus\Omega),$$ and take any $k$. Then $$\begin{split} {\mathcal L}_s(E\cap\Omega,{\mathcal C}E\setminus\Omega)&={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega_k^+)+{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap({\mathcal C}\Omega\setminus D_k))\\ & \leq{\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega_k^+)+\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & \leq{\mathcal L}_s(E\cap\Omega_k^-,{\mathcal C}E\cap\Omega_k^+)+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & \leq{\mathcal L}_s(E\cap(\Omega_k^-\cup\Omega_k^+),{\mathcal C}E\cap(\Omega_k^-\cup\Omega_k^+))+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}\\ & =P^L_s(E,T_k)+2\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}. \end{split}$$ Since we can bound the other term in the same way, we get $$P^{NL}_s(E,\Omega)\leq2P^L_s(E,T_k)+4\frac{n\omega_n}{s}|\Omega|\frac{1}{d_k^s}.$$ By hypothesis we know that $T_k$ is a bounded open set with Lipschitz boundary $$\partial T_k=\partial A_k\cup\partial D_k.$$ Therefore using $(\ref{asymptotics_local_part})$ we have $$\lim_{s\to1}(1-s)P^L_s(E,T_k)=\omega_{n-1}P(E,T_k),$$ and hence $$\limsup_{s\to1}(1-s)P_s^{NL}(E,\Omega) \leq 2\omega_{n-1}P(E,T_k).$$ Since this holds true for any $k$, we get the claim. [Convergence in almost every $\Omega_\rho$]{} Having a “continuous” approximating sequence (the $\Omega_\rho$) rather than numerable ones allows us to improve the previous result and obtain the second part of Theorem $\ref{asymptotics_teo}$. We recall that De Giorgi’s structure Theorem for sets of finite perimeter (see e.g. Theorem 15.9 of [@Maggi]) guarantees in particular that $$|D\chi_E|={\mathcal H}^{n-1}\llcorner\partial^*E$$ and hence $$P(E,B)={\mathcal H}^{n-1}(\partial^*E\cap B)\qquad\textrm{for every Borel set }B\subset{\mathbb R}^n,$$ where $\partial^*E$ is the reduced boundary of $E$. Now suppose that $E$ has finite perimeter in $\Omega_\beta$. Then $$P(E,\partial\Omega_\delta)={\mathcal H}^{n-1}(\partial^*E\cap\{\bar{d}_\Omega=\delta\}),$$ for every $\delta\in(-r_0,\beta)$. Therefore, since $$M:={\mathcal H}^{n-1}(\partial^*E\cap(\Omega_\beta\setminus\overline{\Omega_{-r_0}}))\leq P(E,\Omega_\beta)<\infty,$$ the set $$S:=\left\{\delta\in(-r_0,\beta)\,|\,P(E,\partial\Omega_\delta)>0\right\}$$ is at most countable. Indeed, define $$S_k:=\Big\{\delta\in(-r_0,\beta)\,|\,{\mathcal H}^{n-1}(\partial^*E\cap\{\bar{d}_\Omega=\delta\})>\frac{1}{k}\Big\}.$$ Since $${\mathcal H}^{n-1}\Big(\bigcup_{-r_0<\delta<\beta}(\partial^*E\cap\{\bar{d}_\Omega=\delta\})\Big)=M,$$ the number of elements in each $S_k$ is at most $$\sharp S_k\leq M\,k.$$ As a consequence, $S=\bigcup_k S_k$ is at most countable.\ This concludes the proof of Theorem $\ref{asymptotics_teo}$. [Irregularity of the boundary]{} [The measure theoretic boundary as “support” of the local part of the $s$-perimeter]{} First of all we show that the (local part of the) $s$-perimeter does indeed measure a quantity related to the measure theoretic boundary. Let $E\subset{\mathbb R}^n$ be a set of locally finite $s$-perimeter. Then $$\partial^-E=\{x\in{\mathbb R}^n\,|\,P_s^L(E,B_r(x))>0\textrm{ for every }r>0\}.$$ The claim follows from the following observation. Let $A,\,B\subset{\mathbb R}^n$ s.t. $A\cap B=\emptyset$; then $${\mathcal L}_s(A,B)=0\quad\Longleftrightarrow\quad|A|=0\quad\textrm{or}\quad|B|=0.$$ Therefore $$\begin{split} x\in\partial^-E&\quad\Longleftrightarrow\quad |E\cap B_r(x)|>0\textrm{ and }|{\mathcal C}E\cap B_r(x)|>0\quad\forall\,r>0\\ & \quad\Longleftrightarrow\quad {\mathcal L}_s(E\cap B_r(x),{\mathcal C}E\cap B_r(x))>0\quad\forall\,r>0. \end{split}$$ This characterization of $\partial^-E$ can be thought of as a fractional analogue of $(\ref{support_perimeter})$. However we can not really think of $\partial^-E$ as the support of $$P_s^L(E,-):\Omega\longmapsto P_s^L(E,\Omega),$$ in the sense that, in general $$\partial^-E\cap\Omega=\emptyset\quad\not\Rightarrow\quad P_s^L(E,\Omega)=0.$$ For example, consider $E:=\{x_n\leq0\}\subset{\mathbb R}^n$ and notice that $\partial^-E=\{x_n=0\}$. Let $\Omega:=B_1(2e_n)\cup B_1(-2e_n)$. Then $\partial^-E\cap\Omega=\emptyset$, but $$P_s^L(E,\Omega)={\mathcal L}_s(B_1(2e_n),B_1(-2e_n))>0.$$ On the other hand, the only obstacle is the non connectedness of the set $\Omega$ and indeed we obtain the following Let $E\subset{\mathbb R}^n$ be a set of locally finite $s$-perimeter and let $\Omega\subset{\mathbb R}^n$ be an open set. Then $$\partial^-E\cap\Omega\not=\emptyset\quad\Longrightarrow\quad P_s^L(E,\Omega)>0.$$ Moreover, if $\Omega$ is connected $$\partial^-E\cap\Omega=\emptyset\quad\Longrightarrow\quad P_s^L(E,\Omega)=0.$$ Therefore, if $\widehat{\mathcal O}({\mathbb R}^n)$ denotes the family of bounded and connected open sets, then $\partial^-E$ is the “support” of $$\begin{split} P_s^L(E,-):\,&\widehat{\mathcal O}({\mathbb R}^n)\longrightarrow [0,\infty)\\ & \Omega\longmapsto P_s^L(E,\Omega), \end{split}$$ in the sense that, if $\Omega\in\widehat{\mathcal O}({\mathbb R}^n)$, then $$P_s^L(E,\Omega)>0\quad\Longleftrightarrow\quad\partial^-E\cap\Omega\not=\emptyset.$$ Let $x\in\partial^-E\cap\Omega$. Since $\Omega$ is open, we have $B_r(x)\subset\Omega$ for some $r>0$ and hence $$P_s^L(E,\Omega)\geq P_s^L(E,B_r(x))>0.$$ Let $\Omega$ be connected and suppose $\partial^-E\cap\Omega=\emptyset$. We have the partition of ${\mathbb R}^n$ as ${\mathbb R}^n=E_0\cup\partial^-E\cup E_1$ (see Appendix C). Thus we can write $\Omega$ as the disjoint union $$\Omega=(E_0\cap\Omega)\cup(E_1\cap\Omega).$$ However, since $\Omega$ is connected and both $E_0$ and $E_1$ are open, we must have $E_0\cap\Omega=\emptyset$ or $E_1\cap\Omega=\emptyset$. Now, if $E_0\cap\Omega=\emptyset$ (the other case is analogous), then $\Omega\subset E_1$ and hence $|{\mathcal C}E\cap\Omega|=0$. Thus $$P_s^L(E,\Omega)={\mathcal L}_s(E\cap\Omega,{\mathcal C}E\cap\Omega)=0.$$ [A notion of fractal dimension]{} Let $\Omega\subset{\mathbb R}^n$ be an open set. Then $$t>s\qquad\Longrightarrow\qquad W^{t,1}(\Omega)\hookrightarrow W^{s,1}(\Omega),$$ (see e.g. Proposition 2.1 of [@HitGuide]). As a consequence, for every $u:\Omega\longrightarrow{\mathbb R}$ there exists a unique $R(u)\in[0,1]$ s.t. $$[u]_{W^{s,1}(\Omega)}\quad\left\{\begin{array}{cc} <\infty,& \forall\,s\in(0,R(u))\\ =\infty, &\forall\,s\in(R(u),1) \end{array}\right.$$ that is $$\begin{split}\label{frac_range} R(u)&=\sup\left\{s\in(0,1)\,\big|\,[u]_{W^{s,1}(\Omega)}<\infty\right\}\\ & =\inf\left\{s\in(0,1)\,\big|\,[u]_{W^{s,1}(\Omega)}=\infty\right\}. \end{split}$$ In particular, exploiting this result for characteristic functions, in [@Visintin] the author suggested the following definition of fractal dimension. Let $\Omega\subset{\mathbb R}^n$ be an open set and let $E\subset{\mathbb R}^n$. If $\partial^- E\cap\Omega\not=\emptyset$, we define $${\textrm{Dim}}_F(\partial^- E,\Omega):=n-R(\chi_E),$$ the fractal dimension of $\partial^- E$ in $\Omega$, relative to the fractional perimeter.\ If $\Omega={\mathbb R}^n$, we drop it in the formulas. Notice that in the case of sets $(\ref{frac_range})$ becomes $$\begin{split}\label{frac_range_sets} R(\chi_E)&=\sup\left\{s\in(0,1)\,\big|\,P_s^L(E,\Omega)<\infty\right\}\\ & =\inf\left\{s\in(0,1)\,\big|\,P_s^L(E,\Omega)=\infty\right\}. \end{split}$$ In particular we can take $\Omega$ to be the whole of ${\mathbb R}^n$, or a bounded open set with Lipschitz boundary.\ In the first case the local part of the fractional perimeter coincides with the whole fractional perimeter, while in the second case we know that we can bound the nonlocal part with $2P_s(\Omega)<\infty$ for every $s\in(0,1)$. Therefore in both cases in $(\ref{frac_range_sets})$ we can as well take the whole fractional perimeter $P_s(E,\Omega)$ instead of just the local part.\ Now we give a proof of the relation $(\ref{intro_dim_ineq})$ (obtained in [@Visintin]).\ For simplicity, given $\Gamma\subset{\mathbb R}^n$ we set $$\label{neigh_mink_def} \bar{N}_\rho^\Omega(\Gamma):=\overline{N_\rho(\Gamma)}\cap\Omega =\{x\in\Omega\,|\,d(x,\Gamma)\leq\rho\},$$ for any $\rho>0$. \[vis\_prop\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set. Then for every $E\subset{\mathbb R}^n$ s.t. $\partial^- E\cap\Omega\not=\emptyset$ and $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\geq n-1$ we have $${\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega).$$ By hypothesis we have $$\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)=n-\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\},$$ and we need to show that $$\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\} \leq \sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\}.$$ Up to modifying $E$ on a set of Lebesgue measure zero we can suppose that $\partial E=\partial^-E$, as in Remark $\ref{gmt_assumption}$. Notice that this does not affect the $s$-perimeter. Now for any $s\in(0,1)$ $$\begin{split} 2P_s^L(E,\Omega)&=\int_\Omega\,dx\int_\Omega\frac{|\chi_E(x)-\chi_E(y)|}{|x-y|^{n+s}}\,dy\\ & =\int_\Omega dx\int_0^\infty d\rho\int_{\partial B_\rho(x)\cap\Omega}\frac{|\chi_E(x)-\chi_E(y)|}{|x-y|^{n+s}}\,d{\mathcal H}^{n-1}(y)\\ & =\int_\Omega dx\int_0^\infty\frac{d\rho}{\rho^{n+s}}\int_{\partial B_\rho(x)\cap\Omega}|\chi_E(x)-\chi_E(y)|\,d{\mathcal H}^{n-1}(y). \end{split}$$ Notice that $$d(x,\partial E)>\rho\quad\Longrightarrow\quad\chi_E(y)=\chi_E(x),\quad\forall\,y\in\overline{B_\rho(x)},$$ and hence $$\begin{split} \int_{\partial B_\rho(x)\cap\Omega}|\chi_E(x)-\chi_E(y)|\,d{\mathcal H}^{n-1}(y)& \leq\int_{\partial B_\rho(x)\cap\Omega}\chi_{\bar{N}_\rho(\partial E)}(x)\,d{\mathcal H}^{n-1}(y)\\ & \leq n\omega_n\rho^{n-1}\chi_{\bar{N}_\rho(\partial E)}(x). \end{split}$$ Therefore $$\label{visintin_pf} 2P_s^L(E,\Omega)\leq n\omega_n\int_0^\infty\frac{d\rho}{\rho^{1+s}}\int_\Omega \chi_{\bar{N}_\rho(\partial E)}(x) =n\omega_n\int_0^\infty\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1+s}}\,d\rho.$$ We prove the following\ CLAIM $$\label{visintin_proof} \overline{\mathcal{M}}^{n-r}(\partial E,\Omega)<\infty\quad\Longrightarrow\quad P_s^L(E,\Omega)<\infty,\quad\forall\,s\in(0,r).$$ Indeed $$\limsup_{\rho\to0}\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^r}<\infty\quad\Longrightarrow\quad\exists\,C>0\textrm{ s.t. } \sup_{\rho\in(0,C]}\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^r}\leq M<\infty.$$ Then $$\begin{split} 2P_s^L(E,\Omega)&\leq n\omega_n\Big\{\int_0^C\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1-(r-s)+r}}\,d\rho +\int_C^\infty\frac{|\bar{N}^\Omega_\rho(\partial E)|}{\rho^{1+s}}\,d\rho\Big\}\\ & \leq n\omega_n\Big\{ M\int_0^C\frac{1}{\rho^{1-(r-s)}}\,d\rho+|\Omega|\int_C^\infty\frac{1}{\rho^{1+s}}\,d\rho \Big\}\\ & =n\omega_n\Big\{ \frac{M}{r-s}C^{r-s}+\frac{|\Omega|}{sC^s} \Big\}<\infty, \end{split}$$ proving the claim.\ This implies $$r\leq\sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\},$$ for every $r\in(0,1)$ s.t. $\overline{\mathcal{M}}^{n-r}(\partial E,\Omega)<\infty$.\ Thus for $\epsilon>0$ very small, we have $$\inf\big\{r\in(0,1)\,|\,\overline{\mathcal{M}}^{n-r}(\partial^-E,\Omega)=\infty\big\}-\epsilon \leq\sup\{s\in(0,1)\,|\,P_s^L(E,\Omega)<\infty\}.$$ Letting $\epsilon$ tend to zero, we conclude the proof. In particular, if $\Omega$ has Lipschitz boundary we obtain \[fractal\_dim\_coroll\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Let $E\subset{\mathbb R}^n$ s.t. $\partial^-E\cap\Omega\not=\emptyset$ and $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\in[n-1,n)$. Then $$\label{fractal_per} P_s(E,\Omega)<\infty\qquad\textrm{for every }s\in\left(0,n-\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\right).$$ \[fractal\_dim\_rmk\] Actually, previous Proposition and Corollary still work when $\Omega={\mathbb R}^n$, provided the set $E$ we are considering is bounded.\ Indeed, if $E$ is bounded, we can apply previous results with $\Omega=B_R$ s.t. $E\subset\Omega$. Moreover, since $\Omega$ has a regular boundary, as remarked above we can take the whole $s$-perimeter in $(\ref{frac_range_sets})$, instead of just the local part. But then, since $P_s(E,\Omega)=P_s(E)$, we see that $${\textrm{Dim}}_F(\partial^-E,\Omega)={\textrm{Dim}}_F(\partial^-E,{\mathbb R}^n).$$ [The measure theoretic boundary of a set of locally finite $s$-perimeter (in general) is not rectifiable]{} These results show that a set $E$ can have finite fractional perimeter even if its boundary is really irregular, unlike what happens with a Caccioppoli set and its reduced boundary, which is locally $(n-1)$-rectifiable.\ Indeed, if $\Omega\subset{\mathbb R}^n$ is a bounded open set with Lipschitz boundary and $E\subset{\mathbb R}^n$ is s.t. $\emptyset\not=\partial^-E\cap\Omega$ is not $(n-1)$-rectifiable, with $\overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\in(n-1,n)$, thanks to previous Corollary we have $P_s(E,\Omega)<\infty$ for every $s\in(0,\sigma)$. We give some examples of this kind of sets in the following Sections.\ In particular, the von Koch snowflake $S\subset{\mathbb R}^2$ has finite $s$-perimeter for every $s\in(0,\sigma)$, but $\partial^-S=\partial S$ is not locally $(n-1)$-rectifiable.\ Actually, because of the self-similarity of the von Koch curve, there is no part of $\partial S$ which is rectifiable (see below).\ On the other hand, De Giorgi’s structure Theorem (see e.g. Theorem 15.9 and Corollary 16.1 of [@Maggi]) says that if a set $E\subset{\mathbb R}^n$ has locally finite perimeter, then its reduced boundary $\partial^*E$ is locally $(n-1)$-rectifiable.\ Moreover the reduced boundary is dense in the measure theoretic boundary, which is the support of the Radon measure $|D\chi_E|$, $$\overline{\partial^*E}=\partial^-E=\textrm{supp }|D\chi_E|.$$ This underlines a deep difference between the classical perimeter and the $s$-perimeter, which can indeed be thought of as a “fractional” perimeter.\ Namely, having (locally) finite classical perimeter implies the regularity of an “important” portion of the (measure theoretic) boundary. On the other hand, a set can have a fractal, nowhere rectifiable boundary and still have (locally) finite $s$-perimeter. [Remarks about the Minkowski content of $\partial^-E$]{} In the beginning of the proof of Proposition $\ref{vis_prop}$ we chose a particular representative for the class of $E$ in order to have $\partial E=\partial^-E$. This can be done since it does not affect the $s$-perimeter and we are already considering the Minkowski dimension of $\partial^-E$. On the other hand, if we consider a set $F$ s.t. $|E\Delta F|=0$, we can use the same proof to obtain the inequality $${\textrm{Dim}}_F(\partial^-E,\Omega)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\partial F,\Omega).$$ It is then natural to ask whether we can find a “better” representative $F$, whose (topological) boundary $\partial F$ has Minkowski dimension strictly smaller than that of $\partial^-E$. First of all, we remark that the Minkowski content can be influenced by changes in sets of measure zero. Roughly speaking, this is because the Minkowski content is not a purely measure theoretic notion, but rather a combination of metric and measure. For example, let $\Gamma\subset{\mathbb R}^n$ and define $\Gamma':=\Gamma\cup\mathbb Q^n$. Then $|\Gamma\Delta\Gamma'|=0$, but $N_\delta(\Gamma')={\mathbb R}^n$ for every $\delta>0$. In particular, considering different representatives for $E$ we will get different topological boundaries and hence different Minkowski dimensions. However, since the measure theoretic boundary minimizes the size of the topological boundary, that is $$\partial^-E=\bigcap_{|F\Delta E|=0}\partial F,$$ (see Appendix C), it minimizes also the Minkowski dimension.\ Indeed, for every $F$ s.t. $|F\Delta E|=0$ we have $$\begin{split} \partial^-E\subset\partial F&\quad\Longrightarrow\quad \bar N_\rho^\Omega(\partial^-E)\subset \bar N_\rho^\Omega(\partial F)\\ & \quad\Longrightarrow\quad \overline{\mathcal M}^r(\partial^-E,\Omega)\leq \overline{\mathcal M}^r(\partial F,\Omega)\\ & \quad\Longrightarrow\quad \overline{{\textrm{Dim}}}_\mathcal{M}(\partial^-E,\Omega)\leq \overline{{\textrm{Dim}}}_\mathcal{M}(\partial F,\Omega). \end{split}$$ [Fractal dimension of the von Koch snowflake]{} The von Koch snowflake $S\subset{\mathbb R}^2$ is an example of bounded open set with fractal boundary, for which the Minkowski dimension and the fractal dimension introduced above coincide. Moreover its boundary is “nowhere rectifiable”, in the sense that $\partial S\cap B_r(p)$ is not $(n-1)$-rectifiable for any $r>0$ and $p\in\partial S$.\ First of all we construct the von Koch curve. Then the snowflake is made of three von Koch curves. Let $\Gamma_0$ be a line segment of unit length. The set $\Gamma_1$ consists of the four segments obtained by removing the middle third of $\Gamma_0$ and replacing it by the other two sides of the equilateral triangle based on the removed segment.\ We construct $\Gamma_2$ by applying the same procedure to each of the segments in $\Gamma_1$ and so on. Thus $\Gamma_k$ comes from replacing the middle third of each straight line segment of $\Gamma_{k-1}$ by the other two sides of an equilateral triangle. As $k$ tends to infinity, the sequence of polygonal curves $\Gamma_k$ approaches a limiting curve $\Gamma$, called the von Koch curve.\ If we start with an equilateral triangle with unit length side and perform the same construction on all three sides, we obtain the von Koch snowflake $\Sigma$.\ Let $S$ be the bounded region enclosed by $\Sigma$, so that $S$ is open and $\partial S=\Sigma$. We still call $S$ the von Koch snowflake.\ Now we calculate the (Minkowski) dimension of $\Gamma$ using the box-counting dimensions (see Appendix D).\ The idea is to exploit the self-similarity of $\Gamma$ and consider covers made of squares with side $\delta_k=3^{-k}$. The key observation is that $\Gamma$ can be covered by three squares of length $1/3$ (and cannot be covered by only two), so that $\mathcal{N}(\Gamma,1/3)=3$.\ Then consider $\Gamma_1$. We can think of $\Gamma$ as being made of four von Koch curves starting from the set $\Gamma_1$ and with initial segments of length $1/3$ instead of 1. Therefore we can cover each of these four pieces with three squares of side $1/9$, so that $\Gamma$ can be covered with $3\cdot4$ squares of length $1/9$ (and not one less) and $\mathcal{N}(\Gamma,1/9)=4\cdot3$. We can repeat the same argument starting from $\Gamma_2$ to get $\mathcal{N}(\Gamma,1/27)=4^2\cdot3$, and so on. In general we obtain $$\mathcal{N}(\Gamma,3^{-k})=4^{k-1}\cdot3.$$ Then, taking logarithms we get $$\frac{\log\mathcal{N}(\Gamma,3^{-k})}{-\log3^{-k}}=\frac{\log3+(k-1)\log4}{k\log3}\longrightarrow\frac{\log4}{\log3},$$ so that ${\textrm{Dim}}_\mathcal{M}(\Gamma)=\frac{\log4}{\log3}$. Notice that the Minkowski dimensions of the snowflake and of the curve are the same. Moreover it can be shown that the Hausdorff dimension of the von Koch curve is equal to its Minkowski dimension, so we obtain $$\label{dime_Koch_snow} {\textrm{Dim}}_{\mathcal H}(\Sigma)={\textrm{Dim}}_\mathcal M(\Sigma)=\frac{\log4}{\log3}$$ Now we explain how to construct $S$ in a recursive way and we prove that $$\partial^-S=\partial S=\Sigma.$$ As starting point for the snowflake take the equilateral triangle $T$ of side 1, with baricenter in the origin and a vertex on the $y$-axis, $P=(0,t)$ with $t>0$.\ Then $T_1$ is made of three triangles of side $1/3$, $T_2$ of $3\cdot4$ triangles of side $1/3^2$ and so on.\ In general $T_k$ is made of $3\cdot4^{k-1}$ triangles of side $1/3^k$, call them $T_k^1,\ldots,T_k^{3\cdot4^{k-1}}$. Let $x^i_k$ be the baricenter of $T_k^i$ and $P_k^i$ the vertex which does not touch $T_{k-1}$. Then $S=T\cup\bigcup T_k$. Also notice that $T_k$ and $T_{k-1}$ touch only on a set of measure zero. For each triangle $T^i_k$ there exists a rotation $\mathcal{R}_k^i\in SO(n)$ s.t. $$T_k^i=F_k^i(T):=\mathcal{R}_k^i\Big(\frac{1}{3^k}T\Big)+x_k^i.$$ We choose the rotations so that $F_k^i(P)=P_k^i$. Notice that for each triangle $T_k^i$ we can find a small ball which is contained in the complementary of the snowflake, $B_k^i\subset{\mathcal C}S$, and touches the triangle in the vertex $P_k^i$. Actually these balls can be obtained as the images of the affine transformations $F_k^i$ of a fixed ball $B$. To be more precise, fix a small ball contained in the complementary of $T$, which has the center on the $y$-axis and touches $T$ in the vertex $P$, say $B:=B_{1/1000}(0,t+1/1000)$. Then $$\label{koch3} B_k^i:=F_k^i(B)\subset{\mathcal C}S$$ for every $i,\,k$. To see this, imagine constructing the snowflake $S$ using the same affine transformations $F_k^i$ but starting with $T\cup B$ in place of $T$.\ We know that $\partial^-S\subset\partial S$ (see Appendix C).\ On the other hand, let $p\in\partial S$. Then every ball $B_\delta(p)$ contains at least a triangle $T^i_k\subset S$ and its corresponding ball $B^i_k\subset{\mathcal C}S$ (and actually infinitely many). Therefore $0<|B_\delta(p)\cap S|<\omega_n\delta^n$ for every $\delta>0$ and hence $p\in\partial^-S$. Since $S$ is bounded, its boundary is $\partial^-S=\Sigma$, and ${\textrm{Dim}}_\mathcal M(\Sigma)=\frac{\log4}{\log3}$, we obtain $(\ref{koch1})$ from Corollary $\ref{fractal_dim_coroll}$ and Remark $\ref{fractal_dim_rmk}$. Exploiting the construction of $S$ given above and $(\ref{koch3})$ we prove $(\ref{koch2})$.\ We have $$\begin{split} P_s(S)&={\mathcal L}_s(S,{\mathcal C}S)={\mathcal L}_s(T,{\mathcal C}S)+\sum_{k=1}^\infty{\mathcal L}_s(T_k,{\mathcal C}S)\\ & ={\mathcal L}_s(T,{\mathcal C}S)+\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}S) \geq\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}S)\\ & \geq\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(T_k^i,B_k^i)\qquad\textrm{(by }(\ref{koch3}))\\ & =\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}{\mathcal L}_s(F_k^i(T),F_k^i(B))\\ & =\sum_{k=1}^\infty\sum_{i=1}^{3\cdot4^{k-1}}\Big(\frac{1}{3^k}\Big)^{2-s}{\mathcal L}_s(T,B)\qquad\textrm{(by Proposition }\ref{elementary_properties})\\ & =\frac{3}{3^{2-s}}{\mathcal L}_s(T,B)\sum_{k=0}^\infty\Big(\frac{4}{3^{2-s}}\Big)^k. \end{split}$$ We remark that $${\mathcal L}_s(T,B)\leq{\mathcal L}_s(T,{\mathcal C}T)=P_s(T)<\infty,$$ for every $s\in(0,1)$. To conclude, notice that the last series is divergent if $s\geq2-\frac{\log4}{\log3}$. Exploiting the self-similarity of the von Koch curve, we show that the fractal dimension of $S$ is the same in every open set which contains a point of $\partial S$. \[koch\_coroll\] Let $S\subset{\mathbb R}^2$ be the von Koch snowflake. Then $${\textrm{Dim}}_F(\partial S,\Omega)=\frac{\log4}{\log3}$$ for every open set $\Omega$ s.t. $\partial S\cap\Omega\not=\emptyset$. Since $P_s(S,\Omega)\leq P_s(S)$, we have $$P_s(S,\Omega)<\infty,\qquad\forall\,s\in\Big(0,2-\frac{\log4}{\log3}\Big).$$ On the other hand, if $p\in\partial S\cap\Omega$, then $B_r(p)\subset\Omega$ for some $r>0$. Now notice that $B_r(p)$ contains a rescaled version of the von Koch curve, including all the triangles $T_k^i$ which constitute it and the relative balls $B_k^i$. We can thus repeat the argument above to obtain $$P_s(S,\Omega)\geq P_s(S,B_r(p))=\infty,\qquad\forall\,s\in\Big[2-\frac{\log4}{\log3},1\Big).$$ [Self-similar fractal boundaries]{} The von Koch curve is a well known example of a family of rather “regular” fractal sets, the self-similar fractal sets (see e.g. Section 9 of [@Falconer] for the proper definition and the main properties). Many examples of this kind of sets can be constucted in a recursive way similar to that of the von Koch snowflake. To be more precise, we start with a bounded open set $T_0\subset{\mathbb R}^n$ with finite perimeter $P(T_0)<\infty$, which is, roughly speaking, our basic “building block”. Then we go on inductively by adding roto-translations of a scaling of the building block $T_0$, i.e. sets of the form $$T_k^i=F_k^i(T_0):=\mathcal{R}_k^i\big(\lambda^{-k}T_0\big)+x_k^i,$$ where $\lambda>1$, $k\in\mathbb N$, $1\leq i\leq ab^{k-1}$, with $a,\,b\in\mathbb N$, $\mathcal{R}_k^i\in SO(n)$ and $x_k^i\in{\mathbb R}^n$. We ask that these sets do not overlap, i.e. $$|T^i_k\cap T^j_h|=0,\qquad\textrm{if }i\not=j.$$ Then we define $$\label{frac_ind_def} T_k:=\bigcup_{i=1}^{ab^{k-1}}T_k^i\qquad\textrm{and}\qquad T:=\bigcup_{k=1}^\infty T_k.$$ The final set $E$ is either $$E:=T_0\cup\bigcup_{k\geq1}\bigcup_{i=1}^{ab^{k-1}}T^i_k,\quad\textrm{or}\quad E:=T_0\setminus\Big(\bigcup_{k\geq1}\bigcup_{i=1}^{ab^{k-1}}T^i_k\Big).$$ For example, the von-Koch snowflake is obtained by adding pieces. Examples obtained by removing the $T_k^i$’s are the middle Cantor set $E\subset{\mathbb R}$, the Sierpinski triangle $E\subset{\mathbb R}^2$ and Menger sponge $E\subset{\mathbb R}^3$.\ We will consider just the set $T$ and exploit the same argument used for the von Koch snowflake to compute the fractal dimension related to the $s$-perimeter.\ However, the Cantor set, the Sierpinski triangle and the Menger sponge are s.t. $|E|=0$, i.e. $|T_0\Delta T|=0$.\ Therefore both the perimeter and the $s$-perimeter do not notice the fractal nature of the (topological) boundary of $T$ and indeed, since $P(T)=P(T_0)<\infty$, we get $P_s(T)<\infty$ for every $s\in(0,1)$. For example, in the case of the Sierpinski triangle, $T_0$ is an equilateral triangle and $\partial^-T=\partial T_0$, even if $\partial T$ is a self-similar fractal. Roughly speaking, the problem in these cases is that there is not room enough to find a small ball $B_k^i=F_k^i(B)\subset{\mathcal C}T$ near each piece $T_k^i$. Therefore, we will make the additional assumption that $$\label{add_frac_self_hp} \exists\,S_0\subset{\mathcal C}T\quad\textrm{s.t. }|S_0|>0\quad\textrm{and }S_k^i:=F_k^i(S_0)\subset{\mathcal C}T\quad\forall\,k,\,i.$$ We remark that it is not necessary to ask that these sets do not overlap. Below we give some examples on how to construct sets which satisfy this additional hypothesis starting with sets which do not, like the Sierpinski triangle, without altering their “structure”. \[fractal\_bdary\_selfsim\_dim\] Let $T\subset{\mathbb R}^n$ be a set which can be written as in $(\ref{frac_ind_def})$. If $\frac{\log b}{\log\lambda}\in(n-1,n)$ and $(\ref{add_frac_self_hp})$ holds true, then $$P_s(T)<\infty,\qquad\forall\,s\in\Big(0,n-\frac{\log b}{\log\lambda}\Big)$$ and $$P_s(T)=\infty,\qquad\forall\,s\in\Big[n-\frac{\log b}{\log\lambda},1\Big).$$ Thus $${\textrm{Dim}}_F(\partial^-T)=\frac{\log b}{\log\lambda}.$$ Arguing as we did with the von Koch snowflake, we show that $P_s(T)$ is bounded both from above and from below by the series $$\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k,$$ which converges if and only if $s<n-\frac{\log b}{\log\lambda}$. Indeed $$\begin{split} P_s(T)&={\mathcal L}_s(T,{\mathcal C}T)=\sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T)\\ & \leq \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T_k^i) = \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(F_k^i(T_0),F_k^i({\mathcal C}T_0))\\ & =\frac{a}{\lambda^{n-s}}{\mathcal L}_s(T_0,{\mathcal C}T_0)\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k, \end{split}$$ and $$\begin{split} P_s(T)&={\mathcal L}_s(T,{\mathcal C}T)=\sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,{\mathcal C}T)\\ & \geq \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(T_k^i,S_k^i) = \sum_{k=1}^\infty\sum_{i=1}^{ab^{k-1}}{\mathcal L}_s(F_k^i(T_0),F_k^i(S_0))\\ & =\frac{a}{\lambda^{n-s}}{\mathcal L}_s(T_0,S_0)\sum_{k=0}^\infty\Big(\frac{b}{\lambda^{n-s}}\Big)^k. \end{split}$$ Also notice that, since $P(T_0)<\infty$, we have $${\mathcal L}_s(T_0,S_0)\leq{\mathcal L}_s(T_0,{\mathcal C}T_0)=P_s(T_0)<\infty,$$ for every $s\in(0,1)$. Now suppose that $T$ does not satisfy $(\ref{add_frac_self_hp})$. Then we can obtain a set $T'$ which does, simply by removing a part $S_0$ of the building block $T_0$.\ To be more precise, let $S_0\subset T_0$ be s.t. $|S_0|>0$, $|T_0\setminus S_0|>0$ and $P(T_0\setminus S_0)<\infty$. Then define a new building block $T'_0:=T_0\setminus S_0$ and the set $$T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T'_0).$$ This new set has exactly the same structure of $T$, since we are using the same collection $\{F_k^i\}$ of affine maps. Notice that $$S_0\subset T_0\quad\Longrightarrow\quad F_k^i(S_0)\subset F_k^i(T_0),$$ and $$F_k^i(T'_0)=F_k^i(T_0)\setminus F_k^i(S_0),$$ for every $k,\,i$. Thus $$T'=T\setminus\Big(\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(S_0)\Big)$$ satisfies $(\ref{add_frac_self_hp})$. Roughly speaking, what matters is that there exists a bounded open set $T_0$ s.t. $$|F_k^i(T_0)\cap F_h^j(T_0)|=0,\qquad\textrm{if }i\not=j.$$ This can be thought of as a compatibility criterion for the affine maps $\{F_k^i\}$.\ We also need to ask that the ratio of the logarithms of the growth factor and the scaling factor is $\frac{\log b}{\log\lambda}\in(n-1,n)$.\ Then we are free to choose as building block any set $T'_0\subset T_0$ s.t. $$|T'_0|>0,\qquad|T_0\setminus T'_0|>0\qquad\textrm{and }P(T'_0)<\infty,$$ and the set $$T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T'_0).$$ satisfies the hypothesis of previous Theorem. Therefore, even if the Sierpinski triangle and the Menger sponge do not satisfy $(\ref{add_frac_self_hp})$, we can exploit their structure to construct new sets which do. However, we remark that the new boundary $\partial^-T'$ will look very different from the original fractal. Actually, in general it will be a mix of unrectifiable pieces and smooth pieces. In particular, we can not hope to get an analogue of Corollary $\ref{koch_coroll}$. Still, the following Remark shows that the new (measure theoretic) boundary retains at least some of the “fractal nature” of the original set. \[self\_sim\_frac\_bdry\_nat\_rmk\] If the set $T$ of Theorem $\ref{fractal_bdary_selfsim_dim}$ is bounded, exploiting Proposition $\ref{vis_prop}$ and Remark $\ref{fractal_dim_rmk}$ we obtain $$\overline{{\textrm{Dim}}}_{\mathcal M}(\partial^-T)\geq\frac{\log b}{\log\lambda}>n-1.$$ Moreover, notice that if $\Omega$ is a bounded open set with Lipschitz boundary, then $$P(E,\Omega)<\infty\quad\Longrightarrow\quad{\textrm{Dim}}_F(E,\Omega)=n-1.$$ Therefore, if $T\subset\subset B_R$, then $$P(T)=P(T,B_R)=\infty,$$ even if $T$ is bounded (and hence $\partial^-T$ is compact). [Sponge-like sets]{} The simplest way to construct the set $T'$ consists in simply removing a small ball $S_0:=B\subset\subset T_0$ from $T_0$. In particular, suppose that $|T_0\Delta T|=0$, as with the Sierpinski triangle.\ Define $$S:=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(B) \quad\textrm{and}\quad T':=\bigcup_{k=1}^\infty\bigcup_{i=1}^{ab^{k-1}}F_k^i(T_0\setminus B)=T\setminus S.$$ Then $$\label{fractal_spazz_end} |T_0\Delta T|=0\quad\Longrightarrow\quad |T'\Delta (T_0\setminus S)|=0.$$ Now the set $E:=T_0\setminus S$ looks like a sponge, in the sense that it is a bounded open set with an infinite number of holes (each one at a positive, but non-fixed distance from the others). From $(\ref{fractal_spazz_end})$ we get $P_s(E)=P_s(T')$. Thus, since $T'$ satisfies the hypothesis of previous Theorem, we obtain $${\textrm{Dim}}_F(\partial^-E)=\frac{\log b}{\log\lambda}.$$ [Dendrite-like sets]{} Depending on the form of the set $T_0$ and on the affine maps $\{F_k^i\}$, we can define more intricated sets $T'$. As an example we consider the Sierpinski triangle $E\subset{\mathbb R}^2$.\ It is of the form $E=T_0\setminus T$, where the building block $T_0$ is an equilateral triangle, say with side length one, a vertex on the $y$-axis and baricenter in 0. The pieces $T_k^i$ are obtained with a scaling factor $\lambda=2$ and the growth factor is $b=3$ (see e.g. [@Falconer] for the construction). As usual, we consider the set $$T=\bigcup_{k=1}^\infty\bigcup_{i=1}^{3^{k-1}}T_k^i.$$ However, as remarked above, we have $|T\Delta T_0|=0$. Starting from $k=2$ each triangle $T_k^i$ touches with (at least) a vertex (at least) another triangle $T_h^j$. Moreover, each triangle $T_k^i$ gets touched in the middle point of each side (and actually it gets touched in infinitely many points). Exploiting this situation, we can remove from $T_0$ six smaller triangles, so that the new building block $T'_0$ is a star polygon centered in 0, with six vertices, one in each vertex of $T_0$ and one in each middle point of the sides of $T_0$. ![[*Removing the six triangles (in green) to obtain the new “building block” $T'_0$ (on the right)*]{}](Star_Bblock){width="90mm"} The resulting set $$T'=\bigcup_{k=1}^\infty\bigcup_{i=1}^{3^{k-1}}F_k^i(T'_0)$$ will have an infinite number of ramifications. ![[*The third and fourth steps of the iterative construction of the set $T'$*]{}](Ind_steps_star){width="110mm"} Since $T'$ satisfies the hypothesis of previous Theorem, we obtain $${\textrm{Dim}}_F(\partial^-T')=\frac{\log 3}{\log2}.$$ [“Exploded” fractals]{} In all the previous examples, the sets $T_k^i$ are accumulated in a bounded region. On the other hand, imagine making a fractal like the von Koch snowflake or the Sierpinski triangle “explode” and then rearrange the pieces $T_k^i$ in such a way that $d(T_k^i,T_h^j)\geq d$, for some fixed $d>0$. Since the shape of the building block is not important, we can consider $T_0:=B_{1/4}(0)\subset{\mathbb R}^n$, with $n\geq2$. Moreover, since the parameter $a$ does not influence the dimension, we can fix $a=1$. Then we rearrange the pieces obtaining $$\label{exploded_frac_def} E:=\bigcup_{k=1}^\infty\bigcup_{i=1}^{b^{k-1}}B_\frac{1}{4\lambda^k}(k,0,\ldots,0,i).$$ Define for simplicity $$B_k^i:=B_\frac{1}{4\lambda^k}(k,0,\ldots,0,i)\quad\textrm{and}\quad x_k^i:=k\,e_1+i\,e_n,$$ and notice that $$B_k^i=\lambda^{-k}B_\frac{1}{4}(0)+x_k^i.$$ Since for every $k,\,h$ and every $i\not=j$ we have $$d(B_k^i,B_h^j)\geq\frac{1}{2},$$ the boundary of the set $E$ is the disjoint union of $(n-1)$-dimensional spheres $$\partial^-E=\partial E=\bigcup_{k=1}^\infty\bigcup_{i=1}^{b^{k-1}}\partial B_k^i,$$ and in particular is smooth. The (global) perimeter of $E$ is $$P(E)=\sum_{k=1}^\infty\sum_{i=1}^{b^{k-1}}P(B_k^i)=\frac{1}{\lambda}P(B_{1/4}(0))\sum_{k=0}^\infty \Big(\frac{b}{\lambda^{n-1}}\Big)^k=\infty,$$ since $\frac{\log b}{\log\lambda}>n-1$. However $E$ has locally finite perimeter, since its boundary is smooth and every ball $B_R$ intersects only finitely many $B_k^i$’s, $$P(E,B_R)<\infty,\qquad\forall\,R>0.$$ Therefore it also has locally finite $s$-perimeter for every $s\in(0,1)$ $$P_s(E,B_R)<\infty,\qquad\forall\,R>0,\qquad\forall\,s\in(0,1).$$ What is interesting is that the set $E$ satisfies the hypothesis of Theorem $\ref{fractal_bdary_selfsim_dim}$ and hence it also has finite global $s$-perimeter for every $s<\sigma_0:=n-\frac{\log b}{\log\lambda}$, $$P_s(E)<\infty\qquad\forall\,s\in(0,\sigma_0)\quad\textrm{and}\quad P_s(E)=\infty\qquad\forall\,s\in[\sigma_0,1).$$ Thus we obtain Proposition $\ref{expl_farc_prop1}$. It is enough to choose a natural number $b\geq2$ and take $\lambda:=b^\frac{1}{n-\sigma}$. Notice that $\lambda>1$ and $$\frac{\log b}{\log\lambda}=n-\sigma\in(n-1,n).$$ Then we can define $E$ as in $(\ref{exploded_frac_def})$ and we are done. [Elementary properties of the $s$-perimeter]{} \[elementary\_properties\] Let $\Omega\subset{\mathbb R}^n$ be an open set. \(i) (Subadditivity)$\quad$ Let $E,\,F\subset{\mathbb R}^n$ s.t. $|E\cap F|=0$. Then $$\label{subadditive} P_s(E\cup F,\Omega)\leq P_s(E,\Omega)+P_s(F,\Omega).$$ \(ii) (Translation invariance)$\quad$ Let $E\subset{\mathbb R}^n$ and $x\in{\mathbb R}^n$. Then $$\label{translation_invariance} P_s(E+x,\Omega+x)=P_s(E,\Omega).$$ \(iii) (Rotation invariance)$\quad$ Let $E\subset{\mathbb R}^n$ and $\mathcal{R}\in SO(n)$ a rotation. Then $$\label{rotation_invariance} P_s(\mathcal{R}E,\mathcal{R}\Omega)=P_s(E,\Omega).$$ \(iv) (Scaling)$\quad$ Let $E\subset{\mathbb R}^n$ and $\lambda>0$. Then $$\label{scaling} P_s(\lambda E,\lambda\Omega)=\lambda^{n-s}P_s(E,\Omega).$$ \(i) follows from the following observations. Let $A_1,\,A_2,\,B\subset{\mathbb R}^n$. If $|A_1\cap A_2|=0$, then $${\mathcal L}_s(A_1\cup A_2,B) ={\mathcal L}_s(A_1,B)+{\mathcal L}_s(A_2,B).$$ Moreover $$A_1\subset A_2\quad\Longrightarrow\quad{\mathcal L}_s(A_1,B)\leq{\mathcal L}_s(A_2,B),$$ and $${\mathcal L}_s(A,B)={\mathcal L}_s(B,A).$$ Therefore $$\begin{split} P_s(E\cup F,\Omega)&={\mathcal L}_s((E\cup F)\cap\Omega,{\mathcal C}(E\cup F))+{\mathcal L}_s((E\cup F)\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)\\ & ={\mathcal L}_s(E\cap\Omega,{\mathcal C}(E\cup F))+{\mathcal L}_s(F\cap\Omega,{\mathcal C}(E\cup F))\\ & \qquad+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)+{\mathcal L}_s(F\setminus\Omega,{\mathcal C}(E\cup F)\cap\Omega)\\ & \leq{\mathcal L}_s(E\cap\Omega,{\mathcal C}E)+{\mathcal L}_s(F\cap\Omega,{\mathcal C}F)\\ & \qquad+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega)+{\mathcal L}_s(F\setminus\Omega,{\mathcal C}F\cap\Omega)\\ & =P_s(E,\Omega)+P_s(F,\Omega). \end{split}$$ (ii), (iii) and (iv) follow simply by changing variables in ${\mathcal L}_s$ and the following observations: $$\begin{split} &(x+A_1)\cap(x+A_2)=x+A_1\cap A_2,\qquad x+{\mathcal C}A={\mathcal C}(x+A),\\ & \mathcal{R}A_1\cap\mathcal{R}A_2=\mathcal{R}(A_1\cap A_2),\qquad\mathcal{R}({\mathcal C}A)={\mathcal C}(\mathcal{R}A),\\ & (\lambda A_1)\cap(\lambda A_2)=\lambda(A_1\cap A_2),\qquad\lambda({\mathcal C}A)={\mathcal C}(\lambda A). \end{split}$$ For example, for claim (iv) we have $$\begin{split} {\mathcal L}_s(\lambda A,\lambda B)&=\int_{\lambda A}\int_{\lambda B}\frac{dx\,dy}{|x-y|^{n+s}} =\int_A\lambda^n\,dx\int_B\frac{\lambda^n\,dy}{\lambda^{n+s}|x-y|^{n+s}}\\ & =\lambda^{n-s}{\mathcal L}_s(A,B). \end{split}$$ Then $$\begin{split} P_s(\lambda E,\lambda\Omega)&={\mathcal L}_s(\lambda E\cap\lambda\Omega,{\mathcal C}(\lambda E))+ {\mathcal L}_s(\lambda E\cap{\mathcal C}(\lambda\Omega),{\mathcal C}(\lambda E)\cap\lambda\Omega)\\ & ={\mathcal L}_s(\lambda(E\cap\Omega),\lambda{\mathcal C}E)+{\mathcal L}_s(\lambda(E\setminus\Omega),\lambda({\mathcal C}E\cap\Omega))\\ & =\lambda^{n-s}\left({\mathcal L}_s(E\cap\Omega,{\mathcal C}E)+{\mathcal L}_s(E\setminus\Omega,{\mathcal C}E\cap\Omega)\right)\\ & =\lambda^{n-s}P_s(E,\Omega). \end{split}$$ Proof of Example $\ref{inclusion_counterexample}$ ================================================= Note that $E\subset (0,a^2]$. Let $\Omega:=(-1,1)\subset\mathbb{R}$. Then $E\subset\subset\Omega$ and $\textrm{dist}(E,\partial\Omega)=1-a^2=:d>0$. Now $$P_s(E)=\int_E\int_{{\mathcal C}E\cap\Omega}\frac{dxdy}{|x-y|^{1+s}}+ \int_E\int_{{\mathcal C}\Omega}\frac{dxdy}{|x-y|^{1+s}}.$$ As for the second term, we have $$\int_E\int_{{\mathcal C}\Omega}\frac{dxdy}{|x-y|^{1+s}}\leq\frac{2|E|}{sd^s}<\infty.$$ We split the first term into three pieces $$\begin{split} \int_E&\int_{{\mathcal C}E\cap\Omega}\frac{dxdy}{|x-y|^{1+s}}\\ & =\int_E\int_{-1}^0\frac{dxdy}{|x-y|^{1+s}} +\int_E\int_{{\mathcal C}E\cap(0,a)}\frac{dxdy}{|x-y|^{1+s}}+\int_E\int_a^1\frac{dxdy}{|x-y|^{1+s}}\\ & =\mathcal{I}_1+\mathcal{I}_2+\mathcal{I}_3. \end{split}$$ Note that ${\mathcal C}E\cap(0,a)=\bigcup_{k\in\mathbb{N}}I_{2k-1}=\bigcup_{k\in\mathbb{N}}(a^{2k},a^{2k-1})$.\ A simple calculation shows that, if $a<b\leq c<d$, then $$\label{rectangle_integral}\begin{split} \int_a^b&\int_c^d\frac{dxdy}{|x-y|^{1+s}}=\\ & \frac{1}{s(1-s)}\big[(c-a)^{1-s}+(d-b)^{1-s}-(c-b)^{1-s}-(d-a)^{1-s}\big]. \end{split}$$ Also note that, if $n>m\geq1$, then $$\label{derivative_bound}\begin{split} (1-a^n)^{1-s}-(1-a^m)^{1-s}&=\int_m^n\frac{d}{dt}(1-a^t)^{1-s}\,dt\\ & =(s-1)\log a\int_m^n\frac{a^t}{(1-a^t)^s}\,dt\\ & \leq a^m (s-1)\log a\int_m^n\frac{1}{(1-a^t)^s}\,dt\\ & \leq(n-m)a^m\frac{(s-1)\log a}{(1-a)^s}. \end{split}$$ Now consider the first term $$\mathcal{I}_1=\sum_{k=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_{-1}^0\frac{dxdy}{|x-y|^{1+s}}.$$ Use $(\ref{rectangle_integral}$) and notice that $(c-a)^{1-s}-(d-a)^{1-s}\leq0$ to get $$\int_{-1}^0\int_{a^{2k+1}}^{a^{2k}}\frac{dxdy}{|x-y|^{1+s}} \leq\frac{1}{s(1-s)}\big[(a^{2k})^{1-s}-(a^{2k+1})^{1-s}\big]\leq\frac{1}{s(1-s)}(a^{2(1-s)})^k.$$ Then, as $a^{2(1-s)}<1$ we get $$\mathcal{I}_1\leq\frac{1}{s(1-s)}\sum_{k=1}^\infty(a^{2(1-s)})^k<\infty.$$ As for the last term $$\mathcal{I}_3=\sum_{k=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_a^1\frac{dxdy}{|x-y|^{1+s}},$$ use $(\ref{rectangle_integral}$) and notice that $(d-b)^{1-s}-(d-a)^{1-s}\leq0$ to get $$\begin{split} \int_{a^{2k+1}}^{a^{2k}}\int_a^1\frac{dxdy}{|x-y|^{1+s}}& \leq\frac{1}{s(1-s)}\big[(1-a^{2k+1})^{1-s}-(1-a^{2k})^{1-s}\big]\\ & \leq\frac{-\log a}{s(1-a)^s}a^{2k}\quad\textrm{by }(\ref{derivative_bound}). \end{split}$$ Thus $$\mathcal{I}_3\leq\frac{-\log a}{s(1-a)^s}\sum_{k=1}^\infty(a^2)^k<\infty.$$ Finally we split the second term $$\mathcal{I}_2=\sum_{k=1}^\infty\sum_{j=1}^\infty\int_{a^{2k+1}}^{a^{2k}}\int_{a^{2j}}^{a^{2j-1}} \frac{dxdy}{|x-y|^{1+s}}$$ into three pieces according to the cases $j>k$, $j=k$ and $j<k$. If $j=k$, using $(\ref{rectangle_integral})$ we get $$\begin{split} \int_{a^{2k+1}}^{a^{2k}}&\int_{a^{2k}}^{a^{2k-1}} \frac{dxdy}{|x-y|^{1+s}}=\\ & =\frac{1}{s(1-s)}\big[(a^{2k}-a^{2k+1})^{1-s}+(a^{2k-1}-a^{2k})^{1-s}-(a^{2k-1}-a^{2k+1})^{1-s}\big]\\ & =\frac{1}{s(1-s)}\big[a^{2k(1-s)}(1-a)^{1-s}+a^{(2k-1)(1-s)}(1-a)^{1-s}\\ & \quad\quad\quad\quad\quad-a^{(2k-1)(1-s)}(1-a^2)^{1-s}\big]\\ & =\frac{1}{s(1-s)}(a^{2(1-s)})^k\Big[(1-a)^{1-s}+\frac{(1-a)^{1-s}}{a^{1-s}}-\frac{(1-a^2)^{1-s}}{a^{1-s}}\Big]. \end{split}$$ Summing over $k\in\mathbb{N}$ we get $$\begin{split} \sum_{k=1}^\infty&\int_{a^{2k+1}}^{a^{2k}}\int_{a^{2k}}^{a^{2k-1}} \frac{dxdy}{|x-y|^{1+s}}=\\ & =\frac{1}{s(1-s)}\frac{a^{2(1-s)}}{1-a^{2(1-s)}}\Big[(1-a)^{1-s}+\frac{(1-a)^{1-s}}{a^{1-s}}-\frac{(1-a^2)^{1-s}}{a^{1-s}}\Big]<\infty. \end{split}$$ In particular note that $$\begin{split} (1-s)&P_s(E)\geq(1-s)\mathcal{I}_2\\ & \geq\frac{1}{s(1-a^{2(1-s)})}\big[a^{2(1-s)}(1-a)^{1-s}+a^{1-s}(1-a)^{1-s}-a^{1-s}(1-a^2)^{1-s}\big], \end{split}$$ which tends to $+\infty$ when $s\to1$. This shows that $E$ cannot have finite perimeter. To conclude let $j>k$, the case $j<k$ being similar, and consider $$\sum_{k=1}^\infty\sum_{j=k+1}^\infty\int_{a^{2j}}^{a^{2j-1}}\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}.$$ Again, using $(\ref{rectangle_integral}$) and $(d-b)^{1-s}-(d-a)^{1-s}\leq0$, we get $$\begin{split} \int_{a^{2j}}^{a^{2j-1}}&\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}\\ & \leq\frac{1}{s(1-s)}\big[(a^{2k+1}-a^{2j})^{1-s}-(a^{2k+1}-a^{2j-1})^{1-s}\big]\\ & =\frac{a^{1-s}}{s(1-s)}(a^{2(1-s)})^k\big[(1-a^{2(j-k)-1})^{1-s}-(1-a^{2(j-k)-2})^{1-s}\big]\\ & \leq\frac{a^{1-s}}{s(1-s)}(a^{2(1-s)})^k\frac{(s-1)\log a}{(1-a)^s}a^{2(j-k)-2}\quad\quad\textrm{by }(\ref{derivative_bound})\\ & =\frac{-\log a}{s(1-a^s)a^{s+1}}(a^{2(1-s)})^k(a^2)^{j-k}, \end{split}$$ for $j\geq k+2$. Then $$\begin{split} \sum_{k=1}^\infty&\sum_{j=k+2}^\infty\int_{a^{2j}}^{a^{2j-1}}\int_{a^{2k+1}}^{a^{2k}} \frac{dxdy}{|x-y|^{1+s}}\\ & \leq\frac{-\log a}{s(1-a^s)a^{s+1}}\sum_{k=1}^\infty(a^{2(1-s)})^k\sum_{h=2}^\infty(a^2)^h<\infty. \end{split}$$ If $j=k+1$ we get $$\begin{split} \sum_{k=1}^\infty\int_{a^{2k+2}}^{a^{2k+1}}\int_{a^{2k+1}}^{a^{2k}}\frac{dxdy}{|x-y|^{1+s}}& \leq\frac{1}{s(1-s)}\sum_{k=1}^\infty(a^{2k+1}-a^{2k+2})^{1-s}\\ & =\frac{a^{1-s}(1-a)^{1-s}}{s(1-s)}\sum_{k=1}^\infty(a^{2(1-s)})^k<\infty. \end{split}$$ This shows that also $\mathcal{I}_2<\infty$, so that $P_s(E)<\infty$ for every $s\in(0,1)$ as claimed. Signed distance function ======================== Given $\emptyset\not=E\subset{\mathbb R}^n$, the distance function from $E$ is defined as $$d_E(x)=d(x,E):=\inf_{y\in E}|x-y|,\qquad\textrm{for }x\in{\mathbb R}^n.$$ The signed distance function from $\partial E$, negative inside $E$, is then defined as $$\bar{d}_E(x)=\bar{d}(x,E):=d(x,E)-d(x,{\mathcal C}E).$$ For the details of the main properties we refer e.g. to [@Ambrosio] and [@Bellettini]. We also define the sets $$E_r:=\{x\in{\mathbb R}^n\,|\,\bar{d}_E(x)<r\}.$$ Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. By definition we can locally describe $\Omega$ near its boundary as the subgraph of appropriate Lipschitz functions. To be more precise, we can find a finite open covering $\{C_{\rho_i}\}_{i=1}^m$ of $\partial\Omega$ made of cylinders, and Lipschitz functions $\varphi_i:B'_{\rho_i}\longrightarrow{\mathbb R}$ s.t. $\Omega\cap C_{\rho_i}$ is the subgraph of $\varphi_i$. That is, up to rotations and translations, $$C_{\rho_i}=\{(x',x_n)\in{\mathbb R}^n\,|\,|x'|<\rho_i,\,|x_n|<\rho_i\},$$ and $$\begin{split} \Omega\cap C_{\rho_i}&=\{(x',x_n)\in{\mathbb R}^n\,|\,x'\in B'_{\rho_i},\,-\rho_i<x_n<\varphi_i(x')\},\\ & \partial\Omega\cap C_{\rho_i}=\{(x',\varphi_i(x'))\in{\mathbb R}^n\,|\,x'\in B_{\rho_i}'\}. \end{split}$$ Let $L$ be the sup of the Lipschitz constants of the functions $\varphi_i$. Theorem 4.1 of [@LipApprox] guarantees that also the bounded open sets $\Omega_r$ have Lipschitz boundary, when $r$ is small enough, say $|r|<r_0$.\ Moreover these sets $\Omega_r$ can locally be described, in the same cylinders $C_{\rho_i}$ used for $\Omega$, as subgraphs of Lipschitz functions $\varphi_i^r$ which approximate $\varphi_i$ (see [@LipApprox] for the precise statement) and whose Lipschitz constants are less or equal to $L$.\ Notice that $$\partial\Omega_r=\{\bar{d}_\Omega=r\}.$$ Now, since in $C_{\rho_i}$ the set $\Omega_r$ coincides with the subgraph of $\varphi_i^r$, we have $${\mathcal H}^{n-1}(\partial\Omega_r\cap C_{\rho_i})=\int_{B_{\rho_i}'}\sqrt{1+|\nabla\varphi_i^r|^2}\,dx'\leq M_i,$$ with $M_i$ depending on $\rho_i$ and $L$ but not on $r$.\ Therefore $${\mathcal H}^{n-1}(\{\bar{d}_\Omega=r\})\leq\sum_{i=1}^m{\mathcal H}^{n-1}(\partial\Omega_r\cap C_{\rho_i})\leq\sum_{i=1}^mM_i$$ independently on $r$, proving the following \[bound\_perimeter\_unif\] Let $\Omega\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then there exists $r_0>0$ s.t. $\Omega_r$ is a bounded open set with Lipschitz boundary for every $r\in(-r_0,r_0)$ and $$\label{bound_perimeter_unif_eq} \sup_{|r|<r_0}{\mathcal H}^{n-1}(\{\bar{d}_\Omega=r\})<\infty.$$ Measure theoretic boundary ========================== Since $$\label{fin_spazz_basta1} |E\Delta F|=0\quad\Longrightarrow\quad P(E,\Omega)=P(F,\Omega)\quad\textrm{and}\quad P_s(E,\Omega)=P_s(F,\Omega),$$ we can modify a set making its topological boundary as big as we want, without changing its (fractional) perimeter.\ For example, let $E\subset{\mathbb R}^n$ be a bounded open set with Lipschitz boundary. Then, if we set $F:=E\cup(\mathbb Q^n\setminus E)$, we have $|E\Delta F|=0$ and hence we get $(\ref{fin_spazz_basta1})$. However $\partial F={\mathbb R}^n\setminus E$. For this reason one considers measure theoretic notions of interior, exterior and boundary, which solely depend on the class of $\chi_E$ in $L^1_{loc}({\mathbb R}^n)$.\ In some sense, by considering the measure theoretic boundary $\partial^-E$ defined below we can also minimize the size of the topological boundary (see $(\ref{ess_bdry_intersect})$). Moreover, this measure theoretic boundary is actually the topological boundary of a set which is equivalent to $E$. Thus we obtain a “good” representative for the class of $E$. We refer to Section 3.2 of [@Visintin] (see also Proposition 3.1 of [@Giusti]). For some details about the good representative of an $s$-minimal set, see the Appendix of [@graph]. Let $E\subset{\mathbb R}^n$. For every $t\in[0,1]$ define the set $$\label{density_t} E^{(t)}:=\left\{x\in{\mathbb R}^n\,\big|\,\exists\lim_{r\to0}\frac{|E\cap B_r(x)|}{\omega_nr^n}=t\right\},$$ of points density $t$ of $E$. The sets $E^{(0)}$ and $E^{(1)}$ are respectively the measure theoretic exterior and interior of the set $E$. The set $$\label{ess_bdry} \partial_eE:={\mathbb R}^n\setminus(E^{(0)}\cup E^{(1)})$$ is the essential boundary of $E$. Using the Lebesgue points Theorem for the characteristic function $\chi_E$, we see that the limit in $(\ref{density_t})$ exists for a.e. $x\in{\mathbb R}^n$ and $$\lim_{r\to0}\frac{|E\cap B_r(x)|}{\omega_nr^n}=\left\{\begin{array}{cc}1,&\textrm{a.e. }x\in E,\\ 0,&\textrm{a.e. }x\in{\mathcal C}E. \end{array} \right.$$ So $$|E\Delta E^{(1)}|=0,\qquad|{\mathcal C}E\Delta E^{(0)}|=0\qquad\textrm{and }|\partial_eE|=0.$$ In particular every set $E$ is equivalent to its measure theoretic interior.\ However, notice that $E^{(1)}$ in general is not open.\ We have another natural way to define a measure theoretic boundary. Let $E\subset{\mathbb R}^n$ and define the sets $$\begin{split} &E_1:=\{x\in{\mathbb R}^n\,|\,\exists r>0,\,|E\cap B_r(x)|=\omega_nr^n\},\\ & E_0:=\{x\in{\mathbb R}^n\,|\,\exists r>0,\,|E\cap B_r(x)|=0\}. \end{split}$$ Then we define $$\begin{split} \partial^-E&:={\mathbb R}^n\setminus(E_0\cup E_1)\\ & =\{x\in{\mathbb R}^n\,|\,0<|E\cap B_r(x)|<\omega_nr^n\textrm{ for every }r>0\}. \end{split}$$ Notice that $E_0$ and $E_1$ are open sets and hence $\partial^-E$ is closed. Moreover, since $$\label{density_subsets} E_0\subset E^{(0)}\qquad\textrm{and}\qquad E_1\subset E^{(1)},$$ we get $$\partial_eE\subset\partial^-E.$$ We have $$\label{ess_bdry_top1} F\subset{\mathbb R}^n\textrm{ s.t. }|E\Delta F|=0\quad\Longrightarrow\quad\partial^-E\subset\partial F.$$ Indeed, if $|E\Delta F|=0$, then $|F\cap B_r(x)|=|E\cap B_r(x)|$ for every $r>0$. Thus for any $x\in\partial^-E$ we have $$0<|F\cap B_r(x)|<\omega_nr^n,$$ which implies $$F\cap B_r(x)\not=\emptyset\quad\textrm{and}\quad{\mathcal C}F\cap B_r(x)\not=\emptyset\quad\textrm{for every }r>0,$$ and hence $x\in\partial F$. In particular, $\partial^-E\subset\partial E$. Moreover $$\label{ess_bdry_top2} \partial^-E=\partial E^{(1)}.$$ Indeed, since $|E\Delta E^{(1)}|=0$, we already know that $\partial^-E\subset\partial E^{(1)}$. The converse inclusion follows from $(\ref{density_subsets})$ and the fact that both $E_0$ and $E_1$ are open.\ From $(\ref{ess_bdry_top1})$ and $(\ref{ess_bdry_top2})$ we obtain $$\label{ess_bdry_intersect} \partial^-E=\bigcap_{F\sim E}\partial F,$$ where the intersection is taken over all sets $F\subset{\mathbb R}^n$ s.t. $|E\Delta F|=0$, so we can think of $\partial^-E$ as a way to minimize the size of the topological boundary of $E$. In particular $$F\subset{\mathbb R}^n\textrm{ s.t. }|E\Delta F|=0\quad\Longrightarrow\quad\partial^-F=\partial^-E.$$ From $(\ref{density_subsets})$ and $(\ref{ess_bdry_top2})$ we see that we can take $E^{(1)}$ as “good” representative for $E$, obtaining Remark $\ref{gmt_assumption}$.\ Recall that the support of a Radon measure $\mu$ on ${\mathbb R}^n$ is defined as the set $$\textrm{supp }\mu:=\{x\in{\mathbb R}^n\,|\,\mu(B_r(x))>0\textrm{ for every }r>0\}.$$ Notice that, being the complementary of the union of all open sets of measure zero, it is a closed set. In particular, if $E$ is a Caccioppoli set, we have $$\label{support_perimeter} \textrm{supp }|D\chi_E|=\{x\in{\mathbb R}^n\,|\,P(E,B_r(x))>0\textrm{ for every }r>0\},$$ and it is easy to verify that $$\partial^-E=\textrm{supp }|D\chi_E|=\overline{\partial^*E},$$ where $\partial^*E$ denotes the reduced boundary. However notice that in general the inclusions $$\partial^*E\subset\partial_eE\subset\partial^-E\subset\partial E$$ are all strict and in principle we could have $${\mathcal H}^{n-1}(\partial^-E\setminus\partial^*E)>0.$$ Minkowski dimension =================== Let $\Omega\subset{\mathbb R}^n$ be an open set. For any $\Gamma\subset{\mathbb R}^n$ and $r\in[0,n]$ we define the inferior and superior $r$-dimensional Minkowski contents of $\Gamma$ relative to the set $\Omega$ as, respectively $$\underline{\mathcal{M}}^r(\Gamma,\Omega):=\liminf_{\rho\to0}\frac{|\bar{N}_\rho^\Omega(\Gamma)|}{\rho^{n-r}},\qquad \overline{\mathcal{M}}^r(\Gamma,\Omega):=\limsup_{\rho\to0}\frac{|\bar{N}_\rho^\Omega(\Gamma)|}{\rho^{n-r}}.$$ Then we define the lower and upper Minkowski dimensions of $\Gamma$ in $\Omega$ as $$\begin{split} \underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma,\Omega)&:=\inf\big\{r\in[0,n]\,|\,\underline{\mathcal{M}}^r(\Gamma,\Omega)=0\big\}\\ & =n-\sup\big\{r\in[0,n]\,|\,\underline{\mathcal{M}}^{n-r}(\Gamma,\Omega)=0\big\}, \end{split}$$ $$\begin{split} \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma,\Omega)&:=\sup\big\{r\in[0,n]\,|\,\overline{\mathcal{M}}^r(\Gamma,\Omega)=\infty\big\}\\ & =n-\inf\big\{r\in[0,n]\,|\,\overline{\mathcal{M}}^{n-r}(\Gamma,\Omega)=\infty\big\}. \end{split}$$ If they agree, we write $${\textrm{Dim}}_\mathcal{M}(\Gamma,\Omega)$$ for the common value and call it the Minkowski dimension of $\Gamma$ in $\Omega$.\ If $\Omega={\mathbb R}^n$ or $\Gamma\subset\subset\Omega$, we drop the $\Omega$ in the formulas. Let ${\textrm{Dim}}_\mathcal{H}$ denote the Hausdorff dimension. In general one has $${\textrm{Dim}}_\mathcal{H}(\Gamma)\leq\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)\leq\overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma),$$ and all the inequalities might be strict. However for some sets (e.g. self-similar sets with some symmetric and regularity condition) they are all equal. We also recall some equivalent definitions of the Minkowski dimensions, usually referred to as box-counting dimensions, which are easier to compute. For the details and the relation between the Minkowski and the Hausdorff dimensions, see [@Mattila] and [@Falconer] and the references cited therein.\ For simplicity we only consider the case $\Gamma$ bounded and $\Omega={\mathbb R}^n$ (or $\Gamma\subset\subset\Omega$). Given a nonempty bounded set $\Gamma\subset{\mathbb R}^n$, define for every $\delta>0$ $$\mathcal{N}(\Gamma,\delta):=\min\Big\{k\in\mathbb{N}\,\big|\,\Gamma\subset\bigcup_{i=1}^kB_\delta(x_i),\textrm{ for some }x_i\in{\mathbb R}^n\Big\},$$ the smallest number of $\delta$-balls needed to cover $\Gamma$, and $$\mathcal{P}(\Gamma,\delta):=\max\big\{k\in\mathbb{N}\,|\,\exists\,\textrm{disjoint balls }B_\delta(x_i),\,i=1,\ldots,k\textrm{ with }x_i\in \Gamma\big\},$$ the greatest number of disjoint $\delta$-balls with centres in $\Gamma$. Then it is easy to verify that $$\label{counting} \mathcal{N}(\Gamma,2\delta)\leq\mathcal{P}(\Gamma,\delta)\leq\mathcal{N}(\Gamma,\delta/2).$$ Moreover, since any union of $\delta$-balls with centers in $\Gamma$ is contained in $N_\delta(\Gamma)$, and any union of $(2\delta)$-balls covers $N_\delta(\Gamma)$ if the union of the corresponding $\delta$-balls covers $\Gamma$, we get $$\label{counting2} \mathcal{P}(\Gamma,\delta)\omega_n\delta^n\leq|N_\delta(\Gamma)|\leq \mathcal{N}(\Gamma,\delta)\omega_n(2\delta)^n.$$ Using $(\ref{counting})$ and $(\ref{counting2})$ we see that $$\begin{split} &\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\inf\Big\{r\in[0,n]\,\big|\,\liminf_{\delta\to0}\mathcal{N}(\Gamma,\delta)\delta^r=0\Big\},\\ & \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\sup\Big\{r\in[0,n]\,\big|\,\limsup_{\delta\to0}\mathcal{N}(\Gamma,\delta)\delta^r=\infty\Big\}. \end{split}$$ Then it can be proved that $$\label{log_counting}\begin{split} &\underline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\liminf_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta},\\ & \overline{{\textrm{Dim}}}_\mathcal{M}(\Gamma)=\limsup_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}. \end{split}$$ Actually notice that, due to $(\ref{counting})$, we can take $\mathcal{P}(\Gamma,\delta)$ in place of $\mathcal{N}(\Gamma,\delta)$ in the above formulas.\ It is also easy to see that if in the definition of $\mathcal{N}(\Gamma,\delta)$ we take cubes of side $\delta$ instead of balls of radius $\delta$, then we get exactly the same dimensions. Moreover in $(\ref{log_counting})$ it is enough to consider limits as $\delta\to0$ through any decreasing sequence $\delta_k$ s.t. $\delta_{k+1}\geq c\delta_k$ for some constant $c\in(0,1)$; in particular for $\delta_k=c^k$. Indeed if $\delta_{k+1}\leq\delta<\delta_k$, then $$\begin{split} \frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}&\leq\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_k} =\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_{k+1}+\log(\delta_{k+1}/\delta_k)}\\ & \leq\frac{\log\mathcal{N}(\Gamma,\delta_{k+1})}{-\log\delta_{k+1}+\log c}, \end{split}$$ so that $$\limsup_{\delta\to0}\frac{\log\mathcal{N}(\Gamma,\delta)}{-\log\delta}\leq \limsup_{k\to\infty}\frac{\log\mathcal{N}(\Gamma,\delta_k)}{-\log\delta_k}.$$ The opposite inequality is clear and in a similar way we can treat the lower limits. [90]{} L. Ambrosio and N. Dancer, [*Calculus of variations and partial differential equations*]{}. Springer-Verlag, Berlin (2000). L. Ambrosio, G. De Philippis and L. Martinazzi, [*Gamma-convergence of nonlocal perimeter functionals*]{}. Manuscripta Math. 134, no. 3-4, 377$-$403 (2011). G. Bellettini, [*Lecture notes on mean curvature flows, barriers and singular perturbations*]{}. Edizioni della Scuola Normale 13, Pisa (2013). J. Bourgain, H. Brezis and P. Mironescu, [*Limiting embedding theorems for $W^{s,p}$ when $s\to1$ and applications*]{}. J. Anal. Math. 87, 77$-$101 (2002). L. Caffarelli, J.-M. Roquejoffre and O. Savin, [*Nonlocal minimal surfaces*]{}. Comm. pure Appl. Math. 63, no. 9, 1111$-$1144 (2010). L. Caffarelli and E. Valdinoci, [*Uniform estimates and limiting arguments for nonlocal minimal surfaces*]{}. Calc. Var. Partial Differential Equations 41, no. 1-2, 203$-$240 (2011). J. Davila, [*On an open question about functions of bounded variation*]{}. Calc. Var. Partial Differential Equations,15 no. 4, 519$-$527 (2002). E. Di Nezza, G. Palatucci and E. Valdinoci, [*Hitchhiker’s guide to the fractional Sobolev spaces*]{}. Bull. Sci. Math., 136(5):521$-$573 (2012). S. Dipierro, O. Savin and E. Valdinoci, [*Graph properties for nonlocal minimal surfaces*]{}. (2015). P. Doktor, [*Approximation of domains with Lipschitzian boundary*]{}. Cas. Pest. Mat. 101, 237$-$255 (1976). K.J. Falconer, [*Fractal geometry: mathematical foundations and applications*]{}. John Wiley and Sons (1990). E. Giusti, [*Minimal surfaces and functions of bounded variation*]{}. Monographs in Mathematics, 80. Birkhauser Verlag, Basel (1984). F. Maggi, [*Sets of finite perimeter and geometric variational problems*]{}. Cambridge Stud. Adv. Math. 135, Cambridge Univ. Press, Cambridge (2012). P. Mattila, [*Geometry of sets and measures in Euclidean spaces*]{}. Cambridge Stud. Adv. Math. 44, Cambridge Univ. Press, Cambridge (1995). A. Visintin. [*Generalized coarea formula and fractal sets*]{}. Japan J. Indust. Appl. Math., 8(2):175$-$201 (1991).

ArXiv

--- abstract: | The abundant $\psi'$ events have been collected at the Beijing Electron Positron Collider-II (BEPCII) that could undoubtedly provide us with a great opportunity to study the more attractive charmonium decays. As has been noticed before, in the process of $J/\psi$ decaying to the baryonic final states, $p K^- \overline{\Lambda}$, the evident $\Lambda^*$ and $N^*$ bands have been observed. Similarly, by using the product of $\chi_{cJ}$ from $\psi'$ radiative decay, we may confirm it or find some extra new resonances. $\chi_{c0}$’s data samples will be more than $\chi_{c1,2}$ taking into account the larger branching ratio of $\psi'\to\gamma\chi_{c0}$. Here, we provide explicit partial wave analysis formulae for the very interesting channel $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^- \overline{\Lambda}$. author: - 'Xian-Wei Kang$^{1,2}$' - 'Hai-Bo Li$^2$' - 'Gong-Ru Lu$^1$' - 'Bing-Song Zou$^2$' title: | **Partial wave analysis of $\psi'\to\gamma\chi_{c0}\rightarrow \gamma p K^- \overline{\Lambda}$\ being used for searching for baryon resonance** --- Introduction ============ In experiment at BESIII, about $10\times10^9 J/\Psi$ and $3\times10^9 \psi'$ events can be collected per year’s running according to the designed luminosity of BEPCII in Beijing [@besiii] [@bepcii]. These large data samples will provide great opportunities to study the attractive $\chi_{cJ}$ decay and some hyperon interactions. The product of $\chi_{cJ}$ in $\psi'$ radiative decays may provide useful information on two-gluon hadronization dynamics and glueball decays. On the other side, the radiative decays of $\psi'\to\gamma\chi_{cJ}$ are expected to be dominated by electric dipole (E1) transitions, with higher multipoles suppressed by powers of photon energy divided by quark mass [@Karl] and searching for contributions of higher multipoles is promissing. The possibility of anomalous magnetic moments of higher quark being lager than those for light ones may exist [@Geffen]. Thus,$\chi_{cJ}$ decays contain abundant interesting physics. In this paper, the idea is motivated by that there may exist some baryon resonances in the process of $\chi_{cJ}$ decays to baryons. In fact, in our preliminary Monte Carlo study, we found an explicit band structure, most possibly $\Lambda(1520)$, so it necessitate this PWA. In order to get more useful information about the resonance properties such as $J^{PC}$ quantum numbers, mass, width, production and decay rates,[*[etc.]{}*]{}, partial wave analysis (PWA) are necessary. PWA is an effective method for analyzing the experimental data of hadron spectrums. There are two methods of PWA, one is based on the covariant tensor (also named Rarita-Schwinger) formalism [@Schwinger] and the other is based on the original helicity formalism [@Jackson] [@Jacob] and one covariant helicity one developed by Chung [@Chung1] [@Chung2]. For a more basic exposition, the reader may wish to consult the CERN Yellow Report [@yellow; @report]. Ref.[@Filippini] showed the connection between the covariant tensor formalism and helicity one. In this short paper, we will pay more attention to the more popular one: covariant tensor format also append the helicity one for the specific process for $\Lambda(1520)$. The organization of the paper is as follows. In Sec.I, the general formalism is given. In Sec.II, we will present the covariant tensor amplitude for $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$. In Sec.III,the corresponding helicity formula are provides. At last, In Sec.IV,there is the conclusion. General formalism {#Sec:formalism} ================= In this part,the general formalism which will be used in the following have been mentioned in Ref.[@psi; @decay; @to; @mesons] [@N*NM; @couplings] [@gamma; @chicJ], including formalism for $\psi$ radiative decay to mesons(denoted by $M$),$M\to N^* N$,$N^*\to N M$,where $N^*$ and $N$ has the half integer spin. As discussed in Ref.[@psi; @decay; @to; @mesons],we denote the $\psi$ polarization four-vector by $\psi_{\mu}(m_1)$ and the polarization vector of the photon by $e_{\nu}(m_2)$.Then the general form for the decay amplitude is $$A=\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}=\psi_{\mu}(m_1)e^*_{\nu}(m_2)\underset{i}{\sum}\Lambda_i U^{\mu\nu}_i$$ here, $U^{\mu}_{i}$ is the $i-$th partial wave amplitude with coupling strength determined by a complex parameter $\Lambda_i$. Because of massless properties,there are two additional conditions,$(1)$ the usual orthogonality condition $e_{\nu}q^{\nu}=0$,where $q$ is the photon momentum;$(2)$ gauge invariance condition (assuming the Coulomb gauge in $\psi$ rest system) $e_{\nu}p^{\nu}_{\psi}=0$,where $P_{\psi}$ is the momentum of vector meson $\psi$.Then we have $$\begin{aligned} \underset{m}{\sum}e^*_{\mu}(m)e_{\nu}(m)&=&-g_{\mu\nu}+\frac{q_{\mu}K_{\nu}+K_{\mu}q_{\nu}}{q\cdot K}-\frac{K \cdot K}{(q \cdot K)^2}q_{\mu}q_{\nu}\nonumber\\ &\equiv& -g_{\mu\nu}^{(\perp\perp)}\end{aligned}$$ with $K=p_{\psi}-q $ and $ e_{\nu}K^{\nu}=0.$.To compute the differential cross section,we need an expression for $|A|^2$ ,the square modulus of the decay amplitude,which gives the decay probability of a certain configuration should be independent of any particular frame.Thus the radiative cross section is : $$\begin{aligned} \frac{d\sigma}{d\Phi_n}&=&\frac{1}{2}\sum^2_{m_1=1}\sum^2_{m_2=1}\psi_{\mu}(m_1)e^*_{\nu}(m_2)A^{\mu\nu}\psi^*_{\mu'}(m_1)e_{\nu'}(m_2)A^{*\mu'\nu'} \nonumber\\ &=&-\frac{1}{2}\sum^2_{m_1=2}\psi_{\mu}(m_1)\psi_{\mu'}(m_1)g^{(\perp \perp)}_{\nu\nu'}A^{\mu\nu}A^{*\mu'\nu'}\nonumber\\ &=&-\frac{1}{2}\sum^2_{\mu=1}A^{\mu\nu}g^{(\perp\perp)}_{\nu\nu'}A^{*\mu\nu'}\nonumber\\ &=&-\frac{1}{2}\sum_{i,j}\Lambda_i\Lambda_j^*\sum^2_{\mu=1}U^{\mu\nu}_ig^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}\equiv\sum_{i,j}P_{ij}\cdot F_{ij}\end{aligned}$$ with definition $$\begin{aligned} P_{ij}&=&P_{ji}^*=\Lambda_i\Lambda_j^*, \\F_{ij}&=&F_{ji}^* =-\frac{1}{2}\sum^2_{\mu=1}=-\frac{1}{2}U^{\mu\nu}_i g^{(\perp\perp)}_{\nu\nu'}U_j^{*\mu\nu'}.\end{aligned}$$ note the relation $$\sum^2_{m=1}\psi_{\mu}(m)\psi_{\mu'}^*(m)=\delta_{\mu\mu'}(\delta_{\mu1}+\delta_{\mu2}).$$ The partial wave amplitude $U$ in the covariant Rarita-Schwinger tensor formalism [@Schwinger] can be constructed by using pure orbital angular momentum covariant tensor $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ and covariant spin wave functions $\phi_{\mu_1\mu_2\cdots\mu_S}$ together with the metric tensor $g^{\mu\nu}$, the totally antisymmetric Levi-Civita tensor $\epsilon_{\mu\nu\lambda\sigma}$ and the four momenta of participating particles.For a process $a\to bc$ ,if there exists a relative orbital angular momentum $L_{bc}$ between the particle $b$ and $c$ ,then the pure orbital angular momentum $L_{bc}$ state can be represented by the covariant tensor wave function $\widetilde{t}^{(L)}_{\mu_1\mu_2\cdots\mu_L}$ which is built of the relative momentum.Here,we list the amplitude for pure $S-,P-,D-,$ and $F-$ wave orbital angular momentum: $$\begin{aligned} \widetilde{t}^{(0)}&=&1,\\ \widetilde{t}^{(1)}_{\mu}&=&\widetilde{g}_{\mu\nu}(p_a)r^{\nu}B_1(Q_{abc})\equiv \widetilde{r}^{\mu}B_1(Q_{abc}),\\ \widetilde{t}^{(2)}_{\mu\nu}&=&[\widetilde{r}^{\mu}\widetilde{r}^{\nu}-\frac{1}{3}(\widetilde{r}\cdot\widetilde{r})\widetilde{g}_{\mu\nu}(p_a))]B_2(Q_{abc}),\end{aligned}$$ $$\begin{aligned} \widetilde{t}^{(3)}_{\mu\nu\lambda}&=[\widetilde{r}_{\mu}\widetilde{r}_{\nu}\widetilde{r}_{\lambda}-\frac{1}{5}(\widetilde{r}\cdot\widetilde{r})(\widetilde{g}_{\mu\nu}(p_a)\widetilde{r}_{\lambda}\nonumber \\ &\quad+\widetilde{g}_{\nu\lambda}(p_a)\widetilde{r}_{\mu}+\widetilde{g}_{\lambda\mu}(p_a)\widetilde{r}_{\nu})]B_3(Q_{abc}),\end{aligned}$$ where $r=p_b-p_c$ is the relative momentum of the two decay products in the parent particle rest frame,$\widetilde{r}\cdot\widetilde{r}=\vec{r}\cdot\vec{r}$ where $\vec{r}$ is the magnitude of three-vector , with $$\widetilde{g}_{\mu\nu}(p_a)=-g_{\mu\nu}+\frac{p_{a\mu}p_{a\nu}}{p_a^2}$$ which is the vector boson polarization sum relation, and $$Q^2_{abc}=\frac{(s_a+s_b-s_c)^2}{4s_a}-s_b$$ where $s_a=E_a^2-p_a^2$ and $B_l(Q_{abc})$ is the Blatt-Weisskopf barrier factor [@Hippel],explicitly, $$\begin{aligned} B_1(Q_{abc})=&\sqrt{\frac{2}{Q_{abc}^2}+Q_0^2}\\ B_2(Q_{abc})=&\sqrt{\frac{13}{Q_{abc}^4}+3Q_{abc}^2Q_0^2+9Q_0^4},\\ B_3(Q_{abc})=&\sqrt{\frac{277}{Q_{abc}^6}+6Q_{abc}^4Q_0^2+45Q_{abc}^2Q_0^4+225Q_0^6}\end{aligned}$$ Here $Q_0$ is a hadron scale parameter $Q_0=0.197321/R GeV/c$,in which $R$ is the radius of the centrifugal barrier in fm. If $a$ is an intermediate resonance decaying into $bc$,one needs to introduce into the amplitude a Breit-Wigner propergator [@primer] $$f^{(a)}_{(bc)}=\frac{1}{m_a^2-s_{bc}-im_a\Gamma_a}$$ In this equation,$s_{bc}=(p_b+p_c)^2$ is the invariant mass-squared of $b$ and $c$; $m_a,\Gamma_a$ are the resonance mass and width. Additionally,some expressions depend also on the total momentum of the $ij$ pair,$p_{(ij)}=p_i+p_j$.When one wants to combine two angular momenta $j_b$ and $j_c$ into a total angular momentum $j_a$,if $j_a+j_b+j_c$ is an odd number,then a combination $\epsilon_{\mu\nu\lambda\sigma}p^{\mu}_{a}$ with $p_a$ the momentum of the parent particle is needed,otherwise it is not needed. For a given hadronic decay process $A\rightarrow BC$ (B,C are fermions),in the $L-S$ scheme on hadronic level,the initial state is described by its $4-$ momentum $P_{\mu}$ and its spin state $S_A$,the final state is described by the relative orbital angular momentum state of $BC$ system and their spin state $(S_B,S_C)$.The spin states $(S_A,S_B,S_C)$ can be well represented by the relativistic Rarita-Schwinger spin wave functions for particles of arbitrary spin.As is well known that,spin-$\frac{1}{2}$ wavefunction is the standard Dirac spinor $U(p,s)$ and $V(p,s)$ ;spin-$1$ wavefunction is the standard spin-$1$ polarization four-vector $\epsilon^{\mu}(p,s)$ for particle with momentum $p$ and spin projection $s$.(1)For the case of A as a meson,B as $N^*$ with spin $n+\frac{1}{2}$ and C as $\overline{N}$ with spin $\frac{1}{2}$ ,the total spin of BC $(S_{BC})$ can be either $n$ or $n+1$. The two $S_{BC}$ states can be represented as [@N*NM; @couplings] $$\begin{aligned} \psi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\quad \bar{u}_{\mu_1\mu2\cdots\mu_n}(p_B,s_B)\gamma_5 v(p_C,s_C),\\ \Psi^{(n+1)}_{\mu_1\mu_2\cdots\mu_{n+1}}=&\quad \bar{u}_{\mu_1\mu2\cdots\mu_n}(\gamma_{\mu_{n+1}}-\frac{r_{\mu_{n+1}}}{m_A+m_B+m_C}v(p_c,s_C)) \nonumber\\&+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$ (2)For the case of A as $N^*$ with spin $n+\frac{1}{2}$,B as $N$ and C as a meson,one needs to couple $-S_A$ and $S_B$ first to get $S_{AB}=-S_A+S_B$ states,which are $$\begin{aligned} \phi^{(n)}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)u_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A),\\ \Phi^{n+1}_{\mu_1\mu_2\cdots\mu_n}=&\bar{u}(p_b,s_B)\gamma_5\widetilde{\gamma}_{\mu_1\mu_2\cdots\mu_n}(p_A,s_A)\nonumber\\ &+(\mu_1\leftrightarrow\mu_{n+1})+\cdots+(\mu_{n}\leftrightarrow\mu_{n+1})\end{aligned}$$ Up to now,we have introduced all knowledges for constructing the covariant tensor amplitude.In the concrete case,the P parity conservation may be applied,which expression is $$\label{parity} \eta_A=\eta_B\eta_C(-1)^{L}$$ where $\eta_A$,$\eta_B$ and $\eta_C$ are the intrinsic parities of particles A, B, and C, respectively.From this relation,L can be even or odd for one case,which guarantee a pure L final state,which is the soul of covariant $L-S$ coupling scheme. analysis for $\psi'\rightarrow \gamma \chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$ =========================================================================================== From now on,we denote $P$,$K^-$,$\overline{\Lambda}$ by number $1,2,3$.Firstly,for $\psi'\to \gamma \chi_{c0}$,from the helicity formalism,it is easy to show that there is only one independent amplitude for $\psi'$ radiative deca y to a spin $0$ meson.Hense,the amplitude is $$U^{\mu\nu}_{\gamma \chi_{c0}}=g^{\mu\nu}f^{(\chi_{c0})}.$$ For sequential $\chi_{c0}$ decay,there may be the following modes: $\chi_{c0}\to\Lambda_{x}\overline{\Lambda},\Lambda_x\rightarrow P K^-$,where $\Lambda_{x}$ can be\ $\Lambda(1520)\frac{3}{2}^-,\Lambda(1600)\frac{1}{2}^+, \Lambda(1670)\frac{1}{2}^-,\Lambda(1690)\frac{3}{2}^-,\Lambda(1800)\frac{1}{2}^-,\\ \Lambda(1810)\frac{1}{2}^+,\Lambda(1820)\frac{5}{2}^+,\Lambda(1830)\frac{5}{2}^-,\Lambda(1890)\frac{3}{2}^+, \Lambda(2100)\frac{7}{2}^-,\\ \Lambda(2110)\frac{5}{2}^+;$\ $\chi_{c0}\rightarrow\overline{N}\overline{\Lambda},\overline{N}\rightarrow \overline{\Lambda}K^-$,where $\overline{N}$ is the anti-partner of hyperon $N$,with that $N$ can be\ $N(1650)\frac{1}{2}^-$,$N(1675)\frac{5}{2}^-$,$N(1700)\frac{3}{2}^-$,$N(1710)\frac{1}{2}^+$ or $N(1720)\frac{3}{2}^+$.\ Another possibility that $P \overline{\Lambda}$ may be generated from an unknown intermediate resonance $K_x$ is also taken into account.Amplitudes for up to $K_x's$ spin-4 are given. For $\Lambda_x$ being $\Lambda(1520)\frac{3}{2}^-$,the total spin of $\Lambda(1520)$ and $\overline{\Lambda}\frac{1}{2}^-$ can be $1$ or $2$,corresponding to the $P$ wave and $D$ wave respectively,because of the special property of spin-$0$ of $\chi_{c0}$.The parity relation makes $P$ wave impossible.Considering that this channel is recognized as a meson decaying to two fermions,now one can write the covariant amplitude as $$\Phi_{(\mu\nu)}^{(2)}\widetilde{t}^{(2)\mu\nu}.$$ And then considering $\Lambda(1520)\rightarrow P K^-$,the total spin of particle $1$ and $2$ can only be $\frac{1}{2}$,corresponding to the $P$ wave and $D$ wave,after the parity formula being applied,$L$ must be $2$.As it belongs to a fermion decay to a fermion and a meson,the covariant amplitude can be expressed as $\Phi^{(2)_{\mu\nu}}\widetilde{t}^{(2)\mu\nu}$. Here,we list all the amplitudes for the whole decay chain $\psi'\to\gamma\chi_{c0},\chi_{c0}\rightarrow\Lambda_x\overline{\Lambda},\Lambda_x\rightarrow P K^-$ up to spin-$\frac{7}{2}$ for $\Lambda_x$ according to the above principles. $$\begin{aligned} \Lambda_x(\frac{1}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\ \Lambda_x(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{\chi_{c0}}_{(123)}\psi^{(0)}\phi^{(0)} \\\Lambda_x(\frac{3}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\Lambda_x(\frac{3}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\Lambda_x(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\Lambda_x(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\Lambda_x(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\Lambda_x(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta}\end{aligned}$$note that $\widetilde{t}^{(0)}=1$.For channel $\chi_{c0}\rightarrow \overline{N_x}P,\overline{N_x}\rightarrow K^-\overline{\Lambda}$,we can imitate the amplitude up to spin-$\frac{7}{2}$ for $N_x$ without any difficulty,even through the highest spin for $N_x$ decaying into $K^-\Lambda$ can only be $\frac{5}{2}$ presently [@pdg2008]. $$\begin{aligned} \overline{N_x}(\frac{1}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\phi^{(0)} \\\overline{N_x}(\frac{1}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\Phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\overline{N_x}(\frac{3}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\Phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\overline{N_x}(\frac{3}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(1)}_{\lambda}\widetilde{t}^{(1)\lambda}\phi^{(1)}_{\sigma}\widetilde{t}^{(1)\sigma} \\\overline{N_x}(\frac{5}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(2)}_{\lambda\delta}\widetilde{t}^{(2)\lambda\delta}\phi^{(2)}_{\rho\sigma}\widetilde{t}^{(2)\rho\sigma} \\\overline{N_x}(\frac{5}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\Phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta} \\\overline{N_x}(\frac{7}{2}^+)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\Psi^{(4)}_{\lambda\delta\beta\xi}\widetilde{t}^{(4)\lambda\delta\beta\xi}\Phi^{(4)}_{\rho\sigma\eta\zeta}\widetilde{t}^{(4)\rho\sigma\eta\zeta} \\\overline{N_x}(\frac{7}{2}^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(3)}_{\lambda\delta\beta}\widetilde{t}^{(3)\lambda\delta\beta}\phi^{(3)}_{\rho\sigma\eta}\widetilde{t}^{(3)\rho\sigma\eta}\end{aligned}$$ For channel $\chi_{c0}\rightarrow K^+_x K^-,K^+_x\rightarrow P \overline{\Lambda}$,the amplitudes are also given.$J^{P}=0^+,1^-,2^+,3^-,4^+ \cdots $ are forbidden by the parity relation[@Parity].The partial wave amplitude is denoted by $U^{\mu\nu}_{(LS)}$, $L,S$ means the orbital angular momentum number and spin angular momentum number between $P$ and $\overline{\Lambda}$. $$\begin{aligned} K_x^+(0^-)\qquad U^{\mu\nu}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)} \\K_x^+(1^+)\qquad U^{\mu\nu}_{(10)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\psi^{(0)}\widetilde{T}^{(1)\sigma}\phi_{\sigma}\epsilon_{\lambda}\widetilde{t}^{(1)\lambda} \\U^{\mu\nu}_{(11)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\epsilon_{\sigma}\widetilde{t}^{(1)}_{\eta}\Psi^{(1)}_{\zeta}\widetilde{T}^{(1)\sigma}\phi_{\sigma} \\K_x^+(2^-)\qquad U^{\mu\nu}_{(20)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(2)\eta\zeta}\phi_{\eta\zeta}\widetilde{t}^{(2)\rho\sigma}\psi^{(0)}\phi_{\rho\sigma}\nonumber\\ &=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\rho\sigma\eta\zeta}\psi^{(0)}\widetilde{T}^{(2)\eta\zeta}\widetilde{t}^{(2)\rho\sigma} \\U^{\mu\nu}_{(21)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\phi_{\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\phi_{\iota\eta}\Psi^{(1)}_{\zeta} \nonumber\\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(2)}_{\beta\lambda\iota\eta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{T}^{(2)\beta\lambda}\widetilde{t}^{(2)\iota}_{\sigma}\Psi^{(1)}_{\zeta} \\K_x^+(3^+)\qquad U^{\mu\nu}_{(30)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta}\phi_{\rho\sigma\eta}\nonumber \\&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\rho\sigma\eta}\widetilde{T}^{(3)\lambda\delta\beta}\psi^{(0)}\widetilde{t}^{(3)\rho\sigma\eta} \\U^{\mu\nu}_{(31)}&=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}\widetilde{T}^{(3)\lambda\delta\beta}\phi_{\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\cdot\nonumber \\ &\quad\phi^{(3)}_{\zeta\kappa\xi}\nonumber\\ &=g^{\mu\nu}f^{(\chi_{c0})}_{(123)}P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}\widetilde{T}^{(3)\lambda\delta\beta}\epsilon^{\rho\sigma\eta\zeta}p_{K_x\rho}\widetilde{t}^{(3)\kappa\xi}_{\sigma}\Psi^{(1)}_{\eta}\end{aligned}$$ In the above formulas,$\phi$ implies the spin wave functions for $K_x$,for example,$\phi_{\sigma}$ corresponding to the spin-1,$\phi_{\rho\sigma}$ corresponding to spin-2.There is a general wave fuction for a particle of spin $J$ [@Chung1],which is a rank-$J$ tensor $$\begin{aligned} \phi^{\alpha_1\alpha_2\cdots\alpha_J}(m)&=\underset{m_1m_2\cdots}{\sum}\langle 1m_11m_2|2n_1\rangle\langle 2n_11m_3|3n_2\rangle\nonumber\\ &\quad\cdots\langle J-1n_{J-2}1m_J|Jm\rangle\phi^{\alpha_1}(m_1)\nonumber\\ &\quad\phi^{\alpha_2}(m_2)\cdots\phi^{\alpha_J}(m_J)\end{aligned}$$ where $\phi^{\alpha}(m)$ is the familiar polarization four-vector of spin-1 particle, $$\phi^{\alpha}(1,-1)=\mp\frac{1}{\sqrt{2}}(0;1,\pm i,0), \phi^{\alpha}(0)=(0;0,0,1).$$ note the following useful relationship: $$\phi(-m)=(-)^m\phi^*(m).$$ It is best to illustrate these formulas with some examples.For $J=1$,one finds that it reduces to identities for $\phi(1)$ and $\phi(0)$.For $J=2$,one has $$\begin{gathered} \qquad\qquad\phi^{\alpha\beta}(+2)=\phi^{\alpha}(1)\phi^{\beta}(1) \\ \phi^{\alpha\beta}(+1)=\frac{1}{\sqrt{2}}[\phi^{\alpha}(1)\phi^{\beta}(0)+\phi^{\alpha}(0)\phi^{\beta}(1)]\\ \phi^{\alpha\beta}(0)=\frac{1}{\sqrt{6}}[\phi^{\alpha}(1)\phi^{\beta}(-1)+\phi^{\alpha}(-1)\phi^{\beta}(1)+\sqrt{\frac{2}{3}}\phi^{\alpha}(0)\phi^{\beta}(0)]\end{gathered}$$ $\phi\cdot\phi$ is in reality its spin projection operator $P^{(S)}$. $$\begin{aligned} P^{(2)}_{\rho\sigma\eta\zeta}(p_{K_x})&=\underset{m}{\sum}\phi_{\rho\sigma}(p_{K_x},m)\phi^*_{\eta\zeta}(p_{K_x,m})\nonumber\\&=\frac{1}{2}(\widetilde{g}_{\rho\eta} \widetilde{g}_{\sigma\zeta}+\widetilde{g}_{\rho\zeta}\widetilde{g}_{\sigma\eta})-\frac{1}{3}\widetilde{g}_{\rho\sigma}\widetilde{g}_{\eta\zeta} \\\quad P^{(3)}_{\lambda\delta\beta\zeta\kappa\xi}(p_{K_x})&=\underset{m}{\sum}\phi_{\lambda\delta\beta}(p_{K_x},m)\phi^*{\zeta\kappa\xi}(p_{K_x},m)\nonumber\\&=\frac{1}{6}(\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\kappa}\widetilde{g}_{\beta\xi} +\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\kappa}+\widetilde{g}_{\lambda\kappa}\widetilde{g}_{\delta\zeta}\widetilde{g}_{\beta\xi} \nonumber\\&\quad\quad+\widetilde{g}_{\lambda\zeta}\widetilde{g}_{\delta\xi}\widetilde{g}_ {\beta\kappa}+\widetilde{g}_{\delta\xi}\widetilde{g}_{\beta\zeta}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\kappa}\widetilde{g}_{\lambda\xi}\widetilde{g}_{\beta\zeta}) \nonumber\\&\quad-\frac{1}{15}(\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\beta\xi}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\beta\zeta}+\widetilde{g}_{\lambda\delta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\beta\kappa} \nonumber\\&\quad\quad+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\xi}\widetilde{g}_{\delta\kappa}+\widetilde{g}_{\lambda\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\delta\xi} +\widetilde{g}_{\lambda\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\delta\zeta} \nonumber\\&\quad\quad+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\zeta\kappa}\widetilde{g}_{\lambda\xi}+\widetilde{g}_{\delta\beta}\widetilde{g}_{\kappa\xi}\widetilde{g}_{\lambda\kappa})\end{aligned}$$ So far,we have given the covariant tensor amplitude formula for the process $\psi'\to\gamma\chi_{c0}\rightarrow \gamma P K^- \overline{\Lambda}$. helicity formula ================ For completeness,we also give the helicity format in comparison with tensor formula.Helicity formalism has an explicit advantage, the angular dependence can be easily seen.In this section ,we will give the amplitude for $\chi_{c0}\to\Lambda(1520)\overline{\Lambda},\Lambda(1520)\to P K^-,\overline{\Lambda}\to\overline{P}\pi^+$.$\Lambda(1520)$ the most possible resononse,has been mentioned above. Firstly,we want to introduce the general helicity formula expression.Consider a state with spin(parity) $=J(\eta_J)$ decaying into two states with $S(\eta_s)$ and $\sigma(\eta_\sigma)$.The decay amplitudes are given,in the rest frame of $J$ [@helicity; @guide] [@Jacob], $$\mathcal{M}^{J\to s\sigma}_{\lambda\nu}=\sqrt{\frac{2J+1}{4\pi}}D^{J*}_{M\delta}(\phi,\theta,0)H^J_{\lambda\nu},$$ where $\lambda$ and $\nu$ are the helicities of the two final state particles $s$ and $\sigma$ with $\delta=\lambda-\nu$.The symbol $M$ stands for the $z$ component of the spin $J$ in a coordinate system fixed by production process.The helicities $\lambda$ and $\nu$ are rotational invariants by definition.The direction of the break-up momentum of the decaying particle $s$ is given by the angles $\theta$ and $\phi$ in the $J$ rest frame.Let $\hat{x},\hat{y}$ and $\hat{z}$ be the coordinate system fixed in the $J$ rest frame.It is important to recognize,for applications to sequential decays,the exact nature of the body-fixed (helicity) coordinate system implied by the arguments of the $D$ function given above.Let $\hat{x}_h,\hat{y}_h$ and $\hat{z}_h$ be the helicity coordinate system fixed by the $s$ decay.Then by definition $\hat{z}_h$ describes the direction of $s$ in the $J$ rest frame (also termed the helicity axis) and the $y$ axis is given by $\hat{y}_h=\hat{z}\times\hat{z}_h$ and $\hat{x}_h=\hat{y}_h\times\hat{z}_h$.Parity conservation in the decay leads to the relationship $$H^{J}_{\lambda\nu}=\eta_J\eta_s\eta_\sigma(-)^{J-s-\sigma}H^{J}_{-\lambda-\nu}$$ Let us consider a full process $A\to B+C$ where $B$ and $C$ are also unstable particles decaying to $B_1+B_2$ and $C_1+C_2$ respectively.The decay amplitude is simply [@PhD; @thesis] $$\begin{aligned} \mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})=\underset{\lambda_B,\lambda_C}{\sum}& \mathcal{M}^{A\to B+C}_{\lambda_B,\lambda_C}\cdot \mathcal{M}^{B\to B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}\cdot\nonumber \\&\mathcal{M}^{C\to C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}\end{aligned}$$ with $$\begin{aligned} \mathcal{M}^{A\to B+C}_{\lambda_B,\lambda_C}&=\sqrt{\frac{2J_A+1}{4\pi}}D^{J_A*}_{M_A,\lambda_B-\lambda_C}(\phi_A,\theta_A,0)H^A_{\lambda_B,\lambda_C}, \\ \mathcal{M}^{B\to B_1+B_2}_{\lambda_{B_1},\lambda_{B_2}}&=\sqrt{\frac{2J_B+1}{4\pi}}D^{J_B*}_{\lambda_B,\lambda_{B_1}-\lambda_{B_2}}(\phi_B,\theta_B,0)H^B_{\lambda_{B_1},\lambda_{B_2}}, \\ \mathcal{M}^{C\to C_1+C_2}_{\lambda_{C_1},\lambda_{C_2}}&=\sqrt{\frac{2J_C+1}{4\pi}}D^{J_C*}_{-\lambda_C,\lambda_{C_1}-\lambda_{C_2}}(\phi_C,\theta_C,0)H^C_{\lambda_{C_1},\lambda_{C_2}}, \label{sub3}\end{aligned}$$ Please note in the first subscript of $D^{J_C*}$ is $-\lambda_C$ and NOT $\Lambda_C$ although it also gives the correct result,because the quantization axis is along the direction of the momentum of particle $B$ so that the spin-quantization projection $M_C$ in the particle $C$ rest frame verifies $M_C=-\lambda_C$. The unpolarized angular distribution is then given by averaging over initial spins and by summing over final spins: $$\begin{aligned} \frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B d\Omega_C}=\frac{1}{2S_A+1}&\underset{\lambda_{B_1},\lambda_{C_1},\lambda_{B_2},\lambda_{C_2}}{\sum}\nonumber \\&|\mathcal{M}(\lambda_{B_1},\lambda_{B_2},\lambda_{C_1},\lambda_{C_2})|^2.\end{aligned}$$ where $\mathcal{N}$ is the normalization factor. Using the parity conservation formula,one has $H^{\overline{\Lambda}}_{\frac{1}{2}0}=H^{\overline{\Lambda}}_{-\frac{1}{2}0}$ and $H^{\Lambda(1520)}_{\frac{1}{2}0}=-H^{\Lambda(1520)}_{-\frac{1}{2}0}$.By applying the above amplitude expression,after a lengthy,tedious but trivial evaluating,one can get the helicity amplitude, $$\begin{aligned} \label{helicity distribution} \frac{d^3\Gamma}{\mathcal{N}d\Omega_A d\omega_B d\Omega_C}=&[\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}\nonumber \\&+\frac{9}{2}\cos^2\theta_{\Lambda(1520)}\sin\theta_{\Lambda(1520)}\cos\phi_{\Lambda(1520)}\nonumber \\&+\frac{\sqrt{3}}{2}\cos^2\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber \\&-\frac{\sqrt{3}}{4}\cos\theta_{\Lambda(1520)}\cos2\phi_{\Lambda(1520)}\nonumber \\&-\frac{3\sqrt{3}}{4}\cos2\phi_{\Lambda(1520)}+1]|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2\end{aligned}$$ where the subscript $\Lambda(1520)$ denotes that the angle defined in the rest frame of $\Lambda(1520)$. After integrating $\phi_{\Lambda(1520)}$’s from $[0,2\pi]$,the Eq. becomes $$\frac{d^3\Gamma}{\mathcal{N'}d\Omega_A d\omega_B d\Omega_C}=\frac{3}{2}\cos^2\theta_{\Lambda(1520)}-\frac{3}{2}\cos\theta_{\Lambda(1520)}+1$$ where $\mathcal{N'}=\mathcal{N}|H^{\overline{\Lambda}}_{\frac{1}{2}0}|^2H^{\Lambda(1520)}_{\frac{1}{2}0}|^2$ is redefined normalization factor. conclusion ========== In this short note,firstly,the relevant general tensor formalism hac been introduced,and then give the covariant tensor amplitudes.At last,for completeness and some experimental reasons,the helicity amplitude expression has also been provided and a figure attached. acknowledgments =============== The author acknowledges greatly helpful discussions with B. S. Zou.This work is supported in part by the National Natural Science Foundation of China under contracts Nos. 10521003,10821063, the 100 Talents program of CAS, and the Knowledge Innovation Project of CAS under contract Nos. U-612 and U-530 (IHEP). 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ArXiv

--- abstract: | As the number of possible predictors generated by high-throughput experiments continues to increase, methods are needed to quickly screen out unimportant covariates. Model-based screening methods have been proposed and theoretically justified, but only for a few specific models. Model-free screening methods have also recently been studied, but can have lower power to detect important covariates. In this paper we propose EEScreen, a screening procedure that can be used with any model that can be fit using estimating equations, and provide unified results on its finite-sample screening performance. EEScreen thus generalizes many recently proposed model-based and model-free screening procedures. We also propose iEEScreen, an iterative version of EEScreen, and show that it is closely related to a recently studied boosting method for estimating equations. We show via simulations for two different estimating equations that EEScreen and iEEScreen are useful and flexible screening procedures, and demonstrate our methods on data from a multiple myeloma study. Keywords: Estimating equations; Ultra-high-dimensional data; Sure independence screening; Variable selection author: - | Sihai Dave Zhao, Department of Biostatistics, Harvard School of Public Health\ Yi Li, Department of Biostatistics, University of Michigan bibliography: - 'refs.bib' title: 'Sure screening for estimating equations in ultra-high dimensions' --- Introduction {#sec:intro} ============ Modern high-throughput experiments are producing high-dimensional datasets with extremely large numbers of covariates. Traditional regression modeling strategies work poorly in such situations, leading to recent interest in regularized regression methods such as the lasso [@Tibshirani1996], the Dantzig selector [@CandesTao2007], and SCAD [@FanLi2001]. These procedures can perform well in estimation and prediction even when the number of covariates $p_n$ is larger than the sample size $n$, where here we are allowing $p_n$ to grow with $n$. However, when $p_n$ is extremely large compared to $n$, these methods can become inaccurate and computationally infeasible [@FanLv2008]. Thus there is a need for methods to quickly screen out unimportant covariates before using regularization methods. A number of screening strategies have so far been proposed, and choosing which one to use depends on what model we believe is most suitable for the data. Under the ordinary linear model, @FanLv2008 proposed a procedure with the sure screening property, where the covariates retained after screening will contain the truly important covariates with probability approaching one, even in the ultra-high-dimensional realm where $p_n$ grows exponentially with $n$. @FanSong2010 and @ZhaoLi2012 subsequently proposed procedures that maintain this property for generalized linear models and the Cox model, respectively. Screening methods have also been proposed for nonparametric additive models [@FanFengSong2011], linear transformation models [@Li2011], and single-index hazard models [@GorstRasmussenScheike2011]. In a recent development, @Zhu2011 proposed a screening method valid for any single-index model, a class so large that their screening procedure is nearly model-free. They used a new measure of dependence which can detect a wide variety of functional relationships between the covariates and the outcome, and proved that their method has the sure screening property for any single-index model. They also showed in simulations that it could significantly outperform model-based screening methods when the models were incorrectly specified. On the other hand, model-based screening can have greater power to detect important covariates, a consequence of the bias-variance tradeoff. However, there are often situations where we wish to use some model other than the ones mentioned above. For example, studies involving clustered observations, missing data, or censored outcomes are frequently encountered in genomic medicine, and are often analyzed with more complicated regression models for which no screening methods have yet been developed. In theory it is not difficult to propose a screening procedure for any given model: fit $p_n$ marginal regressions, one for each covariate, and retain those covariates with the largest marginal estimates, in absolute value. But fitting $p_n$ marginal regressions can still be time-consuming, especially if $p_n$ is very large and the fitting procedure is slow, and theoretical properties such as sure screening must still be studied on a case-by-case basis. In this paper we propose EEScreen, a unified approach to screening which can be used with any statistical model that can be fit using estimating equations. This is convenient because estimating equations are frequently used to analyze the previously mentioned correlated, missing, or censored data situations. EEScreen is also fundamentally different from most other screening procedures in that it only requires evaluating $p_n$ estimating equations at a fixed parameter value, rather than solving for $p_n$ marginal regressione estimates, making it exceedingly computationally convenient. We prove theoretical results about the screening properties of EEScreen that hold for any model that can be fit using U-statistic-based estimating equations. Furthermore, because we can design estimating equations to incorporate more or fewer modeling assumptions, we can use our EEScreen framework to span the range between model-based and model-free screening. In particular, we show that EEScreen can provide a screening method very similar to that of @Zhu2011 when used with a particular estimating equation. This estimating equation actually cannot be used for estimation in practice because it involves unknown parameters, but interestingly can still be used to derive a useful screening procedure. Finally, when covariates are highly correlated, @FanLv2008 suggested an iterative version of their screening procedure, which they found to outperform marginal screening in some cases. In this paper we provide an iterative version of EEScreen (iEEScreen), and we also demonstrate a novel connection between iEEScreen and EEBoost, a recently proposed boosting algorithm for estimation and variable selection in estimating equations [@Wolfson2011]. This connection may provide a means for a theoretical analysis of iterative screening methods, something which so far has been difficult to study. We introduce EEScreen in Section \[sec:eescreen\], where we also give some examples, establish its theoretical properties, and briefly discuss how to choose the number of covariates to retain after screening. We derive a new screening method similar to that of @Zhu2011 in Section \[sec:zhu2011\], and discuss iEEScreen in Section \[sec:ieescreen\]. We conduct a thorough simulation study in Section \[sec:sims\], using two different estimating equations, before applying our methods to analyze an issue in multiple myeloma in Section \[sec:data\]. We conclude with a discussion in Section \[sec:discussion\], and provide proofs in the Appendix. EEScreen: sure screening for estimating equations {#sec:eescreen} ================================================= Method {#sec:method} ------ Let $Y_i=(Y_{i1},\ldots,Y_{iK_i})^T$ be a $K_i\times1$ outcome vector and $\bX_i=(\bX_{i1},\ldots,\bX_{iK_i})^T$ be a $K_i\times p_n$ matrix of covariates for units $i=1,\ldots,n$. Then let $\bY=(\bY_1,\ldots,\bY_n)^T$ be a $\sum_iK_i\times1$ vector and $\bX=(\bX_1^T,\ldots,\bX_n^T)^T$ be a $\sum_iK_i\times p_n$ matrix. Assuming some regression model, we can construct a $p_n\times1$ estimating equation $\bU(\bbeta)$ that depends on $\bY_i$ and $\bX_i$ such that $\E\{\bU(\bbeta_0)\}=\b0$, where $\bbeta_0$ is the true $p_n\times1$ parameter vector. Let the set of true regression parameters $\mathcal{M}=\{j:\beta_{0j}\ne0\}$ have size $\vert\mathcal{M}\vert=s_n$, where $\beta_{0j}$ is the $j^{th}$ component of $\bbeta_0$. It is commonly assumed that $s_n$ is small and fixed or growing slowly. When $p_n<n$, $\bbeta_0$ is estimated by finding the $\hat{\bbeta}$ such that $\bU(\hat{\bbeta})=0$, but when $p_n>n$ there are an infinite number of solutions for $\hat{\bbeta}$, in which case regularized regression is used [@Fu2003; @JohnsonLinZeng2008; @Wolfson2011]. However, when $p_n$ is much greater than $n$, these methods can lose accuracy and be too computationally demanding, hence the need for screening methods to quickly reduce $p_n$. Most previously proposed screening methods proceed by fitting $p_n$ regression models, one covariate at a time, to get $p_n$ marginal estimates $\hat{\alpha}_j$. They then retain the covariates with $\vert\hat{\alpha}_j\vert$ above some threshold. This is akin to conducting $p_n$ Wald tests, though without standardizing the $\hat{\alpha}_j$ by their variances. However, in the case of estimating equations, even this procedure can be time-consuming if $p_n$ is large or $\bU$ is cumbersome to fit. Here, instead of marginal Wald tests, we construct marginal score tests for the $\beta_{0j}$ using $\bU$. To motivate our procedure, we first consider the case where the marginal model is correct for $\beta_{01}$. In other words, $\beta_{01}\ne0$ while $\beta_{0j}=0$ for all $j\ne1$. Then $\E[\bU\{(\beta_{01},0,\ldots,0)\}]=\b0$, so that each component of $\bU$ is a valid estimating equation for $\beta_{01}$. This implies that each component of $\bU(\b0)$ is the numerator of a score test for the null hypothesis $\beta_{01}=0$. If the marginal model is correct for $\beta_{01}$, then to achieve sure screening we must reject the score test. Therefore we use as our screening statistic the component of $\bU(\b0)$ that gives the most powerful test, which we denote $U_1(\b0)$. For each $j$, we can identify the component $U_j(\b0)$ of $\bU(\b0)$ that is most powerful for testing $\beta_{0j}=0$ under the marginal model that $\beta_{0j}$ is the only non-zero parameter. In many situations the first component of $\bU(\b0)$ will be associated with $\beta_{01}$, the second with $\beta_{02}$, and so on. When this is not the case, we can follow the construction above to relabel the components of $\bU(\b0)$ appropriately. We propose using the relabeled $U_j(\b0)$ as surrogate measures of association between the outcome and the $j^{th}$ covariate, after first standardizing the covariates to have equal variances. Instead of just taking the numerators of the score tests we could divide each $U_j(\b0)$ by an estimate of its standard deviation, but this would add computational complexity to our procedure, and even without doing so we will be able to achieve good results and prove finite-sample performance guarantees. One advantage to using score tests is that they do not require parameter estimation and so are more computationally convenient than performing $p_n$ marginal regressions. Furthermore, this framework will also allow us to give a unified treatment of the theoretical results for a large class of estimating equations. Specifically, we propose the following screening procedure: 1. Standardize the $p_n$ covariates to have variance 1. 2. For the $j^{th}$ parameter identify the marginal estimating equations $U_j$ as described above. 3. Set a threshold $\gamma_n$. 4. Retain the parameters $\{j:\vert U_j(\b0)\vert\geq\gamma_n\}$. We will denote the set of retained parameters by $\hat{\mathcal{M}}$. Note that this procedure only requires evaluating $p_n$ estimating equations at $\b0$, which can be computed very quickly. The convenience of score tests, however, comes at the price of ambiguity in the proper treatment of nuisance parameters, such as the intercept term in a regression model. Without loss of generality, let $\beta_{01}$ be the intercept term. We can first fit the intercept without any covariates in the model to get an estimate $\hat{\beta}_{01}$. This only needs to be done once, since $\hat{\beta}_{01}$ will remain the same for each $U_j$. We then screen by evaluating each $U_j$ at $\boldeta=(\hat{\beta}_{01},\b0)$ instead of at $\b0$. Our score test idea was motivated by the EEBoost algorithm [@Wolfson2011], a boosting procedure for estimating equations which uses components of the estimating equation $\bU$ as a surrogate measure of association. We therefore refer to our method as EEScreen, and we will draw more connections between EEScreen and EEBoost in Section \[sec:ieescreen\]. Examples {#sec:examples} -------- Here we provide some examples of EEScreen for various estimating equations, assuming throughout that $\E(\bX_i)=\b0$ and $\var(X_{ij})=1$. For the linear model with $K_i=1$, the usual linear regression score equation is $\bU(\bbeta)=\bX^T(\bY-\bX\bbeta)$, so $\bU(\b0)=\bX^T\bY$. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, the $j^{th}$ component of $\E\{\bU(\b0)\}$ equals $\cor(X_{ij},X_{ij'})\beta_{0j'}$, where $X_{ij}$ is the $j^{th}$ component of the $i^{th}$ covariate vector. Clearly this is maximized when $j=j'$ for any value of $\beta_{0j'}$, so the component of $\bU(\b0)$ that gives the most powerful test is $U_{j'}(\b0)$. EEScreen then retains the parameters $\{j:\vert\sum_iX_{ij}Y_i\vert\geq\gamma_n\}$. Note that this is equivalent to the original screening procedure proposed by @FanLv2008. Under the Cox model, when $K_i=1$ with survival outcomes, let $T_i$ be the survival time, $C_i$ the censoring time, $Y_i=\min(T_i,C_i)$, and $\delta_i=I(T_i\leq C_i)$. The Cox model score equation is $$\bU(\bbeta)=\sum^n_{i=1}\int\left\{\bX_i-\frac{\sum^n_{i=1}\bX_i\tilde{Y}_i(x)\exp(\bX_i^T\bbeta)}{\sum^n_{i=1}\tilde{Y}_i(x)\exp(\bX_i^T\bbeta)}\right\}d\tilde{N}_i(x),$$ where $\tilde{N}_i(x)=I(T_i\leq x,\delta_i)$ is the observed failure process and $\tilde{Y}_i(x)=I(Y_i\geq x)$ is the at-risk process. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, @GorstRasmussenScheike2011 show that the largest component of the limiting estimating equation evaluated at $\b0$ is found for the $j$ that maximizes $\int\cor\{X_{ij},F(t\mid X_{ij'})\}$, where $F(t\mid X_{ij'})$ is the distribution function of $T_i$, conditional on $X_{ij'}$. Thus the component of $\bU(\b0)$ that gives the most powerful test is again the $j'^{th}$ component. EEScreen then retains the parameters $$\left[j:\left\vert\sum^n_{i=1}\int\left\{X_{ij}-\frac{\sum^n_{i=1}X_{ij}\tilde{Y}_i(x)}{\sum^n_{i=1}\tilde{Y}_i(x)}\right\}d\tilde{N}_i(x)\right\vert\geq\gamma_n\right].$$ This is exactly the screening statistic of @GorstRasmussenScheike2011. This example illustrates the computational advantages that EEScreen can enjoy. @ZhaoLi2012 proposed screening for the Cox model based on fitting marginal Cox regressions, which requires $p_n$ applications of the Newton-Raphson algorithm. In contrast, @GorstRasmussenScheike2011 and EEScreen only require evaluating the $U_j(\b0)$. The ordinary linear model and the Cox model have already been studied in the screening literature, but EEScreen is most useful for models for which no screening procedures exist yet. In Sections \[sec:sims\] we study its performance on two such models: a $t$-year survival model [@Jung1996] and the accelerated failure time model [@Tsiatis1996; @Jin2003]. Theoretical properties {#sec:theory} ---------------------- One advantage of our EEScreen framework is that we can provide very general theoretical guarantees on its screening performance that hold for a large class of models, without needing to study each model on a case-by-case basis. We require three assumptions on the marginal estimating equations $U_j$ to prove that EEScreen has the sure screening property, where the probability that the retained parameters $\hat{\mathcal{M}}$ contains the true parameters $\mathcal{M}$ approaches 1. Let the expected full estimating equations be denoted $\bu(\bbeta)=\E\{\bU(\bbeta)\}$, so that the expected marginal estimating equations are $u_j(\bbeta)$. \[ass:ustat\] Let $\bX_{ij}$ be the $K_i\times1$ vector of the $j^{th}$ covariate for the $i^{th}$ unit. Each estimating equation $U_j$ has the form $$U_j(\bbeta)=\binom{n}{m}^{-1}\sum_{1\leq i_1<\ldots<i_m}h_j\{\bbeta;(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})\}$$ for all $j$, where $n\geq m$ and $h_j$ is a real-valued kernel function that depends on $\beta$ and is symmetric in the $(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})$. \[ass:bernstein\] There exist some constants $b>0$ and $\Sigma^2>0$ such that for all $j$, $\vert U_j(\b0)-u_j(\b0)\vert\leq b$ and $\var[h_j\{\b0;(\bY_{i_1},\bX_{i_1}),\ldots,(\bY_{i_m},\bX_{i_m})\}]\leq\Sigma^2$. Assumption \[ass:ustat\] requires that each $U_j$ be a U-statistic of order $m$, which encompasses a large number of important estimating equations. Assumption \[ass:bernstein\] amounts to conditions on the moments of the $U_j$, and they can often be satisfied by assuming bounded outcomes and covariates. These conditions are necessary for stating a Bernstein-type inequality for the $U_j$, which gives the probability bounds in Theorems \[thm:surescreening\] and \[thm:size\]. They can therefore be relaxed as long as there exists a similar probability inequality for $U_j$. For example, Bernstein-type inequalities exist for martingales [@vandeGeer1995], which would allow $U_j$ to be the Cox model score equations. \[ass:signal\] There exists some constant $c_1>0$ such that $\min_{j\in\mathcal{M}}\vert u_j(\b0)\vert\geq c_1[n/m]^{-\kappa}$ with $0<\kappa<1/2$, where $m$ is defined in Assumption \[ass:ustat\] and $[n/m]$ is the largest integer less than $n/m$. Assumption \[ass:signal\] is an assumption on the marginal signal strengths of the covariates in $\mathcal{M}$. In EEScreen these signals are quantified by the $u_j(\b0)$, and Assumption \[ass:signal\] requires them to be of at least a certain order so that they are detectable given a sample size $n$. An assumption of this type is always needed in a theoretical analysis of a screening procedure. For example, in the generalized linear model setting, our Assumption \[ass:signal\] is exactly equivalent to the assumption of @FanSong2010 that the magnitude of the covariance between $\E(\bY_i\mid\bX_i)$ and the $j^{th}$ covariate be of order $n^{-\kappa}$. Since EEScreen is similar to conducting $p_n$ score tests, Assumption \[ass:signal\] is similar to requiring that the expected value of the marginal score test statistic for $j\in\mathcal{M}$ be of a certain order. As previously mentioned, we could standardize the screening statistic $\vert u_j(\b0)\vert$ by its variance, in which case the score test analogy would be exact. It is very reasonable to use the marginal score test statistic as a proxy for the marginal association of the covariates. Under these assumptions, we can show that EEScreen possesses the sure screening property. \[thm:surescreening\] Under Assumptions \[ass:ustat\]–\[ass:signal\], if $\gamma_n=c_1[n/m]^{-\kappa}/2$ for $0<\kappa<1/2$, with $m$ defined in Assumption \[ass:ustat\], then $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-2s_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/4}{2\Sigma^2+bc_1[n/m]^{-\kappa}/3}\right\},$$ with $\Sigma^2$ and $b$ defined in Assumption \[ass:bernstein\]. Theorem \[thm:surescreening\] guarantees that all important covariates will be retained by EEScreen with high probability. Similar to previous work, we find that this probability bound depends only on $s_n$ and not on $p_n$. The bound also depends on $m$, the order of the U-statistic, so that EEScreen may not perform as well for larger $m$. Theorem \[thm:surescreening\] is almost an immediate consequence of properties of U-statistics, and the simplicity of the proof is due to the fact that EEScreen uses score tests instead of Wald tests. We therefore do not need to estimate any parameters, nor prove probability inequalities for those estimates, which is a major source of technical difficulty in previous work on screening. Theorem \[thm:surescreening\] is most useful if the size of the $\hat{\mathcal{M}}$ produced by EEScreen is small. In other words, we hope that $\hat{\mathcal{M}}$ does not contain too many false positives. With two more assumptions, we can provide a bound on $\vert\hat{\mathcal{M}}\vert$ that holds with high probability. \[ass:inf\] The expected full estimating equation $\bu(\bbeta)$ is differentiable with respect to $\bbeta$. Let the negative Jacobian $-\partial\bu/\partial\bbeta$ be denoted $\bi(\bbeta)$. \[ass:beta0\] There exists some constant $c_2>0$ such that $\Vert\bbeta_0\Vert_2\leq c_2$. Assumption \[ass:inf\] can hold even if the sample estimating equation $\bU$ is nondifferentiable. Assumption \[ass:beta0\] merely requires that there exist an upper bound on the size of the true $\bbeta_0$ that does not grow with $n$, which is a reasonable condition. \[thm:size\] Under Assumptions \[ass:ustat\]–\[ass:beta0\], if $\gamma_n=c_1[n/m]^{-\kappa}/2$ as in Theorem \[thm:surescreening\], then $$\pr\left[\vert\hat{\mathcal{M}}\vert\leq \frac{16c_2^2\sigma_{\max}^{*2}}{c_1^2[n/m]^{-2\kappa}}\right] \geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\},$$ where $\Sigma^2$ and $b$ are defined in Assumption \[ass:bernstein\] and $\sigma_{\max}^*=\sup_{0<t<1}\sigma_{\max}\{\bi(t\bbeta_0)\}$, where $\sigma_{\max}(\bA)$ denotes the largest singular value of the matrix $\bA$. Like Theorem \[thm:surescreening\], Theorem \[thm:size\] is also almost a simple consequence of properties of U-statistics. Theorem \[thm:size\] provides a finite-sample probability bound on $\vert\hat{\mathcal{M}}\vert$, but asymptotically we would need assumptions on $\bi(\bbeta^*)$ to guarantee that $\sigma_{\max}^*$ will not increase too quickly. In particular, if $\sigma_{\max}^*$ increased only polynomially in $n$, $\vert\hat{\mathcal{M}}\vert$ would increase polynomially. At the same time, the probability that the bound holds tends to one even if $\log p_n=o([n/m]^{1-2\kappa})$, so the false positive rate would decrease quickly to zero with probability approaching one even in ultra-high dimensions. A similar phenomenon was found by @FanFengSong2011. The presence of $\sigma_{\max}^*$ in Theorem \[thm:size\] reflects the dependence of $\vert\hat{\mathcal{M}}\vert$ on the degree of collinearity of our data. For general estimating equations, collinearity not only depends on the design matrix, but also varies across the parameter space. For example, @MackinnonPuterman1989 and @LesaffreMarx1993 showed that generalized linear models can be collinear even if their design matrices are not, and vice versa. In our situation, we are concerned with collinearity along the line segment between $\bbeta_0$ and $\b0$. Note that because $\sigma_{\max}^*$ depends only on $\bi$, $\bbeta_0$, and $\b0$, which are all nonrandom quantities, $\sigma_{\max}^*$ is nonrandom as well. Choosing $\gamma_n$ {#sec:gamma} ------------------- Theorems \[thm:surescreening\] and \[thm:size\] specify optimal rates for $\gamma_n$, and a number of methods have been proposed for choosing $\gamma_n$ in practice. @FanLv2008 suggested choosing $\gamma_n$ such that $\vert\hat{\mathcal{M}}\vert=n-1$ or $n/\log n$. Because these values are hard to interpret, @ZhaoLi2012 showed that $\gamma_n$ is related to the expected false positive rate of screening. @Zhu2011 also recently proposed a thresholding method based on adding artificial auxiliary variables, and provided a bound relating the number of added variables to the probability of including an unimportant covariate. These methods offer more interpretable ways of choosing how many covariates to retain with EEScreen. A related strategy is to set a desired false discovery rate. @BuneaWegkampAuguste2006 showed that FDR methods can achieve the sure screening property in the ordinary linear model, and @Sarkar2004 proposed an FDR method than can also control the false negative rate. It would be interesting to pursue this type of idea for EEScreen. In practice, however, we are often concerned with the prediction error of the estimator obtained by fitting a regularized regression method after EEScreen. If we used the methods above we would still need to choose a false positive rate or false discovery rate, but so far it is not clear what choices would give optimal prediction. In this case another option is to retain different numbers of covariates, fit the regularized regression for each screened model $\hat{\mathcal{M}}$, and select the $\hat{\mathcal{M}}$ that gives the lowest cross-validated estimate of prediction error. This is the approach we take in Section \[sec:data\], where we use EEScreen to analyze data from a multiple myeloma clinical trail. Model-free screening {#sec:zhu2011} ==================== @Zhu2011 recently proposed a screening statistic that can achieve sure screening for any single-index model. Specifically, for a completely observed response $\tilde{Y}_i$ and a $p$-dimensional covariate vector $\bX_i$, they assumed that $F(y\mid\bX_i)=F_0(y\mid\bX_i^T\bbeta_0)$, where $F(y\mid\bX_i)=\pr(\tilde{Y}_i<y\mid\bX_i)$ and $F_0$ is some distribution function that depends on $\bX_i$ only through the index $\bX_i^T\bbeta_0$, so that $j\in\mathcal{M}$ if and only if $\beta_{0j}\ne0$. This is a very mild assumption that holds for a large class of models, making the screening method of @Zhu2011 almost model-free. To simplify things, they assumed that $\E(\bX_i)=\b0$ and $\var(\bX_i)=\bI_{p_n}$, where $\bI_{p_n}$ is the $p_n\times p_n$ identity matrix. They quantified the marginal relationship between the covariates and an outcome $y$ by using the novel statistic $$\bOmega(y)=\E\{\bX_iF(y\mid\bX_i)\}=\cov\{\bX_i,F(y\mid\bX_i)\}=\cov\{\bX_i,I(\tilde{Y}_i<y)\}.$$ Intuitively, the covariance between $X_{ij}$ and $F(\tilde{Y}_i\mid\bX_i)$, where $X_{ij}$ is the $j^{th}$ component of $\bX_i$, should be large in magnitude if $j\in\mathcal{M}$. They therefore used $\omega_j=\E\{\Omega_j(\tilde{Y}_i)^2\}$ as a measure of marginal association, where $\Omega_j(y)$ is the $j^{th}$ component of $\bOmega(y)$, leading to the screening statistic $$\tilde{\omega}_j=n^{-1}\sum_{k=1}^n\left\{n^{-1}\sum_{i=1}^nX_{ij}I(\tilde{Y}_i<\tilde{Y}_k)\right\}^2.$$ This derivation of the screening procedure of @Zhu2011 makes no mention of estimation of $\bbeta_0$, making it seemingly irreconcilable with our EEScreen, which requires an estimating equation. However, we can actually show that EEScreen, combined with a particular estimating equation, leads to a very similar screening procedure. This further illustrates the flexibility and wide applicability of our proposed screening strategy. Note that conditional on $\bX_i$ and $\bX_k$, $F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)$ and $F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)$ are independent and identically distributed uniform random variables. Therefore, we know that $$\begin{aligned} &\pr\left\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)\right\}=\\ &\E\left[\pr\left\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta_0)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta_0)\mid\bX_i,\bX_k\right\}\right]= \frac{1}{2}.\end{aligned}$$ This fact can be used to construct the marginal estimating equations. Consider $$\label{eq:zhuesteq} \bU(\bbeta)=n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left[I\{F_0(\tilde{Y}_i\mid\bX_i^T\bbeta)<F_0(\tilde{Y}_k\mid\bX_k^T\bbeta)\}-\frac{1}{2}\right].$$ Since $\E\{\bU(\bbeta_0)\}=\b0$, (\[eq:zhuesteq\]) is an unbiased estimating equation for $\bbeta_0$. Furthermore, it is a U-statistic of order $m=2$, which is covered by the framework of Section \[sec:theory\]. It is important to note that (\[eq:zhuesteq\]) cannot be implemented in practice, because the functional form of $F_0(y\mid\bX^T\bbeta)$ is unknown, yet it is still useful for constructing a screening procedure. Recall that EEScreen uses the statistic $\bU(\b0)$, and for (\[eq:zhuesteq\]), $$\begin{aligned} \bU(\b0) &= n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left[I\{F_0(\tilde{Y}_i\mid\bX_i^T\b0)<F_0(\tilde{Y}_k\mid\bX_k^T\b0)\}-\frac{1}{2}\right]\\ &= n^{-2}\sum_{k=1}^n\sum_{i=1}^n\bX_i\left\{I(\tilde{Y}_i<\tilde{Y}_k)-\frac{1}{2}\right\},\end{aligned}$$ because $F_0(y\mid\bX_i^T\b0)=F_0(y\mid\bX_k^T\b0)=F_0(y\mid\b0)$, which is a monotonic function since $F_0$ is a distribution function. Under the marginal model that $\beta_{0j'}$ is the only non-zero parameter, the $j^{th}$ component of $\E\{\bU(\b0)\}$ is $\cor\{X_{ij},F(\tilde{Y}_i\mid X_{ij'})\}$. Thus the $j'^{th}$ component of $\bU(\b0)$ gives the most powerful score test, so EEScreen with (\[eq:zhuesteq\]) retains parameters $$\left[j:\left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}\left\{I(\tilde{Y}_i<\tilde{Y}_k)-\frac{1}{2}\right\}\right\vert\geq\gamma_n\right],$$ or equivalently, $$\left\{j:\left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}I(\tilde{Y}_i<\tilde{Y}_k)\right\vert\geq\gamma_n\right\},$$ because the $\bX_i$ are standardized to have mean $\b0$. In the notation of @Zhu2011, this is equivalent to using $\vert E\{\Omega_j(\tilde{Y}_i)\}\vert$ as the screening statistic for the $j^{th}$ covariate, rather than $\E\{\Omega_j(\tilde{Y}_i)^2\}$. The $\tilde{Y}_i$ may not be fully observed in the presence of censoring. If $C_i$ are the censoring times, let $Y_i=\min(\tilde{Y}_i,C_i)$ and $\delta_i=I(\tilde{Y}_i\leq C_i)$. Then if we assume that the $C_i$ are independent of the $\tilde{Y}_i$ and $\bX_i$, we can see that $$\begin{aligned} \E\left\{\frac{\delta_iI(Y_i<Y_k)}{S_C^2(Y_i)}\bigg\vert\bX_i,\bX_k\right\} &= \E\left[\E\left\{\frac{I(\tilde{Y}_i\leq C_i)I(\tilde{Y}_i\leq C_k)I(\tilde{Y}_i\leq\tilde{Y}_k)}{S_C^2(\tilde{Y}_i)}\bigg\vert\tilde{Y}_i,\bX_i,\bX_k\right\}\right]\\ &= \E\left[\E\left\{\frac{S_C^2(\tilde{Y}_k)I(\tilde{Y}_i\leq\tilde{Y}_k)}{S_C^2(\tilde{Y}_i)}\bigg\vert\tilde{Y}_i,\bX_i,\bX_k\right\}\right]\\ &= \E\{I(\tilde{Y}_k<\tilde{Y}_k)\mid\bX_i,\bX_k\},\end{aligned}$$ where $S_C$ is the survival function of the $C_i$. If the support of the $C_i$ is less than that of the $\tilde{Y}_i$, the $S_C(Y_i)$ term above could equal 0 for some $Y_i$. Thus this method of accommodating censoring could cause difficulty if it were used in the estimating equation (\[eq:zhuesteq\]) and could lead to inconsistent estimation of $\bbeta_0$ [@FineYingWei1998]. For simplicity, we will assume here that the support of $C_i$ is greater than or equal to that of $\tilde{Y}_i$. This then suggests that in the presence of censoring, the screening statistic of @Zhu2011 should become $$n^{-1}\sum_{k=1}^n\left\{n^{-1}\sum_{i=1}^nX_{ij}\frac{\delta_iI(Y_i<Y_k)}{\hat{S}_C^2(Y_i)}\right\}^2,$$ and the screening statistic derived using EEScreen should become $$\label{eq:zhueescreen} \left\vert n^{-2}\sum_{k=1}^n\sum_{i=1}^nX_{ij}\frac{\delta_iI(Y_i<Y_k)}{\hat{S}_C^2(Y_i)}\right\vert,$$ where $\hat{S}_C$ is the Kaplan-Meier estimate of $S_C$. This illustrates that the EEScreen framework is flexible enough to allow us to derive something similar to the approach of @Zhu2011, which was originally motivated by very different considerations. It also suggests that EEScreen can provide a sensible screening procedure for a particular model, such as the single-index model, even if the associated estimating equation (\[eq:zhuesteq\]) is not implementable in practice. iEEScreen {#sec:ieescreen} ========= Though the simplicity of EEScreen and related screening procedures is appealing, if the covariates are highly correlated, then in finite samples these univariate screening methods may not be able to achieve sure screening without incurring a large number of false positives. To address this issue, @FanLv2008 and @FanSamworthWu2009 proposed iterative screening, where the general idea is as follows. Below, $\mathcal{M}_l$ and $\mathcal{A}_l$ denote sets of covariate indices. In other words, $\mathcal{M}_l,\mathcal{A}_l\subseteq\{1,\ldots,p_n\}$. 1. Set $\mathcal{M}_0$ to be the empty set. 2. For $l=1:L$, 1. controlling for the variables in $\mathcal{M}_{l-1}$, screen the remaining covariates 2. select a set $\mathcal{A}_l$ of the most important of these covariates 3. use a multivariate variable selection method, such as lasso or SCAD, on the covariates in $\mathcal{M}_{l-1}\cup\mathcal{A}_l$ to get a reduced set $\mathcal{M}_l$ We can adapt these ideas to develop an iterative version of EEScreen, which we will call iEEScreen. However, to operationalize iEEScreen and iterative screening algorithms in general, we must first specify a number of parameters, such as how large $\vert\mathcal{A}_l\vert$ and $\vert\mathcal{M}_l\vert$ should be, what multivariate variable selection procedure to use, and how many iterations to run. @FanFengSong2011 recommended choosing the $\mathcal{A}_l$ using a permutation-based procedure, and the $\mathcal{M}_l$ using a SCAD-type variable selector [@FanLi2001] with cross-validation. Their iterations stop when either $\vert\mathcal{M}_l\vert>\vert\mathcal{A}_1\vert$, or $\mathcal{M}_l=\mathcal{M}_{l-1}$. These are sensible choices, but the many different layers of this procedure make it difficult to analyze. Instead, here we will show that the EEBoost method of @Wolfson2011, viewed as a variable selector rather than an estimation procedure, can actually be thought of as a version of iEEScreen. By linking iterative screening and boosting, we embed iEEScreen in the theoretical framework already developed for EEBoost and other boosting methods. In the future, this theoretical framework could in turn be applied to analyze the properties of iterative screening. We first briefly describe the EEBoost algorithm [@Wolfson2011]. For some small $\epsilon>0$ and the full estimating equation $\bU$, 1. Set $\bbeta^{(0)}=\b0$. 2. For $t=1:T$, 1. compute $\bDelta=\vert\bU(\bbeta^{(t-1)})\vert$ 2. identify $j_t=\argmax_j\Delta_j$, where $\Delta_j$ is the $j^{th}$ component of $\bDelta$ 3. set $\beta^{(t)}_{j_t}=\beta^{(t-1)}_{j_t}-\epsilon\cdot\sign(\Delta_{j_t})$, where $\beta^{(t)}_{j_t}$ is the $j_t^{th}$ component of $\bbeta^{(t)}$ Here, $T$ serves as the regularization parameter, and for a given $T$ only a certain number of $\bbeta^{(t)}_{j_t}$ will have been updated from their initial values of zero, effecting variable selection. @Wolfson2011 recommends choosing $\epsilon$ in the range \[0.001,0.05\], and $T$ can be chosen with some tuning procedure. To express EEBoost as an iterative version of EEScreen, note that at the beginning of EEBoost, $\Delta_j$ corresponds to the screening statistic $\vert U_j(0)\vert$ used in EEScreen. Evaluating $\bU$ at subsequent $\bbeta^{(t-1)}$ is a way of controlling for the variables that have already been selected into the model by EEBoost, which is step 2(a) of iterative screening. In particular, for $i=0,1,\ldots$ define $t_i$ such that $\Vert\bbeta^{(t_i)}\Vert_0\ne\Vert\bbeta^{(t_i+1)}\Vert_0$. In other words, $t_0$ is the first time that the number of nonzero components of $\bbeta^{(t)}$ changes, $t_1$ is the second time this happens, and so on. Then looking back at the iterative screening algorithm, for $l=1,\ldots,L$ we can identify $\mathcal{M}_{l-1}$ to be $\{j:\beta^{(t_{l-1})}_j\ne0\}$, $\mathcal{A}_l$ to be $\{j_{t_l}\}$, and $\mathcal{M}_l$ as being obtained by running EEBoost for $t_l$ iterations starting from the covariates in $\mathcal{M}_{t_l-1}\cup\mathcal{A}_l$. We can choose $L$ by tuning EEBoost with a generalized cross-validation-type criterion. We will thus implement iEEScreen using the EEBoost algorithm. In the remainder of this paper we study the effects of using EEScreen and iEEScreen as preprocessing steps before fitting regularized regression models. In particular, we will use EEBoost to fit the regressions, for two reasons. First, we would like to compare the effects of retaining different numbers of covariates after screening, from keeping only one or two covariates to keeping tens of thousands. Therefore we require a regularization method for estimating equations that can handle an arbitrarily large number of covariates. Second, in Section \[sec:aftsim\] we study a discrete estimating equation, so we require a regularization method which works well in that situation. To our knowledge, EEBoost is the only procedure that meets both of these criteria. However, this leads to a unique problem. We would naturally like to compare the effects of using EEScreen versus iEEScreen. But a careful inspection of the EEBoost algorithm reveals that running EEBoost twice, in other words first selecting covariates using EEBoost, and then using only those covariates in another instance of EEBoost, is actually identical to using EEBoost only once. This means that screening with the version of iEEScreen described in this section has no effect if EEBoost is then used for model-fitting. This behavior is different from, say, the lasso, where running two iterations of the lasso has been termed the relaxed lasso [@Meinshausen2007] and can give different results from the regular lasso. Therefore while we will be able to compare the variable selection properties of EEScreen and iEEScreen in simulations, where we will know the true model, we will not be able to compare EEScreen+EEBoost versus iEEScreen+EEBoost. We would like to address this issue in future work. Simulations {#sec:sims} =========== In our simulation studies, we evaluated the performances of EEScreen and iEEScreen with two different estimating equations, one for a $t$-year survival model and the other for an accelerated failure time model. We implemented iEEScreen by using EEBoost, as described in Section \[sec:ieescreen\], with $\epsilon=0.01$. We compared these to the naive approach of fitting $p_n$ marginal regressions, as well as to the method of @Zhu2011 and our EEScreen-derived method (\[eq:zhueescreen\]) from Section \[sec:zhu2011\]. We studied $p_n=20000$ covariates and set the true parameter vector $\bbeta_0$ to be such that $\beta_{0j}=1.5,j=1,\ldots,10$, $\beta_{0j}=-0.8,j=11,\ldots,20$, and $\beta_{0j}=0,j=21,\ldots,p_n$. We generated covariates $\bX_i$ from a $p_n$-dimensional zero-mean multivariate normal. To simulate an easy setting we used a covariance matrix that satisfied the partial orthogonality condition of @FanSong2010, where the important covariates were independent of the unimportant covariates. The covariance matrix consisted of 9 blocks of 10 covariates, 1 block of 910 covariates, and 19 blocks of 1000 covariates. Each block had a compound symmetry structure with the same correlation parameter $\rho$, which was equal to either 0.5 or 0.9, and the blocks were independent from each other. We matched the non-zero components of $\bbeta_0$ with two of the 10-dimensional blocks. To simulate a more difficult setting we let the entire covariance matrix have a compound symmetry structure with $\rho$ equal to either 0.3 or 0.5. The $t$-year survival model {#sec:tyearsim} --------------------------- We first considered a $t$-year survival model. Let $T_i$ and $\bX_i$ be the survival time and the covariate vector of the $i^{th}$ patient, respectively. We modeled the probability of surviving beyond some time $t_0$ conditional on covariates as $$\logit\{\pr(T_i\geq t_0\mid\bX_i)\}=\bX_i^T\bbeta_0.$$ This model is very useful in clinical investigations, and in fact we apply it to data from clinical trials of multiple myeloma therapies in Section \[sec:data\]. However, we cannot use the logistic regression because the $T_i$ are not directly observed. Let $C_i$ be the censoring time, such that we only observe $Y_i=\min(T_i,C_i)$ and $\delta_i=I(T_i\leq C_i)$. Without modeling the $C_i$, it is difficult to specify a full likelihood model for this data, so we instead turn to estimating equations. To account for the censored data, @Jung1996 assumed that the $C_i$ were independent of the $T_i$ and the $\bX_i$ and proposed using the estimating equation $$\label{eq:tyear} \bU(\bbeta) = n^{-1}\sum_{i=1}^n\frac{\bX_i\pi'(\bX_i^T\bbeta)}{\pi(\bX_i^T\bbeta)\{1-\pi(\bX_i^T\bbeta)\}}\left\{\frac{I(Y_i\geq t_0)}{\hat{S}_C(t_0)}-\pi(\bX_i^T\bbeta)\right\},$$ where $\pi(\eta)=\logit^{-1}(\eta)$, $\pi'(\eta)=\partial\pi/\partial\eta$, and $\hat{S}_C(t)$ is the Kaplan-Meier estimate of the survival function of the $C_i$. According to our procedure, after some simplification we see that EEScreen will retain the parameters $$\left[j:\lAbs\sum_{i=1}^nX_{ij}\frac{I(Y_i\geq t_0)}{\hat{S}_C(t_0)}\rAbs\geq\gamma_n\right]$$ Though $U_j$ does not satisfy Assumption \[ass:ustat\] because of the $\hat{S}_C(t)$ term, @Jung1996 showed that it can be written in the appropriate form, plus a negligible $o_P(1)$ term. To fit the $p_n$ regressions for the marginal screening method we used a simple Newton-Raphson procedure to solve $U_j$. Tuning EEBoost and iEEScreen was difficult because commonly used criteria such as AIC or BIC are not defined in the absence of a likelihood. We instead chose to minimize the GCV-type criterion $\widehat{BS}/(1-n^{-1}\Vert\hat{\bbeta}\Vert_0)^2$, where $\Vert\hat{\bbeta}\Vert_0$ is the number of nonzero components of $\hat{\bbeta}$, and $\widehat{BS}$ is the estimate of the Brier score at $t_0$. If $\hat{\pi}(t_0\mid\bX_i)$ is the predicted survival probability of patient $i$ at $t_0$, then $\widehat{BS}$ is defined by @Graf1999 as $$\widehat{BS}=n^{-1}\sum_i\left[\frac{\{0-\hat{\pi}(t_0\mid\bX_i)\}^2}{\hat{S}_C(X_i)}I(Y_i\leq t_0,\delta_i=1)+\frac{\{1-\hat{\pi}(t_0\mid\bX_i)\}^2}{\hat{S}_C(t_0)}I(Y_i\geq t_0)\right].$$ We generated survival times for $n=100$ subjects from $\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i$ with $\varepsilon_i$ having a logistic distribution with mean -0.5 and scale 1. Under this scheme the model of @Jung1996 is correctly specified. We generated $C_i$ from an exponential distribution to give approximately 50% censoring. We observed that the $20^{th}$ percentile of the simulated survival times was roughly $t_0=0.005$, so we used this $t_0$ when implementing the estimating equation. We simulated 200 such datasets. ------------------- ------------------- ----------------- ------------------ ----------------- $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 2849 (6180) 22 (249.5) 19666.5 (610.5) 19676 (559.5) Marginal 2908 (6278) 22 (228.75) 19659 (611.5) 19696 (550.5) Zhu et al. (2011) 9614.5 (9497.75) 2043.5 (7687) 19647.5 (655.5) 19531.5 (737) Method (2) 7559.5 (11737.75) 944.5 (4121.25) 19614.5 (716.75) 19545.5 (726.5) ------------------- ------------------- ----------------- ------------------ ----------------- : \[tab:tyearmms\]Median minimum model size (interquartile range) for the $t$-year survival model ------------------- ---------------- ------------------- ------------------- ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 1.29 (0.09) 1.38 (0.47) 1.38 (0.36) 1.32 (0.16) Marginal 617.79 (61.99) 1023.79 (1405.58) 1608.09 (2594.27) 1054.86 (252.32) Zhu et al. (2011) 1.52 (0.08) 1.58 (0.45) 1.88 (4.47) 1.49 (0.2) Method (2) 1.54 (0.09) 1.58 (0.45) 2.13 (8.02) 1.48 (0.18) ------------------- ---------------- ------------------- ------------------- ------------------ : \[tab:tyeartiming\]Average runtime in seconds (standard deviation) for the $t$-year survival model Table \[tab:tyearmms\] reports the median sizes of the smallest models $\hat{\mathcal{M}}$ found by the different screening methods that still contained the true model $\mathcal{M}$. The performance is best under the partial orthogonality setting when $\rho=0.9$, which is not surprising because this setting leads to the greatest separation between the important and unimportant covariates. EEScreen and marginal screening show similar performances, while our method (\[eq:zhueescreen\]) appears to actually outperform the method of @Zhu2011 in the partial orthogonality setting. Though EEScreen and marginal screening produce similar results, Table \[tab:tyeartiming\] shows that marginal screening, at least for this $t$-year survival model, can take much longer. These simulations were run on the Orchestra cluster supported by the Harvard Medical School Research Information Technology Group, on machines with 3.6 GHz Intel Xeon processors with at least 12GB of memory, and marginal screening took at least 10 minutes. On the other hand, the EEScreen-type methods and the method of @Zhu2011 were completed in a few seconds, showing the EEScreen can be much more computationally efficient than standard screening methods. ![\[fig:tyearROCs\]Screening performances for the $t$-year survival model](tyearROCs.eps) To better understand the performances of these various screening methods, we studied in Figure \[fig:tyearROCs\] the average number of false positives corresponding to a given number of false negatives achieved by the screened model $\hat{\mathcal{M}}$. We again see that the methods perform best in the partial orthogonality setting when the correlation is high. Furthermore, given the same setting, EEScreen performs better than the model-free methods. This is most likely because the model used by EEScreen is correctly specified, and thus should be more powerful than the model-free methods. This type of phenomenon was also pointed out by @Zhu2011. As in Table \[tab:tyearmms\], our method (\[eq:zhueescreen\]) again appears to outperform that of @Zhu2011. Figure \[fig:tyearROCs\] also shows that in all cases, the variable selection performance of iEEScreen far outperforms the other methods, particularly in the compound symmetry setting. However, we found that iEEScreen is not able to include all of the important covariates. In the partial orthogonality setting, it can only include up to 17 or 18 of the important covariates, while in the compound symmetry setting it cannot achieve fewer than 15 false negatives. It turns out that the boosting procedure we use to implement iEEScreen saturates at some point in its fitting, perhaps due to the fact that there are more parameters than covariates, or perhaps because our choice for the boosting parameter $\epsilon=0.01$ might be too large. ![\[fig:tyearMSEs\]Mean squared errors for the $t$-year survival model](tyearMSEs.eps) Next we studied the effect on estimation accuracy of using screening before fitting a regularized regression model with EEBoost. Figure \[fig:tyearMSEs\] reports the average mean squared error of estimation (MSE) as a function of $\vert\hat{\mathcal{M}}\vert$, the number of variables kept after screening. Here we defined MSE as $\Vert\hat{\bbeta}-\bbeta_0\Vert^2_2$, where $\hat{\bbeta}$ is the estimate obtained by EEBoost after screening. It is clear that using EEScreen first can improve the estimation accuracy of EEBoost, especially in the compound symmetry setting. Screening with the model-free methods does not appear to reduce the MSE, perhaps because they need to retain a large number of covariates before they include the important variables (Table \[tab:tyearmms\]). ![\[fig:tyearPEs\]Out-of-sample AUCs for the $t$-year survival model](tyearPEs.eps) On the other hand, estimation error is not so meaningful in the absence of a correctly specified model. We therefore considered the out-of-sample predictive ability, as measured by the AUC statistic [@Uno2007] at time $t_0$, of the models fit by EEBoost after screening in Figure \[fig:tyearPEs\]. In the partial orthogonality settings, using EEScreen first does not appear to have much of an effect on the AUC, while in the compound symmetry setting it does improve the predictive ability of the subsequent fitted model. Our model-free method (\[eq:zhueescreen\]) does not seem to have much of an effect on AUC in either setting, but appears to perform slightly better than the method of @Zhu2011. The accelerated failure time model {#sec:aftsim} ---------------------------------- The $t$-year survival model is useful when we are interested in a fixed event time. To study the entire survival distribution, one useful approach is the accelerated failure time (AFT) model, which posits that $$\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i,$$ where the $\varepsilon_i$ are independent and identically distributed, and the $\varepsilon_i$ can have an arbitrary distribution. The $\bbeta$ can be estimated using the U-statistic-based estimating equation $$\label{eq:aft} \bU(\bbeta)=n^{-2}\sum_{i=1}^n\sum_{k=1}^n(\bX_k-\bX_i)I\{e_i(\bbeta)\leq e_k(\bbeta)\}\delta_i,$$ where $e_i(\bbeta)=\log(Y_i)-\bX_i^T\bbeta$ [@Tsiatis1996; @Jin2003; @CaiHuangTian2009]. Following our procedure, after some simplification we see that EEScreen will retain the parameters $$\left\{j:\lAbs\sum_{i=1}^n\sum_{k=1}^n(X_{kj}-X_{ij})I(Y_i\leq Y_k)\delta_i\rAbs\geq\gamma_n\right\}.$$ This is a U-statistic of order $m=2$ and therefore satisfies our assumptions in Section \[sec:theory\]. Despite being a discrete estimating equation, (\[eq:aft\]) poses no additional problems to EEScreen or iEEScreen. To fit the $p_n$ regressions for the marginal screening method we used the method of @Jin2003, available in the R package `lss`. To tune EEBoost and iEEScreen, consider the function $$\label{eq:aftobj} L(\bbeta)=n^{-2}\sum_{i=1}^n\sum_{j=1}^n\{e_j(\bbeta)-e_i(\bbeta)\}I\{e_i(\bbeta)\leq e_j(\bbeta)\}\delta_i.$$ @CaiHuangTian2009, in their work on regularized estimation for the AFT model, argued that $L(\bbeta)$ is an adequate measure of the accuracy of estimation. They and @Jin2003 also noted that $\bU(\bbeta)$ is the “quasiderivative” of $-L(\bbeta)$. For these reasons, we tuned EEBoost by minimizing the GCV-type criterion $$L(\hat{\bbeta})/(1-n^{-1}\Vert\hat{\bbeta}\Vert_0)^2,$$ where we used $L(\bbeta)$ in place of a negative log-likelihood. We generated $n=100$ survival times from $\log(T_i)=\bX_i^T\bbeta_0+\varepsilon_i$ with $\varepsilon_i$ having a standard normal distribution. We generated $C_i$ independently from an exponential distribution to give approximately 50% censoring, and we simulated 200 datasets. ----------------------------- ------------------- ---------------- ------------------ ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 997 (2968.75) 20 (2) 19829.5 (316.25) 19822.5 (401.25) Marginal 1750.5 (3742.25) 21 (144) 19835 (353) 19764 (436) Zhu et al. (2011) 10761.5 (9416) 747 (3804.5) 19482 (854) 19464.5 (922.5) Method (\[eq:zhueescreen\]) 7940.5 (11962.75) 282.5 (2230.5) 19501.5 (800.25) 19522 (785.75) ----------------------------- ------------------- ---------------- ------------------ ------------------ : \[tab:aftmms\]Median minimum model size (interquartile range) for the AFT model ----------------------------- ----------------- --------------- ------------------ ------------------ $\rho=0.5$ $\rho=0.9$ $\rho=0.3$ $\rho=0.5$ EEScreen 1.58 (0.15) 1.53 (0.1) 1.51 (0.1) 1.51 (0.11) Marginal 1024.71 (114.2) 971.85 (82.5) 1081.56 (149.64) 1203.19 (106.81) Zhu et al. (2011) 1.6 (0.16) 1.46 (0.11) 1.44 (0.1) 1.46 (0.11) Method (\[eq:zhueescreen\]) 1.6 (0.15) 1.46 (0.11) 1.44 (0.09) 1.45 (0.11) ----------------------------- ----------------- --------------- ------------------ ------------------ : \[tab:afttiming\]Average runtime in seconds (standard deviation) for the AFT model We report for the different screening methods the smallest $\hat{\mathcal{M}}$ that still contained $\mathcal{M}$ in Table \[tab:aftmms\]. As with the $t$-year survival model, the methods perform best in the partial orthogonality setting with $\rho=0.9$. We also again see that our method (\[eq:zhueescreen\]) outperforms the method of @Zhu2011. In addition, Table \[tab:afttiming\] shows that marginal screening is much more time-consuming than the EEScreen-based methods or the procedure of @Zhu2011. ![\[fig:aftROCs\]Screening performances for the AFT model](aftROCs.eps) Figure \[fig:aftROCs\] reports the average number of false positives contained in $\hat{\mathcal{M}}$ as the number of allowed false negatives is varied. As in the $t$-year survival model simulations, iEEScreen performs better than non-iterative EEScreen, though in the compound symmetry case it also saturates before it can select all of the important covariates. We also see that the EEScreen outperforms the model-free methods again, and that our method (\[eq:zhueescreen\]) somewhat outperforms the method of @Zhu2011. The plots in Figure \[fig:aftROCs\] for the model-free methods look very similar to the corresponding ones in Figure \[fig:tyearROCs\], and this is because the models used to generate both survival times were both AFT models, differing only in the distributions of the error terms. ![\[fig:aftMSEs\]Mean squared errors for the AFT model](aftMSEs.eps) The average mean square errors of the models fit after screening are plotted in Figure \[fig:aftMSEs\]. Similar to the results for the $t$-year survival model, we see that screening using model-free methods does not improve the estimation accuracy of the subsequent regularized regression fit. Interestingly, for the AFT model it appears that screening with EEScreen only barely decreases the MSE under partial orthogonality, and is actually detrimental to the MSE in the compound symmetry setting, in contrast to the results for the $t$-year survival model. ![\[fig:aftPEs\]Out-of-sample C-statistics for the AFT model](aftPEs.eps) We see something similar when we examine the out-of-sample predictive abilities of the models fit by EEBoost after screening. We calculated the C-statistics [@Uno2011] of the fitted models on independently generated datasets and report them in Figure \[fig:aftPEs\]. EEScreen does not have much of an effect on the C-statistic, while using the model-free methods tend to decrease the predictive ability of the fitted model. The results in Figures \[fig:aftMSEs\] and \[fig:aftPEs\] are in contrast to the corresponding $t$-year survival simulation results, which showed the EEScreen can indeed improve MSE and prediction. This may be due to the way these figures were generated: to plot these figures we varied the size of $\hat{\mathcal{M}}$ from between 400 to 20000 in increments of 400. However, the advantages of screening in the AFT setting perhaps may only be seen if fewer than 400 covariates are retained. Data example {#sec:data} ============ We illustrate our methods on data from a multiple myeloma clinical trial. Multiple myeloma is the second-most common hematological cancer, but despite recent advances in therapy the sickest patients have seen little improvement in their prognoses. It is of great interest to explore whether genomic data can be used to predict which patients will fall into this high-risk subgroup, so that they might be targeted for more aggressive or experimental therapies. ![\[fig:km\]Kaplan-Meier estimates from multiple myeloma clinical trials](km.eps) The MicroArray Quality Control Consortium II (MAQC-II) study posed exactly this question to 36 teams of analysts representing academic, government, and industrial institutions [@maqcii]. It used data from newly diagnosed multiple myeloma patients who were recruited into clinical trials UARK 98-026 and UARK 2003-33, which studied the treatment regimes total therapy II (TT2) and total therapy III (TT3), respectively [@Zhan2006; @Shaughnessy2007]. Teams were asked to predict the probability of surviving past $t_0=24$ months, which is roughly the median survival time of high-risk myeloma patients [@KyleRajkumar2008], using the TT2 arm as the training set and the TT3 arm as the testing set. There were 340 patients in TT2, with 126 events and an average follow-up time of 55.82 months, and 214 patients in TT3, with 43 events and an average follow-up of 37.03 months. The Kaplan-Meier estimates of the survival curves are given in Figure \[fig:km\]. Gene expression values for 54675 probesets were measured for each subject using Affymetrix U133Plus2.0 microarrays, and 13 clinical variables were also recorded, including age, gender, race, and serum $\beta_2$-microglobulin and albumin levels. Figure \[fig:km\] shows that there was a patient in TT2 censored before 24 months, so we cannot model these data using simple logistic regression. We therefore considered the $t$-year survival model with estimating equation (\[eq:tyear\]), from Section \[sec:tyearsim\]. Because we had a total of 54688 covariates and only 340 patients in TT2, we first implemented a screening step, where we considered EEScreen, our model-free method (\[eq:zhueescreen\]), and the method of @Zhu2011. We then fit the screened variables using EEBoost, with the generalized cross-validation criterion described in Section \[sec:tyearsim\]. To choose the size of $\hat{\mathcal{M}}$, we used 5-fold cross-validation and selected the value of $\vert\hat{\mathcal{M}}\vert$ that gave the best average AUC statistic. The values we considered were 10, 50, 100, 500, 1000, and the numbers from 5000 to 54688 in increments of 5000. Finally, we validated our model in the TT3 arm. Method Optimal $\vert\hat{\mathcal{M}}\vert$ 5-fold CV AUC (SD) AUC in TT3 ----------------------------- --------------------------------------- -------------------- ------------ EEScreen ($t$-year) 5000 0.61 (0.03) 0.61 Method (\[eq:zhueescreen\]) 10 0.63 (0.06) 0.58 Zhu et al. (2011) 100 0.67 (0.08) 0.59 EEScreen (AFT) 100 0.65 (0.08) 0.70 : \[tab:aucs\]AUCs for probability of surviving past $t_0=24$ months Table \[tab:aucs\] summarizes our results. We first focused on the AUCs estimated using five-fold cross-validation. Surprisingly, we found that EEScreen gave us the lowest AUC, and that the model-free methods required fewer covariates while giving better prediction. However, note that screening using the $t$-year survival estimating equation (\[eq:tyear\]) essentially dichotomizes the observed times to binary outcomes, because we are only modeling whether they are larger than $t_0$. In contrast, we can see from the forms of method (\[eq:zhueescreen\]) and the procedure of @Zhu2011 that they use continuous outcomes. We therefore hypothesized that the model-free methods had more power than EEScreen based on equation (\[eq:tyear\]) to detect covariate effects, even though they did not incorporate any modeling assumptions. To test this hypothesis we examined the performance of using EEScreen based on the AFT model estimating equation (\[eq:aft\]). This strategy does not dichotomize the survival outcomes and is also a more restrictive model than the $t$-year model because it makes a global assumption on the distribution of the survival times. After screening we still used the $t$-year survival model to fit the retained covariates. Indeed, Table \[tab:aucs\] shows that with this strategy, we needed to retain only 100 covariates to achieve a high AUC. Turning now to the validation AUCs calculated in the TT3 arm, we found that though the model-free methods gave higher AUCs in cross-validation, their validation AUCs were essentially comparable to that of EEScreen based on the $t$-year survival model. This might perhaps indicate that the model-free methods actually overfit to patients in the TT2 arm, and thus their results didn’t generalize well to patients treated with TT3. In contrast, the EEScreen method based on the AFT model gave a much higher validation AUC of 70%. The final fitted model contained 37 covariates, which in addition to various gene expression levels also included $\beta_2$-microglobulin, albumin, and lactate dehydrogenase levels. Thus our method was able to select important clinical predictors in addition to identifying potentially important genomic factors. Discussion {#sec:discussion} ========== In this paper we introduced EEScreen, a new computationally convenient screening method that can be used with any estimating equation-based regression method. We proved finite-sample performance guarantees that hold for any model that can be fit with U-statistic-based estimating equations, and in addition showed that our approach could be used to derive a model-free screening procedure very similar to one proposed by @Zhu2011. Finally, we have drawn a connection between screening and boosting methods, showing that the EEBoost algorithm of @Wolfson2011 can be viewed as a form of iterative screening. Our simulation results, conducted using a $t$-year survival model as well as the AFT model, support the use of EEScreen in practice. They suggest that EEScreen is capable of retaining most of the important covariates without also including too many false positives, unless the covariates are very highly correlated. In terms of estimation and prediction, when the working model is correctly specified, using EEScreen will usually not give worse results than not using screening at all, and at the very least will dramatically reduce the required computation time. This does not always appear to be true of the model-free methods. On the other hand, in our multiple myeloma example we saw that using different models for the screening step and the regression step can offer better performance than keeping to one model throughout. This illustrates the difficulty in choosing a default screening procedure that works well in all cases. However, our myeloma results suggest that one key consideration is the power of the screening step. The AFT model-based screening appeared to have greater power than the $t$-year model, and perhaps its modeling assumptions prevented it from overfitting to the TT2 arm, as the model-free methods seemed to do. This insight implies that different situations will require choosing different screening methods in order to achieve the greatest power. Estimating equations give us access to a wide range of models to choose from, with more parametric models offering lower variance but higher bias, and models with fewer assumptions offering the opposite tradeoff. Thus our EEScreen approach is perfectly suited to this screening strategy, offering quick computation and good theoretical properties for whichever model we decide to use. Acknowledgments {#acknowledgments .unnumbered} =============== We thank Professors Lee Dicker and Julian Wolfson for reading an earlier version of this manuscript. We also thank Professors Tianxi Cai, Tony Cai, Jianqing Fan, Hongzhe Li, and Xihong Lin for their many helpful comments and suggestions. Sihai Zhao is grateful for the support provided by NIH-NIGMS training grant T32-GM074897. Proof of Theorem \[thm:surescreening\] {#pf:surescreening} ====================================== The event $\{\mathcal{M}\subseteq\hat{\mathcal{M}}\}$ equals $\{\min_{j\in\mathcal{M}}\vert U_j(0)\vert\geq\gamma_n\}$, so it is easy to see that $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-\sum_{j\in\mathcal{M}}\pr(\vert U_j(0)\vert<\gamma_n).$$ By the triangle inequality, we know that for all $j$, $\vert u_j(0)\vert\leq\vert U_j(0)-u_j(0)\vert+\vert U_j(0)\vert$, and by Assumption \[ass:signal\] we see that $c_1[n/m]^{-\kappa}-\vert U_j(0)\vert\leq\vert U_j(0)-u_j(0)\vert$ for all $j\in\mathcal{M}$. Therefore, $\vert U_j(0)\vert<\gamma_n$ for $j\in\mathcal{M}$ implies $\vert U_j(0)-u_j(0)\vert\geq c_1[n/m]^{1-\kappa}/2$. We can conclude from Assumptions \[ass:ustat\] and \[ass:bernstein\] and Bernstein’s inequality for U-statistics [@Hoeffding1963] that $$\pr(\mathcal{M}\subseteq\hat{\mathcal{M}}) \geq 1-2s_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/4}{2\Sigma^2+bc_1[n/m]^{-\kappa}/3}\right\}$$ Proof of Theorem \[thm:size\] {#pf:size} ============================= For the marginal estimating equations $U_j$ and their expected values $u_j$, we know from Assumptions \[ass:ustat\] and \[ass:bernstein\] and Bernstein’s inequality for U-statistics [@Hoeffding1963] that $$\pr\{\max_j\vert U_j(0)-u_j(0)\vert\leq c_1[n/m]^{-\kappa}/4\} \geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\}.$$ Also, if $\max_j\vert U_j(0)-u_j(0)\vert\leq c_1[n/m]^{-\kappa}/4$, then $\vert U_j(0)\vert\geq\gamma_n$ implies that $\vert u_j(0)\vert\geq c_1[n/m]^{-\kappa}/4$. This means that $$\vert\hat{\mathcal{M}}\vert = \vert\{j:\vert U_j(0)\vert\geq\gamma_n\}\vert \leq \vert\{j:\vert u_j(0)\vert\geq c_1[n/m]^{-\kappa}/4\}\vert \leq \frac{16}{c_1^2[n/m]^{-2\kappa}}\sum_ju_j(\b0)^2.$$ From our EEScreen procedure described in Section \[sec:method\], we see that the $u_j(\b0)$ are the possibly relabeled components of the expected full estimating equation $\bu(\b0)$. Thus $\sum_ju_j(\b0)^2=\Vert\bu(\b0)\Vert_2^2$, and by the generalization of the mean value theorem to vector-valued functions [@HallNewell1979] and Assumptions \[ass:beta0\] and \[ass:inf\], $$\Vert\bu(\b0)\Vert_2 = \Vert\bu(\bbeta_0)-\bu(\b0)\Vert_2 \leq \sup_{0<t<1}\Vert\bi(t\bbeta_0)\Vert_2\Vert\bbeta_0\Vert_2 \leq c_2\sup_{0<t<1}\sigma_{\max}\{\bi(t\bbeta_0)\} = c_2\sigma_{\max}^*,$$ so that $$\begin{aligned} \pr\left[\vert\hat{\mathcal{M}}\vert\leq \frac{16c_2^2\sigma_{\max}^{*2}}{c_1^2[n/m]^{-2\kappa}}\right] &\geq \pr\{\max_j\Vert U_j(0)-u_j(0)\Vert_\infty\leq c_1[n/m]^{-\kappa}/4\}\\ &\geq 1-2p_n\exp\left\{-\frac{c_1^2[n/m]^{1-2\kappa}/16}{2\Sigma^2+bc_1[n/m]^{-\kappa}/6}\right\}.\end{aligned}$$

ArXiv

--- abstract: 'Disentanglement is the process which transforms a state $\rho$ of two subsystems into an unentangled state, while not effecting the reduced density matrices of each of the two subsystems. Recently Terno [@Terno98] showed that an arbitrary state cannot be disentangled into a [*tensor product*]{} of its reduced density matrices. In this letter we present various novel results regarding disentanglement of states. Our main result is that there are sets of states which cannot be successfuly disentangled (not even into a separable state). Thus, we prove that a universal disentangling machine cannot exist.' author: - 'Tal Mor[^1]' title: On the Disentanglement of States --- [2]{} Entanglement plays an important role in quantum physics [@Peres93]. Due to its peculiar non-local properties, entanglement is one of the main pillars of non-classicality. The creation of entanglement and the destruction of entanglement via general operations are still under extensive study [@entanglement]. Here we concentrate on the process of disentanglement of states. For simplicity, we concentrate on qubits in this letter, and on the disentanglement of two subsystems. Let there be two two-level systems “X” and “Y”. The state of each such system is called a quantum bit (qubit). A pure state which is a tensor product of two qubits can always be written as $|0({\rm X})0({\rm Y})\rangle$ by an appropriate choice of basis, $|0\rangle$ and $|1\rangle $ for each qubit. For convenience, we drop the index of the subsystem (whenever it is possible), and order them so that “X” is at the left side. By an appropriate choice of the basis $|0\rangle$ and $|1\rangle$, and using the Schmidt decomposition (see [@Peres93]), an entangled pure state of two qubits can always be written as $ | \psi \rangle = \cos \phi |00\rangle + \sin \phi |11\rangle $ or using a density matrix notation $\rho = |\psi\rangle \langle \psi|$ $$\rho = [ \cos \phi |00\rangle + \sin \phi |11\rangle ] [ \cos \phi \langle 00| + \sin \phi \langle 11|] \ .$$ The reduced density matrix of each of the qubits is $\rho_{\rm X} = {\rm Tr}_{\rm Y} [\rho({\rm XY})] $ and $\rho_{\rm Y} = {\rm Tr}_{\rm X} [\rho({\rm XY})] $. In the basis used for the Schmidt decomposition the two reduced density matrices are $$\label{reduced-state} \rho_{\rm X} = \rho_{\rm Y} = \left( \begin{array}{cc} \cos^2\phi & 0 \\ 0 & \sin^2 \phi \end{array} \right) \ .$$ Following Terno [@Terno98] and Fuchs [@Fuchs] let us provide the following two definitions (note that the second is an interesting special case of the first): [*Definition*]{}.— Disentanglement is the process that transforms a state of two (or more) subsystems into an unentangled state (in general, a mixture of product states) such that the reduced density matrices of each of the subsystems are uneffected. [*Definition*]{}.— Disentanglement into a tensor product state is the process that transforms a state of two (or more) subsystems into a tensor product of the two reduced density matrices. We noticed that according to these definitions, when a successful disentanglement is applied onto any pure product state, the state must be left unmodified. That is, $$\label{pure-ps} |00\rangle \longrightarrow |00\rangle$$ (in an appropriate basis). This fact proved very useful in the analysis we report here. The main goal of this letter is to show that a universal disentangling machine cannot exist. A universal disentangling machine is a machine that could disentangle any state which is given to it as an input. In order to prove that such a machine cannot exist, it is enough to find [*one*]{} set of states that cannot be disentangled if the data (regarding which state is used) is not available. To analyze the process of disentanglement consider the following experiment involving two subsystems “X” and “Y”, and a sender who sends [*both systems*]{} to the receiver who wishes to disentangle the state of these two subsystems: Let the sender (Alice) and the disentangler (Eve) define a finite set of states $|\psi_i\rangle$; let Alice choose one of the states at random, and let it be the input of the disentangling machine designed by Eve. Eve does not get from Alice the data regarding [*which*]{} of the states Alice chose, so Eve’s aim is to design a machine that will succeed to disentangle any of the possible states $|\psi_i\rangle$. In the same sense that an arbitrary state cannot be cloned (a universal cloning machine does not exist [@WZ82]), it was recently shown by Terno [@Terno98] that an arbitrary state cannot be disentangled into a tensor product of its reduced density matrices. Note that this novel result of [@Terno98] proves that [*universal disentanglement into product states*]{} is impossible, and it leaves open the more general question of whether a [*universal disentanglement*]{} is impossible (that is, disentanglement into separable states). We extend the investigation of the process of disentanglement well beyond Terno’s novel analysis in several ways. First, we find a larger class (then the one found by Terno) of states which cannot be disentangled into product states. Then, we show that there are non-trivial sets of states that [*can*]{} be disentangled. In particular, we present a set of states that cannot be disentangled into tensor product states, [*but*]{} can be disentangled into separable states. Finally, we present our most important result; a set of states that [*cannot be disentangled*]{}. The existence of such a set of states proves that a universal disentangling machine cannot exist. Using the terminology of [@WZ82] we can say that our letter shows that [*a single quantum can not be disentangled*]{}. Consider a set of states containing only one state. Since the state is known, obviously it can be disentangled. E.g., it is replaced by the appropriate tensor product state. We first prove that there are infinitely many sets of states that [*cannot*]{} be disentangled into product states. Our proof here follows from Terno’s method, with the addition of using the Schmidt decomposition to analyze a larger class of states. The most general form of two entangled states can always be presented (by an appropriate choice of bases) as: $$\begin{aligned} \label{the-states} |\psi_0 \rangle &=& \cos \phi_0 |00\rangle + \sin \phi_0 |11\rangle \nonumber \\ |\psi_1 \rangle &=& \cos \phi_1 |0'0'\rangle + \sin \phi_1 |1'1'\rangle \ .\end{aligned}$$ To prove that there are states for which disentanglement into tensor product states is impossible, let us restrict ourselves to the simpler subclass $$\begin{aligned} |\psi_0 \rangle &=& \cos \phi |00\rangle + \sin \phi |11\rangle \nonumber \\ |\psi_1 \rangle &=& \cos \phi |0'0'\rangle + \sin \phi |1'1'\rangle \ .\end{aligned}$$ There exists some basis $$|0''\rangle = {1 \choose 0} ; |1''\rangle = {0 \choose 1}$$ such that the bases vectors $|0\rangle;|1\rangle$ and $|0'\rangle;|1'\rangle$ become $$|0\rangle = {\cos \theta \choose \sin \theta} ; |1\rangle = {\sin \theta \choose -\cos \theta} \ ,$$ and $$|0'\rangle = {\cos \theta \choose -\sin \theta} ; |1'\rangle = {\sin \theta \choose \cos \theta}$$ respectively, in that basis. The states (\[the-states\]) are now $$\begin{aligned} |\psi_0 \rangle &=& c_\phi {c_\theta \choose s_\theta} {c_\theta \choose s_\theta} + s_\phi {s_\theta \choose - c_\theta} {s_\theta \choose - c_\theta} \nonumber \\ |\psi_1 \rangle &=& c_\phi {c_\theta \choose - s_\theta} {c_\theta \choose - s_\theta} + s_\phi {s_\theta \choose c_\theta} {s_\theta \choose c_\theta} \ ,\end{aligned}$$ with $c_\phi \equiv \cos \phi$, etc. The overlap of the two states is ${\rm OL}= \langle \psi_0 | \psi_1 \rangle = \cos^2 2\theta + \sin 2\phi \sin^2 2 \theta$. The reduced states are given by $$\hat{\rho_0} = {c_\phi}^2 \left( \begin{array}{cc} {c_\theta}^2 & c_\theta s_\theta \\ c_\theta s_\theta & {s_\theta}^2 \end{array} \right) + {s_\phi}^2 \left( \begin{array}{cc} {s_\theta}^2 & - c_\theta s_\theta \\ - c_\theta s_\theta & {c_\theta}^2 \end{array} \right) \ ,$$ and $$\hat{\rho_1} = {c_\phi}^2 \left( \begin{array}{cc} {c_\theta}^2 & - c_\theta s_\theta \\ - c_\theta s_\theta & {s_\theta}^2 \end{array} \right) + {s_\phi}^2 \left( \begin{array}{cc} {s_\theta}^2 & c_\theta s_\theta \\ c_\theta s_\theta & {c_\theta}^2 \end{array} \right) \ .$$ Thus, the state after the disentanglement into tensor product states is $ (\rho_{\rm disent})_0 = \hat{\rho_0} \hat{\rho_0}$ or $ (\rho_{\rm disent})_1 = \hat{\rho_1} \hat{\rho_1}$. The minimal probability of error for distinguishing two states [@Helstrom76] is given by $ {\rm PE} = \frac{1}{2} - \frac{1}{4} {\rm Tr}| \rho_0 - \rho_1| $. For two pure states there is a simpler expression: $ {\rm PE} = \frac{1}{2} - \frac{1}{2} \sqrt{[1 - OL^2]}$. Thus, $${\rm PE}_{\ \!\rm ent} = \frac{1}{2} - \frac{1}{2} \sqrt{[1 - ({c_{2\theta}}^2 + s_{2 \phi} {s_{2 \theta}}^2)^2]}$$ for the two initial entangled states. This probability of error is minimal, hence it cannot be reduced by any physical process. Therefore, if, for some $\theta$ and $\phi$, the disentanglement into the tensor product states [*reduces*]{} the ${\rm PE}$, then that process is non-physical. The difference of the states obtained after disentangling into tensor product states is $ \Delta_{\rm disent} = \hat{\rho_0} \hat{\rho_0}-\hat{\rho_1} \hat{\rho_1}$ This matrix is $$\Delta_{\rm disent} = \cos 2 \phi \sin 2 \theta \left( \begin{array}{cccc} 0 & a & a & 0 \\ a & 0 & 0 & b \\ a & 0 & 0 & b \\ 0 & b & b & 0 \end{array} \right) \ ,$$ with $a = \cos^2 \phi \cos^2 \theta + \sin^2 \phi \sin^2 \theta$ and $b = \cos^2 \phi \sin^2 \theta + \sin^2 \phi \cos^2 \theta$. After diagonalization, we can calculate the Trace-Norm, so finally we get $$\begin{aligned} {\rm PE}_{\rm \ \!disent} &=& \frac{1}{2} - \frac{1}{\sqrt2}\sin 2 \theta \cos 2 \phi \sqrt{a^2 + b^2} \nonumber \\ &=& \frac{1}{2} - \frac{1}{2} s_{ 2 \theta} c_{ 2 \phi} \sqrt{1 + {c_{2 \phi}}^2 {c_{2 \theta}}^2 } \ .\end{aligned}$$ We can now observe that there are values of $\theta$ and $\phi$, e.g., $\theta = \phi = \pi/8$, for which the outcomes of the disentanglement process are illegitimate since they satisfy ${\rm PE}_{\rm disent} < {\rm PE}_{\rm ent} $. Once these outcomes are illegitimate the disentanglement process leading to these outcomes is non-physical, proving that a disentangling machine which disentangle the states $|\psi_0\rangle$ and $|\psi_1\rangle $ cannot exist for these values of $\theta $ and $\phi$. Therefore, this analysis provides a proof (similar to Terno’s proof [@Terno98]) that a universal machine performing disentanglement into tensor product states cannot exist. The following set of states can easily be disentangled: $$|\psi_0\rangle = \frac{1}{\sqrt2} [ |00\rangle + |11\rangle ] \ ; \quad |\psi_1\rangle = \frac{1}{\sqrt2} [ |00\rangle - |11\rangle ]$$ To disentangle them, Eve’s machine uses an ancilla which is another pair of particles in a maximally entangled state (e.g., the singlet state) in any basis. Eve’s machine swaps one of the above particles with one of the members of the added pair, and traces out the ancillary particles. As a result, the state of the remaining two particles (one from each entangled pair) is $$\label{cms} (1/4)[ |00 \rangle \langle 00| + |01 \rangle \langle 01| + |10 \rangle \langle 10| + |11 \rangle \langle 11| ] \ ,$$ the completely mixed state in four dimensions. This set provides a trivial example of the ability to perform the disentanglement process. It is a trivial case of disentanglement, since the two states are orthogonal: they can first be measured and distinguished, and then, once the state is known, clearly it can be disentangled. However, exactly the same disentanglement process can be used to successfully disentangle a non-trivial set of states. Let the basis used for the two states be a different basis (and not the same basis), so the first state is still $|\psi_0\rangle$, and the second state is $$|\psi_1'\rangle = \frac{1}{\sqrt2} [ |0'0'\rangle - |1'1'\rangle ] \ .$$ The same process of disentanglement still works, while now the states are non-orthogonal, and cannot always be successfully distinguished. Hence, this disentanglement process is non-trivial. Note that the same process successfuly works also when more than two maximally entangled states are used as the possible inputs. Before we continue, let us recall some proofs of the no-cloning argument, since the methods we use here are quite similar the those used in the no-cloning argument. Let the cloner obtain an unknown state and try to clone it. To prove that this is impossible, it is enough to provide one set of states for which the cloner cannot clone any state in this set. Let the sender (Alice) and the cloner (Eve) use three states $|0\rangle$, $|1\rangle$, and $|+\rangle = (1/\sqrt2)[|0\rangle + |1\rangle]$. The most general process which can be used here in the attempt of cloning the unknown state from this set is to attach an ancilla in an arbitrary dimension and in a known state (say $|E\rangle$), to tranform the entire system using an arbitrary unitary transformation, and to trace out the unrequired parts of the ancilla. In order to clone the states $|0 \rangle$ and $|1\rangle$ the transformations are restricted to be $$|E0\rangle \longrightarrow |E_0 00\rangle \ ; \quad |E1\rangle \longrightarrow |E_1 11\rangle$$ and once the remaining ancilla is traced out, the cloning process is completed. Due to linearity, this fully determine the transformation of the last state to be $$|E+\rangle] \longrightarrow \frac{1}{\sqrt2} [|E_0 00\rangle + |E_1 11\rangle \ ,$$ while a cloning process should yield $$|E+\rangle \longrightarrow |E_+\! +\!+\rangle \ .$$ The contradiction is clearly apparent since, once the remaining ancilla is traced out, the second expression has a non-zero amplitude for the term $|01\rangle $ while the first expression does not. The conventional way [@WZ82] of proving the no-cloning theorem (using only two states, say $|0\rangle$ and $|+\rangle$) is to compare the overlap before and after the transformation (it must be equal due to the unitarity of quantum mechanics): We obtain that $\langle E|E\rangle \langle 0 | + \rangle = \langle E_0|E_+\rangle \langle 0 | + \rangle \langle 0 | + \rangle$. Hence $ 1 = \langle E_0|E_+\rangle \langle 0 | + \rangle $ which is obviously wrong since all the terms on the right hand side are smaller than one. We shall now use the linearity of quantum mechanics to show that there are states that cannot be disentangled into tensor product states, but can only be disentangled into a mixture of tensor product states. Surprisingly, our proof is [*mainly*]{} based on the [*disentanglement of product states*]{}, that is, on the disentanglement of states which are anyhow not entangled even before the disentanglement process. The reason for the usefulness of such states is that they provide strict restrictions on the allowed transformations. The following set of states cannot be disentangled into product states: $$\begin{aligned} |\psi_0 \rangle &=& |00\rangle \nonumber \\ |\psi_1 \rangle &=& |11\rangle \nonumber \\ |\psi_2 \rangle &=& |00\rangle + |11\rangle \end{aligned}$$ We shall assume that these states can be disentangled into product states and we shall reach a contradiction. Note that the resulting states should be $|\psi_0 \rangle$ and $|\psi_1\rangle $ in the first two cases (see Eq. \[pure-ps\]), and the resulting state should be the completely mixed state (in 4 dimensions) in the last case (see Eq. \[cms\]). The most general process which can be used here is to attach an ancilla in an arbitrary dimension and in a known state (say $|E\rangle$), to transform the entire system using an arbitrary unitary transformation, and to trace out the ancilla. In order to avoid changing the states $|\psi_0 \rangle$ and $|\psi_1\rangle$ the transformations are restricted to be $$\begin{aligned} \label{two-states} |E\psi_0\rangle &=& |E00\rangle \longrightarrow |E_0 00\rangle \nonumber \\ |E\psi_1\rangle &=& |E11\rangle \longrightarrow |E_1 11\rangle \ .\end{aligned}$$ As in the no-cloning argument, these transformations fully determine the transformation of the last state to be $$|E\psi_2\rangle \longrightarrow \frac{1}{\sqrt2} [|E_0 00\rangle + |E_1 11\rangle \ .$$ Once we trace out the ancilla, the resulting state is still entangled unless $|E_0\rangle$ and $E_1\rangle$ are orthogonal. The proof of that statement is as follows: Without loss of generality the states $|E_0\rangle $ and $|E_1\rangle $ can be written as $|E_0 \rangle = |0 \rangle$ and $|E_1 \rangle = \alpha |0 \rangle + \beta |1 \rangle$ with $|\alpha^2| + |\beta^2| = 1$. Thus, $|E\psi_2\rangle \longrightarrow \frac{1}{\sqrt2} |0\rangle ( |00\rangle + \alpha |11\rangle ) + \frac{\beta}{\sqrt2} |111\rangle$. When the ancilla is traced out the remaining state is $$\label{resulting} \left( \begin{array}{cccc} 1/2 & 0 & 0 & \alpha^*/2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \alpha/2 & 0 & 0 &1/2 \end{array} \right) \ .$$ The resulting state is entangled unless $\alpha = 0$. Thus, in a successful disentanglemet process $\alpha = 0$ and hence, $|E_1\rangle = e^{i\theta} |1\rangle$. This state, however, is not a tensor product state, thus the above set of states [*cannot*]{} be disentangled into tensor product states. At the same time, this example also shows that the above set of states [*can*]{} be disentangled into a mixture of tensor product states. The resulting state (\[resulting\]) still has the correct reduced density matrices for each subsystem—the completely mixed state in two dimensions. With $\alpha=0$, the resulting state is $(1/2) [|00\rangle \langle 00| + |11\rangle \langle 11|] $, so we succeeded in showing an example where the states can only be disentangled into a separable state, but not into a tensor product state. Our result resembles a result regarding two commuting mixed states [@broadcast]: these states cannot be cloned, but they can be broadcast. That is, the resulting state of the cloning device cannot be a tensor product of states which are equal to the original states, but can be a separable state whose reduced density matrices are equal to the original states [@Bennett]. At that stage, the main question (raised by [@Terno98] and [@Fuchs]) is still left open: Can there be a universal disentangling machine? That is, can there exist a machine that disentangles any set of states into separable states? We shall now show that such a machine cannot exist. Our result is obtained by combining several of the previous techniques: the use of linearity, unitarity, and the disentanglement of product state. Consider the following set of states $$\begin{aligned} |\psi_0 \rangle &=& |00\rangle \nonumber \\ |\psi_1 \rangle &=& |11\rangle \nonumber \\ |\psi_2 \rangle &=& (1/\sqrt2)[ |00\rangle + |11\rangle ] \nonumber \\ |\psi_3 \rangle &=& |++\rangle \nonumber \\\end{aligned}$$ in which we added the states $|\psi_3 \rangle$ to the previous set. This set of states cannot be disentangled. The allowed transformations are now more restricted since, in addition to (Eq. \[two-states\]), the state $|\psi_3\rangle$ must also not be changed by the disentangling machine: $$|E \psi_3\rangle = |E\!+\!+\rangle \longrightarrow |E_+\! +\!+\rangle \ .$$ Due to unitarity, $\langle E|E\rangle \langle 0 | + \rangle \langle 0 | + \rangle = \langle E_0|E_+\rangle \langle 0 | + \rangle \langle 0 | + \rangle$, and also $\langle E|E\rangle \langle 1 | + \rangle \langle 1 | + \rangle = \langle E_1|E_+\rangle \langle 1| + \rangle \langle 1| + \rangle$. These expressions yield $ 1 = \langle E_0|E_+\rangle $, and $ 1 = \langle E_1|E_+\rangle $, from which we must conclude that $|E_+\rangle = |E_0\rangle = |E_1\rangle$. Recall that we already found that $|E_0\rangle = |0 \rangle$ and $|E_1\rangle = e^{i\theta} |1 \rangle$, due to the disentanglement of $|\psi_2\rangle$. Since the two requirements contradict each other, the proof that the above set of states cannot be disentangled (not even to a separable state) is completed. Thus, we have proved that a universal disentangling machine cannot exist. In otehr words—a single quantum cannot be disentangled. This result resembles a result regarding two non-commuting mixed states [@broadcast]: these states cannot be cloned, and furthermore, they cannot be broadcast. To summarize, we provided a thorough analysis of disentanglement processes, and we proved that a single quantum cannot be disentangle. Interestingly, we used a set of four states to prove this, but we conjecture that there are smaller sets that could be used to establish the same conclusion. The no-cloning of states of composite systems were investigated recently [@BDFMRSSW; @Mor98], and it seems that several interesting connections between these works and the idea of disentanglement can be further explored. For instance, one can probably find systems where the states can only be disentangled (or only be disentangled into product states) if the two subsystems are available together, but [*cannot*]{} be disentangled if the subsystems are available one at a time (with similarity to [@Mor98]), or [*cannot*]{} be disentangled if only bilocal superoperators can be used for the disentanglement process (with similarity to [@BDFMRSSW]). I would like to thank Charles Bennett, Oscar Boykin, and Danny Terno for very helpful remarks and discussions. [99]{} D. Terno, [*Non-linear operations in quantum information theory*]{}, submitted to Phys. Rev. A. Los-Alamos archive, Quant-Ph 9811036. A. Peres, [*Quantum Theory: Concepts and Methods*]{} (Kluwer, Dordrecht, 1993). C. H. Bennett, D. P. DiVincenzo, J. S. Smolin and W. K. Wootters, Phys. Rev. A [**54**]{}, 3824 (1996); V. Vedral and M. Plenio, Phys. Rev. A [**57**]{}, 1619 (1998). C. A. Fuchs. Cited in [@Terno98] for private communication. W. K. Wootters and W. H. Zurek, Nature [**299**]{}, 802 (1982). C. W. Helstrom, [*Quantum Detection and Estimation Theory*]{} (Academic Press, New York, 1976). H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, Phys. Rev Lett. [**76**]{}, 2818 (1996). The observation of this similarity is due to C. H. Bennett, private communication. C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. S. Smolin and W. K. Wootters, [*Quantum nonlocality without entanglement*]{}, to be publish in Phys. Rev. A. Los-Alamos archive, Quant-Ph 9804053. T. Mor, Phys. Rev Lett. [**80**]{}, 3137 (1998). [^1]: Electrical Engineering, UCLA, Los Angeles, Cal., USA

ArXiv

NIKHEF 2015-026\ DAMTP-2015-45 1.5cm [**Quantum corrections in Higgs inflation: the Standard Model case**]{} [ **Damien P. George$^{1,2}$[^1], Sander Mooij$^{3}$and Marieke Postma$^4$**]{} [*$^1$ -.1truecm Department of Applied Mathematics and Theoretical Physics,\ Centre for Mathematical Sciences, University of Cambridge,\ Wilberforce Road, Cambridge CB3 0WA, United Kingdom* ]{} [*$^2$ -.1truecm Cavendish Laboratory, University of Cambridge,\ JJ Thomson Avenue, Cambridge CB3 0HE, United Kingdom* ]{} [*$^3$ -.1truecm FCFM, Universidad de Chile,\ Av. Blanco Encalada 2008,\ 837.0415 Santiago, Chile* ]{} [*$^4$ -.1truecm Nikhef,\ Science Park 105,\ 1098 XG Amsterdam, The Netherlands* ]{} 0.5cm [**ABSTRACT**]{}\ We compute the one-loop renormalization group equations for Standard Model Higgs inflation. The calculation is done in the Einstein frame, using a covariant formalism for the multi-field system. All counterterms, and thus the betafunctions, can be extracted from the radiative corrections to the two-point functions; the calculation of higher n-point functions then serves as a consistency check of the approach. We find that the theory is renormalizable in the effective field theory sense in the small, mid and large field regime. In the large field regime our results differ slightly from those found in the literature, due to a different treatment of the Goldstone bosons. Introduction ============ In Higgs inflation the Higgs field of the Standard Model (SM) is coupled non-minimally to gravity [@Salopek:1988qh; @bezrukov1; @bezrukov2; @bezrukov_loop; @bezrukov3; @bezrukov4]. Apart from this single non-minimal coupling, no new physics is needed to describe inflation and the subsequent period of reheating, and the theory seems to be extremely predictive. However, this presupposes the parameters of the theory at high and low scale are related by renormalization group (RG) flow. The betafunctions in the low scale regime are the usual SM ones; in this paper we calculate the RG flow in both the mid field and the high scale (inflationary) regime. The idea of using the Higgs as the inflaton is an attractive one, not least because it allows one to connect collider observables with measurements of the early universe. Since its inception the model itself has come under a lot of scrutiny and criticism. First, unitarity is lost at high energies and the perturbative theory can only be trusted for energies below the unitarity cutoff [@burgess1; @barbon; @burgess2; @Hertzberg1; @cliffnew]. Although it is uncertain how to interpret this result as the cutoff is field dependent (according to [@bezrukov4; @linde_higgs; @moss; @He1], all relevant physical scales are always below the unitarity bound), it is clear that any new physics living at this scale may affect the inflationary predictions [@cliffnew; @Hertzberg2]. Second, given the currently measured central values for the top and Higgs mass, the Higgs potential becomes unstable around $10^{11}$ GeV, which would be disastrous for Higgs inflation. However, the verdict is not yet out, as it only takes $2-3\sigma$ deviations to push the instability bound all the way to the Planck scale [@disc1; @disc2; @branchina; @branchina2; @archil; @alexss] (in the very recent note [@kniehl] absolute stability of the electroweak vacuum is reported to be excluded by only 1.3 $\sigma$). Even though these claims are still debated, and SM Higgs inflation may still be alive, it is worth noting that constraints may be avoided in modified set-ups with an extended Higgs sector. Our results apply for large non-minimal coupling, but apart from that they are equally applicable to the various implementations of Higgs inflation. The renormalization group equations (RGEs) in Higgs inflation have been derived by several groups [@bezrukov_loop; @bezrukov3; @wilczek; @barvinsky; @barvinsky2; @barvinsky3], but they differ in details. The main source of disagreement comes from the choice of frame, and the treatment of the Higgs sector (Does the Higgs decouple from all fields? And the Goldstone bosons?). In previous work [@damien; @volpe] we have shown that the Jordan and Einstein frame describe exactly the same physics, and that any difference stems from an erroneous comparison of quantities defined in different frames. In this work we will work in the Einstein frame. Although dimensional analysis indicates that some of the Goldstone boson (GB)-loop corrections are large, and seem to spoil renormalization, gauge symmetry kicks in leading to cancellations of these large contributions. We find Higgs inflation is renormalizable in the effective field theory (EFT) sense [^2], and for energies below the unitarity cutoff. The small-field regime of Higgs inflation is where $\phi_0 \ll {m_{\rm p}}/\xi$, with $\phi_0$ the value of the background Higgs field, $\xi$ the non-minimal Higgs-gravity coupling which is of order $10^4$ (well below experimental bounds [@Calmet; @He2]), and ${m_{\rm p}}$ the Planck mass. In this regime the theory is effectively like the SM and therefore renormalizable in the EFT sense. In the large field regime ($\phi_0 \gg {m_{\rm p}}/\sqrt{\xi}$, corresponding to inflation) the potential has an approximate shift symmetry, which restricts the form of the loop corrections. As a result, all one-loop corrections can be absorbed in the parameters of the classical theory, and the EFT is renormalizable. Somewhat surprisingly, we find the same in the mid-field regime (${m_{\rm p}}/\xi < \phi_0< {m_{\rm p}}/\sqrt{\xi}$), even though it is far away from both an IR fixed point and the region in which the shift symmetry applies. In [@damien] we have studied the renormalization of the non-minimally coupled Higgs field in isolation, without any gauge or fermion fields, and our findings were in line with the literature. In this work we want to extend this previous analysis to the full SM. At first glance this does not seem to be problematic. Due to the non-minimal coupling to gravity, the coupling of the radial Higgs to both gauge field and fermions is suppressed in the large field regime. One can simply neglect all diagrams with these couplings. For example, loop diagrams with a fermion or gauge boson loop always dominate over the corresponding diagram with a Higgs loop. Effectively the Higgs decouples from the theory. However, the situation for the Goldstone bosons (GBs) is more complex: their coupling to the gauge fields is also suppressed, while the GB-fermion coupling is not. Upon going to unitary gauge, this corresponds to a coupling of the fermion to the longitudinal polarization of the gauge fields, and both the transverse and longitudinal polarizations couple with the usual SM strength to the fermions. All calculations are performed in the Einstein frame. For a discussion of the equivalence of Einstein and Jordan frame, see [@damien; @volpe]. One of the main complications in the calculation is that after transforming to the Einstein frame one ends up with non-canonical kinetic terms for the Higgs and Goldstone field. Due to the nonzero curvature of the field space, it is impossible to make a field transformation that brings the kinetic terms to their canonical form. Our approach here is to expand the action around a large classical background value for the inflaton field, and use the formalism of [@jinnouk; @seery; @Kaiser2] so that this background expansion can be done maintaining covariance in the field space metric. In our calculation we have neglected the time-dependence of the background field, as well as FLRW corrections and the backreaction from gravity; we argue that these corrections are at most subleading. (The inclusion of gravity corrections to the Higgs part of the theory has been addressed, in a covariant way, in [@moss].) Moreover, we are neglecting higher order kinetic terms by evaluating the field metric on the background. It would be an interesting but equally challenging task to develop a framework that can get around this latter limitation. Our main results are the SM Higgs inflation RGEs in the three regimes, where we included only the top-Yukawa coupling $y_t$. We find that Higgs/GB self-interactions and Higgs-fermion-interactions (but not GB-fermion) can be neglected in the mid and large field regime; Higgs/GB-gauge interactions decouple in the large field regime. This gives the following betafunctions: $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 24 \lambda^2\fac + A +(4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{\fac}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_3} &= -7 g_3^3,\nn\\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - \fac)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + \fac )}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= {\left[}\frac32 \fac y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t{\right]}+(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi\big|_{\rm mid, large} &= (4\pi)^2 \cdot2 \gamma_\phi \xi\end{aligned}$$ with $A= (3/8)(2g_2^4 + (g_2^2+g_1^2)^2) -6y_t^4$ and = { [ll]{} 1, & [small]{} ,\ 0, &[mid]{} , . . \[fac\] These betafunctions break down at the boundary of the regimes, where the EFT expansion in a small parameter is no longer valid; this gives additional threshold corrections which we have not calculated. All sign conventions used in this paper follow the QFT textbook by Srednicki [@srednicki], except for the sign of the Yukawa interaction terms, which is opposite to Srednicki’s. Higgs inflation {#s:hi} =============== In this section we give a brief overview of Higgs inflation and set our notation. Lagrangian ---------- The Jordan frame Lagrangian[^3] is (using $-+++$ metric signature) Ł\^J = $$-\frac12 {m_{\rm p}}^2 {\left(}1+ \frac{2\xi \Phi^\dagger \Phi}{{m_{\rm p}}^2} {\right)}R[g^J] + \L^J_{\rm SM}$$, with $$\begin{aligned} \L^J_{\rm SM} &= -\frac14 (f^a_{\mu\nu})^2 -\frac14 (F^a_{\mu\nu})^2 -\frac14 B_{\mu\nu}^2 -(D_\mu \Phi)^\dagger (D^\mu \Phi) - \lambda(\Phi^\dagger \Phi - v^2/2)^2 \nn\\ &\hspace{0.45cm}+ \bar Q_L (i\slashed{D}) Q_L +\bar u_R (i\slashed{D}) u_R +\bar d_R (i\slashed{D}) d_R - (y_d \bar Q_L \cdot \Phi d_R + y_u \bar u_R (i\sigma^2) \Phi^\dagger Q_L +{\rm h.c} ), \label{L_SM}\end{aligned}$$ where $Q_L =(u \; d)^\top_L$. Further $f^a_{{\mu\nu}},F^a_{{\mu\nu}},B_{{\mu\nu}}$ are the SU(3), SU(2) and U(1) field strengths respectively. The Higgs field is SU(2) complex doublet, which we parameterize = 1 $ \begin{array}{c} {\varphi}^+ \\ \phi_0+{\varphi}+ i \theta_3 \end{array} $ , \[Hfields\] with $\phi_0$ the classical background, ${\varphi}$ the Higgs field and ${\varphi}^+ = \theta_1+i\theta_2, \theta_3$ the GBs. The covariant derivative acts on the Higgs field and fermions as $$\begin{aligned} D_\mu \Phi &= {\left(}\partial_\mu -i g_2 A^a_\mu \tau^a - i Y_\phi g_1 B_\mu{\right)}\Phi ,\nn \\ D_\mu Q_L &=(\partial_\mu - i g_3 f^a_\mu t^a- i g_2 A^a_\mu \tau^a - i Y_Q g_1 B_\mu)Q_L, \nn \\ D_\mu u_R &=(\partial_\mu - i g_3 f^a_\mu t^a - i Y_u g_1 B_\mu)u_R, \label{DH}\end{aligned}$$ with $\tau^a = \sigma/2$ for the spinor representation. The hypercharges are $Y_\phi =1/2$, $Y_Q = 1/6$ and $Y_u=2/3$. At leading order, the only fermion that matters to find the running of the SM couplings is the top quark. We reach the Einstein frame after a conformal transformation: $ g^E_{\mu\nu} = \Omega^2 g^J_{\mu\nu}$, with \^2 = $1+ \frac{2\xi \Phi^\dagger \Phi}{{m_{\rm p}}^2}$. \[Omega\] The Einstein frame Lagrangian becomes Ł\^E =$$-\frac12 {m_{\rm p}}^2 R[g^E] + \L_{\rm mat}^E$$. We neglect the expansion of the universe, and take a Minkowski metric. The gauge kinetic terms are conformally invariant. The fermionic kinetic terms can be made canonical via a rescaling $\psi^E = \psi/\Omega^{3/2}$; the net effect is then a rescaling of the Yukawa interaction. All non-trivial effects of the non-minimal coupling are in the Higgs sector. $$\begin{aligned} \L^E_{\rm mat} &= -\frac14 (f^a_{\mu\nu})^2 -\frac14 (F^a_{\mu\nu})^2 -\frac14 B_{\mu\nu}^2 -\frac{1}{\Omega^2}(D_\mu \Phi)^\dagger (D^\mu \Phi) - \frac{3 \xi^2}{{m_{\rm p}}^2\Omega^4} \partial_\mu (\Phi^\dagger \Phi) \partial^\mu (\Phi^\dagger \Phi) \nn \\ &\hspace{0.45cm}+ \bar Q^E_L (i\slashed{D}) Q^E_L +\bar u^E_R (i\slashed{D}) u_R^E +\bar d^E_R (i\slashed{D}) d^E_R \nn\\ &\hspace{.43cm} - \frac{\lambda}{\Omega^4}(\Phi^\dagger \Phi - v^2/2)^2 - (\frac{y_d}{\Omega} \bar Q^E_L \cdot \Phi d^E_R + \frac{y_u}{\Omega} \bar u^E_R (i\sigma^2) \Phi^\dagger Q^E_L +{\rm h.c.} )\end{aligned}$$ The Higgs kinetic term is non-minimal. Let $\phi^i = \{\phi_R=\phi_0 + \varphi,\theta_i\}$ run over the Higgs field and Goldstone bosons. Then the metric in field space in component form is Ł\^E\_[mat]{} - 12 \_[ij]{} \_i \_j = -12 $$\frac{\delta_{ij}}{ \Omega^2} + \frac{ 6 \xi^2}{{m_{\rm p}}^2\Omega^4} \chi_i \chi_j$$ \_i \_j. \[non\_can\] The curvature on field space $R[\gamma_{ij}] \neq 0$ (this point was made, and further generalized, in [@Kaiser1], and the kinetic terms cannot be diagonalized. At most one can diagonalize the quadratic kinetic terms at one specific point in field space. Consider the electroweak sector. For the gauge bosons the kinetic terms remain canonical in the Einstein frame. As far as the quadratic action is concerned the action for the massive gauge bosons and Goldstone bosons is simply three times the action of a U(1) theory. To see this explicitly, consider the Higgs kinetic terms $$\begin{aligned} \L_{\rm higgs} &\supset -\frac{1}{\Omega^2}(D_\mu \Phi)^\dagger (D^\mu \Phi) \nn \\ &=-\frac{1}{2\Omega^2}{\left[}\partial_\mu {\varphi}\partial^\mu {\varphi}+ \sum_{a=1}^3 (\partial_\mu \theta_a \partial^\mu \theta_a -2g_a A^a_\mu (\phi \partial^\mu \theta_a -\theta_a \partial^\mu \phi) + g_a^2 \phi^2A^a_\mu A_a^\mu) +...{\right]}. $$ The gauge boson mass eigenstates are $\{A_1,A_2, Z,A_\gamma\}$ with Z= (g\_2 A\_3-g\_1 B),A\_= (g\_1 A\_3+g\_2B) , and couplings g\_a = 12 {g\_2,g\_2,,0}. \[ga\] This corresponds to three massive and one massless field. Note that we took the mass eigenstates as real gauge fields, and used the real and imaginary parts of $W_+$, rather than the complex states $W_\pm$. From now on we will work in the Einstein frame. For convenience we drop the superscript $E$, and work in Planck units ${m_{\rm p}}=1$. Three regimes {#s:regimes} ------------- Higgs inflation is non-renormalizable as the field space metric and potential are non-polynomial. But this does not exclude that the theory is renormalizable in the EFT sense over a limited field space. Our demands are that in a given field regime the theory can be expanded in a small parameter $\delta$, and that all loop corrections can be absorbed in counterterms order by order. Truncating the theory at some finite order in $\delta$ gives a renormalizable EFT with a finite number of counterterms. #### Small field regime The small field regime corresponds to $\delta_s \equiv \xi \phi_0 \ll 1$. To leading order in the expansion parameter $\delta_s$, the Lagrangian reduces to the SM Lagrangian. #### Mid field regime The mid field regime corresponds to $1/\xi < \phi_0 < 1/\sqrt{\xi}$. In this regime we rescale $\xi \to \delta_m^{-2}\xi$ and $\phi_0 \to \delta_m^{3/2} \phi_0$, such that both $\xi \phi_0^2 \propto \delta_m$ and $1/(\xi \phi_0)^2 \propto \delta_m$, and use $\delta_m$ as our expansion parameter. (We should admit that formally this expansion can only be trusted in the middle of this regime.) #### Large field regime Inflation takes place for field values $\delta_l \equiv 1/(\xi \phi_0^2) \ll 1$. The expansion in $\delta$ is equivalent to an expansion in slow-roll parameters, since $\eta=\mathcal{O}\left(\delta\right)$ and ${\epsilon}=\mathcal{O}\left(\delta^2\right)$. Covariant formalism and counterterms {#s:cov} ==================================== We want to investigate how the loop corrections and counterterms change in the small, mid and large field regime. For simplicity, we first focus on a U(1) Abelian Higgs model coupled to a left- and right-handed fermion. The generalization to the full SM Higgs inflation is postponed till sec. \[s:abelian\_beta\]. This way the effects of the non-minimal coupling can be studied in a simple set-up, without all intricacies of the chiral SM. Another advantage of the U(1) model is that gauge invariance and the Ward identities assure that many counterterms are independent of the gauge choice, which makes it easier to check the calculation. Lagrangian in covariant fields ------------------------------ This subsection reviews the covariant formalism introduced in [@jinnouk] and further worked out in [@seery; @Kaiser2]. Given the curvature of field space, it is very convenient to adopt an approach that maintains the covariance of the equations. For a U(1) theory with a complex Higgs field and a left- and right-handed Weyl fermion the Einstein frame matter Lagrangian is $$\begin{aligned} \L &= -\frac14 F_{{\mu\nu}}F^{{\mu\nu}}-\frac12 \gamma_{ab} \partial_\mu \phi^a \partial^\mu \phi^b +i\bar \psi \slashed{\partial}\psi- V(\phi^a) - \bar \psi F(\phi^a) \psi \nn\\ & \hspace{.43cm} -g A (G^\theta \partial \phi-G^\phi \partial \theta) -\frac12 g^2 A^2 G + {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}, \label{LU1}\end{aligned}$$ with $$\begin{aligned} V(\phi^a) &= \frac{\lambda}{4} \frac{|\phi_0+\varphi+i\theta |^4}{\Omega^4},\qquad F(\phi^a) = \frac{y}{\sqrt{2}}\frac{\phi_0+ \varphi+ i\gamma^5\theta}{\Omega}, \nn\\ G^\phi &= \frac{\phi}{\Omega^2},\qquad G^\theta = \frac{\theta}{\Omega^2},\qquad G = \frac{(\phi^2 + \theta^2)}{\Omega^2}. \label{VF}\end{aligned}$$ Now expand the Lagrangian around the background $\phi^a = (\phi_0(t) +\varphi(x,t), \theta(x,t))$. The fluctuation fields $\delta\phi^a=(\varphi,\theta)$ are not in the tangent space at $\phi_0^a$, and therefore do not transform as a tensor. We are led to introduce the covariant fluctuation $Q^a = (h,\chi)$, which is related to $\delta\phi^a$ via \^a = Q\^a -1[2!]{} \^a\_[bc]{} Q\^b Q\^c + 1[3!]{}$\Gamma_{be}^a \Gamma^e_{cd} -\Gamma^a_{bc,d}$ Q\^b Q\^c Q\^d + ... \[Qdef\] This is the notation we will use throughout this paper: $({\varphi},\theta)$ are the fluctuations of the original Jordan frame field (with $\phi_0$ the classical background field), and $(h,\chi)$ are the covariant fields. Further we define the covariant time derivative D\_t = \_a. Note that in the limit $\dot \phi_0 =0$ this reduces to the usual derivative $D_t = \partial_t$. Now we can expand the action in covariant fluctuations. We neglect FLRW corrections and the backreaction from gravity, as well as the time-dependence of the background field $\phi_0$; we come back to this in Sec. \[s:checks\]. The result for the interaction Lagrangian is $$\begin{aligned} \L_{\rm int} &= -\ (V + V_{;a} Q^a +\frac1{2!} V_{;ab} Q^a Q^b +...) -\bar\psi \ (F + F_{;a} Q^a +\frac1{2!} F_{;ab} Q^a Q^b +...) \psi \nn\\ & \hspace{.43cm} - g A \partial h ( G^\theta_{;a} Q^a +\frac1{2!} G^\theta_{;ab} Q^a Q^b +...) + g A \partial \chi ( G^\phi_{;a} Q^a +\frac1{2!} G^\phi_{;ab} Q^a Q^b +...) \nn\\ & \hspace{.43cm} -\frac12 g^2 A^2 ( G_{;a} Q^a +\frac1{2!} G_{;ab} Q^a Q^b +...) + {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}. \label{L_cov}\end{aligned}$$ All coefficients are evaluated on the background. The subscript with a semi-colon denotes the covariant derivative. We just found the Lagrangian for the covariant fields by Taylor expanding using covariant derivatives. An equivalent way of deriving the same Lagrangian is solving the relation $\phi^i(Q^j)$ [(\[Qdef\])]{} explicitly, substituting in the Lagrangian [(\[LU1\])]{}, [^4] and then Taylor expand in the fields $Q^i$ (using partial derivatives). This point of view will be useful when defining the counterterms in the next section. Here we just give the explicit form of [(\[Qdef\])]{} relating the original Langragian fields $({\varphi},\theta)$ to the covariant fields $(h,\chi)$: $$\begin{aligned} {\varphi}&= {\left(}h + \frac{ (h^2-\chi^2)}{2\phi_0} + ..{\right)}- \frac1\xi{\left(}\frac{h^2}{\phi_0^3} + \frac{h^3}{3\phi_0^4} + .... {\right)}+ \frac{1}{\xi^2} {\left(}\frac{h^2+\chi^2}{12\phi_0^3} + .... {\right)}.\nn \\ \theta & = {\left(}\chi + \frac{h \chi}{\phi_0} +..{\right)}- \frac1\xi{\left(}\frac{h \chi}{\phi_0^3} + \frac{4h^2 \chi}{3\phi_0^4} + .... {\right)}+ \frac{1}{\xi^2} {\left(}\frac{h \chi}{\phi_0^5} + \frac{h^2 \chi}{12\phi_0^4} + .... {\right)}. \label{Q_expl}\end{aligned}$$ We checked that substituting this in the Lagrangian and expanding, we indeed retrieve [(\[L\_cov\])]{}. Gauge fixing ------------ We have to add a gauge fixing and ghost Lagrangian, which can also be expanded in covariant fields. We fix the gauge via Ł\^E\_[GF]{} =-1[2\_G]{} $\partial^\mu A_\mu - g G^\phi(\phi_0) \xi_G \theta $\^2 . \[L\_GF\] This removes the quadratic $A\partial \theta$ couplings from the Lagrangian. In the small field regime $\Omega_0 \equiv \Omega(\phi_0) =1$ and we retrieve the standard $R_\xi$-gauge. We choose to write the gauge fixing term in terms of the Jordan frame fields (as opposed to the covariant fields) as these have a well defined gauge transformation. We work in Landau gauge ${{\xi_G}}= 0$. Then the ghost field decouples Ł\_[FP]{}\^E |\_[[[\_G]{}]{}=0]{} = -\_|c \^c. Feynman rules ------------- Now we can derive the Feynman rules from the above action. First we define the effective couplings $$\begin{aligned} \L_{\rm int}& = - \lambda_{m h n\chi} h^m \chi^n - y_{m h n \chi} h^m \chi^n \bar \psi (i\gamma^5)^\alpha \psi - (g_{ A \partial h mh n \chi } \partial h - g_{ A \partial \chi mh n\chi} \partial \chi ) A h^m \chi^n \nn \\ & \hspace{.43cm} -g_{2A mh n\chi} A^2 h^m \chi^n +g_L \bar \psi \slashed{A} P_L \psi + g_R \bar \psi \slashed{A} P_R \psi \label{vertices1}\end{aligned}$$ with $\alpha =1$ if the number $n= $ odd, and $\alpha=0$ otherwise (signs are absorbed in the couplings). All interactions are defined with a minus sign (the only exception is for one of the derivative interactions and the fermion-gauge interaction), and without numerical factors. This means that for a vertex with $m$ $h$-fields and $n$ $\chi$-fields and with or without fermion/gauge lines we have, respectively: $$\begin{aligned} V^{(m hn\chi)}&= (-i) m! n! \lambda_{m hn\chi},\nn\\ V^{(m hn\chi2\psi)}&= (-i) m! n! y_{m\phi n\chi}(i\gamma^5)^\alpha,\nn \\ V^{(m hn\chi2A)}&= (-i) 2! m! n! g_{2Am hn\chi}. \label{vertices2}\end{aligned}$$ For the derivative interaction we get V\^[(A h m h n)]{} = -i g\_[(A h m h n)]{} (-i k\^),V\^[(A m h n)]{} = i g\_[(A m h n)]{} (-i k\^), with $k$ the momentum running through the vertex. The fermion, scalar and gauge propagators are given by: $$\begin{aligned} -i D_\psi(k) &= \frac{-i(-\slashed{k}+m_\psi)}{k^2+m_\psi^2-i{\epsilon}}, \nn \\ -i D_{Q^a}(k) &= \gamma^{aa} \frac{-i}{k^2 + (m^2)_a^a-i{\epsilon}}, \nn \\ -i D_{\mu\nu}(k) &\stackrel{\xi_G = 0}{ =}-i \frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}}{k^2 +m_A^2 -i{\epsilon}}, \label{propagators}\end{aligned}$$ with masses $m_\psi = F(\phi_0)$, $m_A = G(\phi_0)$, and $(m^2)^a_b = \gamma^{ac} V_{;cb}(\phi_0)$. The scalar mass is diagonal, which we used in the scalar propagator. The propagators are standard except for the metric factor in the scalar propagator. The ghost field decouples. The above expansion of the Lagrangian, and corresponding Feynman rules are equally valid in all field regimes (although in the small field regime the notation is overkill). The explicit expressions for the vertices are given in Appendix \[s:vertices\]. The Higgs/GB mass and self-interactions are suppressed in the mid and large field regime, and we can neglect Higgs/GB loops in the quantum corrections. Likewise, the interaction with the gauge field is suppressed in the large field regime. The Higgs-fermion coupling is small in the mid and large field regime, but the GB-fermion coupling is not: it has the standard SM strength. This does not come as a surprise. The gauge-fermion interactions are unaffected by the non-minimal coupling. And since the GB is eaten to become the longitudinal polarization of the gauge boson, it should have the same interaction strength as the transverse polarizations. A priori it is not clear what the effects are on the betafunctions. The explicit computation is done in the next section. Counterterms ------------ In this section we introduce counterterms. It proves convenient, or maybe even necessary, to define the wave functions $Z_i$ in terms of the original Jordan frame fields, rather than the covariant fields. The usual U(1) symmetry relations between the various $Z_i$-factors then apply. To define the counterterms we start with the Einstein frame Lagrangian in terms of the [*Jordan frame*]{} fields [(\[LU1\])]{} and rescale the bare fields (with label “b") to physical fields (without label) via $$\begin{aligned} \phi_b &= \sqrt{ Z_\phi} \phi, & \theta_b &= \sqrt{ Z_\theta} \theta, & \psi_b &= \sqrt{Z_\psi} \psi,& A^\mu_b& = \sqrt{Z_A} A^\mu, \nn \\ \lambda_b &= Z_\lambda \lambda, & \xi_b &= Z_\xi \xi, & y_b &= Z_y y, & g_b &= Z_g g, \label{Zelement}\end{aligned}$$ with $\phi = \phi_0 + {\varphi}$. In Landau gauge $\xi_G=0$ the wavefunctions are $Z_{\varphi}=Z_{\phi_0} =Z_\theta$. We further define Z\_i = 1+\_i. We can then split the Lagrangian $\L =\L_{\rm renormalized} + \L_{\rm ct}$ with the counterterms proportional to $\delta_i$. The total Lagrangian is $$\begin{aligned} \L &= -\frac12 (Z_{g_{ab}} g_{ab}) Z_\phi \partial \phi^a \partial \phi^b + Z_\psi \bar \psi i\slashed{\partial}\psi -\frac{1}{4}Z_A F_{\mu\nu} F^{\mu\nu} \nn \\ & - \frac{Z_\lambda Z_\phi^2 \lambda ( \phi^2 + \theta^2)^2}{ 4(1+ Z_\xi Z_\phi \xi \phi^2)^2} - \frac{Z_\psi \sqrt{Z_\phi} Z_y y}{\sqrt{2} (1+ Z_\xi Z_\phi \xi \phi_R^2)} \bar \psi (\phi + i\theta \gamma^5) \psi \nn\\ & -\frac1{ (1+ Z_\xi Z_\phi \xi \phi_R^2)} {\left(}\sqrt{Z_A} Z_\phi Z_g g A (\phi \partial \theta -\theta \partial \phi) + Z_A Z_\phi Z_g^2 \frac12 g^2 A^2 (\phi^2+\theta^2) {\right)}\nn \\ &+Z_\psi Z_g Z_A^{1/2} {\left(}g q_L \bar \psi \slashed{A} P_L \psi +g q_R \bar \psi \slashed{A} P_R \psi{\right)}. \label{Lstart}\end{aligned}$$ This can be simplified using the U(1) Ward identity Z\_g = Z\_A\^[-1/2]{}. \[ward\] Moreover, as we will derive in the next section, we have in the mid and large field regime (in the small field regime, $\xi$ drops out of the Lagrangian at leading order, and no counterterm can be determined at this order) Z\_= 1[Z\_]{}, &gt; . \[Z\_xi\] This means $\Omega^2 = 1 + \xi \phi^2$ does not run (in the large field regime it means $\delta = 1/(\xi \phi_0^2) \sim {{\rm e}}^{-h_0}$ does not run). We can now use $\phi^i(Q^a)$ in [(\[Q\_expl\])]{} to rewrite the Lagrangian in terms of the covariant fields. It is then clear that all interactions arising from expanding the potential have the same renormalization factor, namely $Z_\lambda Z_\phi^2$. Similar statements can be made for the gauge and Yukawa interactions. We can also define “composite” wavefunctions for the interactions of the covariant fields via $$\begin{aligned} \L \supset&= - Z_{\lambda; mh n\chi} \lambda_{mh n\chi} h^m \chi^n - Z_{y; m h n \chi} y_{m h n \chi} h^m \chi^n \bar \psi (i\gamma^5)^\alpha \psi \nn \\ & - Z_{2A; m h n \chi} g_{2A m h n \chi} A^2 h^m \bar \chi^n - Z_{A \partial Q^a; m h n \chi} g_{A \partial Q^a m h n \chi} A \partial Q^a h^m \bar \chi^n . \label{Lstart2}\end{aligned}$$ It is straightforward to rewrite the composite wavefunctions in terms of elementary ones [(\[Zelement\])]{}, by comparing terms in the two actions above (\[Lstart\],\[Lstart2\]), when both written out in covariant fields. As an explicit example, consider the potential in the large field regime; using [(\[Lstart\])]{} it can be expanded as V = + $ - \frac{1}{\phi_0^4 \xi^3} h^2 - \frac{1}{3\phi_0^6 \xi^3} h^4 + \frac{1}{12 \phi_0^6 \xi^5} \chi^2 - \frac{1}{108 \phi_0^{10} \xi^7} \chi^4 + ...$ \[Zcompo\] from which we read off Z\_[V\_0]{} = ,Z\_[2h]{} = Z\_[4h]{} = Z\_[2]{} = Z\_[4]{} = . \[Z\_lambda\] Ignoring $\dot \phi_0$ corrections the Higgs kinetic terms are Ł-12 Z\_[2h]{} g\_[h h]{} (h)\^2 = -12 Z\_[2h]{} (h)\^2 = -12 (h)\^2. \[Z\_kin1\] The leading counterterm vanishes. The kinetic term for the GBs is Ł-12 Z\_[2]{} g\_ ()\^2 = -12 Z\_[2]{} ()\^2 = -12 Z\_h ()\^2. \[Z\_kin2\] And thus Z\_[2h]{} -1 =Ø(), Z\_[2]{} = Z\_. To derive these results we have used $Z_{h} = Z_{\chi} =Z_\phi$, i.e. the same counterterms for the Jordan frame and covariant fields. As we will discuss below, although this approximation is valid for the kinetic terms, it is not in general. ### On the wavefunctions of the covariant fields {#s:cov_wave} Instead of defining the wavefunctions of the Jordan frame fields, we could have tried to work with those of the covariant fields. Rather than beginning from [(\[Zelement\])]{}, we would then introduce $Z_{\phi_0}$, $Z_h$ and $Z_\chi$. Consider the large field regime. The potential is expanded in covariant fields as in [(\[Zcompo\])]{}. We can read off the counterterms from the parametric dependence of the various terms. For the quadratic Higgs and GB interactions we find $Z_h^{-1} Z_\xi^{-3}$ and $Z_h^{-2} Z_\xi^{-5}$ respectively, which should be equal by gauge invariance. This excludes setting $Z_{h} = Z_{\chi} =Z_\phi$ in the potential, as it gives inconsistent results.[^5] As we will now discuss, this approach breaks down for the GB interactions. We can understand where this stems from. To derive the $h^2$ and $h^4$ interactions at leading order, it is enough to only keep the first term in the expansion [(\[Q\_expl\])]{}. Then taking the relevant terms in [(\[Q\_expl\])]{}, and setting $Q_b^i = \sqrt{Z_Q} Q^i$ we get $$\begin{aligned} \sqrt{Z_\phi} {\varphi}&= \sqrt{Z_{h}} {\left(}h + \frac{ (h^2-\chi^2)}{2\phi_0} + ..{\right)}, \\ \sqrt{Z_\phi} \theta & =\sqrt{Z_{\chi}} {\left(}\chi + \frac{h \chi}{\phi_0} +..{\right)}.\end{aligned}$$ Thus for $h$ interactions we can take $Z_{h} = Z_{\chi} = Z_\phi$. Similarly, for the kinetic terms, only the first order expansion is needed, which is what we used above. However, to derive the GB interactions, the leading and subleading terms cancel, and to get the correct interaction one needs to expand $\phi^i(Q^j)$ to sufficient high order in $\delta$. In particular, one needs also to take into account the last term in the expansion in [(\[Q\_expl\])]{}. But then $Z_{h} = Z_{\chi} = Z_\phi =1/Z_\xi$ is no longer a consistent solution; it would give the inconsistent relation = (Q + Q\^2 + ...) + Z\_\^[3/2]{} ( Q\^2+ Q\^3 +...). Thus for GB scattering one cannot really define $Z_{\chi}$, $Z_{h}$ in terms of the elementary wavefunctions [(\[Zelement\])]{}. Fortunately, this is also not necessary, because we can simply use [(\[Zcompo\])]{}. Approximations used {#s:checks} ------------------- Before diving into the calculation we first list here the approximations made. 1. [We have dropped the time-dependence of the background field: $\dot \phi_0 \to 0$]{} 2. [We have neglected FLRW corrections and the backreaction of gravity]{} 3. [We have evaluated the field metric on the classical background.]{} 1\. In [@mp] we calculated the effective action in the SM regime, taking into account the rolling of the classical background field $\dot \phi_0$. Generalizing standard techniques to calculate the effective action to the time-dependent situation, we found the radiative corrections to both the classical potential and the kinetic terms. This allowed us to extract both the $\delta_\lambda$ and $\delta_\phi$ counterterms from the effective action. We retrieved the standard results. The time-dependence does not affect the form of the counterterms. Had we done the calculation in a time-independent way, by neglecting $\dot \phi_0$, we would have found the same $\delta_\lambda$ counterterm. In the large field regime the time-dependence enters also the kinetic terms, which are non-minimal, and it may not be obvious that we can neglect these effects. However, the large field regime is the inflationary regime, and all time-dependent corrections are slow roll suppressed. Working at leading order in the expansion parameter, as we do, they can be neglected. 2\. In [@damien; @GMP] we calculated the effective action in the SM regime, in a FLRW background. We showed that when working in the [*Einstein frame*]{}, the backreaction from gravity can be neglected. The reason is that the corrections are of the order of the slow roll parameter ${\epsilon}\sim \delta^2$, which are small compared to $\eta \sim \delta$ and thus can be neglected at leading order. Doing the calculation in a FLRW background will give order $\mathcal{O}(H^2)$ corrections to the scalar masses, to the Higgs and GB mass in our case. However, these masses only appear in diagrams with a Higgs and GB in the loop, which thus also involve suppressed GB/Higgs couplings. There is however one diagram that becomes of leading order in the $\delta$-expansion, which is the last term of [(\[Pi\_fermion\])]{}, giving the GB loop correction to the fermion propagator. Nevertheless, this diagram is still suppressed by $1/\xi$. Hence, to be really sure that FLRW corrections will not affect our results we have to work in the large $\xi \gg 1$ limit. 3\. The kinetic terms for the GB/Higgs field are of the form Ł- 12 \_[ij]{} Q\^i Q\^j = - 12 \_[ij]{}(\_0) Q\^i Q\^j + ... where we have expanded the field space metric around the background. The first term is quadratic and determines the structure of the propagators. The higher order terms, denoted by the ellipses above, can then be treated as additional interaction terms. It is hard to systematically take into account the effects of higher order interactions, and we have neglected them in our calculations in the next section. Unfortunately, it seems that for at least one diagram this is not a good approximation, as we discuss in section \[s:painful\]. One-loop corrections {#s:loop} ==================== To derive the one-loop betafunctions for the gauge, Yukawa and Higgs interactions, $g,y,\lambda$, it is enough to calculate the corrections to the gauge, fermion and scalar propagator, which is what we will do in this section. We will also compute corrections to 3 and 4-point interactions. These will serve as consistency checks on the result, which provide further checks on the validity of our approximations (discussed in the previous section). Another consistency check is the comparison with the Coleman-Weinberg effective action [@CW]. No field independent counterterms can be defined for the whole regime, but it may be possible to define renormalizable EFTs in the three different regimes. Then the hope is that the threshold corrections in patching them together are small. To find the result in a given regime, we plug in the explicit form of the couplings expanded in the expansion parameter valid in this regime. The expansion parameters were defined in subsection \[s:regimes\]; the explicit form of the couplings can be found in Appendix \[s:vertices\]. Coleman-Weinberg effective action --------------------------------- The Coleman-Weinberg calculation for a dynamical background field has been performed in [@mp]. From this we can extract $Z_{V_0}, Z_{2h}$, which should be consistent with the loop corrections to the Higgs/GB propagator and self-scattering. The effective action gets contributions from the bosonic, mixed and fermionic loops respectively, and is for ${{\xi_G}}=0$ $$\begin{aligned} \Gamma_{\rm CW} &= \frac{1}{32\pi^2{\epsilon}} {\left[}m_h^4 +m_\theta^4 +3 m_A^4 -4 m_f^4 + \frac{3}{2} m_{A\theta}^4 + 4 m_f \ddot m_f {\right]},\end{aligned}$$ with $m_{A\theta}^2 = -2 g \dot{\phi}_0/\Omega_0^2$. Adding classical and all one-loop contributions gives $$\begin{aligned} \Gamma & = \frac12 \gamma_{hh} \dot{\phi}_0^2{\left[}-Z_{2h} + \frac{1}{8\pi^2{\epsilon}} {\left(}\frac{3 g^2}{1+ \xi \phi_0^2(1+6\xi)} - \frac{y^2}{\Omega_0^2 (1+ \xi \phi_0^2(1+6\xi))} {\right)}{\right]}\nn \\ &\hspace{0.45cm} + \frac{\lambda \phi_0^4}{4\Omega_0^4} {\left[}-Z_{V_0} + \frac{1}{8\pi^2{\epsilon}} {\left(}\lambda s(\phi_0) +3 \frac{g^4}{\lambda} - \frac{y^4}{\lambda}{\right)}{\right]}, \label{Gamma}\end{aligned}$$ with s(\_0) = +. It is clear that no field-independent counterterms can be defined over the whole regime. Expanding the corrections in the respective regimes we find (still applying the notation $Z_i=1+\delta_i$) \_[V\_0]{}= 1[8\^2]{}$$10 \fac\lambda + 3\frac{g^4}{\lambda} - \frac{y^4}{\lambda}$$, \[dV0\] where we used notation [(\[fac\])]{}. Note that $\delta_{V_0} = \delta_\lambda + 2\delta_\phi$ in the small and mid field regime, but $\delta_{V_0} = \delta_\lambda -2\delta_\xi$ for large field. As we will see, consistency with Higgs/GB n-point functions requires $\delta_\phi = -\delta_\xi$, as in [(\[Z\_xi\])]{}, and thus $\delta_{V_0}$ constrains the same elementary counterterms in the whole regime. Furthermore, we find \_[[2h]{}]{} = 1[8\^2]{}$$\fac 3g^2 - \fac y^2 +\O(\delta)$$. \[d2h\] In the large field regime $\delta_{{2h}} = \O(\delta)$ is a consistency check, but does not put any constraints on the elementary counterterms [(\[Zelement\])]{}. In the mid field regime we find $\delta_{2h} =2(\delta_\phi+\delta_\xi) = \O(\delta)$, and thus to lowest order [(\[Z\_xi\])]{} is satisfied. In the small field regime $\delta_{{2h}} = \delta_\phi$, and we find an answer consistent with [(\[Zphi\])]{} below. Higgs/GB interactions --------------------- We start with the corrections to the Higgs propagator. Compared to the standard small-field calculation, the fermion loop is different because of the presence of new fermion-Higgs/GB couplings. The gauge loop proceeds as in the small field regime, with the only exception that the diagram with derivative interactions cancels (at first order in the expansion parameter) as in the large field regime $g_{A \partial h \chi} =- g_{A\partial \chi h}$ have opposite sign, instead of being equal. The result for the counterterm, fermion and gauge, mixed gauge-GB and Higgs/GB loops is $$\begin{aligned} \Pi^{h} &= -\delta_{2h} \gamma_{hh} k^2 -\delta_{\lambda_{2h}} 2\lambda_{2h} + \frac1{8\pi^2{\epsilon}} \bigg[ - 12 y_h^2 m_\psi^2 - 8 y_{2h} m_\psi^3 -2y_h^2 k^2 \nn \\ &\hspace{0.45cm} +6 g_{2A 2h} m_A^2 +6 g_{2A h}^2 +3 k^2 \gamma^{\chi\chi}{\left(}\frac{g_{A \partial h \chi}+ g_{A\partial \chi h}}{2} {\right)}^2 \nn \\ &\hspace{0.45cm} +12 \gamma^{hh}\lambda_{4h} m_h^2 +2 \gamma^{\chi\chi} \lambda_{2h2\chi} m_\chi^2 + 18 (\gamma^{hh})^2\lambda_{3h}^2 + 2 (\gamma^{\chi\chi})^2 \lambda_{h2\chi}^2 \bigg] . \label{Pi_h}\end{aligned}$$ The $\gamma^{aa}$ factors stem from the Higgs and GB propagators. Further, we used $m_h^2 = \gamma^{hh} (2\lambda_{2h})$ and similar for the GB mass. This yields \_[\_[2h]{}]{} = 1[8\^2]{}$$10 \fac \lambda + 3\frac{g^4}{\lambda} - \frac{y^4}{\lambda}$$, while for the kinetic term we retrieve [(\[d2h\])]{}. Comparing with the CW result we find that $\delta_{\lambda_{2h}} =\delta_{V_0}$. In the large field regime this gives the equality $\delta_\lambda -\delta_\phi -3\delta_\xi =\delta_\lambda -2 \delta_\xi$, from which we get \_= -\_, \[delta\_xi\] which assures that $\Omega$ does not run. This is the same as derived in the mid field regime from $\delta_{2h} = \O(\delta)$. In the small field regime $\xi$ drops out of the Lagrangian at leading order, and no relation for $\delta_\xi$ can be derived at this order. The correction to the GB propagator is $$\begin{aligned} \Pi^{\chi} &= - \delta_{2\chi} \gamma_{\chi\chi} k^2 -\delta_{\lambda_{2\chi}} 2\lambda_{2\chi} + \frac1{8\pi^2{\epsilon}} \bigg[ - 4 y_\chi^2 m_\psi^2 - 8 y_{2\chi} m_\psi^3 -2y_\chi^2 k^2 \nn \\ &\hspace{0.45cm} +6 g_{2A 2\chi} m_A^2 +3 k^2 \gamma^{hh}{\left(}\frac{g_{A \partial h \chi}+ g_{A\partial \chi h}}{2} {\right)}^2 \nn \\ &\hspace{0.45cm} +12 \gamma^{\chi\chi}\lambda_{4\chi} m_\chi^2 +2 \gamma^{hh} \lambda_{2h2\chi} m_h^2 + 4 \gamma^{hh}\gamma^{\chi\chi} \lambda_{h2\chi}^2 \bigg] . \label{Pi_chi}\end{aligned}$$ We find $\delta_{\lambda_{2h}} =\delta_{\lambda_{2\chi}}$ in all three regimes, as required by gauge invariance. Further we have \_[2]{} = \_= 1[8\^2]{}$$3\fac g^2 -y^2$$. \[Zphi\] It is interesting to note that in the large field regime the counterterm $\delta_{\lambda 2\chi} = \O(\delta^3)$ whereas the individual fermion and gauge loop diagrams in [(\[Pi\_chi\])]{} are $\O(\delta^2)$. Renormalizability thus requires the two fermion diagrams to cancel at leading order, to end up with an $\O(\delta^3)$ loop correction. This is indeed what happens. This intricate cancellation is even more pronounced when we consider corrections to higher $n$-point GB scattering. For example, the structure of the fermion contribution to the four-point GB vertex is $$\begin{aligned} V^{(4\chi)} &= -\delta_{\lambda 4\chi} 4! \lambda_{4\chi} \nn \\ &\hspace{0.45cm} + \frac{ 4!} {8\pi^2{\epsilon}} \Bigg[ 36 \lambda_{4\chi}^2(\gamma^{\chi\chi})^2 + \lambda_{2h2\chi}^2(\ghh)^2 -m_h^2 \lambda_{2h4\chi} \ghh -15 m_\chi^2 \lambda_{6\chi}\gamma^{\chi\chi} + 8\lambda_{h4\chi} \lambda_{h2\chi} \ghh \gamma^{\chi\chi} \nn \\ & \hspace{1 cm} - 4 y_{4\chi} m_\psi^3 - 4y_{3\chi} y_\chi (m_\psi^2+\frac12 k^2)- 6 y_{2\chi}^2 (m_\psi^2+\frac16 k^2) -4 y_{2\chi} y_\chi^2 m_\psi - y_\chi^4 \nn \\ & \hspace{1 cm} + 3(g_{2A2\chi}^2 + g_{2A4\chi} m_A^2) \Bigg] . \label{4theta}\end{aligned}$$ Now the counterterm on the first line is $\delta_{\lambda 4\chi} = \O(\delta^5)$. The GB and Higgs loop diagrams on the second line above give $\O(\delta^6)$ corrections and can be neglected. All individual fermion loop diagrams — the terms on the third line — and all individual gauge loop diagrams — the terms on the fourth line — are $\O(\delta^3)$, much larger than the counterterm. Thus both the leading and subleading contributions need to cancel when adding the diagrams. This is indeed what happens and we find $\delta_{\lambda_{2h}} = \delta_{\lambda_{4\chi}}$ as required by gauge invariance. This intricate cancellation, and the need to go to sub-sub-leading order in the $\delta$-expansion, is the reason we cannot easily define the wavefunctions for the covariant fields, as discussed in section [(\[s:cov\_wave\])]{}. Note, however, that the $k^2$-term in [(\[4theta\])]{} above does not cancel, and gives a correction that cannot be absorbed. [^6] For this we have to add a new dimension-6 counterterm which is a four-point $\chi$-interaction with two derivatives; very schematically Ł \^2 ()\^2, with a cutoff \~(y\_[3]{}y\_+12 y\_[2]{}\^2)\^[-1/2]{} \_[unitarity]{} \[CU1\] that is equal to or larger than the unitarity cutoff \_[unitarity]{} \~ {1[ ]{} , \_0 , } \[unitarity\] in the small, mid and large field regime regime. The EFT breaks down for energy scales beyond the unitarity cutoff. The new counterterms needed to absorb divergencies enter at even higher scales. As such they do not put further constraints on the domain of validity of the EFT. Yukawa interactions ------------------- We first calculate the corrections to the fermion propagator. The fermion-gauge coupling is standard over the whole field range, and the gauge loop gives the same result in all three regimes. This is not the case for the Higgs and GB loop, as the former is suppressed in the mid and large field regime. The fermion two-point function is $$\begin{aligned} \Pi^{(2\psi)} &= -\delta_\psi \slashed{k} - \delta_{m_\psi} m_\psi + \frac1{8\pi^2{\epsilon}} \bigg[ -3 m_\psi g^2 q_L q_R \nn \\ &\hspace{0.45cm} +y_h^2 \gamma^{h h} (m_\psi -\frac12 \slashed{k}) -y_\chi^2 \gamma^{\chi\chi} (m_\psi +\frac12 \slashed{k}) +y_{2h}\gamma^{h h} m_h^2 +y_{2\chi}\gamma^{\chi\chi} m_\chi^2 \bigg], \label{Pi_fermion}\end{aligned}$$ where we used $g_{A\bar \psi_{L,R} \psi_{L,R}} = g q_{L,R}$ in all three regimes. There is a minus sign difference between the two $\slashed{k}$ terms, which originates from the $(i\gamma^5)$ in vertices with an odd number of GBs. This gives for the counterterms $$\begin{aligned} \delta_\psi &= -\frac1{8\pi^2{\epsilon}} {\left[}\frac{y^2}{4}(\fac+1) {\right]}, \nn \\ \delta_{m_\psi}&=\delta_y +\delta_\psi+ \frac12 \delta_\phi =\frac1{8\pi^2{\epsilon}}{\left(}-3 g^2 q_L q_R + \frac12(\fac-1) y^2{\right)}. \label{Z_2psi}\end{aligned}$$ The GB and Higgs contribution add in $\delta_\psi$ and cancel in $\delta_{m_\psi}$ in the small field regime; in the mid and large field regime only the GB contribution survives. The Yukawa interactions, both Higgs-fermion and GB-fermion, should give consistent results. Indeed we find $$\begin{aligned} V^{\Psi \bar{\Psi}h}_{\rm tot} &=- \frac{1}{8\pi^2{\epsilon}}\Bigl( -y_h^3\ghh +y_h y_\chi^2\gamma^{\chi \chi} -3m_h^2 y_{3h}\ghh -m_\chi^2 y_{h2\chi}\gamma^{\chi \chi} -2y_h y_{2h}\ghh \left(\slashed{k}+2m_\Psi\right) \nn\\ & \quad -\frac{y_\chi y_{h\chi}\gamma^{\chi \chi}}{2} \left(\slashed{k}-2m_\Psi\right) -6 y_{2h} \lambda_{3h} (\ghh)^2 -2 y_{2\chi} \lambda_{h2\chi}(\gamma^{\chi \chi})^2 + 3 q_L q_R g_{\bar\Psi A\Psi}^2 y_h\Bigr) -\delta_{y_h}y_h,\end{aligned}$$ and $$\begin{aligned} V^{\Psi \bar{\Psi}\chi}_{\rm tot} &=- \frac{i \gamma^5}{8\pi^2{\epsilon}}\Bigg( y_h^2 y_\chi \ghh - y_\chi^3\gamma^{\chi \chi} -m_h^2 y_{2h\chi}\ghh -3 m_\chi^2 y_{3\chi}\gamma^{\chi \chi} -\frac{y_h y_{h\chi}\ghh}{2} \left(\slashed{k}+2m_\Psi\right) \nn\\ & \quad -2y_\chi y_{2\chi}\gamma^{\chi \chi} \left(\slashed{k}+2m_\Psi\right) - 2y_{h\chi} \lambda_{h2\chi}\ghh\gamma^{\chi \chi} +3q_L q_R g_{\bar\Psi A\Psi}^2 y_\chi \Biggr) - i\gamma_5\delta_{y_\chi} y_\chi. \end{aligned}$$ This indeed gives $\delta_{y_h} = \delta_{y_\chi} = \delta _{m_\psi}$, with the latter given in [(\[Z\_2psi\])]{}. However, once again there is a small glitch as the $\slashed{k}$ terms in both expressions do not cancel. New non-renormalizable counterterms need to be added, schematically of the form Ł |(h + i\^5 ) with cutoff \~(\^y\_y\_[2]{})\^[-1]{} \_[unitarity]{}. \[CU2\] Since the cutoff exceeds the unitarity cutoff, these terms do no affect the range of validity of the EFT in the three regimes. Gauge interactions ------------------ We begin with the gauge boson propagator. $$\begin{aligned} \Pi^A_{{\mu\nu}}&= -\delta_A( k^2 g_{\mu\nu} - k_\mu k_\nu) -\delta_{m_A} m_A^2 g_{{\mu\nu}}\nn \\ &\hspace{0.45cm} + \frac1{8\pi^2 {\epsilon}}\Bigg[ {\left(}3 \gamma^{hh} g_{h2A}^2 + 2 \gamma^{hh} g_{2h2A} m_h^2 + 2\gamma^{\chi\chi} g_{2\chi 2A} m_\chi^2{\right)}g^{{\mu\nu}}\nn \\ &\hspace{0.45cm}+ \gamma^{hh} \gamma^{\chi \chi} {\left[}-\frac14 (g_{A\partial h \chi}+g_{A\partial \chi h})^2 {\left(}\frac{k^2}{3} +m_h^2 +m_\theta^2{\right)}g^{{\mu\nu}}+ (g_{A\partial h \chi}^2-g_{A\partial h \chi}g_{A\partial \chi h}+g_{A\partial \chi h}^2)\frac13k^\mu k^\nu{\right]}\nn \\ &\hspace{0.45cm} -\frac23 (k^2 g^{{\mu\nu}}-k^\mu k^\nu){\left(}g_L^2 +g_R^2{\right)}-2 (g_L -g_R)^2 m_\psi^2 \Bigg]\end{aligned}$$ It should be remembered that we normalized $q_\phi =1$. The counterterms are \_A = -1[8\^2 ]{} g\^2$ \fac \frac13 +\frac23 q_L^2 +\frac23 q_R^2$. Using the Ward identity $2\delta_g= -\delta_A $ [(\[ward\])]{}, it follows that $\delta_{m_A} = 2\delta_g + \delta_\phi + \delta_A = \delta_\phi$. Reading off $\delta_{m_A} $ from the above expression, and comparing with our earlier result [(\[Zphi\])]{} for $\delta_\phi$, we indeed find agreement. In the large field regime there is also a derivative interaction at leading order that is not transversal and cannot be absorbed in $\delta_A$. We find a term \^A 1[8\^2 ]{} k\^k\^. \[transversal\] This term can be neglected only for $\xi \gg 1$. This is the only place where this extra condition is needed. The transverse term breaks the Ward identities in the Landau gauge, and should not be there. It arises as a consequence of our approximations, discussed in more detail in section \[s:painful\]. We are not too worried about this term, as it is absent in the large $\xi$ limit. But moreover, it is also a gauge dependent term. We could have chosen a gauge fixing Ł\^E\_[GF]{} =-1[2\_G]{} $\partial^\mu A_\mu - g \frac{\phi_0}{\Omega_0} \xi_G \chi $\^2 defined in terms of the covariant fields rather than the Jordan frame fields [(\[L\_GF\])]{}. In Landau gauge, this gauge fixing gives the same results for all other diagrams, but now also the transversal part [(\[transversal\])]{} vanishes. As a consistency test we also calculated the $2A2h$ interaction, which gives $$\begin{aligned} V^{2A2h} &=\frac{g^{\mu\nu}}{8\pi^2{\epsilon}} \Bigl( 48 \lambda_{4h} g_{2A2h}q_\Phi^2(\ghh)^2 +8 \lambda_{2h2\chi} g_{2A2\chi} (\gamma^{\chi\chi})^2 - (g_{A\partial h \chi } +g_{A\partial \chi h })^2\ghh (\gamma^{\chi\chi})^2 \lambda_{2h2\chi} \nn\\ & \qquad - 6 (g_{A\partial h \chi } +g_{A\partial \chi h })^2(\ghh)^2 \gamma^{\chi\chi} \lambda_{4h} \ +24 g_{2A2h}^2 \ghh - 4 (q_L-q_R)^2y_h^2 g_{\bar \Psi A \Psi}^2 \nn\\ & \qquad -8 (q_L-q_R)^2 y_{2h}g_{\bar \Psi A\Psi}^2 m_\Psi \Bigr) -4 \delta_{g_{2A2h}} g_{2A2h} g^{\mu\nu},\end{aligned}$$ yielding $\delta_{m_A} = \delta_{g_{2A2h}}$, as it should. Gauge-fermion vertex {#s:painful} -------------------- There is one interaction that does not give a consistent result, which is the fermion-gauge coupling. To calculate it the important terms in the Lagrangian are Ł= - \_(\^) \^(\^) - + \_[i=L,R]{}i|\_i \_i - . \[oerlag\] There are three one-loop diagrams: 1) a GB loop with the photon attached to the fermion line, 2) a Higgs loop with the photon attached to the fermion line, and 3) a mixed Higgs-GB loop with the photon attached via a derivative interaction to $\Phi$. The result is $$\begin{aligned} V_{\rm loop}^{ \bar \Psi A_\mu \Psi} &= -\frac{\gamma^\mu}{16\pi^2 {\epsilon}} \Bigg[ g_{\bar \Psi A \Psi} (y_\chi^2 \gamma^{\chi\chi} +y_h^2 \ghh) \left(q_L P_R+q_R P_L\right) \nn \\ & \qquad + \left(g_{A\partial h \chi }+g_{A\partial \chi h }\right) y_h y_\chi \ghh \gamma^{\chi\chi} \left(q_\Phi P_L - q_\Phi P_R\right) \Bigg], \nn \\ V_{\rm CT}^{ \bar \Psi A_\mu \Psi} &= -\delta_{ \bar \Psi A_\mu \Psi}g_{\bar \Psi A)\mu \Psi} \gamma^\mu(q_L P_L + q_R P_R) .\end{aligned}$$ In the small field regime this reduces to V\_[loop]{}\^[ |A\_]{} =-(q\_L P\_L + q\_R P\_R ), where we used gauge invariance: $ q_\Phi - q_L + q_R =0$. This expression can be absorbed in the counterterm, which is proportional to $\left(q_L P_L + q_R P_R \right)$ as well. In the large field regime, however, the diagrams with a Higgs loop are suppressed and we get V\_[loop]{}\^[ |A\_]{} = . We cannot combine the two parts, and will not get something proportional to $\left(q_L P_L + q_R P_R \right)$. The same problem arises in the mid-field regime. This result in the large field regime breaks gauge invariance explicitly. How did it arise? When we repeat the calculation without the first term in [(\[oerlag\])]{}, the Higgs and GB field still have the same propagator. As a result all three diagrams contribute and the result adds up to something gauge invariant. However, when we include the first term, things go wrong as the Higgs propagator is now suppressed compared to the GB propagator. Note however, that the first term is explicitly gauge invariant. It is our approximation that breaks the gauge invariance, when we evaluate the metric on the background $\gamma_{ij} (\phi,\theta) = \gamma_{ij} (\phi_0)$. In particular, for the first term we set Ł- \_(\^) \^(\^) = - (\_)\^2+ ... where the ellipses denote neglected higher order derivative interactions (to be precise: higher n-point interactions with two derivatives). We listed this as the third approximation in subsection \[s:checks\]. These higher order terms need to be included to obtain a gauge invariant result. Unfortunately, it does not seem straightforward to do so. We would like to postpone the setup of a framework able to handle higher order derivative terms to future work, leaving a loose thread to our current calculation. However, since the calculation of the two-point function involves lower order vertices, we expect it to be less prone to our approximation. RGE equations {#s:beta} ============= First we give the betafunctions for the Abelian-Higgs model with a non-minimal coupling, then in subsection \[s:SM\_beta\] we generalize to full SM Higgs inflation. Abelian Higgs model {#s:abelian_beta} ------------------- First we list all the counterterms, found in the previous section: $$\begin{aligned} \delta_\phi =-\delta_\xi& =\frac1{8\pi^2{\epsilon}} {\left(}3g^2 \fac - y^2{\right)},\nn \\ \delta_{\lambda_{2h}} = \delta_{\lambda_{4h}}=\delta_{\lambda_{2\chi}} = \delta_{\lambda_{4\chi}}&=\frac1{8\pi^2{\epsilon}}{\left(}10 \lambda \fac +3 \frac{g^4}{\lambda} -\frac{y^4}{\lambda} {\right)},\nn\\ \delta_\psi &= -\frac{1}{8\pi^2{\epsilon}}\left(\frac{y^2}{4} (\fac+1) \right),\nn\\ \delta_{m_\psi} = \delta_{y_{h\bar \psi \psi}} =\delta_{y_{\chi\bar \psi \psi}} & =\frac1{8\pi^2{\epsilon}}{\left(}-3 g^2 q_L q_R+\frac12 y^2 (\fac-1) {\right)},\nn\\ \delta_A &= -\frac{1}{8\pi^2{\epsilon}}g^2 (\frac{1}{3}q_\phi^2 \fac +\frac{2}{3}q_L^2+\frac{2}{3}q_R^2). \nn\\\end{aligned}$$ The counterterms for the couplings in the Lagrangian are then given by $\delta_\lambda=\delta_{\lambda_{4h}} -2 \delta_\phi$, $\delta_y = \delta_{m_\psi} -1/2 \delta_\phi -\delta_\psi$, and the Ward identiy $\delta_g =-1/2 \delta_A$ respectively. From this the beta-functions can be found via $\beta_\lambda = \lambda ({\epsilon}\delta_\lambda)$, and similarly for the Yukawa, gauge and non-minimal Higgs-gravity coupling. This gives $$\begin{aligned} \beta_\lambda & = \frac1{8\pi^2}{\left(}10 \fac \lambda^2 +3 g^4 -y^4 -6 \fac g^2 \lambda +2 y^2\lambda {\right)},\nn \\ \beta_y & = \frac1{8\pi^2}{\left(}-3 q_L q_R g^2 y-\frac32 \fac (q_L-q_R)^2g^2y +\frac14(1+3\fac) y^3{\right)}\nn \\ \beta_g&=\frac{1}{8\pi^2} g^3 {\left(}\fac\frac{1}{6}q_\phi^2 +\frac{1}{3}q_L^2+\frac{1}{3}q_R^2{\right)},\nn \\ \beta_\xi \big|_{\rm mid, large} &=-\frac1{8\pi^2} (\fac 3 g^2-y^2) \xi. \label{betaU1}\end{aligned}$$ We have not derived $\beta_\xi$ in the small field regime. At leading order all $\xi$ dependence drops out of the Lagrangian in the SM regime. Since our computation relies on the approximations listed in section \[s:checks\], which fail beyond the leading order, we have to leave the computation of $\beta_\xi$ in the small field regime open. SM Higgs inflation {#s:SM_beta} ------------------ Our results for a U(1) theory can be extended to the full Standard Model (SM) Higgs inflation. Working in background field gauge [^7], the symmetries of the classical effective action are similar to those of the U(1) theory. The main difference is that now there are 3 GBs, the top quark has three colors, and one needs to sum over the strong, weak and hypercharge interactions. #### Higgs coupling First we extend the U(1) results in the small field regime to the full SM beta-functions, which can be found for example in [@sher]. The SM betafunction for the Higgs self-coupling is a straightforward generalization of the U(1) result: $$\begin{aligned} \beta_\lambda &= \frac{1}{8\pi^2} {\left[}(9+n_\theta) \lambda^2 +3 \sum_a g_a^4 - n_c y_t^4 - 2\lambda ( 3\sum_a g_a^2 -n_c y_t^2) {\right]}\nn \\ &= \frac{1}{8\pi^2} {\left[}12 \lambda^2 + \frac{3}{16} {\left(}2 g_2^4 + (g_2^2+g_1^2)^2{\right)}-3 y_t^4 - 2\lambda (\frac94 g_2^2 +\frac34 g_1^2 -3y_t^2) {\right]},\end{aligned}$$ with $n_\theta =3$ the number of GBs, $n_c =3$ the number of colors and the $g_a$ are given in [(\[ga\])]{}. We have only included the running of the top Yukawa. Generalizing from the U(1) model, it follows that the $\lambda^2$ and the $\lambda g_{1,2}^2 $ terms are suppressed in the mid and large field regime. This gives \_ = $$24 \lambda^2\fac + \frac{3}{8} {\left(}2 g_2^4 + (g_2^2+g_1^2)^2{\right)}-6 y_t^4 - \lambda (9 g_2^2 +3{g_1}^2)\fac +12 y_t^2 \lambda)$$. #### Gauge coupling For an $SU(N)$ group the betafunction is (g) |\_[SU(N)]{}= - $ 22N - 2 n_f - n_H$, with $n_f$ the number of Weyl fermions and $n_s$ the number of complex Higgs fields, both in the fundamental representation. For an Abelian group there is no contribution from the gauge field, and the formula becomes (g) |\_[U(N)]{}= $4 \sum q_f^2 +\sum q_s^2$, with $q_f,q_s$ the charges of the Weyl fermion and real scalars respectively. This reproduces our result in the small field regime. In the mid and large field the Higgs and GB contributions to the running are absent; this does not affect QCD, but for the EW sector we get $$\begin{aligned} \beta_{g_3} &= -\frac{7}{(4\pi)^2} g_3^3,\nn\\ \beta_{g_2} &= -\frac{1}{(4\pi)^2} \frac{(20 - \fac)}{6}g_2^3, \nn\\ \beta_{g_1} &= \frac{1}{(4\pi)^2} \frac{(40 + \fac)}{6}g_1^3. \nn\end{aligned}$$ #### Yukawa coupling The running of the top Yukawa follows from the counterterm $\delta_{y} = \delta_{y_{h\bar \psi \psi}}-1/2(\delta_\phi +\delta_{t_L} +\delta_{t_R})$. Explicit expressions can e.g. be found in [@zhou; @tubitak]. For the top quark the SM counterterms are: $$\begin{aligned} \delta_{t_L} &=-\frac{1}{8\pi^2{\epsilon}} {\left(}\frac{\fac}{4} y_t^2 + \frac14 y_t^2 + \frac12 y_b^2{\right)}, \nn \\ \delta_{t_R} &=-\frac{1}{8\pi^2{\epsilon}} {\left(}\frac{\fac}{4} y_t^2 + \frac14 y_t^2 + \frac12 y_t^2{\right)}.\end{aligned}$$ The $y^2$ contributions stem from loops with $h$, $\chi$ and $\phi^+ = \chi_1-i\chi_2$ respectively. For $t_L$ the $\phi^+$ loop can only be made with a bottom quark in the loop, which gives a contribution to $y_b^2$ — which we neglect in the following. In our U(1) toy model we only had the first two contributions from the $h$ and $\chi$-loop; indeed this matches the counterterm we found before. In the mid and large field regime the Higgs loop is suppressed, but not the GB loop. Likewise, we expect the $\phi^+$-loop to contribute in the large field regime, as these are GBs, with the same structure of interactions as $\chi$. The Higgs counterterm is \_=- $$n_c y_t^2 -3 \fac\sum_a g_a^2$$ =- $$3 y_t^2 -\frac34 \fac ( 3 g_2^2 +g_1^2)$$, which generalizes our previous results. Here $n_c$ is the number of colors. The gauge contribution is suppressed in the large field regime. The vertex correction is \_[y\_[h|]{}]{}= - $$-\frac12y_t^2 (\fac-1) +3 Y_{tL} Y_{tR} g_1^2+ 3 C_2(R_t)g_3^2$$, \[delta\_PhiQ\] with $C_2(R_t) =4/3$ for the fundamental in SU(3), and $Y_{tL}=1/6,\, Y_{tR}=2/3$. For the U(1) model we found that the $y^2$-correction from the $h$ and $\chi$-loop cancels in the small field regime. However, in the large field regime the Higgs contribution is negligible, and there is a net contribution from the GB $\chi$. The $\phi^+$ loop gives a contribution $\propto y_b^2$ and can be neglected. The gauge terms stem from top-top-gauge loops, and since the fermion-gauge couplings are standard in the large field regime, these are unaffected. This gives for the betafunction \_[y\_t]{} = 1[(4)\^2]{} $$\frac32 (2+\fac) y_t^3 -{\left(}\frac{8+9\fac }{12} g_1^2 +\frac{9\fac}{4} g_2^2 +8g_3^2 {\right)}y_t$$. In the small field regime $\fac =1$ and we get the standard result. #### Non-minimal coupling Further, we found in the large and mid field regime that $\delta_\phi =-\delta_\xi$. This gives the betafunction for the non-minimal coupling \_|\_[mid, large]{} = $$6 y^2 -\frac32 \fac ( 3 g_2^2 +g_1^2)$$. ### End results $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 24 \lambda^2\fac + A +(4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{\fac}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_3} &= -7 g_3^3,\nn\\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - \fac)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + \fac )}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= \frac32 \fac y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t+(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi\big|_{\rm mid, large} &= (4\pi)^2 \cdot2 \gamma_\phi \xi \label{final}\end{aligned}$$ with $A= (3/8)(2g_2^4 + (g_2^2+g_1^2)^2) -6y_t^4$. Discussion {#s:concl} ========== Comparison with literature. --------------------------- In recent years several groups have presented renormalization group equations for SM Higgs inflation. The disagreement between these results has been a major motivation to write this paper which follows, in our opinion, the most systematic approach so far. In this section we compare our findings to some encountered in the recent literature. Let us first quickly compare this work to our own previous work [@damien]. There we have studied the renormalization of just a (complex) non-minimally coupled scalar, leaving the inclusion of fermions and gauge fields and the generalization to full SM Higgs inflation to this work. Our findings here generalize those in [@damien]. Note in particular that in that work, we concluded that we could not say anything about the RG flow in the mid-field regime, as the corrections were an order of $\delta$ smaller than the counterterms. However, the fermionic and some of the gauge corrections that we have found now are of the same order as the counterterms. That is why we can now present expressions for the running couplings in the mid field regime (barring threshold corrections) without getting in contradiction with our previous work. Now for the comparison to other authors. In general, it seems that all other approaches follow some predefined treatment for Higgs and Goldstone bosons. In some cases only the Higgs contributions are kept, in other cases only the GBs, in yet other cases only loop contributions are excluded, etcetera. Our result does not respect any of these guidelines. For example, we find that the GB contribution to the effective potential is suppressed, while the GB loops do contribute to the Yukawa corrections. To see exactly which field contributes to which loop correction all correction diagrams need to be properly computed. (Although the differences are never very dramatic.) We first compare to reference [@bezrukov3], which states that in large field the action is just the action of the chiral SM with $v = {m_{\rm p}}/\sqrt{\xi}$, and the Higgs (but not the GBs) decouples. The RGEs quoted for the large field regime are $$\begin{aligned} (4\pi)^2 \beta_\lambda &= A + (4\pi)^2 \cdot4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{1}{4} (3g_1^2+6 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - 1/2)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + 1/2)}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t +(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= (4\pi)^2 \cdot 2 \gamma_\phi \xi.\end{aligned}$$ Here a gauge contribution in $\gamma_\phi$ has been included, which explains the difference in $\beta_\lambda$, $\beta_\xi$ and $\beta_{y_t}$ with our work. In the betafunctions for the gauge coupling only the real Higgs field is excluded, whereas we also exclude the GB. The chiral SM is non-renormalizable, and new operators have to be included. At one-loop level there is a correction to the Z-boson mass, which depends on the running coefficients of two of these new operators. There is no such thing in our set-up, which is renormalizable in the EFT sense. We furthermore note that in this work the running of $\xi$ is computed with an approach very different from ours, via the running of the SM vev $v$. However, apart from our disagreement over the gauge contribution to $\gamma_\phi$, we find the same answer. Reference [@wilczek] states that in the inflationary regime quantum loops involving the Higgs field are heavily suppressed. The proposed prescription (originally introduced in [@Salopek:1988qh]) is to assign one factor of $s(\phi)$ for every off-shell Higgs that runs in a quantum loop, with s() = 1[\^2]{} \_ = . Although the metric factor in $s$ above is for the real Higgs field, judging from the RGEs presented the prescription has been applied to the complex field (nothing is said explicitly about Higgs and GBs). In large field, the quoted RGEs reduce to: $$\begin{aligned} (4\pi)^2 \beta_\lambda &= A + (4\pi)^2 \cdot4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= -\frac{1}{4} (3g_1^2+9 g_2^2) + 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{20}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{40 }{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= -3 y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t + (4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= 2 ((4\pi)^2 \cdot\gamma_\phi + 6\lambda) ( \xi+1/6).\end{aligned}$$ There is a gauge contribution to $\gamma_\phi$, which explains the difference in $\beta_\lambda$, $\beta_\xi$ and partly $\beta_{y_t}$ with our result. In $\beta_{g_i}$ the full Higgs doublet is taken out in the large field regime, in agreement with our result. $\beta_\xi$ is found by taking gravity as a classical background, following the pioneering work in [@Odintsov1; @Odintsov2; @Odintsov3]. We think that this is not a good approximation in the Jordan frame. This same s-factor formalism was followed in References [@rose; @kyle], with the modification that now only the loops of the real Higgs field are excluded, and not those of the GBs. However our final answers agree with neither of the results obtained there. Reference [@barvinsky3] writes that Goldstone modes, in contrast to the Higgs particle, do not have mixing with gravitons in the kinetic term. Therefore, their contribution is not suppressed by the $s$-factor. We disagree with this. The GBs cannot be treated as usual, as in polar coordinates $\rho^2 (\partial \theta)^2$ the radial field is not the canonical one. In Cartesian coordinates, all fields are equally coupled to the Ricci tensor via $\xi R \sum (\phi^i)^2$. The quoted results are $$\begin{aligned} (4\pi)^2 \beta_\lambda &= 6 \lambda^2 + A - (4\pi)^2 \cdot 4\lambda \gamma_\phi \nn \\ (4\pi)^2\gamma_\phi &= \frac{1}{4} (3g_1^2+9 g_2^2) - 3y_t^2 \nn \\ (4\pi)^2\beta_{g_2} &= - \frac{(20 - 1/2)}{6}g_2^3, \nn\\ (4\pi)^2\beta_{g_1} &=\frac{(40 + 1/2)}{6}g_1^3 \nn \\ (4\pi)^2\beta_{y_t} &= {\left[}y_t^3 -{\left(}\frac{2}{3} g_1^2 +8g_3^2 {\right)}y_t{\right]}-(4\pi)^2 \cdot\gamma_\phi y_t\nn\\ (4\pi)^2\beta_\xi &= 6\xi \lambda -(4\pi)^2 \cdot2\gamma_\phi \xi.\end{aligned}$$ A gauge contribution to $\gamma_\phi$ is included, which partly explains the difference in $\beta_\lambda$, $\beta_\xi$ and $\beta_{y_t}$ with our results. For $\beta_{y_t}$ the contribution of one GB $y_t^2$-term has been excluded instead of 3GB $y_t^2$-terms. In $\beta_{g_i}$ only the GB is taken out in the large field regime, in disagreement with our result. This should be the identical to the chiral model of [@bezrukov3], as the Higgs field is decoupled in the large field limit. However the RGEs are still different. Lastly, [@bezrukov3; @wilczek] use two different normalization conditions, one with a field independent cutoff in the Jordan frame, or with a field dependent cutoff in the Jordan frame. However, two frames give identical physics. It is often quoted that a field independent cutoff in the Jordan frame corresponds to a field dependent cutoff in the Einstein frame, and vice versa. However, dimensionful quantities by themselves have no invariant meaning, their values depend on the unit system. If we express the cutoff in Planck units (the Planck mass is frame dependent), a constant cutoff in the one frame is equivalent to a constant cutoff in the other frame. The conformal rescaling only rescales [*all*]{} length scales, which does not change the physics. See also our discussion in [@damien]. On-shell equivalence between the frames has also been established in [@christian]. The question about a field dependent or independent cutoff is a frame invariant question when expressed in Planck units. The choice of cutoff has thus nothing to do with a choice of frame. One can still debate whether the results depend on the (field dependent) choice of cutoff in the Einstein frame. A priori, this is not expected; the cutoff is only introduced to regularize the divergent integrals, but is at the end taken to infinity. Requiring the counterterms to be field independent, the different field dependent and independent cutoffs lead to different normalization conditions. In practice one can only relate physical measurements at different energy scales. The translation between the observable and the coupling defined in the normalization condition will be different in each case. The end result is that when comparing physical observables at different scales, the cutoff dependence drops out. Conclusions ----------- We have calculated the one-loop corrections to Higgs inflation in the small, mid and large field regime. We have done the calculations for the Abelian Higgs model; the results can then rather straightforwardly be generalized to full Standard Model Higgs inflation. We have found that in all three regimes the model is renormalizable in the effective field theory sense. The RGEs for SM Higgs inflation we found are given in [(\[final\])]{}. The results for the mid field regime are new. The running of the non-minimal coupling can be derived in the mid and large field regime, and follows from the consistency of the radiative corrections to the potential and to the two-point functions. In the small field regime all dependence on the non-minimal coupling drops out of the equations at leading order in the small field expansion, and nothing can be said about its running. The computation of the radiative corrections was done in the Einstein frame, in the Landau gauge, using a covariant formalism for the multi-field system. The one-loop corrections to the propagators are sufficient to determine the full set of counterterms, and thus the betafunctions. As extra checks, we have calculated many higher n-point functions as well. Especially the results for four-point scattering of the Goldstone bosons are impressive in this regard: in the large (and mid) field regime both the leading and subleading divergencies exactly cancel, yielding a consistent counterterm. However, we have stumbled on some potential problematic outcomes as well. First of all, to cancel all divergencies, new non-renormalizable counterterms need to be added, see (\[CU1\], \[CU2\]). However, the cutoff implied by these new counterterms always exceeds the unitarity cutoff (\[unitarity\]). Therefore they do not put further constraints on the validity of the EFTs in the various regimes. Second, in (\[transversal\]) we have seen that the gauge boson propagator picks up a transverse part, that should be absent by the Ward identities. This term vanishes in the large $\xi$ limit. Moreover, it is gauge dependent, and we believe it should vanish in a full calculation. Thirdly, the one-loop gauge-fermion vertex gives a gauge symmetry breaking result as well. We can trace it back to the explicit symmetry breaking in our approximation of the non-minimal kinetic terms. In a full calculation the symmetry should be restored, yielding a result consistent with the gauge propagator corrections that we have found, but we leave this for further work. In conclusion, we have computed the full set of RGE equations for Standard Model Higgs inflation. The Higgs-fermion part has withstood an impressive set of consistency checks. When including the gauge symmetry our result obtained from propagator corrections has failed one consistency test, which we think can be ascribed to the intrinsic limitations in our approach (neglecting higher order kinetic terms by evaluating the field metric on the background). It would be an interesting but equally challenging task to develop a framework that can get around these limitations. Acknowledgments {#acknowledgments .unnumbered} =============== DG is funded by a Herchel Smith fellowship. SM is supported by the Fondecyt 2015 Postdoctoral Grant 3150126 and by the “Anillo” project ACT1122, funded by the “Programa de Investigación Asociativa." MP is funded by the Netherlands Foundation for Fundamental Research of Matter (FOM) and the Netherlands Organisation for Scientific Research (NWO). We thank Mikhail Shaposhnikov and Sergey Sibiryakov for illuminating discussions. Couplings {#s:vertices} ========= We list the couplings for the Abelian U(1) model. The explicit values in the small, mid, and large field regime are found expanding in $\delta_i$: $$\begin{aligned} & \delta_s = \xi \phi_0, & {\rm small} \nn \\ &\xi \to \delta_m^{-2}\xi,\;\phi_0 \to \delta_m^{3/2} \phi_0, & {\rm mid} \nn \\ & \delta_l = 1/(\xi \phi_0^2) . & {\rm large}\end{aligned}$$ Note that contrary to the small and large field regime, the $\delta_m$ parameter is just a rescaling parameter. In the small field we express the results in $\phi_0$ rather than $\delta_s$, as this form is more familiar. Below we give the leading expression for the metric and for the relevant couplings; the three values between the braces correspond to the small, mid and large field regime. The metric is: $$\begin{aligned} \gamma_{hh} & =\left\{1,~\frac{6 \phi_0^2 \xi ^2}{\delta _m}, ~(6 \xi +1) \delta _l\right\} \nn \\ \gamma_{\chi\chi} & =\left\{1,~1,~\delta _l\right\}.\end{aligned}$$ The Higgs and GB self-interactions are $$\begin{aligned} \lambda_{2h} &= \frac{1}{2!} V_{;\phi\phi} \Big|_{\rm bg} =\lambda \left\{ \frac32 \phi_0^2, ~ \phi_0^2 \delta_m^3, ~ -\frac1\xi \delta_l^2 \right\} \nn\\ \lambda_{2\chi} &= \frac{1}{2!} V_{;\theta\theta} \Big|_{\rm bg} =\lambda \left\{ \frac12 \phi_0^2, ~ \frac{1}{12\xi^2} \delta_m^4, ~ \frac{ \delta_l^3}{2\xi(1+6\xi)} \right\} \nn\\ \lambda_{3h} &= \frac{1}{3!} V_{;\phi\phi\phi} \Big|_{\rm bg} =\lambda \left\{ \phi_0,~ {\left(}\frac1{18\phi_0 \xi^2} -2\phi_0^3 \xi{\right)}\delta_m^{5/2}, ~ \frac{2\delta_l^{5/2}}{3\sqrt{\xi}} \right \} \nn\\ \lambda_{h2\chi} &= \frac{1}{3!} \left(V_{;\phi\theta\theta} + V_{;\theta \phi \theta} + V_{;\theta \theta \phi} \right) \Big|_{\rm bg} =\lambda\left\{\phi_0, ~ \frac{\delta_m^{5/2}}{18 \phi_0 \xi^2},~ -\frac{4 \delta_l^{7/2}}{3\sqrt{\xi}(1+6\xi)} \right\} \nn\\ \lambda_{4h} &= \frac{1}{4!}V_{;\phi\phi\phi\phi} \Big|_{\rm bg} = \lambda\left\{ \frac14,~ -\frac{\delta_m}{18\phi_0^2 \xi^2}, ~ ~- \frac{\delta_l^3}{3} \right\} \nn\\ \lambda_{2h2\chi} &= \frac{1}{4!} \left( V_{;\phi\phi\theta\theta} + {\rm 5 perms} \right) \Big|_{\rm bg} =\lambda \left\{ \frac12, ~ -\frac{\delta_m}{18\phi_0^2\xi^2},~ \frac{11 \delta_l^4}{6(1+6\xi)} \right\} \nn\\ \lambda_{4\chi} &= \frac{1}{4!}V_{;\theta\theta\theta\theta} \Big|_{\rm bg} =\lambda \left\{\frac14,~ \frac{\delta_m^2}{432 \phi_0^4 \xi^4},~ -\frac{\delta_l^5}{3(1+6\xi)^2} \right\} \nn\\ \lambda_{5h} &= {\rm etc.}\end{aligned}$$ The Yukawa interactions are $$\begin{aligned} m_\psi & =F^\phi \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{\phi_0 ,~\phi_0 \delta _m^{3/2},~\frac{1}{\sqrt{\xi }}\right\} \nn \\ y_h & =F^\phi_{;\phi} \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{1,~1,~\delta_l^{3/2}\right\} \nn\\ y_{2h} &= \frac1{2!}F^\phi_{;\phi\phi} \Big|_{\rm bg} =\frac{y}{\sqrt{2}} \left\{-(3 \xi^2 +\xi) \phi_0,~ -\frac{1}{2\phi_0 \delta _m^{3/2}}, -\sqrt{\xi } \delta _l^2 \right\} \nn \\ y_{3h} & =\frac{1}{3!}F^\phi_{;\phi\phi\phi} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{-\frac13 (3 \xi^2 +\xi),~-\frac{1}{2 \phi_0^2 \delta _m^{3}},~\frac23 \xi \delta_l^{5/2} \right\} \nn \\ y_{4h} &= {\rm etc.} \nn \\ y_\chi &=F^\theta_{;\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}}\left\{1,~1,~ \sqrt{\delta_l} \right\} \nn \\ y_{2\chi} &=\frac1{2!} F^\phi_{;\theta\theta} \Big|_{\rm bg} =\frac{y}{\sqrt{2}}\left\{-(3 \xi^2 +\xi) \phi_0 ,~-\frac{1}{2 \phi_0 \delta _m^{3/2}},~-\frac12\sqrt{\xi } \delta _l \right\} \nn \\ y_{3\chi} &=\frac{1}{3!}F^\theta_{;\theta\theta\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{-\frac13 (3 \xi^2 +\xi),~-\frac{1}{6 \phi_0^2 \delta _m^{3}},~-\frac16 \xi \delta_l^{3/2} \right\} \nn \\ y_{4\chi} &=\frac1{4!}F^\phi_{;\theta\theta\theta\theta} \Big|_{\rm bg} = \frac{y}{\sqrt{2}} \left\{{\left(}\frac{\xi^2}{3} +3 \xi^3 +\frac{15\xi^4}{2}{\right)}\phi_0,~\frac{1}{24 \phi_0^3 \delta _m^{9/2}},~\frac1{24} \xi^{3/2} \delta_l^{2} \right\}\end{aligned}$$ with F\^= , F\^= , as follows from [(\[VF\])]{}. Finally, the gauge interactions are $$\begin{aligned} m_A^2 & =g^2 G \Big|_{\rm bg} = g^2\left\{\ \phi_0^2, ~\phi_0^2 \delta_m^3, ~ \frac1\xi \right\} \nn \\ g_{2Ah} & =g^2 G_{;\phi} \Big|_{\rm bg} = g^2\left\{\ \phi_0, ~\phi_0 \delta_m^{3/2}, ~ \frac{\delta_l^{3/2}}{\sqrt{\xi}} \right\} \nn \\ g_{2A2h} & =\frac1{2!} g^2 G_{;\phi\phi} \Big|_{\rm bg} = g^2\left\{\ \frac12, ~{\left(}\frac1{12\phi_0^2 \xi^2} -\phi_0^2\xi{\right)}\delta_m, ~ -\delta_l^{2} \right\} \nn\\ g_{2A2\chi} & =\frac1{2!} g^2 G_{;\theta\theta} \Big|_{\rm bg} = g^2\left\{\ \frac12, ~\frac{\delta_m}{12\phi_0^2 \xi^2}, ~ \frac{\delta_l^{3}}{2(1+6\xi) }\right\} \nn\\ g_{A \chi \partial h } &= g G^\phi_{;\theta} \Big|_{\rm bg} =g\left\{1,~1,~\delta _l \right\} \nn \\ g_{Ah \partial \chi } &= g G^\theta_{;\phi} \Big|_{\rm bg} =g\left\{1,~1,~-\delta _l\right\} \nn\\ g_{A \bar \psi_{L,R} \psi_{L,R}} & = g q_{L,R} \end{aligned}$$ with $G,G^\phi,G^\theta$ defined in [(\[VF\])]{}. 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ArXiv

--- abstract: 'Based on a new approach on modeling the magnetically dominated outflows from AGNs (Li et al. 2006), we study the propagation of magnetic tower jets in gravitationally stratified atmospheres (such as a galaxy cluster environment) in large scales ($>$ tens of kpc) by performing three-dimensional magnetohydrodynamic (MHD) simulations. We present the detailed analysis of the MHD waves, the cylindrical radial force balance, and the collimation of magnetic tower jets. As magnetic energy is injected into a small central volume over a finite amount of time, the magnetic fields expand down the background density gradient, forming a collimated jet and an expanded “lobe” due to the gradually decreasing background density and pressure. Both the jet and lobes are magnetically dominated. In addition, the injection and expansion produce a hydrodynamic shock wave that is moving ahead of and enclosing the magnetic tower jet. This shock can eventually break the hydrostatic equilibrium in the ambient medium and cause a global gravitational contraction. This contraction produces a strong compression at the head of the magnetic tower front and helps to collimate radially to produce a slender-shaped jet. At the outer edge of the jet, the magnetic pressure is balanced by the background (modified) gas pressure, without any significant contribution from the hoop stress. On the other hand, along the central axis of the jet, hoop stress is the dominant force in shaping the central collimation of the poloidal current. The system, which possesses a highly wound helical magnetic configuration, never quite reaches a force-free equilibrium state though the evolution becomes much slower at late stages. The simulations were performed without any initial perturbations so the overall structures of the jet remain mostly axisymmetric.' author: - 'Masanori Nakamura, Hui Li, and Shengtai Li' title: Structure of Magnetic Tower Jets in Stratified Atmospheres --- INTRODUCTION ============ A number of astronomical systems have been discovered to eject tightly collimated and hyper-sonic plasma beams and large amounts of magnetic fields into the interstellar, intracluster and intergalactic medium from the central objects during their initial/final (often violent) stages. Magnetohydrodynamic (MHD) mechanisms are frequently invoked to model the launching, acceleration and collimation of jets from Young Stellar Objects (YSOs), X-ray binaries (XRBs), Active Galactic Nuclei (AGNs), Microquasars, and Quasars (QSOs) [see, [*e.g.*]{}, @M01 and references therein]. Theory of magnetically driven outflows in the electromagnetic regime has been proposed by @B76 and @L76 and subsequently applied to rotating black holes [@BZ77] and to magnetized accretion disks [@BP82]. By definition, these outflows initially are dominated by electromagnetic forces close to the central engine. In these and subsequent models of magnetically driven outflows (jets/winds), the plasma velocity passes successively through the hydrodynamic (HD) sonic, slow-magnetosonic, Alfvénic, and fast-magnetosonic critical surfaces. The first attempt to investigate the nonlinear (time-dependent) behavior of magnetically driven outflows from accretion disks was performed by @US85. The differential rotation in the system (central star/black hole and the accretion disk) creates a magnetic coil that simultaneously expels and pinches some of the infalling material. The buildup of the azimuthal (toroidal) field component in the accretion disk is released along the poloidal field lines as large-amplitude torsional Alfvén waves (“sweeping magnetic twist”). After their pioneering work, a number of numerical simulations to study the MHD jets have been done [see, [*e.g.*]{}, @F98 and references therein]. An underlying large-scale poloidal field for producing the magnetically driven jets is almost universally assumed in many theoretical/numerical models. However, the origin and existence of such a galactic magnetic field are still poorly understood. In contrast with the large-scale field models, Lynden-Bell [@LB94; @L96; @L03; @L06] examined the expansion of the local force-free magnetic loops anchored to the star and the accretion disk by using the semi-analytic approach. Twisted magnetic fluxes due to the disk rotation make the magnetic loops unstable and splay out at a semi-angle 60 from the rotational axis of the disk. Global magnetostatic solutions of magnetic towers with external thermal pressure were also computed by @Li01 using the Grad-Shafranov equation in axisymmetry [see also, @L02; @LR03; @UM06]. Full MHD numerical simulations of magnetic towers have been performed in two-dimension (axisymmetric) [@R98; @T99; @U00; @K02; @K04a] and three-dimension [@K04b]. Magnetic towers are also observed in the laboratory experiments [@HB02; @L05]. This paper describes the nonlinear dynamics of propagating magnetic tower jets in large scales ($>$ tens of kpc) based on three-dimensional MHD simulations. We follow closely the approach described in Li et al. (2006; hereafter Paper I). Different from Paper I, which studied the dynamics of magnetic field evolution in a uniform background medium, we present results on the injection and the subsequent expansion of magnetic fields in a stably stratified background medium that is described by an iso-thermal King model [@K62]. Since the simulated magnetic structures traverse several scale heights of the background medium, we regard that our simulations can be compared with the radio sources inside the galaxy cluster core regions. Due to limited numerical dynamic range, however, the injection region (see Paper I) assumed in this paper will be large (a few kpc). Our goal here is to provide the detailed analysis of the magnetic tower jets, in terms of its MHD wave structures, its cylindrical radial force balance, and collimation. The paper is organized as follows: In §2, we outline the model and numerical methods. In §3, we describe the simulation results. Discussions and conclusions are given in §4 and §5. MODEL ASSUMPTIONS AND NUMERICAL METHODS ======================================= The basic model assumptions and numerical treatments we adopt here are essentially the same as those in Paper I. Magnetic fluxes and energy are injected into a characteristic central volume over a finite duration. The injected fluxes are not force-free so that Lorentz forces cause them to expand, interacting with the background medium. For the sake of completeness, we show the basic equations and other essential numerical setup here again, and refer readers to Paper I for more details. Basic Equations --------------- We solve the nonlinear system of time-dependent ideal MHD equations numerically in a 3-D Cartesian coordinate system ($x,\,y,\,z$): $$\begin{aligned} \label{eq:mass} \frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\mbox{\bf V}) &=& \dot{\rho}_{\rm inj} \\ \label{eq:momentum} \frac{\partial (\rho {\bf V})}{\partial t} + \nabla\cdot \left( \rho {\bf V V} + p + B^2/2 - {\bf B B}\right) &=& - \rho \nabla\psi \\ \label{eq:energy} \frac{\partial E}{\partial t} + \nabla\cdot\left[\left(E +p + B^2/2 \right) {\bf v}-{\bf B}({\bf v}\cdot{\bf B})\right] &=& -\rho {{\bm V}}\cdot \nabla \psi \nonumber \\ &+& \dot{E}_{\rm inj}\\ \label{eq:induction} \frac{\partial {\bf B}}{\partial t} - \nabla\times( {\bf V} \times {\bf B}) &=& {\dot {\bf B}}_{\rm inj},\end{aligned}$$ Here $\rho$, $p$, ${{\bm V}}$, ${{\bm B}}$, and $E$ denote the mass density, hydrodynamic (gas) pressure, fluid velocity, magnetic field, and total energy, respectively. The total energy $E$ is defined as $E=p/(\gamma-1)+\rho V^2/2+B^2/2$, where $\gamma$ is the ratio of the specific heats (a value of $5/3$ is used). The Newtonian gravity is $-\nabla \psi$. Quantities $\dot{\rho}_{\rm inj}$, $\dot{{{\bm B}}}_{\rm inj}$, and $ \dot{E}_{\rm inj}$ represent the time-dependent injections of mass, magnetic flux, and energy, whose expressions are given in Paper I. We normalize physical quantities with the unit length scale $R_{0}$, density $\rho_{0}$, velocity $V_{0}$ in the system, and other quantities derived from their combinations, [*e.g.*]{}, time as $R_0/V_0$, etc. These normalizing factors are summarized in Table \[tbl:unit\]. Hereafter, we will use the normalized variables throughout the paper. Note that a factor of $4 \pi$ has been absorbed into the scaling for both the magnetic field ${{{\bm B}}}$ and the corresponding current density ${{{\bm J}}}$. To put our normalized physical quantities in an astrophysical context, we use the parameters derived from the X-ray observations of the Perseus cluster as an example [@C03]. These values are also given in Table \[tbl:unit\]. The system of dimensionless equations is integrated in time by using an upwind scheme [@LL03]. Computations were performed on the parallel Linux clusters at LANL. Initial and Boundary Conditions ------------------------------- One key difference from Paper I is that we now introduce a non-uniform background medium. An iso-thermal model [@K62] has been adopted to model a gravitationally stratified ambient medium. This is applicable, for example, to modeling the magnetic towers from AGNs in galaxy clusters. The initial distributions of the background density $\rho$ and gas pressure $p$ are assumed to be $$\label{eq:king} \rho = p = \left[1 + \left(\frac{R}{R_{\rm c}}\right)^2\right]^{-\kappa},$$ where $R=(x^2+y^2+z^2)^{1/2}$ is the spherical radius and $R_{\rm c}$ the cluster core radius. (In the following discussion, both “transverse” and “radial” have the same meaning, referring to the cylindrical radial direction.) The parameter $\kappa$ controls the gradient of the ambient medium. Furthermore, we assume that the ambient gas is initially in a hydrostatic equilibrium under a fixed (in time and space) yet distributed gravitational field $-\nabla \psi(R)$ (such as that generated by a dark matter potential). From the initial equilibrium, we get $$-\nabla \psi = \frac{\nabla p|_{t=0}}{\rho|_{t=0}} = -\frac{2 \kappa R}{R_{\rm c}^2}\left[1+\left(\frac{R}{R_{\rm c}^2} \right)^2\right]^{-1}.$$ In the present paper, we choose $R_{\rm c}=4.0$ and $\kappa=1.0$. The initial sound speed in the system is constant, $C_{\rm s}|_{t=0}={\gamma}^{1/2} \approx 1.29$, throughout the computational domain. An important time scale is the sound crossing time $\tau (\equiv R/C_{\rm s}) \approx 0.78$, normalized with $\tau_{\rm s0} (\equiv R_{0} / C_{\rm s0}) \approx 10.0$ Myrs. Therefore, a unit time scale $t=1$ corresponds to $12.8$ Myrs. The total computational domain is taken to be $|x| \leq 16$, $|y| \leq 16$, and $|z| \leq 16$ corresponding to a $(80\ {\rm kpc})^3$ box in the actual length scales. The numbers of grid points in the simulations reported here are $N_{x}\times N_{y}\times N_{z}=240 \times 240 \times 240$, where the grid points are assigned uniformly in the $x$, $y$, and $z$ directions. A cell $\Delta x\ (=\Delta y=\Delta z \sim 0.13)$ corresponds to $\sim 0.65\ {\rm kpc}$. We use the outflow boundary conditions at all outer boundaries. Note that for most of the simulation duration, the waves and magnetic fields stay within the simulation box, and all magnetic fields are self-sustained by their internal currents. Injections of Magnetic Flux, Mass, and Energy --------------------------------------------- The injections of magnetic flux, mass and its associated energies are the same as those described in Paper I. The ratio between the toroidal to poloidal fluxes of the injected fields is characterized by a parameter $\alpha = 15$, which corresponds to $\sim 6$ times more toroidal flux than poloidal flux. The magnetic field injection rate is described by $\gamma_b$ and is set to be $\gamma_b=3$. The mass is injected at a rate of $\gamma_\rho =0.1$ over a central volume with a characteristic radius $r_\rho = 0.5$. Magnetic fields and mass are continuously injected for $t_{\rm inj} = 3.1$, after which the injection is turned off. These parameters correspond to an magnetic energy injection rate of $\sim 10^{43}$ ergs/s, a mass injection rate of $\sim 0.046 M_\odot$/yr, and an injection time $\sim 40$ Myrs. In summary, we have set up an initial stratified cluster medium which is in a hydrostatic equilibrium. The magnetic flux and the mass are steadily injected in a central small volume with a radius of 1. Since these magnetic fields are not in a force-free equilibrium, they will evolve, forming a magnetic tower and interacting with the ambient medium. RESULTS ======= In this section we examine the nonlinear evolution and the properties of magnetic tower jets in the gravitationally stratified atmosphere. Overview of Formation and Propagation of a Magnetic Tower Jet ------------------------------------------------------------- Before considering our numerical results in detail, it is instructive to give a brief overview of the time development of the magnetic tower jet system. We achieve this by presenting the selected physical quantities using two-dimensional $x-z$ slices at $y=0$. The distributions of density at various times $(t=2.5,\,5.0,\,7.5,\,{\rm and}\,10.0)$ are shown in Fig. \[fig:de\_evo\]. At the final stage ($t=10.0$), we see the formation of a quasi-axisymmetric (around the jet axis) magnetic tower jet with low density cavities (a factor of $\sim 30$ smaller than the peak density). Inside these cavities, the Alfvén speed is large $V_{\rm A} \gtrsim 5.0$, while plasma $\beta$ is small ($\beta = 2p/B^2 \lesssim 0.1$). However, the temperature $T$ ($\propto C_{\rm s}^2$) becomes large $T \gtrsim 2.5 $ ($\sim$ twice the initial constant value); that is, the hotter gas is confined in the low-$\beta$ magnetic tower. The jet possesses a slender hourglass-shaped structure with a radially confined “body” for $|z| \leq 3$, which is likely due to the background pressure profile having a core radius $R_c = 4$. As the magnetic tower moves into an increasingly lower pressure background, the expanded “lobes” are formed. A quasi-spherical hydrodynamic (HD) shock wave front moves ahead of the magnetic tower. []{} A snapshot of the gas pressure change ratio $\Delta p/p_{\rm i} = p/p_{\rm i} -1$ (where $p_{\rm i}$ represents $p|_{t=0}$) at $t=10.0$ is shown in Fig. \[fig:final\_prdiff\]. Positive $\Delta p$ can be seen at both the post-shock region of the propagating HD shock wave and just ahead of the magnetic tower ($|z| \sim 8-10$). The distribution of $\Delta p$ forms a “[U]{}”-shaped bow-shock-like structure around the head of the magnetic tower. This structure, however, does not appear until $t \sim 7.5$. This is apparently caused by the local compression between the head of the magnetic tower and the reverse MHD slow-mode wave (see discussions in the next section). The gas pressure inside the magnetic tower becomes small ($|\Delta p/p_{\rm i}| \lesssim 0.5$) due to the magnetic flux expansion. Note that the light-blue region between the HD shock and the magnetic tower shows a small pressure decrease ($\Delta p/p_{\rm i} \approx - 0.1$). In later sections, we will discuss the origin of this depression and what role it plays in the dynamics of magnetic tower jets. The magnetic tower jet has a well-ordered helical magnetic configuration. The 3-D view of the selected magnetic lines of force, as illustrated in Fig. \[fig:3Dlines\], indicates that a tightly wound central helix goes up along the central axis and a loosely wound helix comes back at the outer edge of the magnetic tower jet. The magnetic pitch $B_{\phi}/\sqrt{B_r^2+B_z^2}$ has a broad distribution with a maximum of $\sim 15$. Figure \[fig:final\_Jz\] shows a snapshot of the axial current density $J_z$ at $t=10.0$. Clearly, the axial current flow displays a closed circulating current system in which it flows along the central axis (the “forward” current ${{\bm J}}^{\rm F}$) and returns on the conically shaped path that is on the outside (the “return” current ${{\bm J}}^{\rm R}$). It is well known that an axial current-carrying cylindrical plasma column with a helical magnetic field is subject to current-driven instabilities, such as sausage ($m=0$), kink ($m=1$), and the other higher order $m$ modes ($m$ is the azimuthal mode number). We however do not see any visible evolution of the non-axisymmetric features in this magnetic tower jet. ![\[fig:3Dlines\] Three-dimensional configuration view of the selected magnetic lines at $t=10.0$. ](f3.eps) From this overview, we see that the magnetic tower jet can propagate through the stratified background medium while keeping well-ordered structures throughout the time evolution. The magnetic fields push away the background gas, forming magnetically dominated, low-density cavities. This action also drives a HD shock wave which is ahead of and eventually separated from the magnetic structures. The magnetic tower has a slender ”body” from the confinement of the background pressure and an expanded “lobe” when the fields expand into a background with the decreasing pressure. We will now turn to the discussions on the detailed properties of the tower jet, including the HD shock wave and its impact in the axial ($z$) and radial ($x$) directions in §\[sec:A\] and \[sec:B\]. The radial force balance of the jet is examined in §\[sec:C\]. Structure of a Magnetic Tower Jet in the Axial ($z$) Direction {#sec:A} -------------------------------------------------------------- Figure \[fig:t7.5\_linez\_1\] displays several physical quantities along a line with $(x,y)=(1,0)$ in the axial direction at $t=0.75$. Several features can be identified. First, the HD shock wave front can be seen around $z \sim 13.5$ in the profiles of $\rho$ ([*top*]{} panel), $V_{z}$, and $C_{\rm s}$ ([*bottom*]{} panel). $C_{\rm s}$ is higher than the initial background value $1.29$ in the post-shock region due to shock heating and becomes smaller than $1.29$ at $z \sim 10.7$ due to axial expansion. $V_{z}$ has the similar behavior as $C_{\rm s}$. ![\[fig:t7.5\_linez\_1\] Axial profiles of physical quantities, parallel to the $z$-axis with $(x,\,y)=(1,\,0)$ at $t=7.5$. [*Top*]{}: Density $\rho$ and magnetic field components $(B_r,\,B_\phi,\,B_z)$. [*Bottom*]{}: Sound speed $C_{\rm s}$ and velocity components $(V_r,\,V_\phi,\,V_z)$. The positions of the expanding hydrodynamic shock wave front and the magnetic tower front, which is identified as a tangential discontinuity, are shown in both panels. A horizontal [*solid line*]{} in the [*bottom*]{} panel denotes the initial sound speed (constant throughout the computational domain). ](f5.eps) Second, a magnetic tower front (“tower front” in the following discussions) is located at $z \sim 8.0$, beyond which the magnetic field goes to zero, as seen in the [*top*]{} panel. This indicates that the gas within the magnetic tower jet is separated from the non-magnetized ambient gas beyond the tower front. We regard this front as a tangential discontinuity as the magnetic fields are tangential to the front without the normal component. This is consistent with the fact that the radial and azimuthal field components ($B_{r}$ and $B_{\phi}$) are dominant near the tower front ($z \lesssim 8.0$) but the axial field component $B_{z}$ becomes dominant only for $z<6.0$. The density and pressure show smooth transition through this front though the gradients of $\rho$ and $C_{\rm s}$ are slightly changed there. ![\[fig:t7.5\_linez\_2\] Similar to Fig. \[fig:t7.5\_linez\_1\]. [*Top*]{}: Shown are the gas pressure $p$ ([*dashed line*]{}), magnetic pressure ([*solid line*]{}) $p_{\rm m}$, and total pressure $p_{\rm tot}\,(=p+p_{\rm m})$ ([*light gray thick solid line*]{}). [*Bottom*]{}: Shown are the forces in the axial ($z$) direction: the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}=-\partial/\partial z \left[(B_r^2+B_{\phi}^2)/2 \right]$, ([*solid line*]{}), gas pressure gradient force $F_p = -\partial p/\partial z$ ([*dashed line*]{}), gravitational force $F_{\rm g}=-\rho \partial \psi / \partial R \times |z|/R$ ([*dotted line*]{}), and total force $F_{\rm tot}\,(=F_{{{\bm J}}\times {{\bm B}}}+F_p+F_{\rm g})$ ([*light gray thick solid line*]{}). The position of the reverse slow-mode MHD wave front is also shown in the [*top*]{} panel. ](f6.eps) Third, there is another MHD wave front at $z \sim 7.0$ where $B_{r}$, $C_{\rm s}$, and every velocity component have their local maxima ($\rho$ instead has its local minimum), as seen in both panels. To better understand the nature of this MHD wave front, we plot the axial profiles of pressures and various forces along the line $(x,y) = (1,0)$ at $t=7.5$ in Fig. \[fig:t7.5\_linez\_2\]. The total pressure $p_{\rm tot}$ consists of only the gas pressure $p$ beyond the tower front ($z \gtrsim 8.0$) but is dominated by the magnetic pressure $p_{\rm m}$ behind the MHD wave front ($z \lesssim 7.0$), as seen in the [*top*]{} panel. A transition occurs around $7.0 \lesssim z \lesssim 8.0$, where an increase in $p$ is accompanied by a decrease in $p_{\rm m}$. We therefore identify this as a reverse slow-mode compressional MHD wave front. In magnetic towers, the transition region between gas and magnetic pressures can be identified as a reverse slow wave front in the context of MHD wave structures. It does not depend on the resolution and parameters. So, the reverse slow mode wave (sometimes, it can be steepen into a shock) will always be there. In addition, in Fig. \[fig:t7.5\_linez\_3\], we show several snapshots of $V_z$ and $p$ during $t=7.5 \sim 10.0$ (along the same offset axial path with Figs. \[fig:t7.5\_linez\_1\] and \[fig:t7.5\_linez\_2\]). The axial flow is decelerated by the gravitational force in the post-shock region beyond the tower front as seen in the [*top*]{} panel. On the other hand, the narrow region between the tower front and the reverse slow-mode wave front is accelerated by the magnetic pressure gradient (the “magnetic piston” effect). Note that the reverse slow-mode MHD compressional wave could eventually steepen into the reverse slow-mode MHD shock wave via this nonlinear evolution. Consequently, in the frame co-moving with the reverse slow-mode shock, a strong compression occurs behind the shock wave front and causes a local heating, as seen in the [*bottom*]{} panel and also Fig. \[fig:final\_prdiff\]. This heating could have interesting implications for the enhancement of radiation from radio to X-rays at the terminal part of Fanaroff-Riley type II AGN jets, such as lobes and hot spots, which are generally interpreted as heating caused by the jet terminal shock wave [@BR74; @S74]. ![\[fig:t7.5\_linez\_3\] Similar to Fig. \[fig:t7.5\_linez\_1\], but with selected snapshots during $t=7.5 \sim 10.0$ (each time-interval is equal to 0.5). [*Top*]{}: The axial velocity $V_z$. [*Bottom*]{}: The gas pressure $p$. ](f7.eps) Fourth, the HD shock wave breaks the initial background hydrostatic equilibrium. The passage of the shock wave heats the gas and alters its pressure gradient. As shown in the [*bottom*]{} panel of Fig. \[fig:t7.5\_linez\_2\], the gas pressure gradient force $F_{\rm p}$ stays uupositive at the shock front (which pushes the shock forward), but the total (gravity plus pressure gradient) force $F_{\rm tot}$ becomes negative behind the shock, implying a deceleration of the gas in the axial direction in the post-shock region. This is consistent with Fig. \[fig:t7.5\_linez\_3\]. Deformation of the Jet “Body” in the Radial Direction {#sec:B} ----------------------------------------------------- We next examine the structure and dynamics of the magnetic tower jet along the radial direction in the equatorial plane with $(y,z)=(0,0)$. Figure \[fig:t6.0\_liner\] shows the radial profiles of physical quantities along the $x$-axis at $t=6.0$. The boundary of the magnetic tower jet (“tower edge” in the following discussions) is located at $x \sim 3.0$ where $F_{{{\bm J}}\times {{\bm B}}}$ becomes zero. Two distinct peaks of $C_{\rm s}$ and $V_{x}$ around $x \sim$ 7.8 and 9.5 are visible in the [*top*]{} panel. The first front ($x \sim 9.5$) is the propagating HD shock wave front as we showed in the previous section. The second front ($x \sim 7.8$) also indicates another expanding HD shock wave front generated by a bounce when the magnetic flux pinches in the radial direction caused by the “hoop stress”. This secondary shock front appears only in the radial direction (see also Fig. \[fig:t7.5\_linez\_1\]). $C_{\rm s}$ decreases gradually towards the jet axis in the post-shock region of the secondary shock and becomes smaller than its initial value $1.29$ at $x \sim 5.7$. We can confirm that these shock fronts are purely powered by the gas pressure gradients $F_{p}$, as seen in the [*bottom*]{} panel (twin peaks of $F_{\rm p}$ are shifted a bit behind that of $C_{\rm s}$ in the [*top*]{} panel). ![\[fig:t6.0\_liner\] Radial profiles of physical quantities along the $x$-axis in the equatorial plane with $(y,\,z)=(0,\,0)$ at $t=6.0$. [*Top*]{}: The sound speed $C_{\rm s}$ ([*light gray thick solid line*]{}) and the radial velocity $V_x$ ([*solid line*]{}). [*Bottom*]{}: The forces in the radial ($x$) direction: the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}=-\partial/\partial r \left[(B_{\phi}^2+B_{z}^2)/2\right]-B_{\phi}^2/r$ ([*solid line*]{}), gas pressure gradient force $F_p = -\partial p / \partial r$ ([*dashed line*]{}), gravitational force $F_{\rm g}=-\rho \partial \psi / \partial R \times |x|/R$ ([*dotted line*]{}), and total force $F_{\rm tot}\,(=F_{{{\bm J}}\times {{\bm B}}}+F_p+F_{\rm g})$ ([*light gray thick solid line*]{}). The positions of two expanding hydrodynamic shock wave fronts and the edge of the magnetic tower (tangential discontinuity) are shown in the [*top*]{} panel. A horizontal [*solid line*]{} in the [*top*]{} panel denotes the initial sound speed (constant throughout the computational domain) and a horizontal [*dashed line*]{} in the [*top*]{} panel represents a level “0” for the transverse velocity. ](f8.eps) Part of the region between the secondary HD shock front and the tower edge has $F_{\rm tot}$ being negative (the [bottom]{} panel), meaning that the gas is undergoing gravitational contraction. This is indicated by $V_x < 0$ in the [*top*]{} panel for $x < 5.1$. This behavior is similar to what we have discussed earlier, i.e., the HD shock waves break the background hydrostatic equilibrium, causing a global contraction. Note that this contraction is occurring along the whole jet “body”, e.g., for $|z| \lesssim 4.0$ when $t=7.5$. The [*bottom*]{} panel also helps to address the question on what forces are confining the magnetic fields in the equatorial plane. Since the total magnetic field $F_{{{\bm J}}\times {{\bm B}}}$ stays positive, this means that the inward hoop stress is not strong enough to confine the magnetic fields. Instead, at the tower edge, it is the background gravity that counters the combined effects of magnetic field pressure and a positive pressure gradient (pushing outward). A bit further into the tower edge (at $x \sim 2.1$), however, the pressure force changes from outwardly directed to inwardly directed. Then, both the pressure gradient and the gravitational forces act to counter the outward ${{{\bm J}}\times {{\bm B}}}$ force. This behavior, which is mostly caused by the magnetic tower expanding in a background gravitational field, is different from the usual MHD models for jets where the inwardly directed hoop stress is balanced by the outwardly directed magnetic pressure gradient [@BP82]. ![\[fig:liner\_de\] The radial profiles of density $\rho$ in the equatorial plane with selected snapshots during $t=6.0 \sim 10.0$ (each time-interval is equal to 0.5). The initial profile ($t=0.0$) is also shown ([*dashed line*]{}). ](f9.eps) To investigate the radius evolution of the magnetic tower jet, we show the radial profile of density at the equatorial plane from $t=6.0$ to $ 10.0$ in Fig. \[fig:liner\_de\]. It shows that the radius has an approximately constant contraction speed for $t=6.5 \sim 9.5$. The time scale for contraction is $\tau_{\rm contr} \sim 6$, which is about 7.5 times longer than the local sound-crossing time scale. Force Balance in the Radial Direction {#sec:C} ------------------------------------- We now discuss the jet properties along the radial direction away from the equatorial plane. Figure \[fig:t7.5\_liner\] shows the radial profiles of physical quantities along the $x$-axis with $(y,z)=(0,4)$ at $t=7.5$. The tower edge is now located at $x \sim 3.0$. The plasma $\beta$ in the core of the tower is $\beta \lesssim 0.1$. The [*top*]{} panel shows that the central total pressure (which is dominated by the magnetic pressure) is much bigger than the background “confining” pressure. This illustrates the original suggestion by Lynden-Bell [@L96; @L03] that the hoop stress of the toroidal field component $B_{\phi}$ can act as a pressure amplifier in the central region of the magnetic tower: the pinch effect amplifies $p_{m}$ near the axis of the tower. At the tower edge, however, a finite (albeit small) gas pressure is required [see also @Li01]. ![\[fig:t7.5\_liner\] The radial profiles of physical quantities along the $x$-axis with $(y,\,z)=(0,\,4)$ at $t=7.5$. [*Top*]{}: Similar to the [*top*]{} panel in Fig. \[fig:t7.5\_linez\_2\]. [*Bottom*]{}: Similar to the [*bottom*]{} panel in Fig. \[fig:t6.0\_liner\], but with the centrifugal force $F_{\rm c}=\rho V_{\phi}^2/r$ ([*dash-dotted line*]{}) and the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}$, which can be separated into the magnetic pressure gradient force $-\partial/\partial r \left[(B_{\phi}^2+B_{z}^2)/2\right]$ ([*light gray thick solid line*]{}) and the hoop stress (magnetic tension force) $-B_{\phi}^2/r$ ([*dark gray thick solid line*]{}). The position of the magnetic tower edge (tangential discontinuity) is shown in both panels. ](f10.eps) ![\[fig:t10.0\_liner\_B\] The radial profiles of the magnetic field components $(B_r,\,B_\phi,\,B_z)$, the poloidal magnetic field $B_{\rm p}=\sqrt{B_r^2+B_{z}^2}$, and the quantity $r B_\phi$ along the $x$-axis with $(y,\,z)=(0,\,7)$ at $t=10.0$. The position of the magnetic tower edge (tangential discontinuity) is shown. ](f11.eps) The [*bottom*]{} panel shows the detailed distributions of forces along the radial direction. They show that: - [(Region: $x \gtrsim 3.0$) Beyond the tower edge, the gravitational force $F_{\rm g}$ is slightly stronger than the outwardly directed gas pressure gradient force $F_{p}$, indicating that this edge is subject to the gravitational contraction, as discussed in the previous section.]{} - [(Region: $2.0 \lesssim x \lesssim 3.0$) Interior to the tower edge, the Lorentz force $F_{{{\bm J}}\times {{\bm B}}}$]{} is dominated by the outwardly directed magnetic pressure gradient force $F_{p_{m}}$, and it is also larger than the inwardly directed $F_{p}$, indicating that the outer shell of the magnetic shell should be expanding at the relatively higher $z$, in contrast to the equatorial region ($z=0$) where the tower radius is contracting. The hoop stress $F_{\rm hp}$ plays a minor role in the force balance around this region. Note that ${{\bm J}}^{\rm R}$ flows inside this region. - [(Region: $x \lesssim 2.0$) Inside the jet “body”, Contributions from both $F_{p_{m}}$ and $F_{\rm hp}$ become comparable and nearly cancel each other. The residual $F_{{{\bm J}}\times {{\bm B}}}$ is balanced by $F_{p}$ everywhere in this region. $F_{\rm hp}$ becomes dominant in the Lorentz force at the core part. Note that ${{\bm J}}^{\rm F}$ flows within $x \lesssim 1.0$.]{} In addition, both the $F_{\rm g}$ and the centrifugal force $F_{\rm c}$ play a minor role in terms of the force balance inside the magnetic tower jet. Thus, the interior of the magnetic tower is magnetically dominated but not exactly force-free, i.e., $- \nabla p + {{\bm J}}\times {{\bm B}}\simeq 0~~$. This small but finite pressure gradient force could potentially provide some stabilizing effects on the traditionally kink-unstable twisted magnetic configurations. The detailed examinations of stability properties in magnetic tower jets will be discussed in a forthcoming paper. DISCUSSIONS =========== On scales of $\sim$ tens of kpc to hundreds of kpc, the background density and pressure profiles have a strong influence on the overall morphology of the magnetic tower jet. Most notably, the transverse size of the magnetic tower grows as the jet propagates into a decreasing pressure environment, showing a similar morphology with the jet/lobe configuration of radio galaxies. The radial size of the lobe $r_{\rm lobe}$ can be estimated from the following consideration: Figure \[fig:t10.0\_liner\_B\] shows the radial profiles of magnetic field components parallel to the $x$-axis with $z=7.0$ at $t=10.0$. Together with Fig. \[fig:final\_Jz\], we see that both the poloidal field and especially the poloidal current $I_z$ maintain well collimated around the central axis even at the late stage of the evolution. This implies that the toroidal magnetic fields in the lobe region are distributed roughly as $B_\phi \sim I_z/r$. This is consistent with the results shown in Fig. \[fig:t10.0\_liner\_B\] where $rB_\phi$ have a plateau from $x\approx 1-2.5$. As indicated in Fig. \[fig:t7.5\_liner\], we see that the magnetic pressure and the background pressure try to balance each other at the tower edge. Thus, we expect that $$\label{eq:balance_edge} \frac{B_{\rm P}^{2}+B_{\phi}^{2}}{8 \pi} \sim p_{\rm e}~,$$ where $p_{\rm e}$ is the external gas pressure at the tower edge. When the lobe experiences sufficient expansion, we expect the poloidal field strength to drop much faster than $1/r_{\rm lobe}$. So we have $$\label{eq:balance_edge2} \frac{B_{\phi}^{2}}{8 \pi} \sim \frac{(I_z/r_{\rm lobe})^2}{8\pi}\sim p_{\rm e}~,$$ which gives $$r_{\rm lobe} \sim I_z~p_e^{-1/2}~~.$$ This is generally consistent with our numerical result shown in Fig. \[fig:de\_evo\] though it is difficult to make a firm quantitative determination since the lobes have not expanded sufficiently. To make direct comparison between our simulations and observations, further analysis is clearly needed. Note that for the magnetic tower model, the Alfvén surface is located at the outer edge of the magnetic tower and flow within the magnetic tower is always sub-Alfvénic. This is quite different from the hydromagnetic models where the MHD flow is accelerated and has passed through several critical velocity surfaces, including the Alfvén surface. CONCLUSION ========== By performing 3-D MHD simulations we have investigated in detail the nonlinear dynamics of magnetic tower jets, which propagate through the stably stratified background that is initially in hydrostatic equilibrium. Our simulations, based on the approach developed in @Li06, confirm a number of the global characteristics developed in Lynden-Bell (1996, 2003) and Li et al. (2001). The results presented here give additional details for a dynamically evolving jet in a stratified background. The magnetic tower is made of helical magnetic fields, with poloidal flux and poloidal current concentrated around the central axis. The “returning” portion of the poloidal flux and current is distributed on the outer shell of the magnetic tower. Together they form a self-contained system with magnetic fields and associated currents, being confined by the ambient pressure and gravity. The overall morphology exhibits a confined magnetic jet “body” with an expanded “lobe”. The confinement of the “body” comes jointly from the external pressure and the gravity. The formation of the lobe is due to the expansion of magnetic fields in a decreasing background pressure. A hydrodynamic shock wave initiated by the injection of magnetic energy/flux propagates ahead of the magnetic tower and can break the hydrostatic equilibrium of the ambient medium, causing a global gravitational contraction. As a result, a strong compression occurs in the axial direction between the magnetic tower front and the reverse slow-mode MHD shock wave front that follows. Furthermore, the magnetic tower jets are deformed radially into a slender-shaped body due to the inward-directed flow of the ambient (non-magnetized) gas. The lobe is magnetically dominated and is likely filled with the toroidal magnetic fields generated by the central poloidal current. At the edge of the magnetic tower jet, the outward-directed magnetic pressure gradient force is balanced by the inward-directed gas pressure gradient force, so the radial width of the magnetic tower can be determined jointly by the magnitude of the poloidal current and the magnitude of the external gas pressure. The highly wound helical magnetic field in the magnetic tower never reaches the force-free equilibrium precisely, but obtains radial force-balance, including the gas pressure gradient inside the magnetic tower. The stability of the magnetic tower jets will be considered in our forthcoming papers. Useful discussions with John Finn, Stirling Colgate and Ken Fowler are gratefully acknowledged. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396. It was supported by the Laboratory Directed Research and Development Program at LANL and by IGPP at LANL. Blandford, R. D., & Rees, M. J. 1974, , 169, 395 Blandford, R. D. 1976, , 199, 883 Blandford, R. D., & Znajek, R. L. 1977, , 179, 433 Blandford, R. D., & Payne, D. G. 1982, , 199, 883 Churazov, E., Forman, W., Jones, C., & Böhringer, H. 2003, , 590, 225 Ferrari, A. 1998, , 36, 539 Hsu, S. C., & Bellan, P. M. 2002, , 334, 257 Kato, Y., Hayashi, M. R., & Matsumoto, R. 2004, , 600, 338 Kato, Y., Mineshige, S., & Shibata, K. 2004, , 605, 307 King, I. 1962, , 67, 471 Kudoh, T., Matsumoto, R., & Shibata, K. 2002, , 54, 26 Lebedev, S. V. et al. 2005, , 361, 97 Li, H., Lovelace, R. V. E., Finn, J. M., & Colgate, S. A. 2001, , 561, 915 Li, H., Lapenta, G., Finn, J. M., Li, S., & Colgate, S. A. 2006,  in press (astro-ph/0604469) Li, S., & Li, H. 2003, Technical Report, LA-UR-03-8935, Los Alamos National Laboratory Lovelace, R. V. E. 1976, , 262, 649 Lovelace, R. V. E., Li, H., Koldoba, A. 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I. 2006, preprint (astro-ph/0602419) [rlll]{} $R\,(= \sqrt{x^2+y^2+z^2})$& Length & $R_{0} $ & 5 Kpc\ ${{{\bm V}}}$& Velocity field & $C_{\rm s0} $ & $4.6 \times 10^7$ cm/s\ $t$& Time & $R_{\rm 0}/C_{\rm s0} $ & $1.0 \times 10^7$ yrs\ $\rho$& Density & $\rho_{0} $ & $5.0 \times 10^{-27}$ g/cm$^3$\ $p$& Pressure & $\rho_{0} C_{\rm s0}^{2} $ & $1.4 \times 10^{-11}$ dyn/cm$^2$\ ${{{\bm B}}}$& Magnetic field & $\sqrt{4 \pi \rho_{0} C^{2}_{\rm s0}} $ & 17.1 $\mu$G\

ArXiv

--- abstract: 'In this note we give a precise formulation of “resistance to arbitrary side information” and show that several relaxations of differential privacy imply it. The formulation follows the ideas originally due to Dwork and McSherry, stated implicitly in [@Dw06]. This is, to our knowledge, the first place such a formulation appears explicitly. The proof that relaxed definitions (and hence the schemes of [@DKMMN06; @NRS07; @MKAGV08]) satisfy the Bayesian formulation is new.' author: - | Shiva Prasad KasiviswanathanAdam Smith\ Department of Computer Science and Engineering\ Pennsylvania State University\ e-mail: [$\{$kasivisw,asmith$\}[email protected]]{}\ bibliography: - '../bibfiles/master.bib' title: | A Note on Differential Privacy:\ Defining Resistance to Arbitrary Side Information --- Introduction ============ Privacy is an increasingly important aspect of data publishing. Reasoning about privacy, however, is fraught with pitfalls. One of the most significant is the auxiliary information (also called external knowledge, background knowledge, or side information) that an adversary gleans from other channels such as the web, public records, or domain knowledge. Schemes that retain privacy guarantees in the presence of independent releases are said to [*compose securely*]{}. The terminology, borrowed from cryptography (which borrowed, in turn, from software engineering), stems from the fact that schemes which compose securely can be designed in a stand-alone fashion without explicitly taking other releases into account. Thus, understanding independent releases is essential for enabling modular design. In fact, one would like schemes that compose securely not only with independent instances of themselves, but with [*arbitrary external knowledge*]{}. Certain randomization-based notions of privacy (such as differential privacy [@DMNS06]) are believed to compose securely even in the presence of arbitrary side information. In this note we give a precise formulation of this statement. First, we provide a Bayesian formulation of differential privacy which makes its resistance to arbitrary side information explicit. Second, we prove that the relaxed definitions of [@DKMMN06; @MKAGV08] still imply the Bayesian formulation. The proof is non-trivial, and relies on the “continuity” of Bayes’ rule with respect to certain distance measures on probability distributions. Our result means that the recent techniques mentioned above [@DKMMN06; @CM06; @NRS07; @MKAGV08] can be used modularly with the same sort of assurances as in the case of strictly differentially-private algorithms. Differential Privacy -------------------- Databases are assumed to be vectors in $\mathcal{D}^n$ for some domain $\mathcal{D}$. The Hamming distance ${{d}}({{\mathrm{x}}},{{\mathrm{y}}})$ on $\mathcal{D}^n$ is the number of positions in which the vectors ${{\mathrm{x}}},{{\mathrm{y}}}$ differ. We let $\Pr[\cdot]$ and ${{\mathbb{E}}}[\cdot]$ denote probability and expectation, respectively. Given a randomized algorithm $\mathcal{A}$, we let $\mathcal{A}({{\mathrm{x}}})$ be the random variable (or, probability distribution on outputs) corresponding to input ${{\mathrm{x}}}$. If $\erert$ and $\Q$ are probability measure on a discrete space $D$, the [*statistical difference*]{} (a.k.a. [*total variation distance*]{}) between $\erert$ and $\Q$ is defined as: $${\mathbf{SD}{\left( {{\erert,\Q}} \right)}}= \max_{S \subset D}|\erert[S]-\Q[S)|.$$ \[def:ind\] A randomized algorithm ${\mathcal{A}}$ is said to be ${\epsilon}$-differentialy private if for all databases ${{\mathrm{x}}},{{\mathrm{y}}}\in \mathcal{D}^n$ at Hamming distance at most 1, and for all subsets $S$ of outputs $$\begin{aligned} \Pr[{\mathcal{A}}({{\mathrm{x}}})\in S] \leq e^{{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})\in S].\end{aligned}$$ This definition states that changing a single individual’s data in the database leads to a small change in the [*distribution*]{} on outputs. Unlike more standard measures of distance such as total variation (also called statistical difference) or Kullback-Leibler divergence, the metric here is multiplicative and so even very unlikely events must have approximately the same probability under the distributions ${\mathcal{A}}({{\mathrm{x}}})$ and ${\mathcal{A}}({{\mathrm{y}}})$. This condition was relaxed somewhat in other papers [@DiNi03; @DwNi04; @BDMN05; @DKMMN06; @CM06; @NRS07; @MKAGV08]. The schemes in all those papers, however, satisfy the following relaxation [@DKMMN06]: \[def:indd\] A randomized algorithm ${\mathcal{A}}$ is $({\epsilon},\delta)$-differentially private if for all databases ${{\mathrm{x}}},{{\mathrm{y}}}\in {{\mathcal{D}}}^n$ that differ in one entry, and for all subsets $S$ of outputs, $\Pr[{\mathcal{A}}({{\mathrm{x}}})\in S] \leq e^{{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})\in S]+\delta\,.$ The relaxations used in [@DwNi04; @BDMN05; @MKAGV08] were in fact stronger (i.e., less relaxed) than [Definition \[def:ind\]]{}. One consequence of the results below is that all the definitions are equivalent up to polynomial changes in the parameters, and so given the space constraints we work only with the simplest notion.[^1] Semantics of Differential Privacy {#sec:bayes} ================================= There is a crisp, semantically-flavored interpretation of differential privacy, due to Dwork and McSherry, and explained in [@Dw06]: [*Regardless of external knowledge, an adversary with access to the sanitized database draws the same conclusions whether or not my data is included in the original data.*]{} (the use of the term “semantic” for such definitions dates back to semantic security of encryption [@GM84]). In this section, we develop a formalization of this interpretation and show that the definition of differential privacy used in the line of work this paper follows ([@DiNi03; @DwNi04; @BDMN05; @DMNS06]) is essential in order to satisfy the intuition. We require a mathematical formulation of “arbitrary external knowledge”, and of “drawing conclusions”. The first is captured via a [*prior*]{} probability distribution $b$ on ${{\mathcal{D}}}^n$ ($b$ is a mnemonic for “beliefs”). Conclusions are modeled by the corresponding posterior distribution: given a transcript $t$, the adversary updates his belief about the database ${{\mathrm{x}}}$ using Bayes’ rule to obtain a posterior $\bar{b}$: $$\begin{aligned} \label{eqn:bel} \bar{b}[{{\mathrm{x}}}| t] = \frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=t] b[{{\mathrm{x}}}]}{\sum_{{\mathrm{y}}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t]b[{{\mathrm{y}}}]}\ . \end{aligned}$$ Note that in an interactive scheme, the definition of ${\mathcal{A}}$ depends on the adversary’s choices; for legibility we omit the dependence on the adversary in the notation. Also, for simplicity, we discuss only discrete probability distributions. Our results extend directly to the interactive, continuous case. For a database ${{\mathrm{x}}}$, define ${{\mathrm{x}}}_{-i}$ to be the same vector where position $i$ has been replaced by some fixed, default value in $D$. Any valid value in $D$ will do for the default value. We can then imagine $n+1$ related games, numbered 0 through $n$. In Game 0, the adversary interacts with ${\mathcal{A}}({{\mathrm{x}}})$. This is the interaction that actually takes place between the adversary and the randomized algorithm ${\mathcal{A}}$. In Game $i$ (for $1\leq i \leq n$), the adversary interacts with ${\mathcal{A}}({{\mathrm{x}}}_{-i})$. Game $i$ describes the hypothetical scenario where person $i$’s data is not included. For a particular belief distribution $b$ and transcript $t$, we can then define $n+1$ [*a posteriori*]{} distributions $\bar{b}_0,\dots,\bar{b}_n$, where the $\bar{b}_0$ is the same as $\bar{b}$ (defined in \[eqn:bel\]) and, for larger $i$, the $i$-th belief distribution is defined with respect to Game $i$: $$\bar{b}_i[{{\mathrm{x}}}| t] = \frac{\Pr[{\mathcal{A}}({{\mathrm{x}}}_{-i})=t] b[{{\mathrm{x}}}]}{\sum_{{\mathrm{y}}}\Pr[{\mathcal{A}}({{\mathrm{y}}}_{-i})=t]b[{{\mathrm{y}}}]}.$$ Given a particular transcript $t$, the privacy has been breached if the adversary would draw different conclusions about the world and, in particular, about a person $i$ depending on whether or not $i$’s data was used. It turns out that the exact measure of “different” here does not matter much. We chose the weakest notion that applies, namely statistical difference. We say there is a problem for transcript $t$ if the distributions $\bar{b}_0[\cdot|t]$ and $\bar{b}_i[\cdot|t]$ are far apart in statistical difference. We would like to avoid this happening for any potential participant. This is captured by the following definition. \[def:sem\] A randomized algorithm ${\mathcal{A}}$ is said to be ${\epsilon}$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, for all databases ${{\mathrm{x}}}\in \mathcal{D}^n$, for all possible transcripts $t$, and for all $i = 1,\ldots,n$$\mathrm{:}$ $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ Dwork and McSherry proposed the notion of semantic privacy, informally, and observed that it is equivalent to differential privacy. We now formally show that the notions of ${\epsilon}$-differential privacy (Definition \[def:ind\]) and ${\epsilon}$-semantic privacy (Definition \[def:sem\]) are very closely related. (Dwork-McSherry) \[thm:eind\] ${\epsilon}$-differential privacy implies $\bar{\epsilon}$-semantic privacy, where $\bar{\epsilon}=e^{{\epsilon}}-1$. $\bar{\epsilon}/2$-semantic privacy implies $2{\epsilon}$-differential privacy. We extend the previous Bayesian formulation to capture situations where bad events can occur with some negligible probability (say, $\delta)$. We relax ${\epsilon}$-semantic privacy to $({\epsilon},\delta)$-semantic privacy and show that it is closely related to $({\epsilon},\delta)$-differential privacy. A randomized algorithm is $({\epsilon},\delta)$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, with probability at least $1-\delta$ over pairs $({{\mathrm{x}}},t)$, where the database ${{\mathrm{x}}}$ is drawn according to $b$, and transcript $t$ is drawn according to ${\mathcal{A}}({{\mathrm{x}}})$, and for all $i =1,\dots,n$: $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ This definition is only interesting when ${\epsilon}>\delta$; otherwise just use statistical difference $2\delta$ and leave ${\epsilon}=0$. Below, we assume ${\epsilon}>\delta$. In fact, in many of the proofs we will be assuming that $\delta$ is a negligible function (of $O(1/n^2)$). In Appendix A, we provide another related definition of $({\epsilon},\delta)$-semantic privacy. \[thm:ind2sdp\] (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy for arbitrary (not necessarily informed) beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$. $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. Some Properties of $({\epsilon},\delta)$-Differential Privacy ============================================================= We now describe some properties of $({\epsilon},\delta)$-differential privacy that would be useful later on. This section could be of independent interest. Instead of restricting ourselves to outputs of randomized algorithms, we consider a more general definition of $({\epsilon},\delta)$-differential privacy. \[$({\epsilon},\delta)$-indistinguishability\] Two random variables $X,Y$ taking values in a set $D$ are $({\epsilon},\delta)$-indistinguishable if for all sets $S\subseteq D$, $$\begin{aligned} \Pr[X\in S] \leq e^{\displaystyle {\epsilon}} \Pr[Y\in S] + \delta \ \ \ \ \mbox{ and } \ \ \ \ \Pr[Y\in S] \leq e^{\displaystyle {\epsilon}} \Pr[X\in S] + \delta.\end{aligned}$$ We will also be using a simpler variant of $({\epsilon},\delta)$-indistinguishability, which we call [*point-wise*]{} $({\epsilon},\delta)$-indistinguishability. Claim \[lem:proof\] (Parts \[it:pw2ind\] and \[it:ind2pw\]) shows that $({\epsilon},\delta)$-indistinguishability and point-wise $({\epsilon},\delta)$-indistinguishability are almost equivalent. \[Point-wise $({\epsilon},\delta)$-indistinguishability\] Two random variables $X$ and $Y$ are point-wise $({\epsilon},\delta)$-indistinguishable if with probability at least $1-\delta$ over $a$ drawn from either $X$ or $Y$, we have: $$e^{-{\epsilon}}\Pr[Y=a] \leq \Pr[X=a] \leq e^{{\epsilon}} \Pr[Y=a].$$ The following are useful facts about indistinguishability.[^2] \[lem:proof\] \[it:pw2ind\] If $X,Y$ are point-wise $({\epsilon},\delta)$-indistinguishable then they are $({\epsilon},\delta)$-indistinguishable. \[it:ind2pw\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable then they are point-wise $(2{\epsilon},\frac{2\delta}{e^{{\epsilon}}{\epsilon}})$-indistinguishable. \[it:ajoint\] Let $X$ be a random variable on $D$. Suppose that for every $a \in D$, ${\mathcal{A}}(a)$ and ${\mathcal{A}}'(a)$ are $({\epsilon},\delta)$-indistinguishable (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then the pairs $(X,{\mathcal{A}}(X))$ and $(X,{\mathcal{A}}'(X))$ are $({\epsilon},\delta)$-indistinguishable. \[it:joint\] Let $X$ be a random variable. Suppose with probability at least $1-\delta$ over $a {\leftarrow}X$ $(a\,$ drawn from $X)$, ${\mathcal{A}}(a)$ and ${\mathcal{A}}'(a)$ are $({\epsilon},\delta)$-indistinguishable (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then the pairs $(X,\ {\mathcal{A}}(X))$ and $(X,\ {\mathcal{A}}'(X))$ are $({\epsilon},2\delta)$-indistinguishable. \[it:prob\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable and ${{\cal G}}$ is some randomized algorithm, then ${{\cal G}}(X)$ and ${{\cal G}}(Y)$ are $({\epsilon},\delta)$-indistinguishable. \[it:sd\] If $X,Y$ are $({\epsilon},\delta)$-indistinguishable, then ${\mathbf{SD}{\left( {{X,Y}} \right)}} \leq \bar{\epsilon}+\delta$, where $\bar{\epsilon}=e^{{\epsilon}}-1$. [*Proof of Part \[it:pw2ind\].*]{} Let $Bad$ be the set of [*bad*]{} values of $a$, that is $$Bad = \{a \,:\, \Pr[X=a] < e^{-{\epsilon}} \Pr[Y=a] \mbox{ or } \Pr[X=a]> e^{{\epsilon}} \Pr[Y=a] \}.$$ By definition, $\Pr[X\in Bad]\leq \delta$. Now consider any set $S$ of outcomes. $$\Pr[X\in S] \leq \Pr[X \in S \setminus Bad] + \Pr[X\in Bad].$$ The first term is at most $e^{\epsilon}\Pr[Y\in S \setminus Bad]\leq e^{\epsilon}\Pr[Y\in S]$. Hence, $\Pr[X \in S] \leq e^{\epsilon}\Pr[Y\in S] + \delta$, as required. The case of $\Pr[Y\in S]$ is symmetric. Therefore, $X$ and $Y$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:ind2pw\].*]{} Let $S= \{a \,:\, \Pr[X=a] > e^{2{\epsilon}} \Pr[Y=a]\}$. Then, $$\Pr[X \in S] > e^{2{\epsilon}} \Pr[Y \in S] > e^{{\epsilon}}(1+{\epsilon})\Pr[Y \in S] \Rightarrow \Pr[X \in S] - e^{{\epsilon}}\Pr[Y \in S] > {\epsilon}e^{{\epsilon}} \Pr[Y \in S].$$ Since, $\Pr[X \in S] - e^{{\epsilon}}\Pr[Y \in S] \leq \delta$, we mush have ${\epsilon}e^{{\epsilon}} \Pr[Y \in S] < \delta$. A similar argument when considering the set $S'=\{a \,:\, \Pr[X=a] < e^{-2{\epsilon}} \Pr[Y=a]\}$ shows that ${\epsilon}e^{{\epsilon}} \Pr[Y \in S'] < \delta$. Putting both arguments together, $\Pr[Y \in S \cup S'] \leq 2\delta/({\epsilon}e^{{\epsilon}})$. Therefore, with probability at least $1-2\delta/(e^{{\epsilon}}{\epsilon})$ for any $a$ drawn from either $X$ or $Y$ we have: $e^{-2{\epsilon}}\Pr[Y=a] \leq \Pr[X=a] \leq e^{2{\epsilon}} \Pr[Y=a]$.\ [*Proof of Part \[it:ajoint\].*]{} Let $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ be random variables on $D \times E$. Let $S$ be an arbitrary subset of $D \times E$ and, for every $a \in D$, define $S_a =\{b \in E\,:\, (a,b) \in S\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X)) \in S] &\leq& \sum_{a \in D}\Pr[{{\cal A}}(X) \in S_a \,:\, X=a]\Pr[X=a] \\ & < & \sum_{a \in D}(e^{{\epsilon}}\Pr[{{\cal A}}'(X) \in S_a \,:\, X=a]+\delta)\Pr[X=a] \\ & < & \delta + e^{{\epsilon}}\Pr[(X,{{\cal A}}'(X)) \in S].\end{aligned}$$ By symmetry, we also have $\Pr[(X,{{\cal A}}'(X)) \in S] < \delta+\Pr[(X,{{\cal A}}(X)) \in S]$. Since $S$ was arbitrary, $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:joint\].*]{} Let $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ be random variables on $D \times E$. Let $T \subset D$ be the set of $a$’s for which ${{\cal A}}(a) \leq e^{{\epsilon}} {{\cal A}}'(a)$. Now, let $S$ be an arbitrary subset of $D \times E$ and, for every $a \in D$, define $S_a =\{b \in E\,:\, (a,b) \in S\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X)) \in S] &\leq& \Pr[X \notin T] + \sum_{a \in T}\Pr[{{\cal A}}(X) \in S_a \,:\, X=a]\Pr[X=a] \\ & < & \delta + \sum_{a \in T}(e^{{\epsilon}}\Pr[{{\cal A}}'(X) \in S_a \,:\, X=a]+\delta)\Pr[X=a] \\ & < & 2\delta + e^{{\epsilon}}\Pr[(X,{{\cal A}}'(X)) \in S].\end{aligned}$$ By symmetry, we also have $\Pr[(X,{{\cal A}}'(X)) \in S] < 2\delta+\Pr[(X,{{\cal A}}(X)) \in S]$. Since $S$ was arbitrary, $(X,{{\cal A}}(X))$ and $(X,{{\cal A}}'(X))$ are $({\epsilon},2\delta)$-indistinguishable.\ [*Proof of Part \[it:prob\].*]{} Let $D$ be some domain. A randomized procedure ${{\cal G}}$ is a pair ${{\cal G}}=(g,R)$, where $R$ is a random variable on some set $E$ and $g$ is a function from $D \times E$ to any set $F$. If $X$ is a random variable on $D$, then ${{\cal G}}(X)$ denotes the random variable on $F$ obtained by sampling $X \otimes R$ and applying $g$ to the result, where the symbol $\otimes$ denotes the tensor product. Now for any set $S \subset F$, $$\begin{aligned} \lefteqn{\Pr[{{\cal G}}(X) \in S] - e^{{\epsilon}}\Pr[{{\cal G}}(Y) \in S]}\\ &=& \Pr[g(X \otimes R) \in S] - e^{{\epsilon}} \Pr[g(Y \otimes R) \in S] \\ &=&\Pr[X\otimes R \in g^{-1}(S)] -e^{{\epsilon}} \Pr[Y\otimes R \in g^{-1}(S)] \\ & \leq & \sum_{r \in E} \Pr[X \in S_r \,:\, R=r]\Pr[R=r] - e^{{\epsilon}} \sum_{r \in E} \Pr[Y \in S_r \,:\, R=r]\Pr[R=r] \\ & =& \sum_{r \in E}(\Pr[X \in S_r \,:\, R=r]-e^{{\epsilon}}\Pr[Y \in S_r \,:\, R=r])\Pr[R=r] \\ &\leq &\sum_{r \in E}\delta \Pr[R=r] =\delta.\end{aligned}$$ By symmetry, we also have $\Pr[{{\cal G}}(Y) \in S] - e^{{\epsilon}}\Pr[{{\cal G}}(X) \in S] \leq \delta$. Since $S$ was arbitrary, ${{\cal G}}(X)$ and ${{\cal G}}(Y)$ are $({\epsilon},\delta)$-indistinguishable.\ [*Proof of Part \[it:sd\].*]{} Let $X$ and $Y$ be random variables on $D$. By definition ${\mathbf{SD}{\left( {{X,Y}} \right)}}=\max_{S \subset D}|\Pr[X \in S] - \Pr[Y \in S]|$. For any set $S \subset D$, $$\begin{aligned} \lefteqn{2|\Pr[X \in S]-\Pr[Y \in S]|} \\ &=& \left |\Pr[X \in S]-\Pr[Y \in S]\right | + \left |\Pr[X \notin S]-\Pr[Y \notin S] \right | \\ &=& \left |\sum_{c \in S}(\Pr[X =c]-\Pr[Y =c])\right | + \left |\sum_{c \notin S}(\Pr[X =c]-\Pr[Y =c])\right | \\ &\leq& \sum_{c \in S}\left |\Pr[X =c]-\Pr[Y =c] \right | + \sum_{c \notin S}\left |\Pr[X =c]-\Pr[Y =c]\right | \\ &=& \sum_{c \in D } \left |\Pr[X =c]-\Pr[Y =c] \right | \\ &\leq& \sum_{c \in D}(e^{{\epsilon}}\Pr[Y =c] + \delta - \Pr[Y =c]) + \sum_{c \in D}(e^{{\epsilon}}\Pr[X =c] +\delta - \Pr[X =c]) \\ &=& 2\delta + (e^{{\epsilon}}-1) \sum_{c \in D}\Pr[Y =c] + (e^{{\epsilon}}-1) \sum_{c \in D}\Pr[X =c] \\ &=& 2(e^{{\epsilon}}-1)+2\delta=2\bar{{\epsilon}}+ 2\delta.\end{aligned}$$ This implies that $|\Pr[X \in S]-\Pr[Y \in S]| \leq \bar{\epsilon}+ \delta$. Since the above inequality holds for every $S \subset D$, it immediately follows that the statistical difference between $X$ and $Y$ is at most $\bar{{\epsilon}}+\delta$. Proofs of Theorems \[thm:eind\] and \[thm:ind2sdp\] =================================================== This section is devoted to proving Theorems \[thm:eind\] and \[thm:ind2sdp\]. For convenience we restate the theorem statements. \[Dwork-McSherry\] ${\epsilon}$-differential privacy implies $\bar{\epsilon}$-semantic privacy, where $\bar{\epsilon}=e^{{\epsilon}}-1$. $\bar{\epsilon}/2$-semantic privacy implies $2{\epsilon}$-differential privacy. Consider any database ${{\mathrm{x}}}$. Consider belief distributions $\bar{b}_0[{{\mathrm{x}}}|t]$ and $\bar{b}_i[{{\mathrm{x}}}|t]$. differential privacy implies that the ratio of $\bar{b}_0[{{\mathrm{x}}}|t]$ and $\bar{b}_i[{{\mathrm{x}}}|t]$ is within $e^{\pm {\epsilon}}$ on every point, i.e., for every $i$ and for every possible transcript $t$: $$e^{-{\epsilon}}\bar{b}_i[{{\mathrm{x}}}|t] \leq \bar{b}_0[{{\mathrm{x}}}|t] \leq e^{{\epsilon}}\bar{b}_i[{{\mathrm{x}}}|t].$$ In the remainder of the proof we fix $i$ and $t$. Substituting $\delta=0$ in Claim \[lem:proof\] (part \[it:sd\]), implies that ${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t],\bar{b}_i[{{\mathrm{x}}}|t]}} \right)}}=\bar{\epsilon}$. To see that $\bar{{\epsilon}}$-semantic privacy implies $2{\epsilon}$-differential privacy, consider a belief distribution $b$ which is uniform over two databases ${{\mathrm{x}}},{{\mathrm{y}}}$ which are at Hamming distance of one. Let $i$ be the position in which ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ differ. The distribution $\bar{b}_i[\cdot|t]$ will be uniform over ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ since they induce the same distribution on transcripts in Game $i$. This means that $\bar{b}_0[\cdot|t]$ will assign probabilities $1/2 \pm \bar{\epsilon}/2$ to each of the two databases (follows from ${\epsilon}$-semantic privacy definition). Working through Bayes’ rule shows that $$\frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=t]}{\Pr[{\mathcal{A}}({{\mathrm{y}}})=t]}= \frac{\Pr[\bar{b}_0[{{\mathrm{x}}}|t]={{\mathrm{x}}}]}{\Pr[\bar{b}_0[{{\mathrm{y}}}|t]={{\mathrm{x}}}]} \leq \frac{{\frac{1}{2}}(1+\bar{\epsilon})}{{\frac{1}{2}}(1-\bar{\epsilon})} \leq e^{2{\epsilon}}.$$ This implies that ${\mathcal{A}}$ is point-wise $2{\epsilon}$-differentialy private. Using Claim \[lem:proof\] (part \[it:pw2ind\]), implies that ${\mathcal{A}}$ is $2{\epsilon}$-differentialy private. We will use the following lemma to establish connections between $({\epsilon},\delta)$-differential privacy and $({\epsilon},\delta)$-semantic privacy. Let $B|_{A=a}$ denote the conditional distribution of $B$ given that $A=a$ for jointly distributed random variables $A$ and $B$. \[lem:bayes\] Suppose two pairs of random variables $(X,{\mathcal{A}}(X))$ and $(Y,{\mathcal{A}}'(Y))$ are $({\epsilon},\delta)$-differentialy private (for some randomized algorithms ${\mathcal{A}}$ and ${\mathcal{A}}'$). Then with probability at least $1-\delta''$ over $t{\leftarrow}{\mathcal{A}}(X)$ (equivalently $t {\leftarrow}{{\cal A}}'(Y)$), the random variables $X|_{{\mathcal{A}}(X) =t}$ and $Y|_{{\mathcal{A}}'(Y)=t}$ are $(\hat{\epsilon},\hat\delta)$-differentialy private with $\hat{\epsilon}=3{\epsilon}$, $\hat\delta=2\sqrt{\delta}$, and $\delta''=\sqrt{\delta}+\frac{2\delta}{{\epsilon}e^{{\epsilon}}}=O(\sqrt{\delta})$. Let $(X,{{\cal A}}(X))$ and $(Y,{{\cal A}}'(Y))$ be random variables on $D \times E$. The first observation is that ${{\cal A}}(X)$ and ${{\cal A}}(Y)$ are $({\epsilon},\delta)$-differentialy private. To prove that consider any set $P \in E$, $$\begin{aligned} \Pr[{{\cal A}}(X) \in P] & =& \Pr[(X,{{\cal A}}(X)) \in D \times P] \leq e^{{\epsilon}}\Pr[(Y,{{\cal A}}'(Y)) \in D \times P] + \delta \\ &=& e^{{\epsilon}}\Pr[{{\cal A}}'(Y) \in P] + \delta.\end{aligned}$$ Since $P$ was arbitrary, ${{\cal A}}(X)$ and ${{\cal A}}'(Y)$ are $({\epsilon},\delta)$-differentialy private. In the remainder of the proof, we will use the notation $X|_t$ for $X|_{{{\cal A}}(X)=t}$ and $Y|_t$ for $Y|_{{{\cal A}}'(Y)=t}$. Define, $$\begin{aligned} &Bad_0 = \{a\,:\, e^{-2{\epsilon}}\Pr[{{\cal A}}'[Y]=a] > \Pr[{{\cal A}}(X)=a] > e^{2{\epsilon}} \Pr[{{\cal A}}'[Y]=a] \}& \\ &Bad_1=\{a \, : \, \exists S \subset D \mbox{ such that } \Pr[X|_a \in S] > e^{\hat{\epsilon}}\Pr[Y|_a \in S] + \hat\delta\}& \\ &Bad_2=\{a \, : \, \exists S \subset D \mbox{ such that } \Pr[Y|_a \in S] > e^{\hat{\epsilon}}\Pr[X|_a \in S] + \hat\delta\}.& \end{aligned}$$ We need an upper bound for the probabilities $\Pr[{{\cal A}}(X) \in Bad_1 \cup Bad_2]$ and $\Pr[{{\cal A}}'(Y) \in Bad_1 \cup Bad_2]$. We know from Claim \[lem:proof\] (part  \[it:ind2pw\]), that $$\Pr[{{\cal A}}(X) \in Bad_0] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}} \ \ \mbox{ and } \ \ \Pr[{{\cal A}}'(Y) \in Bad_0] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}}.$$ Note that from the initial observation ${{\cal A}}(X)$ and ${{\cal A}}'(Y)$ are $({\epsilon},\delta)$-differentialy private, therefore the condition required for applying Claim \[lem:proof\] (part \[it:ind2pw\]) holds. Now define, $$Bad_1'=Bad_1 \setminus Bad_0 \ \ \mbox{ and } \ \ Bad_2'=Bad_2 \setminus Bad_0.$$ For each $a \in Bad_1'$ and $T \subset D \times E$, define $S_a=\{b \in D \,:\, (b,a) \in T\}$. Define $T_1 = S_a \times \bigcup_{a \in Bad_1'} \{a\}$. $$\begin{aligned} \Pr[(X,{{\cal A}}(X))\in T_1] &=& \sum_{a \in Bad_1'} \Pr[X \in S_a \,:\, {{\cal A}}(X)=a] \Pr[{{\cal A}}(X)=a] \\ &>& \sum_{a \in Bad_1'} (e^{\hat{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]+\hat\delta)\Pr[{{\cal A}}(X)=a] \\ &=&\sum_{a \in Bad_1'} e^{\hat{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]\Pr[{{\cal A}}(X)=a] + \hat\delta\sum_{a \in Bad_1'} \Pr[{{\cal A}}(X)=a] \\ &=& \sum_{a \in Bad_1'} e^{3{\epsilon}}\Pr[Y \in S_a \,:\, {{\cal A}}'(Y)=a]e^{-2{\epsilon}} \Pr[{{\cal A}}'(Y)=a] + \hat\delta\Pr[{{\cal A}}(X) \in Bad_1'] \\ &=& e^{{\epsilon}}\Pr[(Y,{{\cal A}}'(Y)) \in T_1] + \hat\delta\Pr[{{\cal A}}(X) \in Bad_1'].\end{aligned}$$ The inequality follows because of the definition of $Bad_1'$. By $({\epsilon},\delta)$-differential privacy, $\Pr[(X,{{\cal A}}(X))\in T_1] \leq e^{{\epsilon}}\Pr[(Y,{{\cal A}}(X)) \in T_1] + \delta$. Therefore, $$\hat\delta\Pr[{{\cal A}}(X) \in Bad_1'] \leq \delta \Rightarrow \Pr[{{\cal A}}(X) \in Bad_1'] \leq \delta/\hat\delta.$$ Similarly, $\Pr[{{\cal A}}(X) \in Bad_2'] \leq \delta/\hat\delta.$ Finally, $$\begin{aligned} \Pr[{{\cal A}}(X) \in Bad_1 \cup Bad_2] &\leq& \Pr[{{\cal A}}(X) \in Bad_0] +\Pr[{{\cal A}}(X) \in Bad_1']+\Pr[{{\cal A}}(X) \in Bad_2'] \\ &=& \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\frac{\delta}{\hat\delta}+\frac{\delta}{\hat\delta} = \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\sqrt{\delta}.\end{aligned}$$ By symmetry, we also have $\Pr[{{\cal A}}'(Y) \in Bad_1 \cup Bad_2] \leq \frac{2\delta}{{\epsilon}e^{{\epsilon}}}+\sqrt{\delta}$. Therefore, with probability at least $1-\delta''$, $X|_{t}$ and $Y|_{t}$ are $(\hat{\epsilon},\hat\delta)$-differentialy private. The following corollary follows by using the above proposition (with $Y=X$) in conjunction with Claim \[lem:proof\] (part \[it:sd\]). \[cor:bayes\] Let $(X,{\mathcal{A}}(X))$ and $(X,{\mathcal{A}}'(X))$ be $({\epsilon},\delta)$-differentialy private. Then, with probability at least $1-\delta''$ over $t{\leftarrow}{\mathcal{A}}(X)$ $($equivalently $t {\leftarrow}{{\cal A}}'(X)$$)$, the statistical difference between $X|_{{\mathcal{A}}(X) =t}$ and $X|_{{\mathcal{A}}'(X)=t}$ is at most $e^{\hat{\epsilon}}-1+\hat\delta$ with $\hat{\epsilon}=3{\epsilon}$, $\hat\delta=2\sqrt{\delta}$, and $\delta''=O(\sqrt{\delta})$. (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy for arbitrary (not necessarily informed) beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$. $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. Let ${\mathcal{A}}$ be a $({\epsilon},\delta)$-differentialy private algorithm. Let $b$ be any belief distribution. From Claim \[lem:proof\] (part \[it:ajoint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},\delta)$-differentialy private. Let $\delta''=O(\sqrt{\delta})$. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, for any ${{\mathrm{x}}}{\leftarrow}b$, with probability at least $(1-\delta'')$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that for any ${{\mathrm{x}}}{\leftarrow}b$ with probability at least $1-n\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for $b$. Since $b$ was arbitrary, we get that (${\epsilon},\delta$)-differential privacy implies $({\epsilon}',\delta')$-semantic privacy. To see that $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy, consider a belief distribution $b$ which is uniform over two databases ${{\mathrm{x}}},{{\mathrm{y}}}$ which are at Hamming distance of one. The proof idea is same as in Theorem \[thm:eind\]. Let $i$ be the position in which ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ differ. Let $\bar{\mathcal{A}}$ be an algorithm that with probability $1/2$ draws an output from ${\mathcal{A}}({{\mathrm{x}}})$ and with probability $1/2$ draws an output from ${\mathcal{A}}({{\mathrm{y}}})$. Consider a transcript $t$ drawn from $\bar{\mathcal{A}}$. The distribution $\bar{b}_i[\cdot|t]$ will be uniform over ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$ since they induce the same distribution on transcripts in Game $i$. This means that with probability at least $1-\delta$ over $t {\leftarrow}\bar{\mathcal{A}}$, $\bar{b}_0[\cdot |t]$ will assign probabilities $1/2\pm \bar{\epsilon}/2$ to each of the two databases. Working through Bayes’ rule as in Theorem \[thm:eind\] shows that $\bar{\mathcal{A}}$ is point-wise $(2{\epsilon},\delta)$-differentialy private (with probability at least at least $1-2\delta$ of $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, $e^{-2{\epsilon}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t] \leq \Pr[{\mathcal{A}}({{\mathrm{x}}})=t] \leq e^{2{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})=t]$). Therefore, with probability at least $1-\delta$ of $t {\leftarrow}\bar{\mathcal{A}}$, $e^{-2{\epsilon}}\Pr[{\mathcal{A}}({{\mathrm{y}}})=t] \leq \Pr[{\mathcal{A}}({{\mathrm{x}}})=t] \leq e^{2{\epsilon}} \Pr[{\mathcal{A}}({{\mathrm{y}}})=t]$. Similarly, for $t {\leftarrow}{\mathcal{A}}({{\mathrm{y}}})$. This implies that ${\mathcal{A}}$ is point-wise $(2{\epsilon},2\delta)$-differentialy private. Using Claim \[lem:proof\] (part \[it:pw2ind\]), implies that ${\mathcal{A}}$ is $(2{\epsilon},2\delta)$-differentialy private. Discussion and Consequences =========================== [Theorem \[thm:ind2sdp\]]{} states that the relaxations notions of differential privacy used in some previous work still imply privacy in the face of arbitrary side information. This is [*not*]{} the case for [*all*]{} possible relaxations, even very natural ones. For example, if one replaced the multiplicative notion of distance used in differential privacy with total variation distance, then the following “sanitizer” would be deemed private: choose an index $i\in\{1,\dots,n\}$ uniformly at random and publish the entire record of individual $i$ together with his or her identity (example 2 in [@DMNS06]). Such a “sanitizer” would not be meaningful at all, regardless of side information. Theorems \[thm:ind2sdp\] and A.3 give some qualitative improvements over existing security statements. Theorem A.3 implies that the claims of [@DiNi03; @DwNi04; @BDMN05] can be strengthened to hold for *all* predicates of the input simultaneously (a switch in the order of quantifiers). The strengthening does come at some loss in parameters since $\delta$ is increased. This incurs a factor of 2 in ${\log{\left( {\tfrac{1}{\delta}} \right)}}$, or a factor of $\sqrt{2}$ in the standard deviation. More significantly, [Theorem \[thm:ind2sdp\]]{} shows that noise processes with negligible probability of bad events have nice differential privacy guarantees even for adversaries who are not necessarily informed. There is a hitch however only adversaries whose beliefs somehow represent reality, i.e. for whom the real database is somehow “representative" of the adversary’s view can be said to learn nothing. Finally, the techniques used to prove [Theorem \[thm:ind2sdp\]]{} can also be used to analyze schemes which do not provide privacy for [*all*]{} pairs of neighboring databases ${{\mathrm{x}}}$ and ${{\mathrm{y}}}$, but rather only for [*most*]{} such pairs (remember that neighboring databases are the ones that differ in one entry). Specifically, it is sufficient that those databases where the “differential privacy” condition fails occur only with small probability. \[thm:dsemantic\] Let ${\mathcal{A}}$ be a randomized algorithm. Let $$\mathcal{E} = \{{{\mathrm{x}}}: \forall \mbox{ neighbors }{{\mathrm{y}}}\mbox{ of } {{\mathrm{x}}}, {\mathcal{A}}({{\mathrm{x}}}) \mbox{ and } {\mathcal{A}}({{\mathrm{y}}}) \mbox{ are } ({\epsilon},\delta)\mbox{-differentialy private}\}.$$ Then ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for any belief distribution $b$ such that $b[\mathcal{E}] = \Pr_{{{\mathrm{x}}}{\leftarrow}b}[{{\mathrm{x}}}\in \mathcal{E}] \geq 1-\delta$ with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta'=O(n\sqrt{\delta})$. Let $b$ be a belief distribution with $b[\mathcal{E}] \geq 1-\delta$. Let $\delta''=O(\sqrt{\delta})$. From Claim \[lem:proof\] (part \[it:joint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},2\delta)$-differentialy private. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, with probability at least $(1-\delta'')$ over pairs $({{\mathrm{x}}},t)$ where ${{\mathrm{x}}}{\leftarrow}b$ and $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that with probability at least $1-n\delta''$ over pairs $({{\mathrm{x}}},t)$ where ${{\mathrm{x}}}{\leftarrow}b$ and $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for belief distribution $b$. Let $LS_f(\cdot)$ denote the local sensitivity of function $f$ (defined in [@NRS07]). Let $Lap(\lambda)$ denote the Laplacian distribution. This distribution has density function $h(y) \propto \exp(-|y|/\lambda)$, mean $0$, and standard deviation $\lambda$. Using the Laplacian noise addition procedure of [@DMNS06; @NRS07], along with Theorem \[thm:dsemantic\] we get, Let $\mathcal{E} =\{ {{\mathrm{x}}}\,:\, LS_f({{\mathrm{x}}}) \leq s \}$. Let ${\mathcal{A}}({{\mathrm{x}}}) = f({{\mathrm{x}}}) + \text{Lap}\left(\frac{s}{{\epsilon}}\right )$. Let $b$ be a belief distribution such that $b[\mathcal{E}] = \Pr_{{{\mathrm{x}}}{\leftarrow}b}[{{\mathrm{x}}}\in \mathcal{E}] \geq 1-\delta$. Then ${\mathcal{A}}$ satisfies $({\epsilon}',\delta')$-semantic privacy for the belief distribution $b$ with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta'=O(n\sqrt{\delta})$. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful for helpful discussions with Cynthia Dwork, Frank McSherry, Moni Naor, Kobbi Nissim, and Sofya Raskhodnikova. Appendix A: Another View of Semantic Privacy {#appendix-a-another-view-of-semantic-privacy .unnumbered} ============================================ In this section, we discuss another possible definition of $({\epsilon},\delta)$-semantic privacy. Even though this definition seems to be the more desirable one, it also seems hard to achieve. \[reality-oblivious (${\epsilon},\delta$)-semantic privacy\] A randomized algorithm is reality-oblivious $({\epsilon},\delta)$-semantically private if for all belief distributions $b$ on $\mathcal{D}^n$, for all databases ${{\mathrm{x}}}\in \mathcal{D}^n$, with probability at least $1-\delta$ over transcripts $t$ drawn from ${{\cal A}}({{\mathrm{x}}})$, and for all $i=1,\dots,n$: $${\mathbf{SD}{\left( {{\bar{b}_0[{{\mathrm{x}}}|t]\ ,\ \bar{b}_i[{{\mathrm{x}}}|t]\ }} \right)}} \leq {\epsilon}.$$ We first prove if the adversary has arbitrary beliefs, then $({\epsilon},\delta)$-differential privacy doesn’t provide any reasonable reality-oblivious $({\epsilon}',\delta')$-semantic privacy guarantee. \[lem:noind2sdp\][^3] (${\epsilon},\delta$)-differential privacy does not imply reality-oblivious $({\epsilon}',\delta')$-semantic privacy for any reasonable values of ${\epsilon}'$ and $\delta'$. This counterexample is due to Dwork and McSherry: suppose that the belief distribution is uniform over $\{(0^n),(1,0^{n-1})\}$, but that real database is $(1^{n})$. Let the database ${{\mathrm{x}}}=(x_1,\dots,x_n)$. Say we want to reveal $f({{\mathrm{x}}})=\sum_ix_i$. Adding Gaussian noise with variance $\sigma^2={\log{\left( {\tfrac{1}{\delta}} \right)}}/{\epsilon}^2$ satisfies $({\epsilon},\delta)$-differential privacy (refer [@DMNS06; @NRS07] for details). However, with overwhelming probability the output will be close to $n$, and this will in turn induce a very non-uniform distribution over $\{(0^n),(1,0^{n-1})\}$ since $(1,0^{n-1})$ is exponentially (in $n$) more likely to generate a value near $n$ than $(0^n)$. More precisely, due to the Gaussian noise added, $$\frac{\Pr[{\mathcal{A}}({{\mathrm{x}}})=n \, | \, {{\mathrm{x}}}=(0^n)]}{\Pr[{\mathcal{A}}({{\mathrm{x}}})= n \, | \, {{\mathrm{x}}}=(1,0^{n-1})]} = \frac{\exp\left (\frac{-n^2}{2\sigma} \right)}{\exp \left(\frac{-(n-1)^2}{2\sigma}\right )} = \exp\left (\frac{-2n+1}{2\sigma} \right).$$ Therefore, given that the output is close to $n$, the posterior distribution of the adversary would be exponentially more biased toward $(1,0^{n-1})$ than $(0^n)$. Hence, it is exponentially far away from the prior distribution which was uniform. On the other hand, if the adversary believes he is seeing ${\mathcal{A}}({{\mathrm{x}}}_{-1})$, then no update will occur and the posterior distribution will remain uniform. Since the posterior distributions in these two situations are exponentially far apart (one exponentially far from uniform, other uniform), it shows that (${\epsilon},\delta$)-differential privacy does not imply any reasonable guarantee on reality-oblivious semantic privacy. However, $({\epsilon},\delta)$-differential privacy does provide a strong reality-oblivious $({\epsilon}',\delta')$-semantic privacy guarantee for [*informed*]{} belief distributions. Using terminology from [@BDMN05; @DMNS06], we say that a belief distribution $b$ is informed if $b$ is constant on $n-1$ coordinates and agrees with the database in those coordinates. This corresponds to the adversary knowing some set of $n-1$ entries in the database before interacting with the algorithm, and then trying to learn the remaining one entry from the interaction. Let ${\mathcal{A}}_i$ be a randomized algorithm such that for all databases ${{\mathrm{x}}}$, ${\mathcal{A}}_i({{\mathrm{x}}})={\mathcal{A}}({{\mathrm{x}}}_{-i})$. \[thm:ind2sdp-inf\] (${\epsilon},\delta$)-differential privacy implies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for informed beliefs with ${\epsilon}'=e^{3{\epsilon}}-1+2\sqrt{\delta}$ and $\delta' = O(n\sqrt{\delta})$.[^4] Let ${\mathcal{A}}$ be a $({\epsilon},\delta)$-differentialy private algorithm. Let ${{\mathrm{x}}}$ be any database. Let $b$ be any informed belief distribution. This means that $b$ is constant on all $n-1$ coordinates, and agrees with ${{\mathrm{x}}}$ in those $n-1$ coordinates. Let $i$ be the coordinate which is not yet fixed in $b$. From Claim \[lem:proof\] (part \[it:ajoint\]), we know that $(b,{\mathcal{A}}(b))$ and $(b,{\mathcal{A}}_i(b))$ are $({\epsilon},\delta)$-differentialy private. Therefore, we can apply Lemma \[lem:bayes\]. Let $\delta''=O(\sqrt{\delta})$. From Corollary \[cor:bayes\], we get that with probability at least $1-\delta''$ over $t {\leftarrow}{\mathcal{A}}(b)$, the statistical difference between $b|_{{\mathcal{A}}(b) =t}$ and $b|_{{\mathcal{A}}_i(b)=t}$ is at most ${\epsilon}'$. Therefore, for ${{\mathrm{x}}}$, with probability at least $(1-\delta'')$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Taking union bound over all coordinates $i$, implies that with probability at least $1-n\delta''$ over $t {\leftarrow}{\mathcal{A}}({{\mathrm{x}}})$, for all $i=1,\dots,n$, we have ${\mathbf{SD}{\left( {{b|_{{\mathcal{A}}({{\mathrm{x}}}) =t},b|_{{\mathcal{A}}_i({{\mathrm{x}}}) =t}}} \right)}} \leq {\epsilon}'$. Therefore, ${\mathcal{A}}$ satisfies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for $b$. Since ${{\mathrm{x}}}$ was arbitrary, we get that (${\epsilon},\delta$)-differential privacy implies reality-oblivious $({\epsilon}',\delta')$-semantic privacy for informed beliefs. [^1]: That said, some of the other relaxations, such as probabilistic differential privacy from  [@MKAGV08], might lead to better parameters in [Theorem \[thm:ind2sdp\]]{}. [^2]: A few similar properties relating to statistical difference were shown in [@SV99]. Note that $({\epsilon},\delta)$-indistinguishability is not a metric, unlike statistical difference. But it does inherit some nice metric like properties. [^3]: Note that adversaries whose belief distribution is very different from the real database (as in the counterexample of Theorem A.2 may think they have learned a lot. But does such “learning" represent a breach of privacy? We do not think so, but leave the final decision to the reader. [^4]: Reality-oblivious $(\bar{\epsilon}/2,\delta)$-semantic privacy implies $(2{\epsilon},2\delta)$-differential privacy with $\bar{\epsilon}=e^{{\epsilon}}-1$. For details see the proof of Theorem \[thm:ind2sdp\].

ArXiv

--- address: - 'The Alan Turing Institute, 96 Euston Road, London NW1 2DB' - 'London School of Economics and Political Science, Houghton Street, London, WC2A 2AE' author: - Nikki Sonenberg - Edward Wheatcroft - Henry Wynn bibliography: - 'References.bib' title: Majorisation as a theory for uncertainty --- Introduction {#sec:intro} ============ Majorisation, also called rearrangement inequalities, yields a type of stochastic ordering in which two or more distributions are rearranged in decreasing order of their probability mass (discrete case) or probability density (continuous case) and then compared. When methods of majorisation are applied to probabilities and probability distributions, they provide a concept of the peakedness. This is independent of the ‘location’ of the probabilities, i.e., of the support of the distribution. This geometry-free property makes majorisation a good candidate as a foundation for the idea of uncertainty which is the focus of this paper. Majorisation is a partial, not total, ordering and implies that one or more of the class of order-preserving functions with respect to the ordering might be used as an entropy or ‘uncertainty metric’. Many well-known metrics fall into this category, one of which is Shannon entropy, widely used in information theory. Consider the question, ‘is uncertainty geometric?’. If we think our friend is in London, Birmingham or Edinburgh with probabilities $p_1,p_2, p_3$ respectively (where $p_1+p_2+p_3=1$), does it make any difference to our uncertainty, however we measure it, if the locations are changed to Reading, Manchester and Glasgow with the same probabilities, respectively? In fact, could we just permute the order of the first three cities? If our answer is no, i.e., that there is no difference, then we are in the realm of entropy and information. In the above cases, the Shannon entropy is $-\{p_1 \log(p_1) + p_2 \log(p_2) + p_3 \log(p_3)\}$ and we see that the subscript simply serves as a way to collect the probabilities, not to locate them in the geography of the UK. Another element of the majorisation approach is that it is, in a well-defined sense, dimension-free. In this paper, we show how this approach enables us to create, for a multivariate distribution, a univariate decreasing rearrangement (DR) by considering a decreasing threshold and ‘squashing’ all of the multivariate mass for which the density is above the threshold to a univariate mass adjacent to the origin. This creates the possibility of comparing multivariate distributions with different numbers of dimensions. We introduce a set of operations that can be applied to study uncertainty in a range of settings and illustrate these with examples. We see this work as a merging of methods used in applied mathematics and statistics with general methodology for the study of uncertainty. The methods discussed provide a foundation for the extension to Bayesian probabilities, a topic for further work. There is a large literature on majorisation. The classical results of Hardy, Littlewood and Polya [@Hardy1988] led to developments in a wide variety of fields. Marshall and Olkin’s [@Marshall2011] key volume on majorisation (later extended in [@Marshall2011]) built on these results. Applications in mathematical economics can be found in portfolio theory and income distributions built on classical work by Lorenz [@Lorenz1905] and Gini [@Gini1914] (see recent work by Arnold and Sarabia [@Arnold2018]). Majorisation has also been used in chemistry for mixing liquids and powders [@Klein1997] and in quantum information [@Partovi2011]. Statistical applications include experimental design [@Giovagnoli1987; @Pukelsheim1987], and in application to testing [@eaton1974monotonicity; @tong1988some]. Majorisation has been employed in the area of proper scoring rules by considering the partial and total ordering in the class of well-calibrated experts [@Degroot1986; @Degroot1985; @Degroot1988]. The common feature of these studies is the need to compare and quantify the degree of variation between distributions. We note that the theory of the decreasing rearrangement of functions, which we have used to a limited extent for probability densities, can be considered an area of functional analysis particularly in the area of inequalities of the kind which say that a rearrangement of a function increases or decreases some special functional [@lieb2001graduate].\ This paper is organised as follows. In the remainder of this section we introduce the concept of majorisation of probabilities and present related concepts and previous work. In Section \[sec:cont\_major\], we present results for the continuous case. In Section \[sec:multivariate\], we present the key idea of reducing multivariate distributions into a one dimensional decreasing rearrangement and illustrate this with examples. In Section \[sec:operations\], we collect together operations for the study of uncertainty and, in Section \[sec:algebra\], a lattice and an algebra for uncertainty. In Section \[sec:empirical\], we discuss empirical applications. Concluding remarks are given in Section \[sec:conclusion\]. Discrete majorisation and related work {#sec:discrete} -------------------------------------- We introduce majorisation for discrete distributions following Marshall *et al.* [@Marshall2011]. Consider two discrete distributions with $n$-vectors of probabilities $$p_1=(p^{(1)}_1,\ldots, p_n^{(1)}) \quad \text{ and } \quad p_2=(p^{(2)}_1,\ldots, p_n^{(2)}),$$ where $\sum_i p_i^{(1)}=\sum_i p_i^{(2)}=1$. Placing the probabilities in decreasing order: $$\tilde{p}^{(1)}_1 \geq \ldots \geq \tilde{p}_n^{(1)}\quad \text{ and } \quad \tilde{p}^{(2)}_1 \geq \ldots \geq \tilde{p}_n^{(2)},$$ it is then said that $p_2$ majorises $p_1$, written $p_1 \preceq p_2$ when, for all $n$, $$\sum_{i=1}^n \tilde{p}_i^{(1)} \leq \sum_{i=1}^n \tilde{p}_i^{(2)}.$$ This definition of majorisation is a partial ordering, that is, not all pairs of vectors are comparable. As argued by Partovi [@Partovi2011], this is not a shortcoming of majorisation, rather a consequence of its rigorous protocol for ordering uncertainty. Marshall *et al* [@Marshall2011] provide several equivalent conditions to $p_1\preceq p_2$. We consider three (A1-A3) of the best known in detail below.\ (A1) There is a doubly stochastic $n\times n$ matrix $P$, such that $$\begin{aligned} \label{equiv:doubly} p_1 = P p_2.\end{aligned}$$ This is a well known result by Hardy, Littlewood and Pólya [@Hardy1988]. The intuition of this result is that a probability vector which is a mixture of the permutations of another is more disordered. The relationship between a stochastic matrix $P$ and the stochastic transformation function in the refinement concept was presented by DeGroot [@Degroot1988].\ (A2) Schur [@Schur1923] demonstrated that, if (A1) holds for some stochastic matrix $P$, this leads to the following equivalent condition. For all continuous convex functions $h( \cdot )$, $$\begin{aligned} \label{condition3} \sum_{i=1}^n h(\tilde{p}_i^{(1)}) \leq \sum_{i=1}^n h(\tilde{p}_i^{(2)}),\end{aligned}$$ for all $n$. The sums in (\[condition3\]) are special cases of the more general Schur-convex functions on probability vectors. Details on the characteristics and properties of Schur-convex functions are provided by Marshall *et al.* [@Marshall2011]. In particular, entropy functions such as Shannon information, for which $h(y)=y\log(y)$, are Schur-convex. We also highlight a special case of the Tsallis information for which $$h(y)=\frac{y^{\gamma}-1}{\gamma}, \quad\gamma>0,$$ where, in the limit $\gamma\rightarrow 0$, Shannon information is obtained. The condition (A2) is equivalent to the majorisation ordering for distributions, and we consider it as a continuous extension to Equation (\[condition3\]) (see Section \[sec:cont\_major\] for details). The condition (A2) indicates that the ordering imposed by majorisation is stronger than the ordering by any single entropic measure and, in a sense, is equivalent to all such (entropic) measures taken collectively [@Partovi2011].\ (A3) Let $\pi(p) = (p_{\pi(1)}, \ldots, p_{\pi(n)})$ be the vector whose entries are a permutation $\pi$ of the entries of a probability vector $p$, with symmetric group $S$, then $$\begin{aligned} p_1 \in \mbox{conv}_{\pi \in S} (\{\pi(p_2)\}).\end{aligned}$$ That is to say, $p_1$ is in the convex hull of all permutations of entries of $p_2$. Majorisation is a special case of group-majorisation (G-majorisation) for the symmetric (permutation) group [@Giovagnoli1985]. The general type of groups for which the theory really works are reflection (Coxeter) groups, e.g. permutation and sign changes in $n$ dimensions (called the $B_n$ series). The main work on G-majorisation was by Eaton and Perlmann [@Eaton1977]. Rearrangements can be viewed as special instances of transportation plans, which move a given mass distribution to another distribution of the same total mass (see Buchard [@Burchard2009]). The recent use of transport mapping in UQ is to improve Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) samplers [@Marzouk2016; @Parno2018]. Continuous majorisation {#sec:cont_major} ======================= Following Hardy *et al.* [@Hardy1988], in this section we describe continuous majorisation, extending on the discrete case presented in Section \[sec:discrete\]. Ryff [@Ryff1965] provided continuous analogues to discrete majorisation by replacing vectors and matrices with integrable functions and linear operators. \[drdefn\] Let $f(x)$ be a (univariate) pdf and define $m(y)=\mu\{z: f(z) \geq y\}$. The decreasing rearrangement of $f(x)$ is then $$\begin{aligned} \tilde{f}(z)=\mbox{sup}\{t: m(t) >z\},\; z >0.\end{aligned}$$ Let $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$ be the DR of two pdfs $f_1(x)$ and $f_2(x)$, respectively and $\tilde{F}_1(z)$ and $\tilde{F}_2(z)$ their corresponding cdfs. We say that $f_2(x)$ majorises $f_1(x)$, written $f_1 \preceq f_2$, if and only if $$\tilde{F}_1(z) \leq \tilde{F}_2(z),\;\; \mbox{for all} \quad z > 0.$$ Similarly to the discrete case, we give three equivalent conditions for majorisation, $\preceq$:\ (B1) For some non-negative doubly stochastic kernel $P(x,t)$, $$\begin{aligned} f_1(x) = \int P(x,t) f_2(t) dt.\end{aligned}$$ (B2) For all continuous convex functions $h(\cdot)$, $$\begin{aligned} \int h(f_1(z)) dz \leq \int h(f_2(z))dz.\end{aligned}$$ (B3) Slice condition: $$\begin{aligned} \int(f_1(x)-c)_+ dx \leq \int(f_2(x)-c)_+dx, \quad c>0. \label{eq:slice}\end{aligned}$$ \[example|\_beta\] Consider the Beta$(a,b)$ distribution with $(a, b)=(3, 2)$ and pdf $p(z)=12(1-z)z^2$. We look for $z_1$ and $z_2$ (where $z_1<z_2$) such that $p(z_1)=p(z_2)=c$, that is the points at which $p(z)$ intersects the line $y=c$. The pdf and the horizontal line $y=c$ are both shown in Figure \[fig:DensityPlot\]. Setting $z=z_2-z_1$, this gives us the system of equations: $$\begin{aligned} \label{example_beta_system} \begin{cases} p(z_1)=12(1-z_1)z_1^2=y ,\\ p(z_2)=12(1-z_2)z_2^2=y,\\ z_2-z_1=z,\\ 0\leq z\leq 1. \end{cases}\end{aligned}$$ ![The identification of $z_{1}$ and $z_{2}$ for the probability density function of Beta(3,2).[]{data-label="fig:DensityPlot"}](DensityFunctionBeta.png){width="34.00000%"} The DR can be obtained from Equation (\[example\_beta\_system\]) by eliminating $z_1$ and $z_2$ and setting $\tilde{f} = y$. The elimination variety is $ 48z^6 - 96z^4 + 9y^2 + 48z^2 - 16y$. We obtain the explicit solution, $$\begin{aligned} \tilde{f}(z) = \left\{ \begin{array}{l} \frac{8}{9} + \frac{4}{9} (-27z^6 + 54z^4 - 27z^2 + 4)^{\frac{1}{2}}, \quad 0 \leq z \leq \frac{1}{\sqrt{3}} \\ \frac{8}{9} - \frac{4}{9} (-27z^6 + 54z^4 - 27z^2 + 4)^{\frac{1}{2}}, \quad \frac{1}{\sqrt{3}} \leq z \leq 1. \end{array} \right.\end{aligned}$$ This variety is shown in Figure \[fig:beta\] and the DR $\tilde{f}(z)$ is the section of the curve over $[0,1]$ decreasing from $\left(0, \frac{16}{9}\right)$ to $(1,0)$. The cdf of the DR of Beta$(2,3)$ is $\tilde{F}(z)=-4z^3 + 6z^2$. If $\tilde{F}(z)$ is the cdf corresponding to $\tilde{f}$ obtained by adjoining the equations $Y = F(z_2)-F(z_1)$, we obtain the second variety $4z^4 + 3Y^2 - 8Yz = 0,$ illustrated in Figure \[fig:beta\], and $\tilde{F}(z)$ is the upper portion of the curve from $(0,0)$ to $(1,1)$, namely $$\tilde{F}(z) = 2\left(\frac{2}{3} + \frac{\sqrt{-3z^2 + 4}}{3}\right)z.$$ It is hard to derive when $f_1(x) \leq f_2(x)$ for the general case in which $(a_1,b_1)$ and $(a_2,b_2)$ and $f_i(x) \sim \mbox{Beta}(a_i,b_i),\; i=1,2$. However, we can prove the following. Assume $a_1 , b_1 , a_2 , b_2 > 1$. If pdfs $f_1 (x)\sim \text{Beta}(a_1 , b_1 )$ and $f_2 (x)\sim\text{Beta}(a_2 , b_2 )$, have the same mode, then $\max_x f_1(x) \leq \max_x f_2(x)$ if and only if $X_1 \preceq X_2.$ We first prove that, under the same mode condition, $f_1(x)$ and $f_2(x)$ intersect at two distinct $x$-values at which the values of $f_1(x)$ and $f_2(x)$ are the same. Setting the modes equal, that is $$\frac{a_1-1}{a_1+b_1-2} = \frac{a_2-1}{a_2+b_2-2},$$ We find that, and assuming without loss of generality, that $a_2 > a_1$, we obtain $$\frac{f_1(x)}{f_2(x)} = \left\{x (1-x)^u \right\}^v C,$$ where $u= a_2-a_1,v = \frac{b_1-1}{b_2-1}$ and $C$ is a constant. If we set this equal to 1, we obtain two solutions given by $$x (1-x)^u = C^{-\frac{1}{v}}.$$ It is then straightforward to verify that the common value of $f_1(x)$ and $f_2(x)$ is the same at the two solutions. The proof of the theorem is completed by using the slice condition in Equation . Multivariate case: matching of uncertainty {#sec:multivariate} ========================================== The following construction shows how to induce a one dimensional DR from a multidimensional distribution. This is in fact the continuous multidimensional version of the following discrete version. Thus, suppose we have a 2-dimensional table with probabilities $p_{i,j}, i=1, \ldots, I; j= 1, \ldots, J$. Line up the probabilities in decreasing (non-increasing) order from $1$ to $n =IJ$ with support on $1, \ldots, n$, respectively. If we now count the number of $\{i,j\}$ such that the probability is greater than or equal to a constant $c$, it is the same as the original table as for the DR. \[multitouni\] A univariate decreasing rearrangement $\tilde{f}(z)$, compatible with $f(x)$, is, for all constants $c\geq 0$, $$\begin{aligned} \label{decreaseRearrangemult} \int_{\{x:f(x)\geq c \}}f(x)dx=\int_{\{z:\tilde{f}(z)\geq c\}}\tilde{f}(z)dz.\end{aligned}$$ Following [@Burchard2009]: as $$\begin{aligned} {\{x:f(x)\geq c \}} = {\{z:\tilde{f}(z)\geq c\}},\end{aligned}$$ then the volume of these sets are consistent. The notation on the lemma will be important to us. We shall use $f(x)$ for the multivariate pdf for a random variable $X = (X_1, \ldots, X_n)$ of dimension $n$, and $\tilde{f}(z)$ will be its one-dimensional pdf and we reserve $\tilde{F}(z)$ for the corresponding cdf, which will be the basis for the majorisation. The following lemma shows that the information/entropy for $X \sim f(x)$ and $Z \sim \tilde{f}(z)$ are the same. This is a crucial result and gives us confidence in the matching. Let $f(x)$ be a multidimensional pdf and $\tilde{f}(z)$ on $[0, \infty]$ its decreasing rearrangement. Then, given a convex function $\varphi(x)$, we have $$\int_S \varphi(f(x)) dx = \int_0^{\infty} \varphi(\tilde{f}(z)) dz,$$ where $S$ is the support of $f(x)$. (sketch) The proof consists of matching volume to length elements in $S$ and $[0,1)$. For $c>0$ and small $\delta c > 0$ we have $$\int_{x: f(x) \geq c, x \in S} f(x)dx - \int_{x: f(x) \geq c + \delta c, x \in S} f(x) dx = \int_{z: \tilde{f}(z) \geq c, z \in [0, \infty)} \tilde{f}(z) dz -\int_{z: \tilde{f}(z) \geq c +\delta c, z \in [0, \infty)} \tilde{f}(z) dz.$$ We can then write, approximately, $$u(c)A(c, \delta c)=u(c)L(c, \delta c),$$ where $A(c, \delta c)$ and $L(c, \delta c)$ are the corresponding increments in volume and length, respectively, as corresponding to the interval $[c,c+\delta c)$, that is $ f^{(-1)}([c,c+ \delta))$ and $ \tilde{f}^{(-1)}([c,c+ \delta))$, respectively. Cancelling $c$, we can equate $A(c, \delta c)$ and $L(c, \delta c)$, and this allows us to recapture and equate the integrals of any measurable function $u(\cdot)$: $$u(c)A(c,\delta c) = u(c) L(c, \delta c).$$ In particular, we can write $u(c) = \varphi(f(c)).$ In Examples \[multi\_norm\_ex\] and \[indep\_exp\_ex\], we demonstrate how to obtain a DR for the standard multivariate Normal distribution and the $n$-fold independent standard exponential distribution. In the examples below, we use the following idea to carry out computations. There may be cases in which, for a given $c$, the inverse set $f^{(-1)}(c)$ is described by some useful quantity $\delta$. Moreover $\delta$, expressed as a function of $x$, then becomes a random variable with a known (univariate) distribution. In that case, we can write Definition \[multitouni\] as $$\tilde{F}\big(\tilde{f}^{-1}(c) \big) =F_{\delta}\big(f_{{X}}^{-1}(c) \big)\nonumber \label{Fstuff},$$ then $$\begin{aligned} \label{eq:valid_DR} \tilde{f}(r) & = f_{\delta} \left( f_X^{(-1)}(\tilde{f}(r)) \right) \frac{\partial}{\partial r}\left(f_X^{(-1)}(\tilde{f}(r))\right).\end{aligned}$$ \[multi\_norm\_ex\] We provide a representation of Definition \[multitouni\] for a two-dimensional multivariate normal in Figure \[fig:DRM2\]. ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultivariateNorm1.png "fig:"){width="30.00000%"} ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultivariateNorm2.png "fig:"){width="30.00000%"} ![*Left panel:* Density plot of a two-dimensional standard multivariate normal. The dashed line and blue shaded region correspond to $f(x)=c$ and $\int_{\{x:f(x)\geq c \}}f(x)dx$ respectively. *Central panel:* A plot to demonstrate the connection between ${X}=(X_1, X_2)^T$ and $R^2$. The radius $r$ of a blue circle corresponds to $x=f^{-1}(c)$. *Right panel:* A DR $\tilde{f}(z)$ obtained for a multivariate normal. The blue shaded region corresponds to $\int_{\{z: \tilde{f}(z)\geq c\}}\tilde{f}(z) dz$.[]{data-label="fig:DRM2"}](MultiVariateNorm3.png "fig:"){width="30.00000%"} Let a real random vector ${X}=(X_1, \dots, X_n)^T$ be an $n$-variate standard normal distribution with density $$f_{{X}}(x_1, \dots, x_n)=\frac{1}{(2\pi)^{\frac{n}{2}}}\exp\Bigg\{-\frac{\sum_{i=1}^n x_i}{2} \Bigg\}.$$ We refer to a real random vector ${X}$ as a spherical Gaussian random vector with ${X}\sim\text{N}_n({0}, I_n)$, where ${0}$ is an $n$-vector of zeros and $I_n$ is an $n\times n$ identity matrix. Define the square of the radius of a spherical Gaussian random vector, that is, $$R^2 = \sum_{i=1}^nX_i^2.$$ To construct a DR, we slice the pdf of a multivariate normal at $f_{{X}}(x_1, \dots, x_n)=c$, then $$\begin{aligned} \label{eq:DR_derivation1} &c = \frac{1}{(2\pi)^{n/2}}\exp\Big\{-\frac{1}{2}\sum_{i=1}^n x_i^2 \Big\}. \end{aligned}$$ Given the relationship between the vector ${X}$, and a random variable $R^2$, define $r^2 =\sum_{i=1}^n x_i^2$, then $$\begin{aligned} r=\Big(-2\log\big((2\pi)^{n/2} c\big) \Big)^{1/2},\end{aligned}$$ where the volume of the $n$-dimensional Euclidean ball of radius $r$ is $$\label{eq:ball_volume} V_n(r)=\frac{\pi^{n/2}}{\Gamma\Big(\frac{n}{2}+1 \Big)}r^n.$$ Substituting Equation (\[eq:DR\_derivation1\]) into Equation (\[eq:ball\_volume\]), we obtain $$\begin{aligned} c=\frac{1}{(2\pi)^{n/2}}\exp\bigg\{-\frac{1}{2}\bigg(\frac{V_n(r)\Gamma(n/2+1)}{\pi^{n/2}} \bigg)^{2/n} \bigg\},\end{aligned}$$ noting the values of $c$ and $V_n(r)$ are dependent on each other. To generalise the expression above, we replace $c$ and $V_n(r)$ with $\tilde{f}(z)$ and $z$, respectively. The final form of the DR is $$\label{eq:DRM_mult_normal} \tilde{f}(z) = \frac{1}{(2\pi)^{n/2}}\exp\bigg\{-\frac{1}{2}\bigg(\frac{z}{V_n} \bigg)^{2/n} \bigg\},$$ where $V_n$ is the volume of the unit sphere in $\mathbb{R}^n$. We validate the form of the DR in Equation (\[eq:DRM\_mult\_normal\]) by employing the construction from Equation (\[eq:valid\_DR\]). Here, $R^2=\sum_{i=1}^n X_i^2$ follows a Chi-squared distribution with $n$ degrees of freedom, $R^2\sim \chi_n^2$, with pdf and cdf, $$\begin{aligned} f_{R^2}(y)=\frac{1}{2^{n/2}\Gamma(n/2)}y^{n/2-1}\exp\left\{-\frac{y}{2} \right\}, \qquad F_{R^2}(y)=\frac{1}{\Gamma(n/2)}\gamma\left(\frac{n}{2}, \frac{y}{2} \right),\end{aligned}$$ respectively. The result for the DR in Equation (\[eq:DRM\_mult\_normal\]) can be verified by employing Equation (\[eq:valid\_DR\]) with $\delta = R^2$ and directly substituting the required functions. \[indep\_exp\_ex\] Let the real random vector ${X}=(X_1, \dots, X_n)^T$ be an $n$-fold independent standard exponential distribution with density $$f_X(x_1, \dots, x_n)=\exp\left\{-\sum_{i=1}^n x_i \right\}.$$ As $f_X(x_1, \dots, x_n) = f_1(x_1) f_2(x_2) \cdots f_n(x_n)$, slicing the pdf at $c=f_{{X}}(x_1, \dots, x_n)$ yields $$\begin{aligned} \label{eq:DR_derivation2} &-\log(c)=\sum_{i=1}^n x_i.\end{aligned}$$ The volume of an $n$-dimensional simplex in which all $n$ variables are greater than $0$ but with sum less than $R$ is $$\label{eq:simplex_volume} V_n = \frac{R^n}{n!}.$$ Substituting Equation (\[eq:DR\_derivation2\]) into Equation (\[eq:simplex\_volume\]), we obtain $$\begin{aligned} c=\exp\left\{-(n!V_n)^{1/n} \right\},\end{aligned}$$ where the values of $c$ and $V_n$ depend on each other. To generalise this expression, we replace $c$ and $V_n$ with $\tilde{f}(z)$ and $z$, respectively. The DR can then be written as $$\label{eq:DRM_mult_exp} \tilde{f}(z)=\exp\left\{-(n!z)^{1/n} \right\}.$$ To verify the form of the DR in Equation (\[eq:DRM\_mult\_exp\]), we employ the construction in Equation (\[eq:valid\_DR\]). Define the random variable $R=\sum_{i=1}^n X_i$, such that $R\sim\text{Gamma}(n, 1)$, with pdf and cdf, $$\begin{aligned} f_R(y)=\frac{1}{\Gamma(n)}y^{n-1}e^{-y}, \qquad F_R(y)=\frac{1}{\Gamma(n)}\gamma(n, y).\end{aligned}$$ Similarly to Example \[multi\_norm\_ex\], we derive the DR, $\tilde{f}(z)$, for a Gamma-distributed random variable $R$, that is, $$\label{eq:DR_mult_exp} \tilde{f}(r)=f_R\big(f_{{X}}^{-1}(\tilde{f}(r))\big)\frac{\partial}{\partial r}(f_{{X}}^{-1}(\tilde{f}(r))).$$ The results for the DR in Equation (\[eq:DRM\_mult\_exp\]) can be verified by employing Equation (\[eq:valid\_DR\]) with the required functions. Some operations with $\preceq$ {#sec:operations} ============================== Inverse Mixing {#subsec:InverseMix} -------------- We present *inverse mixing* as a method for combining uncertainty given two distributions over two different populations. \[def:inversemixing\] Define the inverse mixture $$\tilde{f}_1\; [+] \;\tilde{f}_2= \left(\tilde{f}_1^{(-1)}(z)+\tilde{f}_2^{(-1)}(z) \right)^{(-1)},$$ and the $\alpha$-inverse mixture $$\tilde{f}_1\; [+]_{\alpha}\; \tilde{f}_2=\left(\tilde{f}_1\left(\frac{z}{1-\alpha}\right)^{(-1)} + \tilde{f}_2\left(\frac{z}{\alpha}\right)^{(-1)} \right)^{(-1)},$$ where $0 < \alpha < 1$ is the mixing parameter. For the case in which $\alpha= 1/2$, we claim that, in the discrete case, the *inverse mixture* corresponds to a combination of all the probabilities in both populations scaled by a factor $\alpha=\frac{1}{2}$, i.e., $\frac{1}{2}p_i\cup\frac{1}{2}q_i, i=1, \dots, 5$ and sorting them in a decreasing order. In the following examples, we demonstrate inverse mixing for the discrete and continuous cases. \[exampleworkplace\] Consider two distinct groups of people in a work place. Denote the probability of the $i$-th member of group one and the $i$-th member of group two obtaining a promotion to be $p_i$ and $q_i$, respectively. Let the probabilities $p_{1},...,p_{5}$ and $q_{1},...,q_{5}$ be those denoted in Table \[tab:Probabilities\_MF\], noting that $ p_1\geq p_2\geq\cdots\geq p_5$, $q_1\geq q_2\geq \cdots q_5$ and $\sum_{i}p_i=1$, $\sum_{i}q_i=1$, ----------------- ------- ------- ------- ------- ------- $i$ 1 2 3 4 5 \[0.5ex\] $p_i$ 0.577 0.192 0.128 0.064 0.038 $q_i$ 0.730 0.219 0.036 0.007 0.007 ----------------- ------- ------- ------- ------- ------- : Ordered probabilities for Example \[exampleworkplace\].[]{data-label="tab:Probabilities_MF"} To perform inverse mixing, we take the inverse of each pmf, combine them and sorting them into ascending order (demonstrated in panel (a) of Figure \[fig:inversetilt\]). The inverse is taken to obtain a pmf (panel (b), noting the change of scale on the $y$ axis). The result of direct mixing, obtained by summing ordered probabilities of two populations and scaling by a factor $\alpha$, i.e., $\frac{1}{2}(p_i+q_i), i=1, \dots, 5$, is shown in panel (c). Both mixtures provide information about the joint population, but the inverse mixture also preserves information about the individual subpopulations. ![(a) the addition of two inverse pmfs, (b) inverse mixture distribution with $\alpha=\frac{1}{2}$, (c) direct mixture distribution with $\alpha=\frac{1}{2}$.[]{data-label="fig:inversetilt"}](combine_inv.png){width="100.00000%"} We now consider inverse mixing for the continuous case. In the continuous case, we need to pay attention to the maximum values (modes) of the probability distributions. We demonstrate the importance of this condition in the following two examples. Given univariate and bivariate exponential distributions with the following form of decreasing rearrangements, $$\tilde{f}_1(z)=\exp\{-z\}, \quad \tilde{f}_2(z)=\exp\{-(2z)^{1/2}\},$$ we observe that $0<\tilde{f}_1(z), \tilde{f}_2(z)\leq 1$ and obtain the functional inverses, $$\tilde{f}_1^{(-1)}(z) = -\log(z), \quad \tilde{f}_2^{(-1)}(z)=\frac{1}{2}(\log(z))^2, \quad z\in(0, 1].$$ The left and central panels in Figure \[fig:inverse\_mixing\_con1\] show $\tilde{f}_1(z)$, $\tilde{f}_2(z)$, $\tilde{f}_1^{(-1)}(z)$ and $\tilde{f}_2^{(-1)}(z)$. The maximum value of these two probability functions occurs at the same point, $\tilde{f}_1(0)=\tilde{f}_2(0)=1$, so there is no kink in the inverse mixing of these two distributions. The inverse mixture of the two distributions is then $$f^{(1)}(z) = \left\{ \tilde{f}_1^{(-1)}\left(\frac{z}{1-\alpha}\right) + \tilde{f}_2^{(-1)}\left(\frac{z}{\alpha}\right) \right\}^{(-1)} = \left\{-\log\Big(\frac{z}{1-\alpha}\Big) +\frac{1}{2} \Big(-\log\Big(\frac{z}{\alpha}\Big) \Big)^2 \right\}^{(-1)} ,$$ for $0 \leq \alpha \leq 1$. The direct averaging of $f_1(x)$ and $f_2(x)$ gives: $$f^{(2)}(z) = \left\{ (1-\alpha)\tilde{f}_1^{(-1)}(z)+ \alpha \tilde{f}_2^{(-1)}(z)\right\} ^{(-1)}= \left\{(1-\alpha)(-\log(z))+\frac{\alpha}{2}(-\log(z))^2 \right\}^{(-1)}.$$ We specify $\alpha=1/2$ to obtain the following expression: $$f^{(1)}(z)=\Big\{-\log(2z)+\frac{1}{2}\big[\log(2z)\big]^2 \Big\}^{(-1)}.$$ Since the expression above is a quadratic in $\log(2z)$, we obtain the two solutions $f^{(1)}(z)=\frac{1}{2}\exp\{1+\sqrt{1+2z}\}$ and $f^{(1)}(z)=\frac{1}{2}\exp\{1-\sqrt{1+2z}\}$. Proceeding with the second solution, as the first solution does not integrate to 1, we obtain the mean, variance and Shannon entropy: $\frac{7}{2}, \frac{99}{4}$ and $\frac{3}{2}+\log(2)$, respectively. We perform similar calculations for direct mixing with $\alpha=1/2$. The pdf has the form, $$f^{(2)}(z)=\exp\left\{1-\sqrt{1+4z}\right\},$$ and, in this case, the values of the mean, variance and Shannon entropy are $\frac{7}{4}, \frac{99}{16}$ and $\frac{3}{2}$. Based on the pdfs, we have the following relationship for inverse mixing and direct averaging when $\alpha=1/2$, $$\label{eq:inverse_direct} f^{(2)}(z)=2f^{(1)}(2z),$$ illustrated in the right panel of Figure \[fig:inverse\_mixing\_con1\]. Here, $f^{(1)}(z)$ (red line) stretches the support of the distributions, and lowers the overall maximum, whereas $f^{(2)}(z)$ (blue line) preserves the maximum and shrinks the support, confirmed by Equation (\[eq:inverse\_direct\]). ![*Left panel:* plot of DR functions $\tilde{f}_1(z)$ (solid line) and $\tilde{f}_2(z)$ (dashed line). *Central panel:* plot of functional inverses of the DR functions, i.e. $\tilde{f}_1^{(-1)}(z)$ (solid line) and $\tilde{f}_2^{(-1)}(z)$ (dashed line). *Right panel:* plot of inverse mixing and direct averaging, $f^{(1)}(z)$ (red line) and $f^{(2)}(z)$ (blue line).[]{data-label="fig:inverse_mixing_con1"}](inverse_mixing_con1){width="100.00000%"} \[example:6\] We consider exponential distributions with means 1 and 2 and note that they are already DRs: $$\tilde{f}_1(z)=\exp\{-z\}, \quad\tilde{f}_2(z)=\frac{1}{2}\exp\{-z/2\}.$$ The left panel in Figure \[fig:inverse\_mixing\_con2final\] shows $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$, depicted by solid and dotted lines respectively. We note that $0<\tilde{f}_1(z)\leq 1$ and $0<\tilde{f}_2(z)\leq\frac{1}{2}$, which indicates that there is different support for the functional inverses, i.e. $$\begin{aligned} &\tilde{f}_1^{(-1)}(z)=-\log(z), \quad z\in (0, 1],\\ &\tilde{f}_2^{(-1)}(z)=-2\log(2z), \quad z\in (0, 1/2].\end{aligned}$$ The inverse mixing of these two distributions with $\alpha=\frac{1}{2}$ is defined as $$f^{(1)}(z)=\tilde{f}_1\; [+]_{\frac{1}{2}}\; \tilde{f}_2=\left\{-\log(2z)-2\log(4z) \right\}^{(-1)}.$$ To avoid negative values of the expression inside the functional inverse, we propose the following modification: $$\label{eq:arg_inverse} \tilde{f}_1^{(-1)}(2z)+\tilde{f}_2^{(-1)}(2z) =\max\{0, -\log(2z)\}+\max\{0, -2\log(4z)\}.$$ In the central plot of Figure \[fig:inverse\_mixing\_con2final\], the dotted line corresponds to the function in Equation (\[eq:arg\_inverse\]). We note that the introduced modification results in a kink in $\tilde{f}_1^{(-1)}(2z)+\tilde{f}_2^{(-1)}(2z)$ at $z=0.25$. To obtain the inverse mixture, we are required to take another functional inverse by swapping the abscissa and ordinate. Therefore we observe a kink in $f^{(1)}(z)$ at $z=\log(2)$ in the right panel of Figure \[fig:inverse\_mixing\_con2final\] (blue line). We obtain the final form of the inverse mixing: $$\begin{aligned} f^{(1)}(z)&=\begin{cases} \frac{1}{2}\exp\{-z\}, &\mbox{if } 0<z<\log(2) \\ \frac{1}{2}\exp\{ \frac{-2\log(2)-z}{3} \}, &\mbox{if } z \geq \log(2), \end{cases}\end{aligned}$$ and we obtain the values of the mean, variance and Shannon entropy: $2.85$, $8.91$ and $1 + \frac{3\log(2)}{2}$. Similarly to example 5, we consider the direct averaging of these distributions with $\alpha=\frac{1}{2}$, i.e. $$f^{(2)}(z)=\big\{\frac{1}{2}\tilde{f}_1^{(-1)}(z)+\frac{1}{2}\tilde{f}_2^{(-1)}(z) \big\}^{(-1)}= \left\{-\frac{1}{2}\log(z)-\log(2z) \right\}^{(-1)}.$$ As with inverse mixing, to avoid negative values, we modify the argument of the functional inverse: $$\label{eq:arg_direct} \frac{1}{2}\tilde{f}_1^{(-1)}(z)+\frac{1}{2}\tilde{f}_2^{(-1)}(z)=\max\left\{ 0,-\frac{1}{2}\log(z)\right\}+\max\left\{0, -\log(2z) \right\}.$$ The solid line in the central plot of Figure \[fig:inverse\_mixing\_con2final\] corresponds to the function in Equation (\[eq:arg\_direct\]). We observe a kink in the function at $z=\frac{1}{2}$. As a result, we obtain a kink in $f^{(2)}(z)$ at $z=-\frac{1}{2}\log{\frac{1}{2}}$ in the right panel of Figure \[fig:inverse\_mixing\_con2final\] (red line). The final form of the direct averaging is $$\begin{aligned} f^{(2)}(z) &=\begin{cases} \exp\{-2z\}, &\mbox{if } 0<z<-\frac{1}{2}\log(\frac{1}{2}), \\ \exp\Big(\frac{-2z-2\log(2)}{3}\Big), &\mbox{if } z\geq -\frac{1}{2} \log(\frac{1}{2}), \end{cases}\end{aligned}$$ where the values of the mean, variance and Shannon entropy are $1.42$, $2.23$ and $1+\frac{\log(2)}{2}$. ![*Left panel:* plot showing DR functions $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$. *Central panel:* plot showing the function $\tilde{f}_{12}^{(-1)}(z)$ which is employed in direct averaging and inverse mixing. *Right panel:* plot showing pdfs obtained from inverse mixing and direct averaging, $f^{(1)}(z)$ and $f^{(2)}(z)$.[]{data-label="fig:inverse_mixing_con2final"}](inverse_mixing_con2final){width="100.00000%"} From the representation of inverse mixing and direct averaging in Figure \[fig:inverse\_mixing\_con2final\], we can see that $f^{(1)}(z)$ stretches the support of the distribution, whilst $f^{(2)}(z)$ shrinks it. The maximum (mode) from the direct averaging is double the maximum of the inverse mixing. This example shows how distributions can be approximated using DR from higher dimensional distributions. Consider the pdf $$f(x)=3(1-x)^2,\quad x\in[0, 1],$$ which has the functional inverse $f^{(-1)}(x)=1-\sqrt{ x/3}$. We need to perform an expansion in $t=\log(x)$, substituting $x=\exp\{t\}$ in the test function, defined as $$h(t)=f^{(-1)}(\exp\{t\})=1-\sqrt{\exp\{t\}/3}.$$ We provide the first three terms of the Taylor series, $$h(t)\approx \Big(1-\frac{\sqrt{3}}{3}\Big)-\frac{\sqrt{3}}{6}t-\frac{\sqrt{3}}{24}t^2,$$ and obtain the approximation $$\hat{f}^{(-1)}(x) \approx \left(1-\frac{\sqrt{3}}{3}\right)-\frac{\sqrt{3}}{6}\log(x)-\frac{\sqrt{3}}{24}(\log(x))^2,$$ from which we obtain the functional inverse, $$\hat{f}(x)=\exp\left( -2+2\sqrt{2\sqrt{3}-1-2\sqrt{3}x}\right).$$ From the left panel in Figure \[fig:series\_plot\], we observe that the functional inverse (solid line) and its approximation (dotted line) are close to each other and, from the right panel, that $\hat{f}(x)$ (red line) is a good approximation to $f(x)$ (blue line). ![*Left panel:* plot of the functional inverse, $f^{(-1)}(x)$(solid line), and its approximation, $\hat{f}^{(-1)}(x)$ (dotted line). *Right panel:* plot of $f(x)$ (blue line) and its approximation $\hat{f}(x)$ (red line).[]{data-label="fig:series_plot"}](series_plot){width=".7\textwidth"} Independence, conditional distributions --------------------------------------- It is well known and axiomatic that Shannon information $S$ and entropy $(-S)$ are additive under independence: if $ X$ and $Y$ are independent random variables of the same dimension then, $$S(X+Y) = S(X) + S(Y).$$ It is a natural conjecture that entropy is a maximum in some sense when random variables are independent. This result holds for Shannon entropy. \[lem:lem4\_2\] Within the class of bivariate random variables $(X_1,X_2)$ with given marginal distributions $X_1 \sim f_1(x) $ and $X_2 \sim f_2(x)$, the maximum Shannon entropy is uniquely achieved when $X_1$ and $X_2$ are independent. Let $H( \cdot )$ be Shannon information. For random variables $X,Y$, we have the well-known expansion for the joint information: $$H(X,Y) = H(X) + \mbox{E}_X H(Y |X).$$ Also well known is the inequality, which follows from Jensen’s inequality, for the second term $$\mbox{E}_X H(Y |X) \geq H(Y),$$ with equality, and uniquely for Shannon information. The resulting additivity, $H(X,Y) = H(X) + H(Y)$, characterises Shannon information. We can argue by Lemma \[lem:lem4\_2\] that, within the class with fixed marginals, the independence case cannot be uniformly dominated within the ordering $\preceq$. This example shows that if we change the type of entropy, in this case to Tsallis, then the independent case may no longer achieve the minimum. Take $X_1,X_2 = 0,$ with probabilities $$p_{00} = \alpha \beta,\; p_{10} = (1-\alpha)\beta,\; p_{01} = \alpha (1-\beta),\; p_{11} = (1-\alpha)(1-\beta)$$ We can generate all distributions with the same margins with a perturbation $\epsilon$: $$p_{00} = \alpha \beta + \epsilon,\; p_{10} = (1-\alpha)\beta-\epsilon,\; p_{01} = \alpha (1-\beta)-\epsilon,\; p_{11} = (1-\alpha)(1-\beta)+\epsilon,$$ with the restriction that $|\epsilon| < \min(p_{00}, p_{10}, p_{01}, p_{11}).$ Taking the Tsallis entropy with $\gamma = 1$, we find the minimum when $$\epsilon = - \frac{(2\beta-1)(2\alpha-1)}{4},$$ which is zero if at least one of $\alpha$ or $\beta$ is $\frac{1}{2}$, which, interestingly, is a little less restricted than the case in which the distribution must be uniform i.e., all $p_{ij} = \frac{1}{4}$. Note that independence and inverse mixing are closely related. If $X_1 =\{0,1\}$ is a binary random variable with $\mbox{prob}\{X_1=1\} = \alpha$ and $X_2$ has $J$ levels, the one-dimensional DR of $(X_1,X_2)$ is the inverse mixture with mixing parameter $\alpha$. Thinking in terms of a $2 \times J$ table, we combine the top row of probabilities, the distribution of $X_2$ multiplied by $\alpha$, with the bottom row in which they are multiplied by $1-\alpha$. In the continuous case, if the DR pdfs of $X_1$ and $X_2$ are $\tilde{f}_1(z)$ and $\tilde{f}_2(z)$, respectively then, for the joint distribution, we can think of $\tilde{f}_1$ weighting $\tilde{f}_2$ (or vice versa) in a similar way. This leads to the formula which we write informally: $$\tilde{f}_{12} (z) = \int \tilde{f}_1 \left( \frac{z}{\tilde{f}_2^{-1}(x)} \right) dx = \int \tilde{f}_2 \left( \frac{z}{\tilde{f}_1^{-1}(x)} \right) dx.$$ Iterating this operation, we can cover independent random variables in several dimensions and recapture results, such as those in Section \[sec:cont\_major\]. As with Example \[example:6\], care has to be taken in handling supports and limits of integration. The following results enable us to propagate the ordering $\preceq$ via conditional distributions. Consider two pairs of joint distributed random variable $(X_1, X_3)$ and $(X_2,X_3)$ and suppose that the conditional distributions satisfy $X_1\; \vline \; X_3 \preceq X_2 \; \vline \; X_3$, for all values of the conditioning random variable, $X_3$. Then, for the joint distributions, $(X_1,X_3) \preceq (X_2,X_3)$. This follows since, with simple notation, the joint distributions are $f(x_1,x_3) = f(x_1| x_3) f(x_3)$ and $f(x_2,x_3) = f(x_2| x_3) f(x_3)$. By assumption, we have $\tilde{F}(x_1|x_3) \leq \tilde{F}(x_2|x_3)$, for all $x_3$. An inverse mixing with respect to $f(x_3)$, as mentioned above, completes the proof. This result can be understood easily in the discrete case using tables. It says that if there are two tables of probabilities with the same row margins then if every row of one table dominates the corresponding row of the other table then the whole table dominates the other whole table. Volume-contractive mappings --------------------------- We have seen that the area of the support is a key component of studying $\preceq$. For example, in the discrete case, if $n=0$ and $p= (p_1, p_2, p_3,0)$ are our probabilities with $p_1+p_2+p_3 = 1$, we can say we have support size 3. If we then split $p_3$ to form $q = (p_1, p_2, \frac{p_3}{2}, \frac{p_3}{2})$ then $q \preceq p$. In the continuous case, we refer to such an operation as dilation: locally, we have the same amount of density but stretch the support. This is a dilation in the continuous case via a special kind of transformation of the random variable, whose inverse we can call contractive. The result implies that the volume contractible mappings decrease uncertainty. A volume differential invertible mapping $ h: \mathbb R \rightarrow \mathbb R$, $y = h(x)$ will be called a volume-contractive mapping if the determinant of its Jacobian: $J = \vline \max{det} \left\{ \frac{\partial y_i}{\partial x_j}\right\} \vline$ satisfies $ 0 < J \leq 1$ for $x \in \mathbb R$. If $h( \cdot) $ is a volume contractible inverse mapping $\mathbb R \rightarrow \mathbb R$, then, for any random variable $X \sim f_X(x)$, it holds that: $$X \preceq Y = h(X).$$ We give a proof for the one dimensional case and, in addition, assume $f_X(x)$ and $f_Y(y)$ are invertible. Using the slice condition, we want to show that $$\mbox{prob} \{ f_X(X) \geq c\} \geq \mbox{prob} \{ f_Y(Y) \geq c\}.$$ Developing the left hand side, we see that $$\begin{aligned} \{ f_X (X) \geq C \} & \Leftrightarrow & \{X \geq f_X^{-1}(c) \} \\ & \Leftrightarrow & \{Y \geq h(f_X^{-1}(c)) \} \\ & \Leftrightarrow & \{X \geq f_X^{-1}(c) \}.\end{aligned}$$ Computing the density of $Y$ as $$f_Y(y) = |J^{-1}| f_X(h^{-1} (y)),$$ gives $$f_Y^{-1}(c) = h\left(f_X^{-1} ( J c)\right).$$ We thus need to establish whether $h(f_X^{-1} (c)) \geq h( f_X^{-1} ( J c)).$ We see the statement reduces to $c \geq J^{-1} c,$ which holds by assumption. Contractive flows in sensitivity analysis ----------------------------------------- A motivation for the previous subsection was to provide a method of analysis for systems within the general area of uncertainty quantification, one aspect of which is sensitivity analysis: the study of the propagation of variability through systems from input to output. We have seen above how a volume contractive mapping $Y=h(U)$ can decrease uncertainty. This subsection covers a closely related idea: how to show that, for two different inputs, the outputs may have more or less uncertainty. There are several areas of study in which it is hoped to decrease the variability of $Y$ via different types of intervention on the input $U$. Examples include the theory of antithetic variables in classical simulation, stochastic control, portfolio optimisation and robust design [@bates2002optimisation]. More generically, trying different random inputs $U$ is central to Monte Carlo simulation. We represent the intervention on $u$ as a deterministic transformation and we seek a way to reduce the variability of $Y$ by shifting $U$ in some way. Consider the composition: $$y_0 = G(u_0), \quad u_1 = h(u_0), \quad y_1 = G(u_1).$$ For some function intervention on $u$ given by $h(\cdot)$, we express schematically $$\begin{array}{ccc} y_0 & \leftarrow & u_0 \\ \rotatebox{-90}{$\dashrightarrow$} & & \rotatebox{-90}{$\rightarrow$} \\ y_1 & \leftarrow & u_1 \end{array}$$ Here, we are interested in what kind of interventions will result in volume contractive mappings from $y_0$ to $y_1$, induced by the intervention $h(u)$ in the dashed arrow in the diagram. When this holds, we can say that there is less uncertainty about the stochastic output $Y_1$ than about $Y_0$, for any input $U_0$. A note of caution is that the induced function $y_0 \rightarrow y_1$ needs to be properly defined, in which case we can say that $h(u)$ is compatible. It is somewhat easier in explanation when the $u$-space and $y$-space have the same dimension. In addition, as we now see, local developments are easier. It is convenient to express $h(u)$ as a [*flow*]{} of the form: $$\begin{aligned} h(u,\epsilon) = h(u_0) + \epsilon \xi(u_0) +\mbox{O}(\epsilon^2).\end{aligned}$$ In one dimension, $$\begin{aligned} y_1 = G(u_0 + \epsilon \xi(u_0) ) + \mbox{O}(\epsilon^2),\end{aligned}$$ then $$\begin{aligned} \frac{d y_1}{d y_0} & = G'(u_0 + \epsilon \xi (u_0)) \frac{d u_0}{dy_0} + \mbox{O}(\epsilon^2),\\ & = 1 + \epsilon\frac{\xi(u_0) G'' (u_0) + G'(u_0)\xi'(u_0)}{G'(u_0)} + \mbox{O}(\epsilon^2).\end{aligned}$$ Thus, if $\xi(u) = c > 0 $, that is a constant, then $\frac{d y_1}{d y_0} < 1$ (locally) if and only if $$\begin{aligned} \frac{G''(u_0)}{G'(u_0)} = \frac{d}{du_0} \log G'(u_0) < 1.\end{aligned}$$ For the multivariate linear case, we assume that the dimension of the $u$- and $y$- space are the same, namely $n$, for $n \times n$ matrices $A,B$ write: $$\begin{aligned} y_0 = A u_0, \quad u_1 = u_0 + \epsilon B u_0 + \mbox{O}(\epsilon^2),\quad y_1 = Au_1,\end{aligned}$$ so that $$\begin{aligned} y_1 = y_0 + \epsilon A B A^{-1}y_0 + \mbox{O}(\epsilon^2).\end{aligned}$$ Then locally we want $$\begin{aligned} |\mbox{det} (I + \epsilon A B A^{-1})| < 1,\end{aligned}$$ for small $\epsilon$. If $\{\lambda_i\}$ are the eigenvalues of $ABA^{-1}$ then the condition is $$\begin{aligned} | \prod_{i=1}^n (1+ \epsilon \lambda_i)| < 1.\end{aligned}$$ There is a particular problem when the input space and output space have different dimensions. Thus let the $y$-space above be one dimensional and the $u$-space $n$-dimensional. We can write for an $n$-vector $a$ $$y_0 = a^Tu_0, \quad u_1 = u_0 + \epsilon B u_0 + \mbox{O}(\epsilon^2)\quad y_1 = a^Tu_1,$$ so that $$y_1 = (a^T + \epsilon a^T B)u_0 + \mbox{O}(\epsilon^2).$$ Then, when we consider the random input version, and with the additional condition that the components of the random $U_0$ are independent, we have that $Y_0 \prec Y_1 $, when $B^T a \leq 0$ (componentwise). Geometrically, this says that the vector $a$ must lie in the dual cone generated by the columns of $B$. Consider the problem when the input space and output space have different dimensions. We study the quadratic case, whose importance stems from the fact that, under suitable smoothness conditions, all function are locally quadratic. Let $A$ be an $n \times n$ symmetric positive definite matrix and consider the quadratic form $$y = u^TAu.$$ The function $y$ has a minimum at the origin. Starting at the point $u_0$, the natural shift which would contract the output is the direction of steepest descent which is given by the negative of the gradient at $u_0$, namely $$\frac{\partial y}{\partial u}\; \vline_{_{\;u=u_0}} = 2Au.$$ The flow in that direction would be expressed by $$u_1 = u_0 - \epsilon Au_0 +\mbox{O}(\epsilon^2),\quad y_0=u_0^TAu_0,$$ which gives $$\begin{aligned} y_1 & = & u_1^T A u_1 + O(\epsilon^2),\\ & = & y_0 - 2\epsilon u_0^T A^2 u_0 + \mbox{O} (\epsilon^2), \\ & = & u_0^T (A - 2\epsilon A^2) u_0 + \mbox{O}(\epsilon^2). \end{aligned}$$ Taking the spectral decomposition of $A: A= \sum \lambda_i z_i z_i^T$, where the $\{\lambda_i\}$ are non-negative eigenvalues and the $\{z_i\}$ are standard unit length eigenvectors of $n\times 1$ dimension, we have $$y_1 = \sum_{i=1}^n (\lambda_i - 2\epsilon \lambda_i^2) (u_0^Tz_i)^2 + O(\epsilon^2).$$ If we take the input $U_0$ to be a vector of iid standard $N(0,1)$ random variables, then the variables $\{Z_i = (U_0^Tz_i)^{2}\}$ are independent $\chi^2$ random variables with 1 degree of freedom. Then it is straightforward to show that $$\sum_{i=1}^n \lambda_i Z_i \prec \sum_{i=1}^n (\lambda_i - 2\epsilon \lambda_i^2)\; Z_i,$$ so that approximately $Y_0 \prec Y_1$ where $Y_0 = U_0^T A U_0$, $Y_1 = U_1^T A U_1$ and $U_1 = U_0 - \epsilon AU_0 +\mbox{O}(\epsilon^2)$. Algebra for uncertainty {#sec:algebra} ======================= Let us recall some of the notation we have used. The majorisation ordering $\preceq$ is defined by: $$X_1 \preceq X_2 \Leftrightarrow \tilde{F}_1(z) \leq \tilde{F}_2(z)\; \mbox{for all}\; x>0,$$ where $\tilde{F}$ means the (one dimensional) cdf for the decreasing rearrangement density $\tilde{f}$. We may write, equivalently, $\tilde{F}_1(z) \preceq \tilde{F}_2(z).$ Recall also that we can compare distributions in different dimensions via the construction given in Section \[sec:multivariate\]. We have the general concept of inverse mixing $ \tilde{h}_1\; [+] \; \tilde{h}_2$, as in Definition \[def:inversemixing\], applied to invertible functions $ h_1(x)$ and $h_2(x)$, whether or not they are densities. Denote by $\tilde{F}_1 \otimes \tilde{F}_2$ the associated cdf of the density, $$\begin{aligned} \label{eq:x} \frac{1}{2} ( \tilde{f}_1(z) \; [+] \; \tilde{f}_2(z)) = \left(\tilde{h}_1(2z)^{(-1)} + \tilde{h}_2(2z)^{(-1)} \right)^{(-1)}. \end{aligned}$$ The first type of algebra is based on the following definition. \[def:lattice\] For any two DR cdfs $\tilde{F}_1$ and $\tilde{F}_2 (z)$, we define $$\tilde{F}_1(z) \vee \tilde{F}_2 (z) = \max(\tilde{F}_1(z), \tilde{F}_2(z)),$$ $$\tilde{F}_1(z) \wedge \tilde{F}_2 (z) = \min(\tilde{F}_1(z), \tilde{F}_2(z)),$$ which themselves are cdfs. The partial ordering $\preceq$ under the meet and join $\vee$ and $\wedge$ defines a lattice which we refer to as the [*uncertainty lattice*]{}. It is satisfying that the ‘meet’ and ‘join’ which are defined once $\preceq$ is established can be manifested by the max and min of Definition \[def:lattice\]. We stress that, because we can embed a multidimensional distribution with density $f(x)$ into the one dimensional DR with pdf $\tilde{f}(z)$ and cdf $\tilde{F}(z)$, we can claim that the lattice is universal. The inverse mixing cdf $\otimes$ can be combined with $\vee$ (or $\wedge$). To make the notation more appropriate, we will replace $\vee$ by $\oplus$ and then $\oplus$ and $\otimes$ yield a so-called max-plus (also called tropical) algebra [@maclagan2015introduction]. For this to be valid, we need distributivity. \[dist\] We have $$\tilde{F}_3 \otimes ( \tilde{F}_1 \oplus \tilde{F}_2) = (\tilde{F}_3 \otimes \tilde{F}_1) \oplus (\tilde{F}_3 \otimes \tilde{F}_2),$$ where $\tilde{F}_1,\tilde{F}_2$ and $\tilde{F}_3$ are all cdfs for DR where $\tilde{F}_1(z) \oplus \tilde{F}_2 (z) = \max(\tilde{F}_1(z),\tilde{F}_2(z))$ and $\otimes$ is given in Equation (\[eq:x\]). This follows from the useful expression of inverse mixing for cdfs, namely: $$\tilde{F}_1(z) \otimes \tilde{F}_2(z) = \left( \tilde{F}_1(2z)^{(-1)} + \tilde{F}_2(2z)^{(-1)} \right)^{(-1)}.$$ Swapping min for max, we obtain $$\min\left(\tilde{F}_3(2z)^{(-1)} + \tilde{F}_1(2z)^{(-1)}, \tilde{F}_3(2z)^{(-1)} + \tilde{F}_2(2z)^{(-1)} \right) =$$ $$\tilde{F}_3(2z)^{(-1)} + \min \left(\tilde{F}_1(2z)^{(-1)}, \tilde{F}_2(2z)^{(-1)}\right),$$ which is the key step and common to maxplus (tropical) proofs. We can now define a (semi) ring which we shall call the [*uncertainty ring*]{}. We first note that the $\otimes$ unit element will be $0$ and the $\oplus$ unit element will be $-\infty$. We immediately have the issue that, to define the ring, we need to remove the fact that $\tilde{f}$ is a density and $\tilde{F}$ is a non-negative function with $\tilde{F}(z) \rightarrow 1$ as $z \rightarrow \infty$. The polynomials which comprise the ring require powers and monomials.\ Consider the pdf arising from $\tilde{F}_1 \otimes \tilde{F}_1$. This is: $$\begin{aligned} \tilde{f}(z) & = & \left( \tilde{f}(2z)^{(-1)} + \tilde{f}(2z)^{(-1)} \right)^{(-1)}, \\ & = & \frac{1}{2} \tilde{f}\left(\frac{z}{2} \right), \end{aligned}$$ which is the pdf for the scaled random variable $Y= 2 X$ where $X \sim \tilde{f}(z)$. The pdf for $Y$ is $\tilde{F}\left(\frac{z}{2}\right)$. In general the $k$-th $\otimes$ power is $$\otimes^n \tilde{F} (z) = \tilde{F}\left( \frac{z}{n} \right).$$ The intuition of this expression is that increasing powers represent increasing dilation and form a decreasing chain with respect to our $\preceq$ ordering. A monomial with respect to $\otimes$ takes the form $$\prod_{i=1}^{m} \otimes^{\alpha_i} \tilde{F}_i(z).$$ Adjoining the base field $\mathbb R$, and appealing to Lemma \[dist\], we can define a ring of tropical polynomials [@glicksberg1959convolution]. The uncertainty ring is the toric semi ring of non-decreasing, twice-differentiable functions on $[0, \infty)$ on $\otimes$ and $\oplus$ and with $\oplus$ identity as $-\infty$. To obtain proper decreasing densities we need to impose the additional condition that $\tilde{f}(z)$ is decreasing, $\tilde{F}(0) = 0$ and $\tilde{F}(z) \rightarrow 1$ as $z \rightarrow \infty$. Assuming that we have proper pdfs, we summarise the operations that we have: 1. Scalar multiplication $\tilde{F} \rightarrow \beta\tilde{F}$, $ \beta \in \mathbb R$. 2. Inverse mixing $\tilde{F}_1 \otimes \tilde{F}_2$. 3. Maximum and minimum of $\tilde{F}_1, \tilde{F}_2$, denoted $\vee$ and $\wedge$, respectively. Noting that $\vee$ is written $\oplus$, when discussing the ring. We can also define a min-plus algebra and may use $\oplus$. 4. Powers and monomials. 5. Convolution $\tilde{F}_1 * \tilde{F}_2$. This refers to the DR pdf of the sum of independent random variables $X_1 \sim F_1$ and $X_2 \sim F_2$. Further natural developments using ring concepts such as ideals are the subject of further work. In fact, convolutions themselves form a semi-group [@feller2008introduction], but we do not delve into the relationship between our ring and that semi-group. It is instructive to work over the binary field so that we do not have to use full scales from $\mathbb R$, but only $\{0,1\}$. This also has the advantage that, in every polynomial, we have proper pdfs and cdfs. An analogy is Boolean algebra. In Example \[sec6:example\_exp\] we illustrate these operations and the complexity obtained from a single distribution. \[sec6:example\_exp\] Exponential with unit mean. Let $X_1 \sim \exp\{-x\}, $ with $x>0$. We have, $$\begin{aligned} \tilde{F}_1 & = & 1-e^{-z}, \\ \tilde{F}_2 = \tilde{F}_1 * \tilde{F}_1 & = & 1 - (1+\sqrt{2z}) e^{-\sqrt{2z}}, \\ \tilde{F}_3 = \otimes^2 \tilde{F}_1 & = & \frac{z}{2} \left( 1- e^{-z} \right), \\ \tilde{F}_4 = \otimes^2 \tilde{F}_2 & = & \frac{1}{2} \left( z- e^{-\sqrt{2z}} \right) \\ \tilde{F}_5 = \tilde{F}_1 \otimes \tilde{F}_2 & = & \frac{z}{2} - \left( \frac{z}{2} + \frac{z^{\frac{3}{2}}}{\sqrt{2}} \right) e^{-\sqrt{2z}}.\end{aligned}$$ For the uncertainty lattice in Definition \[def:lattice\], we have $$\tilde{F}_4 \preceq \tilde{F}_5 \preceq \tilde{F}_2 \preceq \tilde{F}_1, \quad \text{and} \quad \tilde{F}_5 \preceq \tilde{F}_3 \preceq \tilde{F}_1,$$ and neither $\tilde{F}_2$ or $\tilde{F}_3$ dominate the other. Under $\vee$ and $\wedge$ we can include $ \tilde{F}_3 \vee \tilde{F}_2$ and $\tilde{F}_3 \wedge \tilde{F}_2,$ and show that $$\tilde{F}_4 \preceq \tilde{F}_3 \wedge \tilde{F}_2 \preceq \tilde{F}_3 \vee \tilde{F}_2 \preceq \tilde{F}_1.$$ Algebra and entropies --------------------- Given the equivalence of the majorisation in Section \[sec:cont\_major\], we study how the above structures affect the manipulation of uncertainty measured by a single metric, $$H(f(x)) = \int h(f(x)) dx,$$ for some convex function $h( \cdot)$. Since $$H(f(x)) = \int_{0}^{\infty} h(\tilde{f}(z)) dz,$$ we can, without loss of generality, consider $H$ as a functional of the DR with the advantage that the operations $\otimes,\oplus$ can be applied. The following, then, is a collection of operations and results, which we can claim as a toolbox for handling uncertainty, which can be applied to any uncertainty ($-H$), in this class. We omit the proofs. 1. $\tilde{f}_1 \preceq \tilde{f_2} \Rightarrow H(\tilde{f}_1) \leq H(\tilde{f}_2).$ 2. $H\left(\tilde{f}_1 \otimes \tilde{f}_2 \right) = H\left(\tilde{\frac{1}{2}f}_1\right) + H\left(\frac{1}{2}\tilde{f}_2\right).$ 3. $H\left(\tilde{f}_1 \oplus \tilde{f}_2 \right) \geq \max \left\{H\left(\tilde{f}_1\right), H\left(\tilde{f}_2\right) \right\}.$ 4. $H\left(\tilde{f}_1 \otimes (\tilde{f}_2 + \tilde{f}_3) \right) \geq H\left(\frac{1}{2}\tilde{f}_1\right) + \max\left\{H\left(\frac{1}{2}\tilde{f}_2\right), H\left(\frac{1}{2}\tilde{f}_3\right)\right\}.$ Uncertainty toolbox: future scenarios ------------------------------------- We present two practical situations in which the results in Section \[sec:algebra\] are employed to handle the uncertainty in practical situations, by combining them in two different ways. Suppose there is initially to be $i=1,2$ races with a different number of horses $n_i$ ranked in order of their probability of winning (in the mind of a punter or bookmaker): $$p_{(i,1)} \geq \cdots \geq p_{(i,n_i)},$$ such that $\sum_{j=1}^{n_i}p_{i,j}=1, i=1,2$. If the two sets of horses should be combined into a single race, the issue then is how to combine the probabilities. The simplest approach is to divide each probability by two and combine all the probabilities, ranking them in the process, i.e, $\tilde{F}_1 \otimes \tilde{F}_2$, that is inverse mixing with $\alpha = \frac{1}{2}$. One can imagine some effect which may lead to having $\alpha \neq \frac{1}{2}$ such as the track being wet and not suited to one set of horses. Consider the case in which we have a single race (say race 1) with two punters who rank the horses in the same order, but with different probabilities, i.e., $p_{(1,1)}\geq \ldots \geq p_{(1,n_1)}$ and $q_{(1,1)} \geq \ldots \geq \ldots q_{(1,n_1)}$, respectively. To define a joint betting strategy, a set of odds combining each of their own odds could take different approaches: an optimistic, more certain, approach with $\max(\tilde{F}_1(z),\tilde{F}_2(z))$ or a more pessimistic, uncertain, approach with $\min(\tilde{F}_1(z),\tilde{F}_2(z))$. We note that the argument in the last paragraph is predicated on the two punters’ initial rank order being the same. If not, there is a danger that the same horse may appear twice in the min or max ordering. The min or max may then refer to a kind of hypothetical race. Nonetheless, we suggest that they are useful notionally. The same issue arises if one considers an average of the two actual probabilities, direct mixing, $\frac{1}{2}(p_{(1,1)} +q_{(1,1)} )\geq \ldots \geq\frac{1}{2}(p_{(1,n_1)}+q_{(1, n_1)}),$ which corresponds to taking $\frac{1}{2}(\tilde{F}_1(z) + \tilde{F}_2(z))$. Now suppose there are two sets of horses of size $n_1$ and $n_2$ and two punters. We see then that the toolbox provides various ways to combine uncertainties within rows and columns. With obvious notation, we have punter 1 punter 2 -------- ------------------- ------------------ race 1 $\tilde{F}_{11}$ $\tilde{F}_{21}$ race 2 $ \tilde{F}_{12}$ $\tilde{F}_{22}$ \[commiteeexample\] Consider a scenario with two committees, each with the same number of members. Each committee covers a different area of oversight, for example different technologies to solve a particular problem, say ‘electricity’ and ‘gas’. We assume that, within each committee, there is a common agreement about a range of possible future scenarios in the committee’s area. We define two future scenario types: 1. an active situation: to choose between the two technologies on some grounds, in which the two committees’ assessments would not be combined. 2. a passive situation: where only the probability of a particular technology being used can be assessed, which may be a consequence of unforeseen events. In the latter case, the inverse mixing is a way to combine the uncertainties into an overall assessment. The actual future would be one gas-based alternative or one electricity alternative, the “horse” that won the combined race. The difficulties that may arise, as pointed out already, are that the private initial order of members of the [*same*]{} committee may not be the same as one familiar in subjects such as choice theory and rater assessment. Note that independence between committees tends to increase uncertainty, and, in our development, the unidimensional DR is equivalent to inverse mixing. This shows, heuristically, that, when two committees act independently, there is more, rather than less, uncertainty. Empirical decreasing rearrangements {#sec:empirical} =================================== We extend the applications of DR to analyse a data set collected from an experiment in a large number of trials or from a product of computer simulations. We present two approaches for deriving the empirical DR and its associated cdf from a data set. In Section \[subsec:Discrete\_Climate\], we assess the uncertainties associated with climate projections, in which we transform continuous variables into discrete ones by grouping values, which is easy to perform in two dimensions. In Section \[subsec:Cont\_Heat\], algorithms are presented to obtain approximations for $\tilde{f}(z)$ and $\tilde{F}(z)$. This allows us to perform majorisation over data sets in higher dimensions, which we demonstrate analysing the risk associated with individual infrastructure decisions in an energy system. Discrete Majorisation - climate projections {#subsec:Discrete_Climate} ------------------------------------------- The UKCP18 climate projections [@UKCP18] consider four different scenarios of greenhouse gas concentrations in the atmosphere, called Representative Concentration Pathways (RCP). We consider variables over the 12km gridboxes that cover the UK. UKCP18 uses ensemble methods in which the model is run multiple times with slightly differing initial conditions and parameter values to account for observational and parametric uncertainty. We consider two variables: 1. Increase in mean air temperature at a height of 1.5 metres, 2. Percentage increase in precipitation, where each variable is relative to the baseline period of 1981-2010. The projections correspond to mean daily values over the period from 2050 to 2079. The data, illustrated in Figure \[fig:climate\_scatter\], is discretised by dividing each variable into ranges and counting the number of ensemble members that fall into each category in the two dimensions. The temperature anomaly is divided into five categories, whilst the increase in precipitation is divided into four categories, therefore an ensemble member falls into one of 20 categories. ![Scatterplot showing how ensemble members are categorised according to the value each variable. Each point represents an ensemble member and each colour represents a different RCP.[]{data-label="fig:climate_scatter"}](climate_scatter.png) We present the joint distribution for two contrasting scenarios for RCP2.6 and RCP8.5, of temperature anomaly and percentage change in precipitation is shown in Table \[table:clim\_dist\]. The ordered probabilities for each RCP are shown in Figure \[fig:climate\_example\_pdf\_cdf1\]. We use this to obtain empirical cdfs of DR together with the maximum and minimum of the cdfs depicted in the left and central panel plots in Figure \[fig:climate\_example\_pdf\_cdf2\]. We observe that the empirical cdf for RCP2.6 lies above that of RCP8.5. The uncertainty is therefore related to the level of assumed greenhouse gas concentration in the atmosphere. In particular, RCP2.6 carries the lowest level of uncertainty among the considered scenarios since its cdf corresponds to $\tilde{F}_1(z) \lor\tilde{F}_2(z)\lor\tilde{F}_3(z)\lor\tilde{F}_4(z)$, where the subscript indicates the scenario. In contrast, RCP8.5 carries the most uncertainty, since its cdf corresponds to $\tilde{F}_1(z) \land\tilde{F}_2(z)\land\tilde{F}_3(z)\land\tilde{F}_4(z)$. [|ll|&gt;cc&gt;cc&gt;cc&gt;cc&gt;cc|]{} & &\ & & & & & &\ & [**$<$0%**]{} & 0.01 & 0.00 & 0.18 & 0.02 & 0.17 & 0.09 & 0.01 & 0.10 & 0.00 & 0.04\ & [**0-5%**]{} & 0.01 & 0.00 & 0.22 & 0.03 & 0.28 & 0.16 & 0.04 & 0.20 & 0.00 & 0.08\ & [**5-10%**]{} & 0.00 & 0.00 & 0.02 & 0.01 & 0.05 & 0.07 & 0.01 & 0.11 & 0.00 & 0.06\ & [**$>$10%**]{} & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.02 & 0.00 & 0.02\ ![Ordered probabilities under each RCP.[]{data-label="fig:climate_example_pdf_cdf1"}](climate_example_pdf_cdf1.pdf) ![*Left panel*: Empirical cdf for the DR for each scenario. *Central panel*: the representation of $\max(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z), \tilde{F}_4(z))$ and $\min(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z), \tilde{F}_4(z))$. *Right panel*: cdf of inverse mixing with equal weights specified for each scenario.[]{data-label="fig:climate_example_pdf_cdf2"}](climate_example_pdf_cdf2.pdf){width="100.00000%"} We can apply inverse mixing, described in Section \[subsec:InverseMix\], to combine the uncertainties in the four scenarios. For the discrete case, we specify an equal weighting for each pdf, and we illustrate cdf of the inverse mixing in the right panel in Figure \[fig:climate\_example\_pdf\_cdf2\]. Majorisation in higher dimensions {#subsec:Cont_Heat} --------------------------------- We denote a data set $x_{ij}, i=1, \dots, m$ and $j=1, \dots, n$, where $m$ and $n$ correspond to the total number of data points and dimensions respectively. To obtain the DR, we require the density function values to construct the distribution function $m(y)$ (see Section \[sec:cont\_major\]). We assume that the observed data are a sample from a population with unknown pdf $f_X(x_1, \dots, x_n)$, from which we estimate the pdf $\hat{f}_X(x_1, \dots, x_n)$. In our examples, we employ kernel density estimation to obtain $\hat{f}_X(x_1, \dots, x_n)$ using the `ks` package in `R`, which automatically selects the bandwidth parameters [@ks2020]. Algorithm 1 describes the approach adopted in this paper to obtain empirical DRs $\tilde{f}_{\hat{f}}(z)$. We note from Definition \[drdefn\] that obtaining the DR is a two-stage process. At the first stage, we are required to obtain a measure (distribution) function $m(y)$. For instance, in 1D, a distribution function $m(y)$ returns the length of the intervals on the $x$-axis at which $f(x)\geq y$. At the second stage, we employ the measure function to derive the DR. Based on data $x_{ij}\in R, i=1, \dots, m$ and $j=1, \dots, n$, fit a pdf $\hat{f}_X(x_1, \dots, x_n)$ using kernel density estimation Produce a uniform and/or space-filling set $S$ of size $N$ across the input space $R$, with “sites” $s\in S$ Plot the estimated measure function values, $m_{\hat{f}}(y)$ against $y$ Swap the abscissa with ordinate, so that $\tilde{f}_{\hat{f}}(z)$ and $z$ correspond to $y$ and $m_{\hat{f}}(y)$. As part of this algorithm, we employ Monte Carlo integration [@Fok1989] to estimate the volume of domain $S_y$ to derive the measure function $m_{\hat{f}}(y)$. In particular, [@Fok1989] proposed specifying another domain $R$ (a hypercube or a hyperplane) of known volume $\text{Vol}(R)$, such that $ S_y\in R. $ The ratio of two volumes, $p=\text{Vol}(S_y)/\text{Vol}(R)$, and the volume $\text{Vol}(S_y)$ are estimated by $$\hat{p}=N_y/N\quad \text{and}\quad \hat{\text{Vol}}(S_y)=\hat{p}\text{Vol}(R).$$ We employ Algorithm 1 on a bivariate data set. We start by generating a random sample of size $m=200$, i.e. $x_{ij}, i=1, \dots, 200$ and $j=1, 2$, from a standard bivariate normal distribution, since we have the closed form expression for DR of a standard 2-dimensional normally distributed random variable $X$: $$\label{eq:DR2D} \tilde{f}(z)=\frac{1}{2\pi}\exp\Big\{-\frac{1}{2}\frac{z}{\pi} \Big\}.$$ To perform the algorithm, we produce a uniform sample of points of size $N=2500$ across the domain $R=[-5, 5]\times [-5, 5]$ of $\text{Vol}(R)=10^2$. Figure \[fig:EmpDRFinal1\] demonstrates the implementation of Algorithm 1 as well as comparing $\tilde{f}_{\hat{f}(z)}$ to $\tilde{f}(z)$. In the left panel, we depict the estimated values of the distribution function $m_{\hat{f}}(y)$ against $y$. Note that the smoothness of the estimated distribution function depends on $M$, i.e. we expect to obtain a smooth representation of the distribution function with a large value of $M$ (cutoffs in density). We observe that the empirical DR (red dashed line) is overlapping with the DR in equation (\[eq:DR2D\]) (blue solid line) in the right panel. ![*Left panel*: A plot of the estimated measure function $m_{\hat{f}}(y)$ against $y$. *Right panel*: A DR plot where the blue solid line and the red dashed line correspond to $\tilde{f}(z)$ and $\tilde{f}_{\hat{f}}(z)$.[]{data-label="fig:EmpDRFinal1"}](EmpDRFinal.pdf){width=".8\textwidth"} We present Algorithm 2 for obtaining an empirical cdf of the DR, an approximation to $\tilde{F}(z)$, denoted as $\tilde{F}_{\hat{f}}(z)$. Specify an equally spaced vector $\boldsymbol{z}^*=(z_1^*, z_2^*, \dots, z_l^*)$ Fit a linear interpolator (spline) through $\{z_i, \tilde{f}_{\hat{f}}(z_i)\}_{i=1}^M$ (these values were derived in Algorithm 1) to obtain values of $\tilde{f}_{\hat{f}}(z_i^*), i=1, \dots, l$ Plot $\tilde{F}_{\hat{f}}(z^*)$ against $z^*$. The weighting of computed probabilities by $$\frac{1}{\sum_{k=1}^{n-1}P(z_k<z<z_{k+1})},$$ in Algorithm 2 comes from the assumption that $z$ is upper bounded and we can only compute probabilities at the values specified in $\boldsymbol{z}^*$. Therefore, we expect $\sum_{k=1}^{n-1}P(z_k<z<z_{k+1})=1$. However, we tend to observe this sum to be slightly less than one due to errors introduced by numerical integration. Our proposed weighting is similar to normalisation performed as part of the construction of a histogram. We proceed to demonstrate the implementation of Algorithm 2 in Figure \[fig:CDFDR2D\] on the bivariate data set used previously. We have a closed form expression for $\tilde{F}(z)$ of a standard 2-dimensional normally distributed random variable $X$: $$\label{eq:CDF_DR2D} \tilde{F}(z)=1-\exp\Big\{-\frac{z}{2\pi} \Big\}.$$ ![*Left panel*: binned probability representation of the empirical DR, $\tilde{f}_{\hat{f}}(z)$, obtained as part of Algorithm 2. *Right panel*: empirical cdf of the DR, $\tilde{F}_{\hat{f}}(z)$ obtained from Algorithm 2. The blue solid line and red dashed line correspond to $\tilde{F}(z)$ and $\tilde{F}_{\hat{f}}(z)$ respectively.[]{data-label="fig:CDFDR2D"}](CDFModifiedScheme2D.pdf){width=".75\textwidth"} Based on the right panel in Figure \[fig:CDFDR2D\], we conclude that by employing Algorithm 2, the empirical cdf $\tilde{F}_{\hat{f}}(z)$ is an accurate representation of $\tilde{F}(z)$. ![Empirical cdf of DRs obtained for random samples from the standard normal distribution with $n=1, \dots, 4$ (dimension).[]{data-label="fig:EmpiricalCDF"}](EmpiricalCDF.png){width="50.00000%"} We used Algorithms 1 and 2 to construct $\tilde{F}_{\hat{f}}(z)$ in Figure \[fig:EmpiricalCDF\] for a random sample from the multivariate standard normal with varying dimensions. We observe that the empirical cdf of the DR based on a sample from the univariate normal distribution majorises the remaining empirical cdfs. This is an expected result and confirms our expectation that uncertainty increases as we increase the number of dimensions. The accuracy of the representation of the DR is sensitive to the number of data points in the sample, $m$, while the smoothness of DR depends on $M$. ### District heating example {#subsubsec:DHE} We compare the uncertainty associated with three potential design options for supplying heat in a model based on a system in Brunswick, Germany [@REUSEHEAT]. District heating networks allow heat from a centralised source to be distributed to buildings through a network of insulated pipes [@werner2013district]. This allows for a range of potential sources of heat to be connected and, in recent years, the idea of ‘reusing’ excess heat produced by nearby sources such as factories has gained traction as part of efforts to decarbonise the energy sector. This is done by using excess heat to heat water which is then distributed through the network. Traditionally, high temperature sources such as from heavy industry have been used, there has been increased interest in recent years in low temperature sources such as data centres, metro systems and sewage which require electric heat pumps to ‘upgrade’ the temperature before being suitable for use in the system. The Brunswick case is a demonstrator on the EU funded ReUseHeat project [@REUSEHEAT] that aims to demonstrate the use of low temperature sources of heat for use in district heating networks. The city’s existing district heating network is powered by a Combined Heat and Power (CHP) plant, which uses natural gas as a fuel and outputs both heat for use in the network and electricity. The network in the newly constructed area of interest will be connected to the CHP and, in addition, there is an option to use excess heat from a data centre to provide at least some of the heat to the district. There are therefore three potential design options to be considered and these are shown in Table \[tab:Design\]. The question of interest is then which design option should be chosen, taking into consideration both cost and carbon emissions. Although not used in actual decision making for the city or in the ReUseHeat project itself, a simple model was described in [@Volodina2020] which outputs both Net Present Cost (NPC) in € and $\text{CO}_2$-equivalent emissions (in metric tonnes). We use these simulations to demonstrate majorisation as a tool for decision making. Local and global sensitivity analysis was performed for each design option by varying a number of inputs to the model, resulting in a total of 81 simulations. In addition, three scenarios were considered and these are shown in Table \[tab:Scenarios\], further details of which are given in [@Volodina2020]. In Figure \[fig:ScatterPlot\], we illustrate the outputs of the model under sensitivity analysis by plotting emissions against NPC for each of the three design options and under each scenario. The highest level of emissions comes from design option 1 due to the use of natural gas whilst design option 3 is shown to be the most environmentally friendly. However, there is an inverse relationship between carbon emissions and cost with design option 3 being the most expensive. ![Net Present Costs against carbon emissions for all three design options under the three scenarios.[]{data-label="fig:ScatterPlot"}](ScatterPlot.png){width="70.00000%"} We demonstrate majorisation and DR in the context of the model outputs under the three scenarios. In particular, we obtain $\tilde{f}_{\hat{f}}(z)$ and $\tilde{F}_{\hat{f}}(z)$ based on the distribution of points under the sensitivity analysis for each design option and under each scenario by employing Algorithms 1 and 2. To apply equal importance to both outputs, we scale the data on $[0, 1]$ and generate a uniform set $S$ across $[0, 1]\times [0, 1]$ of size $N=2500$. To produce a smooth representation of the DR and its cdf, we use a value of $M=5000$, which corresponds to the number of cutoffs in the density values. ----------------- ------------------------------- **Design type** **Description** Design option 1 Combined Heat and Power (CHP) Design option 2 CHP and Heat Pump Design option 3 Heat Pump ----------------- ------------------------------- : Description of design options in the District Heating study [@Volodina2020]. \[tab:Design\] -------------- ---------------------- --------------------- ------------------------------------------ **Scenario** **Emission Penalty** **Consumer demand** **Commodity prices** Green 100€/metric tonne -1% annual change $\uparrow$ gas, $\downarrow$ electricity Neutral 40€/metric tonne small fluctuations small fluctuations Market no penalty +1% annual change $\downarrow$ gas, $\uparrow$ electricity -------------- ---------------------- --------------------- ------------------------------------------ : Description of scenarios in the District Heating study [@Volodina2020]. \[tab:Scenarios\] ![Empirical cdf for decreasing rearrangements $\tilde{F}_{\hat{f}}(z)$ for (i) all three design options plotted together for each individual scenario (*first row*), (ii) all three scenarios plotted together for each individual design option (*second row*).[]{data-label="fig:CDF_HeatExample"}](ScenarioPlotHeat.pdf "fig:"){width="100.00000%"} ![Empirical cdf for decreasing rearrangements $\tilde{F}_{\hat{f}}(z)$ for (i) all three design options plotted together for each individual scenario (*first row*), (ii) all three scenarios plotted together for each individual design option (*second row*).[]{data-label="fig:CDF_HeatExample"}](DesignPlotHeat.pdf "fig:"){width="100.00000%"} Plots of the empirical cdfs of the DR $\tilde{F}_{\hat{f}}(z)$ are shown in Figure \[fig:CDF\_HeatExample\]. In the first row, the cdfs obtained for all three design options are considered together under individual scenarios. A feature here is that, under the green and neutral scenarios, the empirical cdf for design option 3 lies above that for design option 2, which lies above that for design option 1 whilst, under the market scenario, the ordering of the empirical cdfs changes and the empirical cdf for design option 1 lies above that for design option 2. We conclude that under all three scenarios, the (unknown) distribution function associated with design option 3 majorises the cdfs for both design options 1 and 2. We therefore consider that, for the outputs of interest, design option 3 is less uncertain than the alternatives. The second row of plots shows empirical cdfs for each scenario plotted together under individual design options. This gives a slightly different view, allowing us to compare how the uncertainty under each design option varies under the different scenarios. For example, under design option 1, there is clear ordering of the cdfs which implies that the Market scenario is less uncertain than the Neutral scenario which, in turn, is less uncertain than the Green scenario. We now demonstrate the uncertainty tools from Section \[sec:algebra\] in order to combine the uncertainty under different scenarios and produce orderings of design options. In particular, under each design option, we find the minimum of the empirical cdfs associated with individual scenarios to obtain an approximation to $\tilde{F}_1(z)\lor\tilde{F}_2(z)\lor\tilde{F}_3(z)$. This is shown in the left panel of Figure \[fig:maxminDR\] and can be considered to represent an ‘optimistic’ approach. We find that design option 3 majorises the other design options. ![*Left panel*: representation of $\max(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z))$. *Right panel*: representation of $\min(\tilde{F}_1(z), \tilde{F}_2(z), \tilde{F}_3(z))$.[]{data-label="fig:maxminDR"}](maxminDR.pdf){width=".7\textwidth"} We also produce an approximation to $\tilde{F}_1(z)\land\tilde{F}_2(z)\land\tilde{F}_3(z)$, which corresponds to a ‘pessimistic’ approach. The results are shown in the right panel of Figure \[fig:maxminDR\] in which we obtain the maximum of the empirical cdfs associated with individual scenarios. In this case, we observe a clear ordering between design options: design option 3 majorises design option 2, which majorises design option 1. Under both the pessimistic and optimistic outlooks, we therefore conclude that design option 3 is less uncertain than the two alternatives. ![cdfs from inverse mixing with different weightings on each scenario: *Left panel*: equal weights on each scenario. *Central panel*: $\alpha_{G}=0.7$, $\alpha_{N}=0.2$ and $\alpha_{M}=0.1$. *Right panel*: $\alpha_{G}=0.05$, $\alpha_{N}=0.05$ and $\alpha_{M}=0.9$[]{data-label="fig:HeatInverseMixing"}](HeatInverseMixing.pdf){width="100.00000%"} We employ inverse mixing to combine the uncertainty associated with all three scenarios under individual design options. This is done by estimating probabilities from the empirical DR, by Algorithm 2 and ordering the probabilities. We consider the defined weights in inverse mixing as probabilities of the occurrence of each scenario. Let $\alpha_{G}, \alpha_{N}$ and $\alpha_{M}$ be the weights applied to the Green, Neutral and Market scenarios, respectively. We consider three different cases for the weights: (i) equal weights, (ii) $\alpha_G=0.7$, $\alpha_M=0.15$ and $\alpha_N=0.15$ and (iii) $\alpha_G=0.05$, $\alpha_M=0.9$ and $\alpha_N=0.05$. The results of the inverse mixing in the three cases are shown in the left, middle, and right panels of Figure \[fig:HeatInverseMixing\], respectively. In cases (i) and (ii), there are clear orderings in which the cdf of design option 3 lies above the cdf of design option 2 which lies above that of design option 1. In case (iii), however, there is no ordering between the empirical cdfs. In particular, the cdfs for design options 1 and 2 cross. However, the pdf associated with design option 3 majorises the pdfs for both design options 1 and 2 and we conclude that design option 3 is the least risky option in all three cases. It is important to note that, whilst the above results provide useful guidance for comparing uncertainty, the uncertainty is only one aspect of such decisions and one would want to take into account the actual costs and carbon emissions (rather than just their variability) in each case. However, here we have demonstrated majorisation to be an intuitive approach to comparing uncertainty and ultimately aiding informed decisions in such settings. Concluding remarks {#sec:conclusion} ================== The concept of uncertainty is the subject of much discussion, particularly at the technical interface between scientific modelling and statistics. It is our contention that uncertainty is close in spirit to entropy, but that restriction to a limited definition of entropy can be lifted. The fact that there are different types points to the existence of a wider framework which may then widen the scope of uncertainty. The idea presented is that a candidate for a wider framework is a stochastic ordering under for which most, if not all, types of entropy are order preserving. Here, we suggest majorisation, which only compares the rank order of probability mass, continuous or discrete. We have shown that any two distributions can be compared, and consider this to be a principal contribution of the paper. We demonstrated this approach to assess the uncertainty in applications of climate projections and energy systems.

ArXiv

--- abstract: 'This paper presents a class of Dynamic Multi-Armed Bandit problems where the reward can be modeled as the noisy output of a time varying linear stochastic dynamic system that satisfies some boundedness constraints. The class allows many seemingly different problems with time varying option characteristics to be considered in a single framework. It also opens up the possibility of considering many new problems of practical importance. For instance it affords the simultaneous consideration of temporal option unavailabilities and the dependencies between options with time varying option characteristics in a seamless manner. We show that, for this class of problems, the combination of any Upper Confidence Bound type algorithm with any efficient reward estimator for the expected reward ensures the logarithmic bounding of the expected cumulative regret. We demonstrate the versatility of the approach by the explicit consideration of a new example of practical interest.' author: - 'T. W. U. Madhushani$^{1}$ and D. H. S. Maithripala$^2$ and N. E. Leonard$^{3}$ [^1] [^2] [^3]' bibliography: - 'DynamicBandit.bib' title: '**Asymptotic Allocation Rules for a Class of Dynamic Multi-armed Bandit Problems**' --- Introduction {#sect:Introduction} ============ In decision theory Multi-Armed Bandit problems serve as a model that captures the salient features of human decision making strategies. The elementary case of a *1-armed bandit* is a slot machine with one lever that results in a numerical reward after every execution of the action. The reward is assumed to satisfy a specific but unknown probability distribution. A slot machine with multiple levers is known as a *Multi-Armed Bandit* (MAB) [@Sutton; @Robbins]. The problem is analogous to a scenario where an agent is repeatedly faced with several different options and is expected to make suitable choices in such a way that the cumulative reward is maximized [@Gittins]. This is known to be equivalent to minimizing the expected cumulative regret [@LaiRobbins]. Over decades optimal strategies have been developed to realize the above stated objective. In the standard multi-armed bandit problem the reward distributions are stationary. Thus if the mean values of all the options are known to the agent, in order to maximize the cumulative reward, the agent only has to sample from the option with the maximum mean. In reality this information is not available and the agent should choose options to maximize the cumulative reward while gaining sufficient information to estimate the true mean values of the option reward distributions. This is called the exploration-exploitation dilemma. In a case where the agent is faced with these choices with an infinite time horizon exploitation-exploration sampling rules are guaranteed to converge to the optimum option. In their seminal work Lai and Robbins [@LaiRobbins] established a lower bound for the cumulative regret for the finite time horizon case. Specifically, they establish a logarithmic lower bound for the number of times a sub-optimal option needs to be sampled by an optimal sampling rule if the total number of times the sub-optimal arms are sampled satisfies a certain boundedness condition. The pioneering work by [@LaiRobbins] establishes a confidence bound and a sampling rule to achieve logarithmic cumulative regret. These results are further simplified in [@AgrawalSimpl] by establishing a confidence bound using a sample mean based method. Improving on these results, a family of Upper Confidence Bound (UCB) algorithms for achieving asymptotic and uniform logarithmic cumulative regret was proposed in [@Auer]. These algorithms are based on the notion that the desired goal of achieving logarithmic cumulative regret is realized by choosing an appropriate uncertainty model, which results in optimal trade-off between reward gain and information gain through uncertainty. What all these schemes have in common is a three step process: 1) a predication step, that involves the estimation of the expected reward characteristics for each option based on the information of the obtained rewards, 2) an objective function that captures the tradeoff between estimated reward expectation and the uncertainty associated with it and 3) a decision making step that involves formulation of an action execution rule to realize a specified goal. For the standard MAB problem the reward associated with an option is considered as an iid stochastic process. Therefore in the frequentist setting the natural way of estimating the expectation of the reward is to consider the sample average [@LaiRobbins; @AgrawalSimpl; @Auer]. The papers [@Kauffman; @Reverdy] present how to incorporate prior knowledge about reward expectation in the estimation step by leveraging the theory of conditional expectation in the Bayesian setting. We highlight that all these estimators ensure certain asymptotic bounds on the tail probabilities of the estimate of the expected reward. We will call such an estimator an *efficient reward estimator*. Furthermore all these methods with the exception of [@LaiRobbins] rely on UCB type algorithms for the decision making process. An extension to the standard MAB problem is provided in [@Kleinberg2010] to include temporal option unavailabilities where they propose a UCB based algorithm that ensures that the expected regret is upper bounded by a function that grows as the square root of the number of time steps. In all of the previously discussed papers, the option characteristics are assumed to be static. However many real world problems can be modeled as multi-armed bandit problems with dynamic option characteristics [@dacosta2008adaptive; @Slivkins; @granmo2010solving; @Garivier2011; @srivastava2014surveillance; @schulz2015learning; @tekin2010online]. In these problems reward distributions can change deterministically or stochastically. The work [@dacosta2008adaptive; @Garivier2011; @srivastava2014surveillance] present allocation rules and associated regret bounds for a class of problems where the reward distributions change deterministically after an unknown number of time steps. The paper [@dacosta2008adaptive] presents a UCB1 based algorithm where they incorporate the Page-Hinkley change point detection method to identify the the point at which the underlying option characteristics change. A discounted UCB or a sliding-window UCB algorithm is proposed in [@Garivier2011] to solve non stationary MAB problems where the expectation of the reward switches to unknown constants at unknown time points. This work is extended in [@srivastava2014surveillance] by proposing sliding window UCL (SW-UCL) algorithm with adaptive window sizes for correlated Gaussian reward distributions. They incorporate the Page-Hinkley change point detection method to adjust the window size by identifying abrupt changes in the reward mean. Similarly, they also propose a block SW-UCL algorithm to restrict the transitions among arms. A class of MAB problems with gradually changing reward distributions are considered in [@Slivkins; @granmo2010solving]. Specifically [@Slivkins] considered the case where the expectation of the reward follows a random walk while [@granmo2010solving] addresses the problem where, at each time step, the expectation of each reward is modified by an independent Gaussian perturbation of constant variance. In [@schulz2015learning] the expectation of the reward associated with an option is considered to depend on a linear static function of some known variables that characterize the option and propose to estimate the reward based on learning this function. A different class of dynamically and stochastically varying option characteristics is considered in [@tekin2010online] where the reward distribution of each option is modeled as a finite state irreducible, aperiodic, and reversible Markov chain. In this paper we consider a class of *Dynamic Multi-Armed Bandit* problems (DMAB) that will include most of the previously stated dynamic problems as special cases. Specifically we consider a class of DMAB problems where the reward of each option is the noisy output of a multivariate linear time varying stochastic dynamic system that satisfies some boundedness conditions. This formulation allows one to accommodate a wide class of real world problems such as the cases where the option characteristics vary periodically, aperiodically, or gradually in a stochastic way. Furthermore incorporating this dynamic structure allows one to easily capture the underlying characteristic variations of each option as well as allow the possibility of incorporating dependencies between options. To the best of our knowledge this is the first time that such a wide class of dynamic problems have been considered in one general setting. We also incorporate temporal option unavailabilities into our structure that helps broaden the applicability of this model in real world problems. To the best of our knowledge it is the first time that temporal option unavailabilities are incorporated in a setting where the reward distributions are non-stationary. One major advantage of this linear dynamic systems formulation is that it immediately allows us to use the vast body of linear dynamic systems theory including that of switched systems to the problem of classification and solution of different DMAB problems. In this paper we prove that if the system characteristics satisfy certain boundedness conditions and the number of times the optimal arm becomes unavailable is at most logarithmic, then the expected cumulative regret is logarithmically bounded from above when one combines any UCB type decision making algorithm with any efficient reward estimator. We demonstrate the effectiveness of the scheme using an example where an agent intends to maximize the information she gathers under the constraint of option unavailability and periodically varying option characteristics. In section-\[Secn:DMAB\] we formally state the class of DMAB problems that is considered in this paper. We show in section-\[Secn:AsymptoticAllocationRules\] that the combination of any UCB type allocation rule with an efficient estimator guarantees that the expected cumulative regret is bounded above by a logarithmic function of the number of time steps. In section-\[Secn:EfficientEstimators\] we explicitly show, using a Hoeffding type tail bound [@Garivier2011], that the sample mean estimator is an efficient estimator. Finally in section-\[Secn:Example\] we provide a novel DMAB example that deals with unknown periodically and continuously varying options characteristics. Dynamic Multi-Armed Bandit Problem {#Secn:DMAB} ================================== In this paper we consider a wide class of dynamic multi-armed bandit problems where the reward is a noisy measurement of a linear time varying stochastic dynamic process. The ‘noise’ in the measurement and the ‘noise’ in the process are assumed to have a bounded support. This is a reasonable and valid assumption since the rewards in physical problems are bounded and are greater than zero. Consider a *k-armed bandit*. Let the reward associated with each option $i \in \{1,2,3,\ldots,k\}$ at the $t^{\mathrm{th}}$ time step be given by the real valued random variable $X_i^t$. The expectation of this reward depends linearly on a $\mathbb{R}^m$ valued random variable $\theta^t$. The random variable $\theta^t$ represents option characteristics. The dynamics of the option characteristics can be multidimensional and thus we allow provision for $m$ to be larger than $k$. These option characteristics could either evolve deterministically or stochastically. The reward is assumed to depend linearly on the option characteristics. The dependence of the reward on the option characteristics may be precisely known or there could be some uncertainty about it. We model this uncertainty by an additive ‘noise’ term with finite support. We also allow the possibility of incorporating option dependencies and thereby considering the possibility of other options directly or indirectly influencing the reward associated with a given option. In order to capture this behavior in a concrete theoretical setting we assume that the bounded random variables $\theta^t \in \chi_{\theta}\subset \mathbb{R}^m$, with $\chi_{\theta}$ compact, and $X_i^t\in [0,\chi_x]$ with $0\leq \chi_x<\infty$, specifically satisfy a linear time varying stochastic process, $$\begin{aligned} \theta^t&=A^t\theta^{t-1}+B^tn^t_{\theta},\label{eq:Process}\\ X_i^t&=\gamma_i^t\,\left(H_i^t\theta^t+g_i^{t}\,n^t_{xi}\right),\label{eq:NoisyReward}\end{aligned}$$ where $\{n^t_{\theta}\}$ is a bounded $\mathbb{R}^q$ valued stochastic process with zero mean and constant covariance $\Sigma_\theta$ while $\{n^t_{xi}\}$ is a $\mathbb{R}$ valued bounded stochastic process with zero mean and constant variance $\sigma_{xi}$. We also let $\{\gamma_i^t\},\{g_i^t\}$ be real valued deterministically varying sequences while $\{A^t\},\{B^t\},\{H_i^t\}$ are matrix valued deterministic sequences of appropriate dimensions. We allow the variances, $\sigma_{xi}^2$, corresponding to each arm to be different. Letting $\gamma_i^t\in \{0,1\}$ allows us to consider temporal option unavailabilities. Expression-(\[eq:Process\]) describes the collective time varying characteristics of all the options and the absence or presence of $B^tn_{\theta}$ dictates whether these dynamics are deterministic or stochastic. Expression-(\[eq:NoisyReward\]) describes how the reward depends on the option characteristics. The presence of the ‘noise’ term $g_i^{t}\,n_{xi}$ indicates that the rewards that one obtains given the knowledge of the option characteristics involve some bounded uncertainty. The case where $\{A^t\},\{B^t\},\{H_i^t\}$ each has a block diagonal structure represents independent arms and the case where there are off diagonal entries represent situations where the arms depend on each other. Notice that by setting $A^t\equiv I$ and $B^t\equiv 0$ we obtain the standard MAB with temporal option unavailabilities. One major advantage of this linear dynamic systems formulation is that it allows one to use the vast body of linear dynamic systems theory including that of switched systems in the classification and solution of different DMAB problems. From equations (\[eq:Process\]) and (\[eq:NoisyReward\]) we see that the expectations $E(\theta^t),E(X_i^t)$ evolve according to $$\begin{aligned} E(\theta^t)&=A^tE(\theta^{t-1}),\label{eq:EProcess}\\ E(X_i^t)&=\gamma_i^t\,H_i^tE(\theta^t),\label{eq:EReward}\end{aligned}$$ and that the covariances $\Sigma(\theta^t)\triangleq E(\theta^{t}{\theta^{t}}^T)-E(\theta^t){E(\theta^t)}^T$, $\Sigma(X_i^t)\triangleq E(X_i^{t}{X_i^{t}}^T)-E(X_i^t){E(X_i^t)}^T$ evolve according to $$\begin{aligned} \Sigma(\theta^t)&=A^t\Sigma(\theta^{t-1}){A^t}^T+B^t\Sigma_\theta{B^t}^T,\label{eq:VProcess}\\ \Sigma(X_i^t)&=(\gamma_i^t)^2\,\left(H_i^t\Sigma(\theta^t){H_i^t}^T+\sigma_{xi}^2{(g_i^t)}^2\right).\label{eq:VReward}\end{aligned}$$ Boundedness of $\theta_i^t$ implies $E(\theta^{t})$ and $\Sigma(\theta^t)$ should remain bounded. Let $\Phi^t_\tau\triangleq \left(\prod_{j=\tau}^tA^j\right)$ and then since $E(\theta^t)=\Phi^t_1E(\theta^0)$ we find that the expectation and the covariance of the reward become unbounded if $\lim_{t\to\infty}||\Phi^t_1||=\infty$. On the other hand the expectation converges to zero if $\lim_{t\to\infty}||\Phi^t_1||=0$. Thus sequences $\{A^t\}$ that satisfy the conditions $\limsup_{t\to\infty}||\Phi^t_1||=\bar{a}<\infty$ and $\liminf_{t\to\infty}||\Phi^t_1||={a}>0$ are the only ones that correspond to a meaningful DMAB problem. Thus to ensure boundedness of $E(\theta^{t})$ we assume that: \[As:MainAssumption0\] The sequence $\{A^t\}$ satisfies: $$\begin{aligned} \limsup_{t\to\infty}\left|\left|\prod_{j=1}^tA^j\right|\right|&<\infty\\ \liminf_{t\to\infty}\left|\left|\prod_{j=1}^tA^j\right|\right|&>0\end{aligned}$$ and $\exists \:\: a,\bar{a}>0$ such that, $$\begin{aligned} a<\left|\left|\prod_{j=\tau}^tA^j\right|\right|<\bar{a},\:\:\:\:\:\end{aligned}$$ $\forall \:\: t\geq\tau$. Several examples of sequences $\{A^t\}$ of practical significance that ensure this condition are those where $A^t$: 1. is an orthogonal matrix or is a stochastic matrix (i.e. $||A^t||=1$), 2. is a periodic matrix (i.e. $A^t=A^{t+N}$ for some $N>0$), 3. corresponds to a stable switched system. Next we will consider conditions needed for the boundedness of $\Sigma(X_i^t)$. Note that the covariance of $\theta^t$ is given by [$$\begin{aligned} \Sigma(\theta^t)&=\Phi^t_1\Sigma(\theta^{0}){\Phi^t_1}^T+\sum_{\tau=1}^t\Phi^{t}_{\tau+1}B^{\tau}\Sigma_\theta{B^{\tau}}^T(\Phi^{t}_{\tau+1})^T.\label{eq:VProcess1}\end{aligned}$$ ]{} Assumption-\[As:MainAssumption0\] ensures that the first term on the right hand side is bounded and that [$$\begin{aligned} ||\Sigma(\theta^t)||&\leq \bar{a}^2||\Sigma(\theta^{0})||+\bar{a}^2||\Sigma_\theta||^2\sum_{\tau=1}^t||B^\tau||^2.\label{eq:VProcess2}\end{aligned}$$ ]{} Thus it also follows that $||\Sigma(\theta^t)||$ remains bounded in any finite time horizon if $||B^t||$ remains bounded in that period. On the other hand if the sequence $\{B^t\}$ satisfies $||B^t||\leq c/t$ for some $c>0$ or if the number of time steps where the condition $\Phi^{t}_{\tau}B^{\tau-1}\neq 0$ is satisfied remains finite then $||\Sigma(\theta^t)||$ is guaranteed to be bounded for all $t>0$. Therefore from (\[eq:VReward\]) we find that in order to satisfy the boundedness of $X_i^t$ the sequences $\{\gamma_i^t\},\{||H_i^t||\},\{g_i^t\}$ must necessarily be bounded from above in addition to what is specified in Assumption \[As:MainAssumption0\]. In order to define a meaningful DMAB problem the notion of an optimal option should be well defined. That is $i^*\triangleq \arg\limits_{i}\max\{H_i^tE(\theta^t)\}$ is independent of time. The following assumption specifies the conditions necessary for the boundedness of the reward $X_i^t$ as well as the conditions necessary for the existence of an optimal arm. \[As:MainAssumption\] We will assume that the sequences $\{\gamma_i^t\},\{B^t\},\{H_i^t\},\{g_i^t\}$ guarantee the following conditions for all $t>0$: $$\begin{aligned} ||\Sigma(\theta^t)||\leq &\sigma,\label{eq:SigmaBnd}\\ \gamma_i^t\in \{0,1\}&,\\ 0<g_i^t\leq&\bar{g}_i,\\ ||B^t|| \leq \frac{b}{t}\:\:\:\mbox{or}\:\: &||B^t||\neq 0 \:\:\:\mbox{finitely many times},\\ h_i<||H_i^t||\leq&\bar{h}_i,\label{eq:HBnd}\end{aligned}$$ and $\forall \: t\geq 0$ there exists a unique $i^*=i^t_*$ such that $$\begin{aligned} \Delta_{i}\leq \Delta_i^t&\triangleq {H_{i^t_*}}^tE(\theta^t_{i^t_*})-H_i^t E(\theta^t) \leq \bar{\Delta},\label{eq:OptimalArm}\end{aligned}$$ $\forall \:\:i\neq i^t_*$ and while $$\begin{aligned} \sum_{j=2}^t\mathbb{I}_{\{\gamma^j_{i^*}=0\}} \leq \gamma \log{t},\label{eq:LogBndAvailability}\end{aligned}$$ for some $\bar{g}_i,h_i,\bar{h}_i,\bar{\Delta},{\Delta}_i,\gamma,\sigma,b>0$ where $\mathbb{I}_{\{\gamma^j=0\}}$ is the indicator function. Note that condition (\[eq:OptimalArm\]), which implies existence of a well defined optimal arm, is guaranteed if $({h_{i^*}}a||E(\theta^0_{i^*})||-\bar{h}_i\bar{a}||E(\theta^0)||)>0, \:\: \forall \:\: t\geq \tau>0$. Condition (\[eq:LogBndAvailability\]) implies that this optimal arm becomes unavailable at most logarithmically with the number of time steps. Finally the boundedness of $X_i^t$ is guaranteed by the conditions (\[eq:SigmaBnd\]) – (\[eq:HBnd\]). We will now proceed to analyze the regret of the DMAB problem stated above. Consider the probability space $(\Omega,\mathcal{U},\mathcal{P})$ and the increasing sequence of subalgebras $\mathcal{F}_{0}\subset\mathcal{F}_{1}\cdots \subset\mathcal{F}_{t}\cdots \subset\mathcal{F}_{n-1}\subset \mathcal{U}$ for $t=0,1,\cdots,n$ where $\mathcal{P}$ is the probability measure on the sigma algebra $\mathcal{U}$ of $\Omega$. The sigma algebra $\mathcal{F}_{t}$ represents the information that is available at the $t^{\mathrm{th}}$ time step. Let $\{\varphi_t\}_{t=1}^n$ be a sequence of random variables, each defined on $(\Omega,\mathcal{F}_{t-1},\mathcal{P})$ and taking values in $\{1,2,\cdots,k\}$. The random variable $\varphi_t$ models the action taken by the agent at the $t^{\mathrm{th}}$ time step. The value $i\in\{1,2,\cdots,k\}$ of the random variable $\varphi_t$ specifies that the $i^{\mathrm{th}}$ option is chosen at time step $t$. Then $\mathbb{I}_{\{\varphi_t =i\}}$ is the $\mathcal{F}_{t-1}$ measurable indicator random variable that takes a value one if the $i^{\mathrm{th}}$ option is chosen at step $t$ and is zero otherwise. The DMAB problem is to find an allocation rule $\{\varphi_t\}_{t=1}^n$ that maximizes the expected cumulative reward or equivalently that minimizes the cumulative regret. The cumulative reward after the the $n^{\mathrm{th}}$ time step is defined to be the real valued random variable $S_n$ defined on the probability space $(\Omega,\mathcal{F}_{n-1},\mathcal{P})$ that is given by $$\begin{aligned} S_n&=\sum_{t=1}^n\sum_{i=1}^k E(X_i^t\mathbb{I}_{\{\varphi_{t} =i\}}|\mathcal{F}_{t-1})\\ &=\sum_{t=1}^n\sum_{i=1}^k E(X_i^t|\mathcal{F}_{t-1})\mathbb{I}_{\{\varphi_{t} =i\}}. \end{aligned}$$ Thus the expected cumulative reward is, $$\begin{aligned} E(S_n)=\sum_{t=1}^n\sum_{i=1}^k E({X}_i^t)E(\mathbb{I}_{\{\varphi_{t} =i\}}) \end{aligned}$$ where $T_i(n)=\sum_{t=1}^n\mathbb{I}_{\{\varphi_t =i\}}$ is a real valued random variable defined on $(\Omega,\mathcal{F}_{n-1},\mathcal{P})$ that represents the number of times the $i^{\mathrm{th}}$ arm has been sampled in $n$ trials. Note that $E(X_i^t)=\gamma_i^tH_i^tE(\theta^t)$. Let $i^t_*=\max_i\{E(X_i^t)\}$. Then the expected cumulative regret is defined as [ $$\begin{aligned} R_n&\triangleq \sum_{t=1}^n\left(E({X}^t_{i_t^*})- \sum_{i=1}^k\gamma_{i}^tE({X}_i^t)E(\mathbb{I}_{\{\varphi_{t} =i\}}) \right). \label{eq:DynRegret} \end{aligned}$$ ]{} Then from condition (\[eq:OptimalArm\]) we find that [ $$\begin{aligned} R_n &=\sum_{i=1}^k\sum_{t=1}^n\mathbb{I}_{\{\gamma_{i^*}^t=1\}}\left(H_{i^*}^tE(\theta_{i^*}^t)- \gamma_i^tH_{i}^tE(\theta_{i}^t) \right)E(\mathbb{I}_{\{\varphi_{t} =i\}})\nonumber\\ &+\sum_{i=1}^k\sum_{t=1}^n\mathbb{I}_{\{\gamma_{i^*}^t=0\}}\left(H_{i^t_*}^tE(\theta_{i^t_*}^t)- \gamma_i^tH_{i}^tE(\theta_{i}^t) \right)E(\mathbb{I}_{\{\varphi_{t} =i\}})\nonumber\\ & \leq \bar{\Delta}\sum_{i\neq i^*}^kE\left(T_i(n)\right). \end{aligned}$$ ]{} In their seminal work [@LaiRobbins] Lai and Robbins proved that, for the static MAB problem, the regret is bounded below by a logarithmic function of the number of time steps. Asymptotic Allocation Rules for the DMAB Problem {#Secn:AsymptoticAllocationRules} ================================================ In this section we show how to construct *asymptotically efficient* allocation rules for the class of DMAB problems that were formally defined above. Specifically, in the following, we will show that the combination of any *UCB based* decision making process and an *efficient estimator* provides such an allocation rule. In the DMAB problem $\mu_i^t\triangleq E(X_i^t)$ is time varying. Thus one needs to consider a ‘time average’ for $\mu_i^t$. This time average depends on how one samples option $i$. Specifically it is a $\mathcal{F}_{t-1}$ measurable random variable $$\begin{aligned} \widehat{\mu}_i^t&\triangleq \frac{1}{T_i(t)}\sum_{j=1}^{t}E(X_i^j)\mathbb{I}_{\{\psi_j=i\}}. \label{eq:TimeAverageMean}\end{aligned}$$ This random variable can not be estimated using the maximum likelihood principle since $E(X_i^j)$ are unknown and thus will have to be estimated by other means. We will consider a $\mathcal{F}_{t-1}$ measurable random variable $\widehat{X}_i^t$ to be an estimator of $\widehat{\mu}_i^t$ if $E(\widehat{X}_i^t)=E(\widehat{\mu}_i^t)$. Let $\widehat{X}_i^t$ be a $\mathcal{F}_{t-1}$ measurable random variable such that $E(\widehat{X}_i^t)=E(\widehat{\mu}_i^t)$ and $T_i(t)$ be the $\mathcal{F}_{t-1}$ measurable random variable that represents the number of times the $i^{\mathrm{th}}$ option has been sampled up to time $t$. An estimator $\widehat{X}_i^t$ that ensures $$\begin{aligned} \mathcal{P}\left(\widehat{X}_i^t\geq \widehat{\mu}_i^t+\sqrt{\frac{\vartheta}{T_i(t)}}\right)&\leq \frac{\nu\,\log{t}}{\exp\left(2\kappa \vartheta\right)},\label{eq:TailProbBnd1}\\ \mathcal{P}\left(\widehat{X}_i^t\leq \widehat{\mu}_i^t-\sqrt{\frac{\vartheta}{T_i(t)}}\right)&\leq \frac{\nu\,\log{t}}{\exp\left(2\kappa \vartheta\right)}. \label{eq:TailProbBnd2}\end{aligned}$$ for some $\kappa,\vartheta,\nu>0$ will be referred to as an *efficient reward estimator*. In section-\[Secn:EfficientEstimators\] we show that the frequentist average mean estimator satisfies this requirement. Let $\widehat{X}_i^t$ be a $\mathcal{F}_{t-1}$ measurable random variable such that $E(\widehat{X}_i^t)=E(\widehat{\mu}_i^t)$. The allocation rule $\{\varphi_{t}\}_1^{n}$ will be referred to as *UCB based* if it is chosen such that $$\begin{aligned} \mathbb{I}_{\{\varphi_{t+1}=i\}}=\left\{ \begin{array}{cl} 1 & \:\:\:Q^{t}_i=\max\{Q^{t}_1,\cdots,Q^{t}_k\}\label{eq:UCBallocation}\\ 0 & \:\:\: {\mathrm{o.w.}}\end{array}\right. \end{aligned}$$ with $$\begin{aligned} Q^t_i&\triangleq \widehat{X}^t_i+\sigma\sqrt{\frac{\Psi\left(t\right)}{T_i(t)}}\label{eq:UCBQ} \end{aligned}$$ where $\Psi(t)$ is an increasing function of $t$ with $\Psi(1)=0$ and $\sigma>0$. We will also let $T_i(1)=1$ for all $i$. There exists two choices for picking an option at the first time step. If there exists some prior knowledge one can use that as the initial estimate $\widehat{X}^1_i$. On the other hand in the absence of such prior knowledge one can sample every option once and select the the sampled values for $\widehat{X}^1_i$. We will show below that combining any efficient estimator with a UCB based allocation rule ensures that the number of times a suboptimal arm is sampled is bounded above by a logarithmic function of the number of samples. This result is formally stated below and is proved in the appendix. \[Theom:DynamicRegret\] Let conditions specified in Assumption \[As:MainAssumption0\] and \[As:MainAssumption\] hold. Then any efficient estimator combined with an UCB based allocation rule $\{\varphi_t\}_1^{n}$ ensures that for every $i\in\{1,2,\cdots,k\}$ such that $i\neq i^{*}$ and for some $ l\geq 1$. $$\begin{aligned} E(T_i(n))\leq \gamma\log{n}+ \frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(n) +\left(l+\nu\sum_{t=l}^{n-1}\frac{\log{t}}{t^{2\kappa\sigma^2\alpha}}\right).\end{aligned}$$ Thus the cumulative expected regret satisfies: $$\begin{aligned} R_n\leq c_0\log{n} + c_1,\end{aligned}$$ for some constant $c_0,c_1>0$ if $\Psi(t)$ satisfies $$\begin{aligned} \alpha \log{t}\leq \Psi(t) \leq \beta \log{t},\end{aligned}$$ for some constants $3/(2\kappa\sigma^2)<\alpha\leq \beta$. If one selects $\Psi(t)=16\log{t}$ one obtains the standard UCB-Normal algorithm proposed in [@Auer] while if one selects ${\Psi(t)}=\left(\Phi^{-1}\left(1-1/(\sqrt{2\pi e}\,t^2)\right)\right)^2$ where $\Phi^{-1}\left(\cdot\right)$ is the inverse of the cumulative distribution function for the normal distribution one obtains the UCL algorithm proposed in [@Reverdy]. Efficient Estimators {#Secn:EfficientEstimators} -------------------- Let $S_i^t$ be the $\mathcal{F}_{t-1}$ measurable random variable that gives the cumulative reward received by choosing arm $i$ up to the $t^{\mathrm{th}}$ time step that is given by $$\begin{aligned} S_i^t=\sum_{j=1}^{t}E(X_i^j\mathbb{I}_{\{\psi_j=i\}}|\mathcal{F}_{j-1}) =\sum_{j=1}^{t}E(X_i^j|\mathcal{F}_{j-1})\mathbb{I}_{\{\psi_j=i\}}.\end{aligned}$$ Define the $\mathcal{F}_{t-1}$ measurable simple average sample mean estimate $\widehat{X}_i^t$ of the cumulative mean reward received from arm $i$ as $$\begin{aligned} \widehat{X}_i^t&\triangleq \left\{\begin{array}{lc}\widehat{X}_i^1 & \mathrm{if}\:\: T_i(t)=0\\ \frac{S_i^t}{T_i(t)} & \mathrm{o.w.} \end{array}\right..\label{eq:MeanAverage}\end{aligned}$$ Then $$\begin{aligned} E(\widehat{X}_i^t)&=\sum_{j=1}^{t}E\left(\frac{E(X_i^j| \mathcal{F}_{j-1})\mathbb{I}_{\{\psi_j=i\}}}{T_i(t)}\right).\end{aligned}$$ Since $E(X_i^j| \mathcal{F}_{j-1})$ is independent of $\mathbb{I}_{\{\psi_j=i\}}$ and $T_i(t)$ we have that $$\begin{aligned} E(\widehat{X}_i^t)&=\sum_{j=1}^{t}E(X_i^j)E\left(\frac{\mathbb{I}_{\{\psi_j=i\}}}{T_i(t)}\right)\nonumber \\ &=E\left(\sum_{j=1}^{t}\frac{E(X_i^j)\mathbb{I}_{\{\psi_j=i\}}}{T_i(t)}\right)=E(\widehat{\mu}_i^t).\end{aligned}$$ The tail probability distribution for the above sample mean average is given by the following lemma which follows from Theorem 4 of [@Garivier2011]. If the random process $\{{X}_i^t\}$ satisfies ${X}_i^t\in [0,\chi_x], \:\:\forall i\in{1,2,\ldots,k}$ and $t>0$ and $\widehat{X}_i^t,\widehat{\mu}_i^t$ are given by (\[eq:MeanAverage\]) and (\[eq:TimeAverageMean\]) respectively we have that, $$\begin{aligned} \mathcal{P}\left(\widehat{X}_i^t>\widehat{\mu}_i^t+\sqrt{\frac{\vartheta}{T_i(t)}}\right)\leq \frac{\nu\,\log{t}}{\exp\left(2\kappa \vartheta\right)}\end{aligned}$$ where $\kappa=\left(1-\frac{\eta^2}{16}\right)/\chi^2$ and $\nu=1/\log (1+\eta)$ for all $t>0$ and $\eta,\vartheta>0$. Thus we have that (\[eq:TailProbBnd1\]) and (\[eq:TailProbBnd2\]) are satisfied for the sample mean estimate of the reward. Example: Periodically Continuously Varying Option Characteristics {#Secn:Example} ================================================================= In this section we consider a novel example of practical interest. The problem that we consider is that of an agent trying to maximize the reward that depends continuously on certain, periodically and continuously varying, option characteristics. The agent is assumed to be unaware of any information about this periodic behavior. Specifically we consider the problem where an agent is encountered with $k$ number of options. Each option may vary with time and may become unavailable from time to time. For this example we assume that the options do not depend on each other. This is the case, for instance, if the agent is dealing with collecting human behavioral information in a recreational park and has several options for locating herself for the purpose of collecting this information. The average number of people who frequent the park may vary depending on whether it is in the morning, afternoon or evening. Similar circumstances occur if one needs to select the type of optimal crops, highlight a particular product in a store, sample a set of sensors whose characteristics vary with the time of the day, or advertise a particular event in super markets. In each of these cases due to certain external events some of the options may become temporally unavailable as well. This class is characterized by a fixed periodic block diagonal matrix $A^t=\mathrm{diag}(A^t_1,A^t_2,\cdots,A^t_k)$ where each $A_i^t$ is a $p\times p$ matrix that satisfies the property $A_i^{t+N}=A_i^t$ for some $N>1$. The matrix $A_i^t$ encodes how the dynamics of the expected characteristics of the $i^{\mathrm{th}}$ option varies. For the example we consider here the options do not depend on each other thus we will also set $B^t=[B_1,B_2,\cdots,B_k]^T$ where each $B_i$ is a $1\times p$ row matrix and each $H_i^t=H_i$ has a corresponding block structure so that the option dynamics are not coupled. We will consider two cases. One where we will assume that the total number of people who visit a park is always the same for every day and the more realistic case where the number of people that visit a park varies stochastically. For the first case we will set $B^tn^t_{\theta}\equiv 0$ which guarantees the boundedness of the covariance of $\theta^t$ for infinite time horizons as well. In the second case we pick $B^tn^t_{\theta}\neq 0$ where $B^t\equiv B$ is a constant matrix and $n^t_{\theta}$ is a bounded random process with zero mean. In this second case the boundedness of the covariance of $\theta^t$ is guaranteed only for finite time horizons. For illustration we use $k=5$ and $N=3$. We see that this problem can be modeled by selecting $\theta^t=(\theta^t_1,\theta^t_2,\cdots,\theta^t_5)$ where each $\theta^t_i\in \mathbb{R}^3$. We let $A^t=\mathrm{diag}(A^t_1,A^t_2,\cdots,A^t_5)$ where each $A^t_i=(A_i)^t$ with $$\begin{aligned} A_i=\begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 1 & 0 & 0\end{bmatrix}.\end{aligned}$$ The output matrix $H^t_i$ is a constant $1\times 3k$ row matrix and has all zero entries with the exception of the $(3i-2)^{\mathrm{th}}$ entry which is equal to one. The direct noise coupling term corresponding to each reward is chosen to be $g^t_i\equiv 1$ and we will assume that the noise is a bounded random variable $n^t_{x_i}$ with zero mean. Thus we see that the expected reward of the $i^{\mathrm{th}}$ option satisfies $E(X_i^t)=H(A_i)^t\theta_i^0$ where $H=[1\:\:\:0\:\:\:0]$ and hence satisfies the condition $E(X_i^{t+3})=E(X_i^t)$ for all $t>0$. The initial condition is chosen such that $\theta^0=(\theta^0_1,\theta^0_2,\cdots,\theta^0_5)$ where each $\theta_i^0$ takes the form $\theta_i^0=\bar{\theta}_i[\alpha_1\:\:\: \alpha_2\:\:\:\alpha_3]^T$ with $\bar{\theta}_i\in \mathbb{R}$. The real positive constants $\alpha_1, \alpha_2, \alpha_3$ satisfy the condition $\alpha_1\alpha_2\alpha_3=1$ and captures the periodic variation of the number of visitors within a given day. The block diagonal structure of $A^t,B^t,H_i^t$ amounts to the assumption that the number of people that frequent different locations are uncorrelated. The total number of people that visit the park in a particular day is given by $(\bar{\theta}_1+\bar{\theta}_2+\bar{\theta}_3+\bar{\theta}_4+\bar{\theta}_5)(\alpha_1+\alpha_2+\alpha_3)$. In the first case we will assume that each $\bar{\theta}_i$ is fixed for each day and thus set $B^t\equiv 0$. In the other case we will consider the more realistic situation where this value changes stochastically according to a uniform distribution on the support $[\bar{\theta}_i-50,\bar{\theta}_i+50]$. That is, we select $n_\theta^t$ to be uniformly distributed on $[-50,50]$. We also let the noise term $n_{xi}$ for each $i$ to be uniformly distributed on $[-50,50]$. Notice that in this second case the growth condition for the covariance $\Sigma(X_i^t)$ is only satisfied for a finite time horizon. For the simulations we let $\alpha_1=3/4,\alpha_2=1,\alpha_3=4/3$ and $\bar{\theta}_1=400, \bar{\theta}_2=350, \bar{\theta}_3=750, \bar{\theta}_4=1000, \bar{\theta}_5=526$. Thus the optimal option is $i^*=4$ and is well defined for all $t$. For the simulations we use the UCB algorithem (\[eq:UCBallocation\]) – (\[eq:UCBQ\]) with $\Psi(t)=16\log{t}$ and the standard frequentist average estimator (\[eq:MeanAverage\]) for the estimation of the rewards. In compliance with the assumption that the optimal option becomes unavailable at most logarithmically, we let $$\begin{aligned} \gamma_i^t&=\left\{\begin{array}{lc} 0 & \mathrm{if}\:\:\left([\log{(n_i+t+1)}]-[\log{(n_i+t)}]\right)=1\\ 1 & \mathrm{o.w.} \end{array}\right.,\end{aligned}$$ for some integer $n_i>0$ where $[x]$ denotes the nearest integer value of $x$. For convenience of simulation we will let $\gamma_i^t\equiv 1$ for $i\neq i^*$ and $\{\gamma_{i^*}^t\}$ as chosen above. We estimate the expected reward, $E(S_n)$, the expected cumulative regret, $E(R_n)$, and the expected number of times the optimal arm is selected, $E(T_{i^*}(n))$, by simulating the algorithm for each $1\leq n\leq200$ a $1000$ times and by computing the frequentist mean as an estimate for $E(S_n)$, $E(R_n)$, and $E(T_{i^*}(n))$. The expected reward, $E(S_n)$, the expected number of times the optimal arm is sampled, $E(T_{i^*}^n)$, and the expected cumulative regret, $E(R_n)$, are plot against $n$ in Figurers \[Fig:ErewardNoNoise\] – \[Fig:EregretNoNoise\] in the absence of process noise and in Figures \[Fig:ErewardNoise\] – \[Fig:EregretNoise\] for uniformly distributed process noise. We observe that, as expected, the covariance of the regret and the reward increase as the number of time steps increase when the option dynamics are influenced by uncertainty. However since we only consider a finite time horizon they remain bounded during this horizon. Notice that yet the expected number of times the optimal arm is chosen behaves the same as when there is no process noise. ![Expected reward $E(S_n)$ for the time varying DMAB with no process noise.[]{data-label="Fig:ErewardNoNoise"}](PeriodicNoNoiseEreward.jpg){width="50.00000%"} ![Expected number of times, $E(T_{i^*}(n))$, the optimal arm has been sampled for the time varying DMAB with no noise. []{data-label="Fig:EToptNoNoise"}](PeriodicNoNoiseEtopt.jpg){width="50.00000%"} ![Expected cumulative regret $E(R_n)$ for the time varying DMAB with no noise. []{data-label="Fig:EregretNoNoise"}](PeriodicNoNoiseEregret.jpg){width="50.00000%"} ![Expected reward $E(S_n)$ for the time varying DMAB with with uniform process noise.[]{data-label="Fig:ErewardNoise"}](PeriodicNoiseEreward.jpg){width="50.00000%"} ![Expected number of times, $E(T_{i^*}(n))$, the optimal arm has been sampled for the time varying DMAB with with uniform process noise. []{data-label="Fig:EToptNoise"}](PeriodicNoiseEtopt.jpg){width="50.00000%"} ![Expected cumulative regret $E(R_n)$ for the time varying DMAB with uniform process noise. []{data-label="Fig:EregretNoise"}](PeriodicNoiseEregret.jpg){width="50.00000%"} Conclusion ========== This paper presents a novel unifying framework for modeling a wide class of Dynamic Multi-Armed Bandit problems. It allows one to consider option unavailabilities and option correlations in a single setting. The class of problems is characterized by situations where the reward for each option depends uncertainly on a multidimensional parameter that evolves according to a linear stochastic dynamic system that captures the internal and hidden collective behavior of the dynamically changing options. The dynamic system is assumed to satisfy certain boundedness conditions. For this class of problems we show that the combination of any Upper Confidence Bound type algorithm with any efficient estimator guarantees that the expected cumulative regret is bounded above by a logarithmic function of the time steps. We provide a novel practically significant example to demonstrate these ideas. In the following we will proceed to prove the above Theorem \[Theom:DynamicRegret\] by closely following the proof provided in [@Auer; @Reverdy]. Let $C_i^t\triangleq \sqrt{\frac{\Psi(t)}{T_i(t)}}$. Then for $i\neq i^*$ and $l\geq 1$ $$\begin{aligned} E(T_i(n))&=\sum_{t=0}^{n-1}\mathcal{P}({\{\varphi_{t}=i\}})\leq l+\sum_{t=l}^{n-1}\mathcal{P}({\{\varphi_{t}=i\}})\\ &\leq l+\sum_{t=l}^{n-1}\mathcal{P}(\{Q_{i^*}^t< {Q^t_{i}}\}).\end{aligned}$$ Let $$\begin{aligned} \mathcal{A}_i^t&\triangleq\{\widehat{X}_{i^*}^t+C_{i^*}^t\geq \widehat{\mu}^t_{i^*}\},\\ \mathcal{B}_i^t&\triangleq\{\widehat{\mu}^t_{i^*}\geq \widehat{\mu}^t_{i}+2{C^t_{i}}\},\\ \mathcal{C}_i^t&\triangleq\{\widehat{\mu}^t_{i}+2{C_{i}}^t\geq\widehat{X}_{i}^t+{C^t_{i}}\}\\ \mathcal{D}_i^t&\triangleq\{\gamma^t_{i^*}\neq 0\}\end{aligned}$$ Then we have, $$\begin{aligned} \{\mathcal{A}_i^t\cap \mathcal{B}_i^t \cap \mathcal{C}_i^t\}\cap \mathcal{D}_i^t\} \subseteq \{Q_{i^*}^t\geq{Q_{i}}^t\}\end{aligned}$$ Therefore, $$\begin{aligned} \{Q_{i^*}^t< {Q^t_{i}}\} \subseteq {\bar{\mathcal{A}}_i^t}\cup {\bar{\mathcal{B}}_i^t} \cup {\bar{\mathcal{C}}_i^t}\cup {\bar{\mathcal{D}}_i^t}\end{aligned}$$ From above equations we have, $$\begin{aligned} \mathcal{P}(\{Q_{i^*}^t< {Q^t_{i}}\})&\leq \mathcal{P}(\{\widehat{X}_{i^*}^t+C^t_{i^*}<\widehat{\mu}^t_{i^*}\})\\ &\:\:\:\:+\mathcal{P}(\{\widehat{\mu}^t_{i^*}< \widehat{\mu}^t_{i}+2{C^t_{i}}\})\\ &\:\:\:\:+\mathcal{P}(\{\widehat{X}_{i}^t-C^t_{i}>\widehat{\mu}^t_{i}\})\\ &\:\:\:\:+\mathbb{I}_{\{\gamma^t_{i^*}= 0\}}.\end{aligned}$$ Note that the conditions (\[eq:TailProbBnd1\]) and (\[eq:TailProbBnd2\]) of the tail probabilities of the distribution of the estimate $\widehat{X}_i^t$ gives us that, $$\begin{aligned} \mathcal{P}(\{\widehat{X}_{i^*}^t+C_{i^*}^t<\widehat{\mu}^t_{i^*}\})&\leq\:\:\:\:\: \frac{\nu\,\log{t}}{\exp\left(2\kappa \sigma^2\Psi(t)\right)},\\ \mathcal{P}(\{\widehat{X}_{i}^t-C_{i}^t>\widehat{\mu}^t_{i}\}) &\leq\:\:\:\:\: \frac{\nu\,\log{t}}{\exp\left(2\kappa \sigma^2\Psi(t)\right)},\end{aligned}$$ and hence that $$\begin{aligned} \mathcal{P}(\{Q_{i^*}^t< {Q^t_{i}}\})&\leq \mathbb{I}_{\{\gamma^t_{i^*}= 0\}}+\mathcal{P}(\bar{B}^t_i)+\frac{2\nu\,\log{t}}{\exp\left(2\kappa\sigma^2 \Psi(t)\right)}.\end{aligned}$$ From condition (\[eq:LogBndAvailability\]) we have $\sum_{j=2}^t\mathbb{I}_{\{\gamma^j_{i^*} =0\}}\leq \gamma \log{t}$. Thus $$\begin{aligned} E(T_i(n))&\leq l+\gamma \log{n}+\sum_{t=l}^{n-1}\mathcal{P}(\bar{B}^t_i)+\sum_{t=l}^{n-1}\frac{2\nu\,\log{t}}{\exp\left(2\kappa \sigma^2\Psi(t)\right)}.\end{aligned}$$ Let us proceed to find an upper bound for $\sum_{t=1}^n\mathcal{P}(\bar{B}^t_i)$. Since $\bar{B}^t_i=\{\widehat{\mu}^t_{i^*}< \widehat{\mu}_i^t+2{C^t_{i}}\}$. Let $\Delta^t_{i}=\widehat{\mu}^t_{i^*}-\widehat{\mu}_i^t$. Then if $\bar{B}^t_i$ is true then, $$\begin{aligned} \frac{\Delta^t_{i}}{2}&<\sigma\sqrt{\frac{\Psi(t)}{T_i(t)}},\end{aligned}$$ where the last inequality follows from (\[eq:UCBQ\]). Since $0<{\Delta_i}<\Delta^t_{i}$, thus we have that if $\bar{B}^t_i$ is true then $$\begin{aligned} T_i(t)&<\frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(t),\end{aligned}$$ is true. Thus for sufficiently large $t$ $$\begin{aligned} &\bar{B}^t_i\subseteq \left\{T_i(t)<\frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(t)\right\},\\ &\mathcal{P}\left(\{\bar{B}^t_i\}\right)\leq \mathcal{P}\left(\left\{T_i(t)<\frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(t)\right\}\right).\end{aligned}$$ Thus $\mathcal{P}\left(\bar{B}^t_i\right)\neq 0$ only if $$\begin{aligned} T_i(t)&<\frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(t).\end{aligned}$$ Thus we have $$\begin{aligned} \sum_{t=l}^{n-1}\mathcal{P}(\bar{B}^t_i)&=\sum_{t=l}^{\tilde{t}}\mathcal{P}(\bar{B}^t_i)\leq \frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(n),\end{aligned}$$ and hence $$\begin{aligned} E(T_i(n))\leq l+\gamma \log{n}+\frac{4\sigma^2}{{\Delta^2_i}}\,\Psi(n) +\sum_{t=l}^{n-1}\frac{\nu\,\log{t}}{\exp\left(2\kappa \sigma^2\Psi(t)\right)}.\end{aligned}$$ If $\alpha \log{t}\leq \Psi(t) \leq \beta \log{t}$ then $$\begin{aligned} E(T_i(n))\leq l+\gamma\log{n}+\frac{4\sigma^2\beta}{{\Delta^2_i}}\,\log n +\sum_{t=l}^{n-1}\frac{\nu\,\log{t}}{\exp\left(2\kappa\sigma^2 \alpha \log{t}\right)}.\end{aligned}$$ The series on the right converges as long as $\alpha>3/(2\kappa \sigma^2)$. Thus from (\[eq:DynRegret\]) we have that the cumulative expected regret satisfies: $$\begin{aligned} R_n\leq c_1+ \bar{\Delta}\sum_{i\neq i^*}^k\left(\gamma+\frac{4 \sigma^2\beta}{\Delta_i^2}\right)\,\log{n} ,\end{aligned}$$ where $$c_1=k\bar{\Delta}\nu\,\left(\frac{\log{2}}{2^{2\kappa\sigma^2 \alpha}}+\int_2^{n-1}\frac{1}{t^{2\kappa \sigma^2\alpha-1}}dt\right)$$ The integral on the right converges as $n\to \infty$ if $\alpha>3/(2\kappa\sigma^2)$. Thus completing the proof of Theorem \[Theom:DynamicRegret\]. [^1]: $^{1}$Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA. [[email protected]]{} [^2]: $^{2}$Department of Mechanical Engineering, University of Peradeniya, KY 20400, Sri Lanka. [[email protected]]{} [^3]: $^{3}$Department of Mechanical and Aerospace Engineering, Princeton University, NJ 08544, USA. [[email protected]]{}

ArXiv

--- abstract: 'The two time-dependent Schrödinger equations in a potential $V(s,u)$, $u$ denoting time, can be interpreted geometrically as a moving interacting curves whose Fermi-Walker phase density is given by $-(\partial V/\partial s)$. The Manakov model appears as two moving interacting curves using extended da Rios system and two Hasimoto transformations.' address: | $^{1}$ Institute of Electronics, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria\ $^{2}$ Université de Cergy-Pontoise, 2 avenue, A. Chauvin, F-95302, Cergy-Pontoise Cedex, France\ $^{3}$Institute for Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 72 Tsarigradsko chaussee, 1784 Sofia, Bulgaria author: - 'N. A. KOSTOV$^{1}$, R. DANDOLOFF$^{2}$, V. S. GERDJIKOV$^{3}$ and G. G. GRAHOVSKI$^{2,3}$' title: THE MANAKOV SYSTEM AS TWO MOVING INTERACTING CURVES --- Introduction ============ In recent years, there has been a large interest in the applications of the Frenet-Serret equations [@e60; @RogSchief] for a space curve in various contexts, and interesting connections between geometry and integrable nonlinear evolution equations have been revealed. The subject of how space curves evolve in time is of great interest and has been investigated by many authors. Hasimoto [@h72] showed that the evolution of a thin vortex filament regarded as a moving space curve can be mapped to the nonlinear Schrödinger equation (NLSE): $$\begin{aligned} \label{eq:1} i \Psi_{u}+\Psi_{ss}+\frac{1}{2}|\Psi|^{2}\Psi=0,\end{aligned}$$ Here, u and s are time and space variables, respectively, subscripts denote partial derivatives. Lamb [@L77] used Hasimoto transformation to connect other motions of curves to the mKdV and sine-Gordon equations. Langer and Perline [@lr91] showed that the dynamics of non-stretching vortex filament in $\bbbr^{3}$ leads to the NLS hierarchy. Motions of curves on $S^{2}$ and $S^{3}$ were considered by Doliwa and Santini [@DoSa]. Lakshmanan [@Laks] interpreted the dynamics of a nonlinear string of fixed length in $\bbbr^{3}$ through the consideration of the motion of an arbitrary rigid body along it, deriving the AKNS spectral problem without spectral parameter. Recently, Nakayama [@Nakaya] showed that the defocusing nonlinear Schrödinger equation, the Regge-Lund equation, a coupled system of KdV equations and their hyperbolic type arise from motions of curves on hyperbola in the Minkowski space. Recently the connection between motion of space or plane curves and integrable equations has drawn wide interest and many results have been obtained [@FokGel; @dodd; @ciel; @ChouQu; @LanPer; @H02; @CaIv]. Preliminaries {#sec:2} ============= The Manakov model {#sec:2.1} ----------------- Time-dependent Schrödinger equation in potential $V(s,u)$ $$\begin{aligned} i\Psi_{u}+\Psi_{ss}-V(s,u)\Psi=0,\end{aligned}$$ goes into the NLS eq. (\[eq:1\]) if the potential $V(s,u)=-1/2 |\psi(s,u)|^2$. Similarly, a set of two time-dependent Schrödinger equations: $$\begin{aligned} i\Psi_{1,u}+\Psi_{1,ss}-V(s,u)\Psi_{1}=0,\quad i\Psi_{2,u}+\Psi_{2,ss}-V(s,u)\Psi_{2}=0,\end{aligned}$$ where $V(s,u)=-|\Psi_{1}|^{2} -|\Psi_{2}|^{2}$, can be viewed as the Manakov system: $$\begin{aligned} &&i\Psi_{1,u}+\Psi_{1,ss}+(|\Psi_{1}|^{2} +|\Psi_{2}|^{2})\Psi_{1}=0,\label{ManakovSys1}\\ &&i\Psi_{2,u}+\Psi_{2,ss}+(|\Psi_{1}|^{2}+|\Psi_{2}|^{2})\Psi_{2}=0. \label{ManakovSys2}\end{aligned}$$ It is convenient to use two Hasimoto transformations [@h72] $$\begin{aligned} \Psi_{i}=\kappa_{i}(s,u) \exp\left[i\int^{s}\tau_{i}(s',u)d s'\right],\quad i=1,2,\end{aligned}$$ in Eqs. (\[ManakovSys1\]), (\[ManakovSys2\]). Equating imaginary and real parts, this leads to the coupled partial differential equations (extended daRios system [@r1906]) $$\begin{aligned} &&\kappa_{i,u}=-(\kappa_{i}\tau_{i})_{s}-\kappa_{i,s}\tau_{i},\quad i=1,2 ,\label{daRios1}\\ &&\tau_{i,u}=\left[\frac{\kappa_{i,ss}}{\kappa_{i}}-\tau_{i}^{2}\right]_{s}-V(s,u)_{s}, \label{daRios2}\end{aligned}$$ where $$\begin{aligned} V(s,u)=-|\Psi_{1}|^{2} -|\Psi_{2}|^{2}=-\kappa_{1}^{2}-\kappa_{2}^2.\end{aligned}$$ Soliton curves {#sec:2.2} -------------- A three dimensional space curve is described in parametric form by a position vectors ${\bf r}_{i}(s), i=1,2$, where s is the arclength. Let ${\bf t}_{i}=(\partial {\bf r}_{i}/\partial s), i=1,2$ be the two unit tangent vectors along the two curves. At a given instant of time the triads of unit orthonormal vectors $({\bf t}_{i},{\bf n}_{i},{\bf b}_{i})$, where ${\bf n}_{i}$ and ${\bf b}_{i}$ denote the normals and binormals, respectively, satisfy the Frenet-Serret equations for two curves $$\begin{aligned} \label{FSeq} {\bf t}_{i,s}=\kappa_{i} {\bf n_{i}},\quad {\bf n}_{i,s}=-\kappa_{i} {\bf t}_{i}+ \tau_{i}{\bf b}_{i},\quad {\bf b}_{i,s}=-\tau_{i} {\bf n}_{i},\quad i=1,2 ,\end{aligned}$$ $\kappa_{i}$ and $\tau_{i}$ denote, respectively the two curvatures and torsions of the curves. A moving curves are described by $r_{i}(s,u)$, where u denote time. The temporal evolution of two triads corresponding to a given value $s$ can be written in the general form as $$\begin{aligned} {\bf t}_{i,u}=g_{i} {\bf n}_{i}+h_{i} {\bf b}_{i},\quad {\bf n}_{i,u}=-g_{i} {\bf t}_{i} + \tau^{0}_{i}{\bf b}_{i},\quad {\bf b}_{i,u}=-h_{i} {\bf t}_{i} - \tau^{0}_{i}{\bf n}_{i},\end{aligned}$$ where the coefficients $g_{i}$,$h_{i}$ and $\tau_{i}^{0}$, as well as $\kappa_{i}$ and $\tau_{i}$, are functions of $s$ and $u$. $$\begin{aligned} \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)_{s} =\left(% \begin{array}{ccc} 0 & \kappa_{i} & 0 \\ -\kappa_{i} & 0 & \tau_{i} \\ 0 & -\tau_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right),\quad \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)_{u} =\left(% \begin{array}{ccc} 0 & g_{i} & h_{i} \\ -g_{i} & 0 & \tau^{0}_{i} \\ -h_{i} & -\tau^{0}_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right).\nonumber\end{aligned}$$ Introduce $$\begin{aligned} L_{i} =\left(% \begin{array}{ccc} 0 & \kappa_{i} & 0 \\ -\kappa_{i} & 0 & \tau_{i} \\ 0 & -\tau_{i} & 0 \\ \end{array}% \right),\qquad M_{i} =\left(% \begin{array}{ccc} 0 & g_{i} & h_{i} \\ -g_{i} & 0 & \tau^{0}_{i} \\ -h_{i} & -\tau^{0}_{i} & 0 \\ \end{array}% \right)\end{aligned}$$ and $\Delta {\bf t}_{i} \equiv ({\bf t}_{i,su}-{\bf t}_{i,us})$, $ \Delta {\bf n}_{i} \equiv ({\bf n}_{i,su}-{\bf n}_{i,us})$, and $\Delta {\bf b}_{i} \equiv ({\bf b}_{i,su}-{\bf b}_{i,us}) $. Then $$\begin{aligned} \left(% \begin{array}{c} \Delta {\bf t}_{i} \\ \Delta {\bf n}_{i} \\ \Delta {\bf b}_{i} \\ \end{array}% \right) =&&\left( \partial_{s} M_{i}-\partial_{u} L_{i}+[L_{i},M_{i}] \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right)\nonumber\\&& = \left(% \begin{array}{ccc} 0 & \alpha^{1}_{i} & \alpha^{2}_{i} \\ -\alpha^{1}_{i} & 0 & \alpha^{3}_{i} \\ - \alpha^{2}_{i} & -\alpha^{3}_{i} & 0 \\ \end{array}% \right) \left(% \begin{array}{c} {\bf t}_{i} \\ {\bf n}_{i} \\ {\bf b}_{i} \\ \end{array}% \right),\end{aligned}$$ where $$\begin{aligned} \alpha^{1}_{i}=\kappa_{i,u}g_{i,s}+h_{i}\tau_{i},\, \alpha^{2}_{i}=-h_{i,s}+\kappa_{i}\tau_{i}^{0}-g_{i}\tau_{i},\, \alpha^{3}_{i}=\tau_{i,u}-\tau_{i,s}-\kappa_{i} h_{i}.\end{aligned}$$ $$\begin{aligned} \label{KappaTau} \kappa_{i,u}=g_{i,s}-h_{i}\tau_{i},\qquad \tau_{i}^{0}=(h_{i,s}+g_{i}\tau_{i})/\kappa_{i},\end{aligned}$$ A generic curve evolution must satisfy the geometric constraints $$\begin{aligned} \Delta {\bf t}_{i}\cdot( \Delta {\bf n}_{i}\times \Delta {\bf b}_{i})=0, \label{constraint}\end{aligned}$$ i.e., $\Delta {\bf t}_{i}$, $ \Delta {\bf n}_{i}$ and $ \Delta {\bf b}_{i}$ must remain coplanar vectors under time involution. Further, since Eq. (\[constraint\]) is automatically satisfied for $\Delta {\bf t}_{i}=0$, we see that $\Delta {\bf n}_{i}$ and $\Delta {\bf b}_{i}$ need not necessarily vanish. In addition, we see from (\[constraint\]) that $\Delta {\bf t}_{i}=0$ implies $\alpha_{i}^{1}=\alpha_{i}^{2}=0$, so that $$\begin{aligned} \Delta {\bf n}_{i}=\alpha_{i}^{3} \Delta {\bf b}_{i},\quad \Delta {\bf b}_{i}=\alpha_{i}^{3} \Delta {\bf n}_{i}\quad g_{i}=-\kappa_{i} \tau_{i},\qquad h_{i}=\kappa_{i,s}.\end{aligned}$$ Substituting these in the second equation in (\[KappaTau\]) gives $$\begin{aligned} \tau_{i}^{0}=\left[\frac{\kappa_{i,ss}}{\kappa_{i}}-\tau_{i}^{2}\right],\end{aligned}$$ hence Eq. (\[daRios1\]) yields $(\tau_{i,u}-\tau^{0}_{i,s})=-V(s,u)_{s}=(\kappa_{1}^{2}+\kappa_{2}^2)_{s}$. Next there is an underlying angle anholonomy [@bd99; @bbd90] or ’Fermi-Walker phase’ $\delta\Phi^{FW}=(\tau_{i,u}-\tau^{0}_{i,s}) dsdu$ with respect to its original orientation, when $s$ and $u$ change along an infinitesimal closed path of area $dsdu$. Integration of the extended da Rios system ========================================== The coupled nonlinear equations (\[daRios1\]),(\[daRios2\]) constitute the extended Da Rios system as derived in [@r1906] . The solutions of (\[daRios1\]),(\[daRios2\]) with $\kappa(\xi)$ and $\tau=\tau(\xi)$, where $\xi=s-c\,u$ are simple. On substitution, we obtain $$\begin{aligned} &&c\,\kappa_{i,\xi}=2\kappa_{i,\xi}\tau_{i}+\kappa_{i}\tau_{i,\xi},\quad \xi=s-cu, i=1,2 , \\&& -c \tau_{i,\xi}=\left[ -\tau_{i}^2+\frac{\kappa_{i,\xi\xi}}{\kappa_{i}}+\kappa_{1}^{2}+\kappa_{2}^{2} \right]_{\xi}, \qquad\tau_{i}=\frac{c}{2}.\end{aligned}$$ where we use the boundary condition $\kappa_{i}\rightarrow 0, i=1,2$ as $s\rightarrow \infty$. Hence $\kappa_{i}$ obey the nonlinear oscillator equations $$\begin{aligned} \kappa_{i,\xi\xi}+ \left(\sum_{j=1}^{2}\kappa_{j}^{2}\right)\kappa_{i}=a_{i}\kappa_{i},\quad i=1,2. \label{oscillator7}\end{aligned}$$ where $a_{i}, i=1,2$ are arbitrary constants. [**Example 1**]{} One soliton solutions of the Manakov system are given by $$\begin{aligned} &&\Psi_{1}=\sqrt{2a}\,\epsilon_{1}\mbox{e}^{i(\frac{1}{2} c(s-s_{0})+(a-\frac{1}{4} c^2) u)}\mbox{sech}(\sqrt{a}(s-s_{0}-ct))\\ &&\Psi_{2}=\sqrt{2a}\,\epsilon_{2}\mbox{e}^{i(\frac{1}{2} c(s-s_{0})+(a-\frac{1}{4} c^2) u)}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\end{aligned}$$ and $|\epsilon_{1}|^{2}+|\epsilon_{2}|^{2}=1$ We first note that for Manakov system, the expressions for the curvatures $\kappa_{i}, i=1,2$ and the torsions $\tau_{i}, i=1,2$ for the moving curves corresponding to a one soliton solutions of the Manakov system are given by $$\begin{aligned} \kappa^2=\kappa_{1}^{2}+\kappa_{2}^{2}=\sqrt{2a}\,\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\quad \tau_{1}=\tau_{2}=\frac{1}{2}c.\end{aligned}$$ and $$\begin{aligned} \kappa_{1}=\sqrt{2a}\,\epsilon_{1}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)),\quad \kappa_{2}=\sqrt{2a}\,\epsilon_{2}\mbox{sech}(\sqrt{a}(s-s_{0}-ct)).\nonumber\end{aligned}$$ ![Two curves (\[oneSolManSys\]) of one soliton solution of Manakov system, $\epsilon_{1}=\sqrt{2}/\sqrt{3}$, $\epsilon_{2}=1/\sqrt{3}$ []{data-label="fig1"}](solit.eps){width="10cm" height="8cm"} [**Example 2**]{} One special solution of Manakov system is written by $$\begin{aligned} \kappa_{1}=C_{1}\mbox{cn}(\alpha \xi,k),\qquad \kappa_{2}=C_{2}\mbox{cn}(\alpha \xi,k),\end{aligned}$$ where $$\begin{aligned} \alpha^2=\frac{a_{1}}{2k^2-1},\quad C_{1}^{2}+C_{2}^{2}= 2\alpha^2 k^2, \quad a_{1}=a_{2}=a,\end{aligned}$$ In the limit $k\rightarrow 1$ we obtain the well known [*Manakov*]{} soliton solution $$\begin{aligned} &&\Psi_{1}=\frac{ \sqrt{2a} \epsilon_{1} \exp\left\{i\left(\frac{1}{2}c(s-s_{0})+(a-\frac{1}{4} c^{2})u\right) \right\} } {\mbox{ch}(\sqrt{a} (s-s_{0}-ct)) },\nonumber \\ &&\Psi_{2}=\frac{ \sqrt{2a} \epsilon_{2} \exp\left\{i\left(\frac{1}{2}c(s-s_{0})+(a-\frac{1}{4} c^{2})u\right) \right\} } {\mbox{ch}(\sqrt{a} (s-s_{0}-ct)) }. \nonumber\end{aligned}$$ Here we introduce the following notations $$\begin{aligned} |\epsilon_{1}|^2+|\epsilon_{2}|^2=1, \quad \zeta_{1}=\frac{1}{2} c+i\sqrt{a} =\xi_{1}+i\eta_{1},\end{aligned}$$ where $s_{0}$ is the position of soliton, $(\epsilon_{1},\epsilon_{2})$ are the components of polarization vector. The real part of $\zeta_{1}$ i.e. $c/2$ gives us the soliton velocity while the imaginary part of $\zeta_{1}$, i.e. $\sqrt{2a}$, gives the soliton amplitude and width. [**Example 3**]{} Integrating (\[FSeq\]) for two unit tangent vectors along the curves ${\bf t}_{i}=(\partial {\bf r}_{i}/\partial s), i=1,2$ for position vectors ${\bf r}_{i}(s), i=1,2$ we obtain $$\begin{aligned} {\bf r}_{j}=\left(\begin{array}{c} \frac{s}{2}-\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \tanh{\left(\epsilon_{j}(s-cu)\right)} \\ -\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \mbox{sech}{\left(\epsilon_{j}(x-cu)\right)} \cos{\left(\frac{1}{2}cs+(\epsilon_{j}^{2}-\frac{1}{4} c^2)u \right)} \\ -\frac{\epsilon_{j}}{\epsilon_{j}^{2}+\frac{1}{2} c} \mbox{sech}{\left(\epsilon_{j}(x-cu)\right)} \sin{\left(\frac{1}{2}cs+(\epsilon_{j}^{2}-\frac{1}{4} c^2)u \right)} \\ \end{array}\right),\quad j=1,2 , \label{oneSolManSys}\end{aligned}$$ and $\epsilon_{1}=\cos{\alpha},\,\epsilon_{2}=\sin{\alpha}$, where $\alpha$ is arbitrary positive number. [**Example 4**]{} Let $u(x)=6\wp(\xi+\omega^{\prime})$ be the two-gap Lamé potential with simple periodic spectrum (see for example [@ek94]) $$\lambda_{0}=-\sqrt{3g_{2}},\quad \lambda_{1}=-3e_{0}, \quad \lambda_{2}=-3e_{1}, \quad \lambda_{3}=-3e_{2}, \quad \lambda_{4}=\sqrt{ 3g_{2}}.$$ and the corresponding Hermite polynomial have the form $$F(\wp(\xi+\omega^{\prime}),\lambda)=\lambda^{2}- 3\wp(\xi+\omega^{\prime})\lambda+ 9\wp^{2}(\xi+\omega^{\prime})-\frac{9}{4}g_{2} . \label{HerPol}$$ Consider the genus $2$ nonlinear anisotropic oscillator (\[oscillator7\]) with Hamiltonian $$H=\frac{1}{2}(p_{1}^{2}+p_{2}^{2})+\frac{1}{4}(\kappa_{1}^{2}+\kappa_{2}^{2})^{2}- \frac{1}{2}(a_{1}\kappa_{1}^{2}+a_{2}\kappa_{2}^{2}),$$ where $(\kappa_{i},p_{i})$, $i=1,2$ are canonical variables with $p_{i}=\kappa_{i,x}$ and $a_{1},a_{2}$ are arbitrary constants. The simple solutions of these system are given in terms of Hermite polynomial $$\kappa_1^2=2\frac{F(x,\tilde{\lambda}_{1})} {\tilde{\lambda}_{2}-\tilde{\lambda}% _{1}} ,\quad \kappa_2^2=2\frac{F(x,\tilde{\lambda}_{2})} {\tilde{\lambda}_{1}-% \tilde{\lambda}_{2}} ,$$ Let us list the corresponding solutions [**(A)**]{} Periodic solutions in terms of single Jacobian elliptic function The nonlinear anisotropic oscillator admits the following solutions: $$\begin{aligned} \kappa_1 = C_1 \mbox{sn}(\alpha \xi, k), \qquad \kappa_2 = C_2 \mbox{cn}(\alpha \xi, k). \label{onegap}\end{aligned}$$ Here the amplitudes $C_1$, $C_2$ and the temporal pulse-width $1/\alpha$ are defined by the parameters $a_{1}$ and $a_{2}$ as follows: $$\begin{aligned} \alpha^2 k^2 = a_{2}-a_{1} , \quad C^{2}_{1} = a_{2} + \alpha^2 - 2\alpha^2 k^2 , \quad C^{2}_{2} = a_{1} + \alpha^{2}+\alpha^2 k^2 ,\end{aligned}$$ where $0 < k < 1$. Following our spectral method it is clear, that the solutions (\[onegap\]) are associated with eigenvalues $\lambda_2 = - e_2$ and $\lambda_3 = - e_3$ of one – gap Lamé potential. [**(B)**]{} Periodic solutions in terms of products of Jacobian elliptic functions Another solution is defined by [@ft89] $$\begin{aligned} \kappa_1 = C \mbox{dn}(\alpha \xi, k) \mbox{sn}(\alpha \xi, k), \qquad \kappa_2 = C \mbox{dn}(\alpha \xi ,k ) \mbox{cn}(\alpha \xi, k), \label{Flor}\end{aligned}$$ where $\mbox{sn}$,$\mbox{cn}$, $\mbox{dn}$ are the standard Jacobian elliptic functions, $k$ is the modulus of the elliptic functions $ 0 < k < 1$. The wave characteristic parameters: amplitude $C$, temporal pulse-width $1/\alpha$ and $k$ are related to the physical parameters and, $k$ through the following dispersion relations $$\begin{aligned} C^{2} = \frac{2}{5} (4a_{2} - a_{1}) , \quad \alpha^{2} = \frac{1}{15}(4a_{2}-a_{1}) , \quad k^{2} = \frac {5 (a_{2}-a_{1})} {4a_{2}-a_{1}} .\end{aligned}$$ We have found the following solutions of the nonlinear oscillator [@ku92] $$\begin{aligned} \kappa_1 = C\alpha^2 k^2 \mbox{cn}(\alpha \xi,k)\mbox{sn}(\alpha \xi,k), \quad \kappa_2 = C\alpha^2 \mbox{dn}^2 (\alpha \xi, k) + C_{1} \label{UzKos}\end{aligned}$$ where $C$, $C_1$, $\alpha$ and $k$ are expressed through parameters $a_{1}$ and $a_{2}$ by the following relations $$\begin{aligned} C^2 & = & \frac {18} {a_{2}-a_{1}}, \quad \alpha^2 = \frac {1}{10} \left( 2 a_{2}-3a_{1}+ \sqrt{\frac{5}{3}(a_{2}^{2}-a_{1}^2) } \right) \nonumber \\ k^2 & = & \frac {2 \sqrt { \frac{5}{3} (a_{2}^{2}-a_{1}^{2})}} {\sqrt{\frac{5}{3} (a_{2}^{2}-a_{1}^{2})}+2a_{2}-3a_{1}},\quad C_1 = \frac {C}{30} (4a_{1}-a_{2}),.\end{aligned}$$ [**(C)**]{} Periodic solutions associated with the two-gap Treibich-Verdier potentials. Below we construct the two periodic solutions associated with the Treibich-Verdier potential. Let us consider the potential $$u(x)=6\wp(\xi+\omega^{\prime})+2{\frac{(e_1-e_2)(e_1-e_3)}{\wp(\xi+\omega^{% \prime})-e_1}}, \label{tv4}$$ and construct the solution in terms of Lamé polynomials associated with the eigenvalues $\tilde{\lambda}_1,\tilde{\lambda}_2$, $\tilde{\lambda}_1 > \tilde{\lambda}_2$ $$\begin{aligned} \tilde{\lambda}_1=e_2+2e_1+2\sqrt{(e_1-e_2)(7e_1+2e_2)}, \\ \tilde{\lambda}_2=e_3+2e_1+2\sqrt{(e_1-e_3)(7e_1+2e_3)}. \nonumber \label{zz}\end{aligned}$$ The finite and real solutions $q_1,q_2$ have the form $$\begin{aligned} \kappa_1= C_{1}\mbox{sn}(\xi,k)\mbox{dn}(\xi,k) +C_{2}\mbox{sd}(\xi,k) ,\, \kappa_2= C_{3}\mbox{cn}(\xi,k)\mbox{dn}(\xi,k) +C_{4}\mbox{cd}(z,k), \nonumber\end{aligned}$$ where $C_{i}$, $i=1,\ldots 4$ are constants and have important geometrical interpretation [@ek94] and $\mbox{sd}$, $\mbox{cd}$, are standard Jacobian elliptic functions. The concrete expressions in terms of $k,\tilde{ \lambda}_{1},\tilde{\lambda_{2}}$ are given in [@ceek95; @chr:eil:eno:kos] In a similar way we can find the elliptic solution associated with the eigenvalues $$\begin{aligned} \tilde{\lambda}_1&=&e_2+2e_1+2\sqrt{(e_1-e_2)(7e_1+2e_2)},\quad \tilde{% \lambda}_2=-6e_1, \label{zz2}\end{aligned}$$ We have $$\begin{aligned} \kappa_1=\tilde{C}_{1}\mbox{dn}^{2}(\xi,k),\qquad \kappa_2= C_{1}\mbox{sn}(\xi,k)\mbox{dn}(\xi,k) +C_{2}\mbox{sd}(\xi,k) , \label{ee3}\end{aligned}$$ where $\tilde{C}_{1}, C_{1}, C_{2}$ are given in [@ceek95; @chr:eil:eno:kos]. The general formula for elliptic solutions of genus $2$ nonlinear anisotropic oscillator is given by [@ceek95] $$\begin{aligned} \kappa_1^2&=&{\frac{1}{\tilde{\lambda}_2-\tilde{\lambda}_1}} \left(2\tilde{\lambda}_1^2+2\tilde{\lambda}_1\sum_{i=1}^N \wp(\xi-x_i) \right. \nonumber \\ &&+\left.6\sum_{1\leq i< j\leq N}\wp(\xi-x_i)\wp(\xi-x_j)-{\frac{Ng_2}{4}}+ \sum_{1\leq i< j\leq 5}\lambda_i\lambda_j\right), \nonumber \\ \kappa_2^2&=&{\frac{1}{\tilde{\lambda}_1-\tilde{\lambda}_2}} \left(2\tilde{\lambda}_2^2+2\tilde{\lambda}_2\sum_{i=1}^N \wp(\xi-x_i) \right. \nonumber \\ &&+\left.6\sum_{1\leq i< j\leq N}\wp(\xi-x_i)\wp(\xi-x_j)-{\frac{Ng_2}{4}}+ \sum_{1\leq i< j\leq 5}\lambda_i\lambda_j\right), \nonumber\end{aligned}$$ where $x_{i}$ are solutions of equations $\sum_{i\neq j}\wp'(x_{i}-x_{j})=0, j=1,\ldots, N$. Extended da Rios-Betchov system =============================== Following Betchov we can derive the system of equations, which may be reduced to those for a two fictitious gases with negative pressures accompanied with two complicated nonlinear dispersive stresses. Introducing four new variables $\rho_{1}=\kappa_{1}^{2}$, $\rho_{1}=\kappa_{1}^{2}$, $u_{1}=2\tau_{1}$, $u_{2}=2 \tau_{2}$ using extended Da Rios system (\[daRios1\]), (\[daRios2\]) we obtain $$\begin{aligned} \label{Betchov} &&\frac{\partial\rho_{1}}{\partial u}+\frac{\partial (\rho_{1}u_{1})}{\partial s}=0,\qquad \frac{\partial\rho_{2}}{\partial u}+\frac{\partial(\rho_{2}u_{2})}{\partial s}=0, \nonumber \\&& \frac{\partial(\rho_{1} u_{1})}{\partial u}+\frac{\partial }{\partial s}\left[\rho_{1} u_{1}^2-(\rho_{1}^2+\rho_{2}^2)-\rho_{1}\frac{\partial^{2}}{\partial s^2}(\log \rho_{1})\right]=0,\nonumber \\&& \frac{\partial(\rho_{2} u_{2})}{\partial u}+\frac{\partial }{\partial s}\left[\rho_{2} u_{2}^2-(\rho_{1}^2+\rho_{2}^2)-\rho_{2}\frac{\partial^{2}}{\partial s^2}(\log \rho_{2})\right]=0. \nonumber\end{aligned}$$ HF system is gauge equivalent to Manakov system =============================================== The vector nonlinear Schrödinger equation is associated with type ${\bf A.III}$ symmetric space ${\rm SU(n+1)}/{\rm S(U(1)}\otimes {\rm U(n)})$. The special case $n=2$ of such symmetric space is associated with the famous Manakov system [@ma74]. Let us first fix the notations and the normalizations of the basis of ${\frak g} $. By $\Delta _+ $ ($\Delta _- $) we shall denote the set of positive (negative) roots of the algebra with respect to some ordering in the root space. By $\{E_{\alpha }, H_i\} $, $\alpha \in \Delta $, $i=1 \dots r $ we denote the Cartan–Weyl basis of ${\frak g} $ with the standard commutation relations [@Helg]. Here $H_i$ are Cartan generators dual to the basis vectors $e_i$ in the root space. The root system is invariant under the action of the Weyl group ${\frak W}({\frak g}) $ of the simple Lie algebra ${\frak g} $ [@Helg]. Let us now consider the gauge equivalent systems. The notion of gauge equivalence allows us to associate with the vector nonlinear Schrödinger equation an equivalent equation solvable by the ISM for the gauge equivalent linear problem [@ForKu*83]: $$\begin{aligned} \label{eq:2.3} \tilde{L}\tilde{\psi }(x,t,\lambda )= \left(i{d \over dx}-\lambda {\cal S}(x,t) \right) \tilde{\psi }(x,t,\lambda )=0, \nonumber\\ \tilde{M}\tilde{\psi }(x,t,\lambda )= \left(i{d \over dt}-\lambda^2 {\cal S}-\lambda {\cal S}_{x} {\cal S}(x,t) \right) \tilde{\psi }(x,t,\lambda )=0,\end{aligned}$$ where $$\begin{aligned} \label{eq:2.4} &&\tilde{\psi }(x,t,\lambda ) = \psi _0^{-1}\psi (x,t,\lambda ), \quad{\cal S}(x,t)=\sum_{\alpha=1}^{r}(S_{\alpha}E_{\alpha}+S_{\alpha}^{*}E_{-\alpha}) +\sum_{j=1}^{r} S_{j} H_{j} ,\nonumber\\&& {\cal S}(x,t) = \mbox{Ad}_{\hat{\psi }_0} J \equiv \psi _0^{-1}J\psi _0(x,t), \qquad J= \sum_{s=1}^n H_s,\end{aligned}$$ and $\psi _0=\psi (x,t,0) $ is the Jost solution at $\lambda =0 $. 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--- author: - Dejan Lavbič and Marjan Krisper title: Facilitating Ontology Development with Continuous Evaluation --- > **Dejan Lavbič** and Marjan Krisper. 2010. **Facilitating Ontology Development with Continuous Evaluation**, [Informatica **(INFOR)**](https://www.mii.lt/informatica/), 21(4), pp. 533 - 552. Abstract {#abstract .unnumbered} ======== In this paper we propose facilitating ontology development by constant evaluation of steps in the process of ontology development. Existing methodologies for ontology development are complex and they require technical knowledge that business users and developers don’t poses. By introducing ontology completeness indicator developer is guided throughout the development process and constantly aided by recommendations to progress to next step and improve the quality of ontology. In evaluating the ontology, several aspects are considered; from description, partition, consistency, redundancy and to anomaly. The applicability of the approach was demonstrated on Financial Instruments and Trading Strategies (FITS) ontology with comparison to other approaches. Keywords {#keywords .unnumbered} ======== Ontology development methodology, ontology evaluation, ontology completeness, rapid ontology development, semantic web Introduction ============ The adoption of Semantic Web technologies is less than expected and is mainly limited to academic environment. We are still waiting for wide adoption in industry. We could seek reasons for this in technologies itself and also in the process of development, because existence of verified approaches is a good indicator of maturity. As technologies are concerned there are numerous available for all aspects of Semantic Web applications; from languages for capturing the knowledge, persisting data, inferring new knowledge to querying for knowledge etc. In the methodological sense there is also a great variety of methodologies for ontology development available, as it will be further discussed in section \[related-work\], but the simplicity of using approaches for ontology construction is another issue. Current approaches in ontology development are technically very demanding and require long learning curve and are therefore inappropriate for developers with little technical skills and knowledge. In majority of existing approaches an additional role of knowledge engineer is required for mediation between actual knowledge that developers possess and ontology engineers who encode knowledge in one of the selected formalisms. The use of business rules management approach [@smaizys_business_2009] seems like an appropriate way to simplification of development and use of ontologies in business applications. Besides simplifying the process of ontology creation we also have to focus on very important aspect of ontology completeness. The problem of error-free ontologies has been discussed in [@fahad_ontological_2008; @porzel_task-based_2004] and several types of errors were identified - inconsistency, incompleteness, redundancy, design anomalies etc. All of these problems have to already be addressed in the development process and not only after development has reached its final steps. In this paper we propose a Rapid Ontology Development (ROD) approach where ontology evaluation is performed during the whole lifecycle of the development. The idea is to enable developers to rather focus on the content than the formalisms for encoding knowledge. Developer can therefore, based on recommendations, improve the ontology and eliminate the error or bad design. It is also a very important aspect that, before the application, the ontology is error free. Thus we define ROD model that introduces detail steps in ontology manipulation. The starting point was to improve existing approaches in a way of simplifying the process and give developer support throughout the lifecycle with continuous evaluation and not to conclude with developed ontology but enable the use of ontology in various scenarios. By doing that we try to achieve two things: - guide developer through the process of ontology construction and - improve the quality of developed ontology. The remainder of the paper is structured as follows. In the following section \[related-work\] state of the art is presented with the review of existing methodologies for ontology development and approaches for ontology evaluation. After highlighting some drawbacks of current approaches section \[ROD\] presents the ROD approach. Short overview of the process and stages is given with the emphasis on ontology completeness indicator. The details of ontology evaluation and ontology completeness indicator are given in section \[indicator\], where all components (description, partition, redundancy and anomaly) that are evaluated are presented. In section \[evaluation\] evaluation and discussion about the proposed approach according to the results obtained in the experiment of **Financial Instruments and Trading Strategies (FITS)** is presented. Finally in section \[conclusion-and-future-work\] conclusions with future work are given. Related work ============ Review of related approaches ---------------------------- Ontology is a vocabulary that is used for describing and presentation of a domain and also the meaning of that vocabulary. The definition of ontology can be highlighted from several aspects. From taxonomy [@corcho_methodologies_2003; @sanjuan_text_2006; @veale_analogy-oriented_2006] as knowledge with minimal hierarchical structure, vocabulary [@bechhofer_thesaurus_2001; @miller_wordnet:_1995] with words and synonyms, topic maps [@dong_hyo-xtm:_2004; @park_xml_2002] with the support of traversing through large amount of data, conceptual model [@jovanovic_achieving_2005; @mylopoulos_information_1998] that emphasizes more complex knowledge and logic theory [@corcho_methodologies_2003; @dzemyda_optimization_2009; @waterson_verifying_1999] with very complex and consistent knowledge. Ontologies are used for various purposes such as natural language processing [@staab_system_1999], knowledge management [@davies_semantic_2006], information extraction [@wiederhold_mediators_1992], intelligent search engines [@heflin_searching_2000], digital libraries [@kesseler_schema_1996], business process modeling [@brambilla_software_2006; @ciuksys_reusing_2007; @magdalenic_dynamic_2009] etc. While the use of ontologies was primarily in the domain of academia, situation now improves with the advent of several methodologies for ontology manipulation. Existing methodologies for ontology development in general try to define the activities for ontology management, activities for ontology development and support activities. Several methodologies exist for ontology manipulation and will be briefly presented in the following section. CommonKADS [@schreiber_knowledge_1999] is in fact not a methodology for ontology development, but is focused towards knowledge management in information systems with analysis, design and implementation of knowledge. CommonKADS puts an emphasis to early stages of software development for knowledge management. Enterprise Ontology [@uschold_towards_1995] recommends three simple steps: definition of intention; capturing concepts, mutual relation and expressions based on concepts and relations; persisting ontology in one of the languages. This methodology is the groundwork for many other approaches and is also used in several ontology editors. METHONTOLOGY [@fernandez-lopez_building_1999] is a methodology for ontology creation from scratch or by reusing existing ontologies. The framework enables building ontology at conceptual level and this approach is very close to prototyping. Another approach is TOVE [@uschold_ontologies:_1996] where authors suggest using questionnaires that describe questions to which ontology should give answers. That can be very useful in environments where domain experts have very little expertise of knowledge modeling. Moreover authors of HCONE [@kotis_human_2003] present decentralized approach to ontology development by introducing regions where ontology is saved during its lifecycle. OTK Methodology [@sure_methodology_2003] defines steps in ontology development into detail and introduces two processes – Knowledge Meta Process and Knowledge Process. The steps are also supported by a tool. UPON [@nicola_building_2005] is an interesting methodology that is based on Unified Software Development Process and is supported by UML language, but it has not been yet fully tested. The latest proposal is DILIGENT [@davies_semantic_2006] and is focused on different approaches to distributed ontology development. From information systems development point of view there are several methodologies that share similar ideas found in ontology development. Rapid Ontology Development model, presented in this paper follows examples mainly from blended, object-oriented, rapid development and people-oriented methodologies [@avison_information_2006]. In blended methodologies, that are formed from (the best) parts of other methodologies, the most influential for our approach was Information Engineering [@martin_information_1981] that is viewed as a framework within which a variety of techniques are used to develop good quality information systems in an efficient way. In object-oriented approaches there are two representatives – Object-Oriented Analysis (OOA; @booch_object_1993) and Rational Unified Process (RUP; @jacobson_unified_1999). Especially OOA with its five major activities: finding class and objects, identifying structures, indentifying subjects, defining attributes and defining services had profound effect on our research, while it was extended with the support of design and implementation phases that are not included in OOA. The idea of rapid development methodologies is closely related to ROD approach and current approach addresses the issue of rapid ontology development which is based on rapid development methodologies of information systems. James Martin’s RAD [@martin_rapid_1991] is based on well known techniques and tools but adopts prototyping approach and focuses on obtaining commitment from the business users. Another rapid approach is Dynamic Systems Development Method (DSDM; @consortium_dsdm_2005) which has some similarities with Extreme Programming (XP; @beck_extreme_2004). XP attempts to support quicker development of software, particularly for small and medium-sized applications. Comparing to techniques involved in information systems development, the ontology development in ROD approach is mainly based on *holistic techniques* (rich pictures, conceptual models, cognitive mapping), *data techniques* (entity modeling, normalization), *process techniques* (decision trees, decision tables, structured English) and *project management techniques* (estimation techniques). The ROD approach extends reviewed methodologies by simplifying development steps and introducing continuous evaluation of developed ontology. This is achieved by ontology completeness indicator that is based on approaches for ontology evaluation. Based on existing reviews in [@brank_survey_2005; @gangemi_modelling_2006; @gomez-perez_evaluation_1999; @hartmann_d1.2.3_2004] we classify evaluation approaches into following categories: - compare ontology to *“golden standard”* [@maedche_measuring_2002], - using ontology in an *application* and evaluating results [@porzel_task-based_2004], - compare with source of data about the *domain to be covered* by ontology [@brewster_data_2004] and - *evaluation* done *by humans* [@lozano-tello_ontometric:_2004; @noy_user_2005]. Usually evaluation of different levels of ontology separately is more practical than trying to directly evaluate the ontology as whole. Therefore, classification of evaluation approaches based on the level of evaluation is also feasible and is as follows: lexical, vocabulary or data layer, hierarchy or taxonomy, other semantic relations, context or application level, syntactic level, structure, architecture and design. Prior the application of ontologies we have to assure that they are free of errors. The research performed by @fahad_ontological_2008 resulted in classification and consequences of ontology errors. These errors can be divided into inconsistency errors, incompleteness errors, redundancy errors and design anomalies. Problem and proposal for solution --------------------------------- The review of existing approaches for ontology development in this section pointed out that several drawbacks exist. Vast majority of ontology development methodologies define a complex process that demands a long learning curve. The required technical knowledge is very high therefore making ontology development very difficult for nontechnically oriented developers. Among methodologies for ontology development there is a lack of rapid approaches which can be found in traditional software development approaches. On the other hand methodologies for traditional software development also fail to provide sufficient support in ontology development. This fact can be confirmed with the advent of several ontology development methodologies presented at the beginning of this section. Majority of reviewed methodologies also include very limited evaluation support of developed ontologies. If this support exists it is limited to latter stages of development and not included throughout the process. This paper introduces a novel approach in ontology modeling based on good practices and existing approaches [@allemang_semantic_2008; @cardoso_semantic_2007; @fahad_ontological_2008; @fernandez-lopez_building_1999; @sure_methodology_2003; @uschold_towards_1995] while trying to minimize the need of knowing formal syntax required for codifying the ontology and therefore bringing ontology modeling closer to business users who are actual knowledge holders. Based on the findings from the comparison of existing methodologies for ontology development and several evaluation approaches it has been noted that no approach exist that would constantly evaluate ontology during its lifecycle. The idea of proposed ROD approach with ontology completeness evaluation presented in section \[ROD\] is to create a feedback loop between developed ontology and its completeness by introducing indicator for completeness. With ROD approach detailed knowledge of development methodology is also not required as the process guides developers through the steps defined in methodology. By extending existing approaches with constant evaluation the quality of final artifact is improved and the time for development is minimized as discussed in section \[indicator\]. Rapid Ontology Development {#ROD} ========================== Introduction to ROD process --------------------------- The process for ontology development ROD (Rapid Ontology Development) that we propose is based on existing approaches and methodologies (see section \[related-work\]) but is enhanced with continuous ontology evaluation throughout the complete process. It is targeted at domain users that are not familiar with technical background of constructing ontologies. Developers start with capturing concepts, mutual relations and expressions based on concepts and relations. This task can include reusing elements from various resources or defining them from scratch. When the model is defined, schematic part of ontology has to be binded to existing instances of that vocabulary. This includes data from relational databases, text file, other ontologies etc. The last step in bringing ontology into use is creating functional components for employment in other systems. ROD stages {#ROD-stages} ---------- The ROD development process can be divided into the following stages: *pre-development*, *development* and *post-development* as depicted in Figure \[fig:ROD-process\]. Every stage delivers a specific output with the common goal of creating functional component based on ontology that can be used in several systems and scenarios. In pre-development stage the output is feasibility study that is used in subsequent stage development to construct essential model definition. The latter artifact represents the schema of problem domain that has to be coupled with instances from the real world. This is conducted in the last stage post-development which produces functional component for usage in various systems. ![Process of rapid ontology development (ROD)[]{data-label="fig:ROD-process"}](img/ROD-process){width="0.7\linewidth"} The role of constant evaluation as depicted in Figure \[fig:ROD-process\] is to guide developer in progressing through steps of ROD process or it can be used independently of ROD process. In latter case, based on semantic review of ontology, enhancements for ontology improvement are available to the developer in a form of multiple actions of improvement, sorted by their impact. Besides actions and their impacts, detail explanation of action is also available (see Figure \[fig:OC-GUI\]). ![Display of ontology completeness (OC) results and improvement recommendations[]{data-label="fig:OC-GUI"}](img/OC-GUI){width="0.3\linewidth"} In case of following ROD approach, while developer is in a certain step of the process, the OC measurement is adapted to that step by redefinition of weights (see Figure \[fig:OC-weights\] for distribution of weights by ROD steps) for calculation (e.g., in Step 2.1 of ROD process where business vocabulary acquisition is performed, there is no need for semantic checks like instance redundancy, lazy concept existence or inverse property existence, but the emphasis is rather on description of TBox and RBox component and path existence between concepts). When OC measurement reaches a threshold (e.g., $80\%$) developer can progress to the following step (see Figure \[fig:OC-calculation\]). The adapted OC value for every phase is calculated on-the-fly and whenever a threshold value is crossed, a recommendation for progressing to next step is generated. This way developer is aided in progressing through steps of ROD process from business vocabulary acquisition to functional component composition. In case that ontology already exists, with OC measure we can place the completeness of ontology in ROD process and start improving ontology in suggested phase of development (e.g., ontology has taxonomy already defined, so we can continue with step 2.4 where ad hoc binary relations identification takes place). Ontology evaluation and ontology completeness indicator {#indicator} ------------------------------------------------------- ![OC calculation[]{data-label="fig:OC-calculation"}](img/OC-calculation){width="0.7\linewidth"} **Ontology completeness (OC)** indicator used for guiding developer in progressing through steps of ROD process and ensuring the required quality level of developed ontology is defined as $$OC = f \left( C, P, R, I \right) \in [0, 1] \label{eq:OC}$$ where $C$ is set of concepts, $P$ set of properties, $R$ set of rules and $I$ set of instances. Based on these input the output value in an interval $[0, 1]$ is calculated. The higher the value, more complete the ontology is. OC is weighted sum of semantic checks, while weights are being dynamically altered when traversing from one phase in ROD process to another. OC can be further defined as $$OC = \sum_{i=1}^{n} w_i^{'} \cdot leafCondition_i \label{eq:OC-sum}$$ where $n$ is the number of leaf conditions and $leafCondition$ is leaf condition, where semantic check is executed. For relative weights and leaf condition calculation the following restrictions apply $\sum_i w_i^{'} = 1$, $\forall w_i^{'} \in [0, 1]$ and $\forall leafCondition_i \in [0, 1]$. Relative weight $w_i^{'}$ denotes global importance of $leafCondition_i$ and is dependent on all weights from leaf to root concept. The tree of conditions in OC calculation is depicted in Figure \[fig:OC-tree\] and contains semantic checks that are executed against the ontology. The top level is divided into *TBox*, *RBox* and *ABox* components. Subsequent levels are then furthermore divided based on ontology error classification [@fahad_ontological_2008]. Aforementioned sublevels are *description*, *partition*, *redundancy*, *consistency* and *anomaly*. ![Ontology completeness (OC) tree of conditions, semantic checks and corresponding weights[]{data-label="fig:OC-tree"}](img/OC-tree){width="0.7\linewidth"} This proposed structure can be easily adapted and altered for custom use. Leafs in the tree of OC calculation are implemented as semantic checks while all preceding elements are aggregation with appropriate weights. Algorithm for ontology completeness (OC) price is depicted in Definition \[def:OC-evaluation\], where $X$ is condition and $w = w(X, Y)$ is the weight between condition $X$ and condition $Y$. \[Ontology completeness evaluation algorithm\] $$\begin{aligned} &\text{' Evaluation is executed on top condition "OC components" with weight 1} \\ &\textbf{Evaluate } \boldsymbol{(X, w)} \\ &\quad price_{OC} = 0 \\ &\quad \text{mark condition } X \text{ as visited} \\ &\quad \text{if not exists sub-condition of } X \\ &\qquad \text{' Execute semantic check on leaf element} \\ &\qquad \text{return } w \cdot exec(X) \\ &\quad \text{else for all conditions } Y \text{ that are sub-conditions of } X \text{ such that } Y \text{ is not visited} \\ &\qquad \text{' Aggregate ontology evaluation prices} \\ &\qquad \text{if } w(X, Y) \neq 0 \\ &\qquad \quad price_{OC} = price_{OC} + Evaluate(Y, w(X, Y)) \\ &\quad \text{return } w \cdot price_{OC} \\ &\textbf{End}\end{aligned}$$ Each leaf condition implements a semantic check against ontology and returns value $leafCondition \in [0, 1]$. Figure \[fig:OC-weights\] depicts the distribution of OC components (description, partition, redundancy, consistency and anomaly) regarding individual phase in ROD process (see section \[ROD-stages\]). In first two phases 2.1 and 2.2 developer deals with business vocabulary identification and enumeration of concepts’ and properties’ examples. Evidently with aforementioned steps emphasis is on description of ontology, while partition is also taken into consideration. The importance of components description and partition is then in latter steps decreased but it still remains above average. In step 2.3 all other components are introduced (redundancy, consistency and anomaly), because developer is requested to define taxonomy of schematic part of ontology. While progressing to the latter steps of ROD process emphasis is on detail description of classes, properties and complex restriction and rules are also added. At this stage redundancy becomes more important. This trend of distributions of weights remains similarly balanced throughout the last steps 2.5 and 2.6 of development phase. In post-development phase when functional component composition is performed, ontology completeness calculation is mainly involved in redundancy, description and anomaly checking. The details about individual OC components are emphasized and presented in details in the following subsections. ![Impact of weights on OC sublevels in ROD process[]{data-label="fig:OC-weights"}](img/OC-weights){width="0.7\linewidth"} ### Description Description of ontology’s components is very important aspect mainly in early stages of ontology development. As OC calculation is concerned there are several components considered: - *existence of entities* (classes and properties) and *instances*, - (multiple) *natural language descriptions* of TBox and RBox components and - *formal description* of concepts and instances. The notion of existence of entities is very straightforward; if ontology doesn’t contain any entities than we have no artifacts to work with. Therefore the developer is by this metric encouraged to first define schematic part of ontology with classes and properties and then also to add elements of ABox component in a form of individuals. Next aspect is natural language descriptions of entities. This element is despite of its simplicity one of the most important, due to ability to include these descriptions in further definition of complex axioms and rules [@vasilecas_towards_2009]. Following business rules approach [@vasilecas_practical_2008] it’s feasible to create templates for entering this data on-the-fly by employing this natural description of entities. Developer is encouraged to describe all entities (classes and properties) with natural language using readable labels (e.g., `rdfs:label` and `rdfs:comment`) that don’t add to the meaning of captured problem domain but greatly improves human readability of defined ontology. When constructing ontology it is always required to provide labels and description in English, but the use of other languages is also recommended to improve employment of ontology. The last aspect of ontology description is formal description of TBox and ABox components that concerns concepts and instances. When describing classes with properties ontologists tend to forget defining domain and range values. This is evaluated for schematic part of ontology while for instances all required axioms are considered that are defined in TBox or ABox. Ontologists tend to leave out details of instances that are required (e.g., cardinality etc.). ### Partition Partition errors deal with omitting important axioms or information about the classification of concept and therefore reducing reasoning power and inferring mechanisms. In OC calculation several components are considered: - *common classes* and *instances*, - *external instances* of ABox component, - *connectivity of concepts* of TBox component and - *hierarchy of entities*. The notion of common classes deals with the problem of defining a class that is a sub-class of classes that are disjoint. The solution is to check every class $C_i$ if exist super-classes $C_j$ and $C_k$ that are disjoint. Similar is with common instances where situation can occur where instance is member of disjointing classes. When decomposing classes in sub-class hierarchy it is often the case that super-class instance is not a member of any sub-class. In that case we deal with a problem of external instances. The solution is to check every class $C_i$ if exist any instance that is a member of $C_i$, but not a member of any class in set of sub-classes. The aspect of connectivity of concepts deals with ontology as whole and therefore not allowing isolated parts that are mutually disconnected. The first semantic check deals with existence of inverse properties. If we want to contribute to full traversal among classes in ontology the fact that every object property has inverse property defined is very important. The second semantic check deals with existence of path between concepts. Ontology is presented as undirected graph $G = (V, E)$ and we try to identify maximum disconnected graphs. The last aspect of ontology completeness as partition is concerned with hierarchy of entities. We introduce data oriented approach for definition of hierarchy of entities where technical knowledge from domain user is not required. This is based on requirement that for every class and property defined ontologist is requested to insert also few instances (see preliminary steps in ROD process introduced in section \[ROD-stages\]). After this requirement is met, set of competency questions are introduced to the domain user and the result are automatically defined hierarchy axioms (e.g., `rdfs:subClassOf`, `owl:equivalentClass`, `owl:disjointWith`, `rdfs:subPropertyOf` and `rdfs:equivalentProperty`). The approach for disjoint class recommendation is depicted in Definition \[def:disjoint-axiom\], while approach for other hierarchy axioms is analogous. \[Recommend disjoint axiom between classes\] $$\begin{aligned} &\textbf{recommendDisjointWithClasses} \\ &\quad \tau_{\subseteq}^{sibling} = \{ \} \leftarrow \text{ Set of all sub-class pairs } (C, D) \\ &\quad Q_n \leftarrow \text{ Competency questions} \\ &\quad disjointClassRecommend = \{ \} \\ &\quad \text{for each } C_i \in TBox \\ &\qquad \text{add all sub-class pairs of class } C_i \text{ to } \tau_{\subseteq}^{sibling} \\ &\qquad \text{for each sub-class pair } (C_j, C_k) \in TBox \text{ where } C_j \subseteq C_i \wedge C_k \subseteq C_i \wedge C_j \neq C_k \\ &\qquad \quad \text{if } \exists i (C_j), i (C_k) \in ABox : ( \neg Q_1 (C_j, C_k) \wedge \neg Q_3 (C_j, C_k) ) \text{ then} \\ &\qquad \qquad \text{if } C_j \cap C_k \neq \{ \} \text{ then} \\ &\qquad \qquad \quad disjointClassRecommend = disjointClassRecommend \cup (C_j, C_k) \\ &\qquad \qquad \text{end if} \\ &\qquad \quad \text{end if} \\ &\qquad \text{end for} \\ &\quad \text{end for} \\ &\quad price = 1 - \frac{ \left | disjointClassRecommend \right | } { \left | \tau_{\subseteq}^{sibling} \right | } \\ &\quad \text{return } \boldsymbol{disjointClassRecommend} \text{ and } \boldsymbol{price} \\ &\textbf{end}\end{aligned}$$ Using this approach of recommendation, domain users can define axioms in ontology without technical knowledge of ontology language, because with data driven approach (using instances) and competency questions the OC calculation indicator does that automatically. Redundancy occurs when particular information is inferred more than once from entities and instances. When calculating OC we take into consideration following components: - *identical formal definition* and - *redundancy in hierarchy of entities*. When considering identical formal definition, all components (TBox, RBox and ABox) have to be checked. For every entity or instance Ai all belonging axioms are considered. If set of axioms of entity or instance $A_i$ is identical to set of axioms of entity or instance $A_j$ and $A_i \neq A_j$, then entities or instances $A_i$ and $A_j$ have identical formal definition. This signifies that $A_i$ and $A_j$ describe same concept under different names (synonyms). Another common redundancy issue in ontologies is redundancy in hierarchy. This includes sub-class, sub-property and instance redundancy. Redundancy in hierarchy occurs when ontologist specifies classes, properties or instances that have hierarchy relations (`rdfs:subClassOf`, `rdfs:subPropertyOf` and `owl:instanceOf`) directly or indirectly. ### Consistency In consistency checking of developed ontology the emphasis is on finding circulatory errors in TBox component of ontology. Circulatory error occurs when a class is defined as a sub-class or super-class of itself at any level of hierarchy in the ontology. They can occur with distance $0$, $1$ or $n$, depending upon the number of relations involved when traversing the concept down the hierarchy of concepts until we get the same from where we started traversal. The same also applies for properties. To evaluate the quality of ontology regarding circulatory errors the ontology is viewed as graph $G = (V, E)$, where $V$ is set of classes and $E$ set of `rdfs:subClassOf` relations. ### Anomaly Design anomalies prohibit simplicity and maintainability of taxonomic structures within ontology. They don’t cause inaccurate reasoning about concepts, but point to problematic and badly designed areas in ontology. Identification and removal of these anomalies should be necessary for improving the usability and providing better maintainability of ontology. As OC calculation is concerned there are several components considered: - *chain of inheritance* in TBox component, - *property clumps* and - *lazy entities* (classes and properties). The notion of chain of inheritance is considered in class hierarchy, where developer can classify classes as `rdfs:subClassOf` other classes up to any level. When such hierarchy of inheritance is long enough and all classes have no appropriate descriptions in the hierarchy except inherited child, then ontology suffers from chain of inheritance. The algorithm for finding and eliminating chains of inheritance is depicted in Definition \[def:chain-of-inheritance\]. \[Find chain of inheritance\] $$\begin{aligned} &\textbf{findChainOfInheritance} \\ &\quad price = 1 \\ &\quad axiom(C) = [ type, entity, value ] \leftarrow \text{ Axiom of class C} \\ &\quad A(C) = \forall axiom(C) : entity = C \leftarrow \text{ Set of asserted axioms of class C} \\ &\quad A_{\subseteq}^{-} \leftarrow \text{ Set of asserted axioms of class } C \text{ without rdfs:subClassOf axiom} \\ &\quad chainOfInheritance = \{ \} \\ &\quad \text{while } \exists C_i, C_j \in TBox \wedge \exists C_1, C_2, \ldots, C_n \in TBox : (C_j \subseteq C_n \subseteq C_{n-1} \subseteq \ldots \subseteq C_2 \subseteq C_1 \subseteq C_i) \wedge \\ &\qquad ( \forall C_1, C_2, \ldots, C_n : \left | superClass(C_n) \right | = 1 \wedge A_{\subseteq}^{-} (C_n) = \{ \} ) \wedge \left | A_{\subseteq}^{-} (C_i) \right | > 0 \wedge \left | A_{\subseteq}^{-} (C_j) \right | > 0 \text{ then} \\ &\qquad \quad price = price - \frac{n}{n_{\subseteq}^{direct}} \\ &\qquad \quad chainOfInheritance = chainOfInheritance \cup \{ C_i, C_j, \{ C_1, C_2, \ldots, C_n \} \} \\ &\quad \text{end while} \\ &\quad \boldsymbol{chainsOfInheritance} \text{ and } \boldsymbol{price} \\ &\textbf{end}\end{aligned}$$ The next aspect in design anomalies is property clumps. This problem occurs when ontologists badly design ontology by using repeated groups of properties in different class definitions. These groups should be replaced by an abstract concept composing those properties in all class definitions where this clump is used. To identify property clumps the following approach depicted in Definition \[def:property-clumps\] is used. \[Find property clumps\] $$\begin{aligned} &\textbf{findPropertyClumps} \\ &\quad price \leftarrow 1 \\ &\quad n_R \leftarrow \text{ Number of properties (datatype and object)} \\ &\quad V \leftarrow \text{ Classes and properties} \\ &\quad E \leftarrow \text{ Links between classes and properties} \\ &\quad propertyClumps = \{ \} \\ &\quad \text{while exist complete bipartite sub-graph } K_{m,n}^{'} \text{ of graph } G(V,E) \\ &\qquad \text{select } K_{m,n}^{''} \text{ from } K_{m,n}^{'} \text{, where } \max (\frac{m^{''} \cdot n^{''}}{m^{''} + n^{''}}) \\ &\qquad propertyClumps = propertyClumps \cup K_{m,n}^{''} \\ &\qquad \text{remove all edges from } G(V, E) \text{ that appear in } K_{m,n}^{''} \\ &\qquad price = price - \frac{ m^{''} \cdot n^{''} - (m^{''} + n^{''}) }{n_R} \\ &\quad \text{end while} \\ &\quad \text{return } \boldsymbol{propertyClumps} \text{ and } \boldsymbol{price} \\ &\textbf{end}\end{aligned}$$ The last aspect of design anomalies is lazy entities, which is a leaf class or property in the taxonomy that never appears in the application and does not have any instances. Eliminating this problem is quite straightforward; it just requires checking all leaf entities and verifying whether it contains any instances. In case of existence those entities should be removed or generalized or instances should be inserted. Evaluation ========== Method ------ The ROD process was evaluated on Financial Instruments and Trading Strategies (FITS) ontology that is depicted in Figure \[fig:FITS-ontology\]. ![Financial instruments and trading strategies (FITS)[]{data-label="fig:FITS-ontology"}](img/FITS-ontology){width="0.8\linewidth"} When building aforementioned ontology one of the requirements was to follow Semantic Web mantra of achieving as high level of reuse as possible. Therefore the main building blocks of FITS ontology are all common concepts about financial instruments. Furthermore every source of data (e.g., quotes from Yahoo! Finance in a form of CSV files and direct Web access, AmiBroker trading program format etc.) is encapsulated in a form of ontology and integrated into FITS ontology. Within every source of data developer can select which financial instrument is he interested in (e.g., `GOOG`, `AAPL`, `PCX`, `KRKG` etc.). The last and the most important component are financial trading strategies that developers can define. Every strategy was defined in its own ontology (e.g., `FI-Strategy-Simple`, `FI-Strategy-SMA`, `FI-Strategy-Japanese` etc.). The requirement was also to enable open integration of strategies, so developer can select best practices from several developers and add its own modification. Two different approaches in constructing ontology and using it in aforementioned use case were used. The approach of rapid ontology development (ROD) was compared to ad-hoc approach to ontology development, which was based on existing methodologies CommonKADS, OTK and METHONTOLOGY. With ROD approach the proposed method was used with tools IntelliOnto and Protégé. The entire development process was monitored by iteration, where ontology completeness price and number of ontology elements (classes, properties and axioms with rules) were followed. At the end the results included developed ontology, a functional component and information about the development process by iteration. The final version of ontology was reviewed by a domain expert, who confirmed adequateness. At implementation level ontology was expected to contain about $250$ to $350$ axioms of schematic part and numerous instances from various sources. Results and Discussion ---------------------- The process of ontology creation and exporting it as functional component was evaluated on FITS ontology and the results are depicted in Figures \[fig:OC-assessment-ROD\] and \[fig:OC-assessment-ad-hoc\]. Charts represent ontology completeness price and number of ontology elements regarding to iterations in the process. ![OC assessment and number of ontology elements through iterations and phases of ROD process[]{data-label="fig:OC-assessment-ROD"}](img/OC-assessment-ROD){width="1\linewidth"} ![OC assessment and number of ontology elements through iterations of ad-hoc development process[]{data-label="fig:OC-assessment-ad-hoc"}](img/OC-assessment-ad-hoc){width="1\linewidth"} Comparing ROD to ad-hoc approach the following conclusions can be drawn: - the number of iterations to develop required functional component using ROD approach $(30)$ is less than using ad-hoc approach $(37)$ which results in $23\%$ less iterations; - ontology developed with ROD approach is throughout the development process more complete and more appropriate for use than in ad-hoc, due to continuous evaluation and simultaneous alerts for developers. During the process of ontology construction based on ROD approach the developer was continuously supported by ontology evaluation and recommendations for progressing to next steps. When developer entered a phase and started performing tasks associated with the phase, ontology completeness was evaluated as depicted in Figure \[fig:OC-GUI\]. While OC was less than a threshold value, developer followed instructions for improving ontology as depicted in Figure \[fig:OC-calculation\]. Results of OC evaluation are available in a simple view, where basic statistics about ontology is displayed (number of concepts, properties, rules, individuals etc.), progress bar depicting completeness, and details about evaluation, improvement recommendations and history of changes. The core element is progress bar that denotes how complete ontology is and is accompanied with a percentage value. Following are recommendations for ontology improvement and their gains (e.g., remove circulatory errors $(+10\%)$, describe concepts in natural language $(+8\%)$, connect concepts $(+7\%)$ etc.). When improvement is selected (e.g., remove circulatory errors) the details are displayed (gain, task and details). The improvement and planned actions are also clearly graphically depicted on radar chart (see Figure \[fig:OC-GUI\]). The shaded area with strong border lines presents current situation, while red dot shows TO-BE situation if we follow selected improvement. When OC price crosses a threshold value (in this experiment $80\%$) a recommendation to progress to a new phase is generated. We can see from our example that for instance recommendation to progress from phase 2.5 to phase 2.6 was generated in 20th iteration with OC value of $91,3\%$, while in 19th iteration OC value was $76,5\%$. As Figure \[fig:OC-assessment-ROD\] depicts ontology completeness price and number of ontology elements are displayed. While progressing through steps and phases it’s seen that number of ontology elements constantly grow. On the other hand OC price fluctuate – it’s increasing till we reach the threshold to progress to next phase and decreases when entering new phase. Based on recommendations from the system, developer improves the ontology and OC price increases again. With introduction of OC steps in ontology development are constantly measured while enabling developers to focus on content and not technical details (e.g. language syntax, best modeling approach etc.). Conclusions and Future work {#conclusion-and-future-work} =========================== Current methodologies and approaches for ontology development require very experienced users and developers, while we propose ROD approach that is more suitable for less technically oriented users. With constant evaluation of developed ontology that is introduced in this approach, developers get a tool for construction of ontologies with several advantages: - the required technical knowledge for ontology modeling is decreased, - the process of ontology modeling doesn’t end with the last successful iteration, but continues with post-development activities of using ontology as a functional component in several scenarios and - continuous evaluation of developing ontology and recommendations for improvement. In ontology evaluation several components are considered: description, partition, redundancy, consistency and anomaly. Description of ontology’s components is very important aspect mainly in early stages of ontology development and includes existence of entities, natural language descriptions and formal descriptions. This data is furthermore used for advanced axiom construction in latter stages. Partition errors deal with omitting important axioms and can be in a form of common classes, external instances, hierarchy of entities etc. Redundancy deals with multiple information being inferred more than once and includes identical formal definition and redundancy in hierarchy. With consistency the emphasis is on finding circulatory errors, while anomalies do not cause inaccurate reasoning about concepts, but point to badly designed areas in ontology. This includes checking for chain of inheritance, property clumps, lazy entities etc. It has been demonstrated on a case study from financial trading domain that a developer can build Semantic Web application for financial trading based on ontologies that consumes data from various sources and enable interoperability. The solution can easily be packed into a functional component and used in various systems. 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ArXiv

--- abstract: 'We present a framework for obtaining reliable solid-state charge and optical excitations and spectra from optimally-tuned range-separated hybrid density functional theory. , allows for accurate prediction of exciton binding energies. We demonstrate our approach through calculations of one- and two-particle excitations in pentacene, a molecular semiconducting crystal, where our work is in excellent agreement with experiments and prior computations. We further show that with one adjustable parameter, , this method accurately predicts band structures and optical spectra of silicon and lithium flouride, prototypical covalent and ionic solids. Our findings indicate that for a broad range of extended bulk systems, this method may provide a computationally inexpensive alternative to many-body perturbation theory, opening the door to studies of materials of increasing size and complexity.' author: - 'Sivan Refaely-Abramson' - Manish Jain - Sahar Sharifzadeh - 'Jeffrey B. Neaton' - Leeor Kronik bibliography: - 'tdrsh.bib' title: | Solid-state optical absorption from optimally-tuned time-dependent\ range-separated hybrid density functional theory --- Many solid-state systems exhibit strong excitonic effects, notably an optical excitation spectrum that is affected substantially by interaction between excited electron and hole quasiparticle states. The nature of this electron-hole, or [*excitonic*]{}, interaction is of central importance for a variety of applications in, e.g., optoelectronics and photovoltaics [@Savoie2014]. Nevertheless, its accurate theoretical prediction remains a challenging task. It is common to account for such interactions within the framework of ab initio many-body perturbation theory, in which single-particle excitations are well-predicted from Dyson’s equation, typically solved within the GW approximation [@Hedin1965; @Hybertsen1986], and two-particle excitations are well-predicted using the Bethe-Salpeter equation (BSE) [@Rohlfing1998; @*Rohlfing2000; @Strinati1982; @*Strinati1984]. Current GW-BSE calculations are highly demanding and therefore presently impose significant practical limits on the calculated system size and complexity. Density functional theory (DFT), in both its time-independent [@DreizlerGross; @ParrYang] and time-dependent (TDDFT) [@Marques2012; @Casida1995; @Burke2005; @Ullrich_book] forms, is considerably more efficient computationally. However, common (semi-)local approximations to both DFT and TDDFT suffer from serious deficiencies which have precluded their use as a viable alternative to GW-BSE in the prediction of excitonic properties [@Onida2002]. First, quasi-hole and quasi-electron excitation energies are generally underestimated and overestimated, respectively, by the DFT Kohn-Sham eigenvalue spectrum [@Kummel2008; @Kronik2012]. While the same functionals often perform better in the prediction of optical excitation energies of isolated molecular systems, the Kohn-Sham gap is typically similar to the optical gap [@Salzner1997; @Chong2002; @Baerends2013; @Kronik2012; @Kronik2014]. In any case, they still fail in the solid-state limit [@Onida2002; @Ullrich_book; @Izmaylov2008; @Sharma2014; @Ullrich2014]. Therefore, neither one- nor two-particle excitations are well-predicted in the solid-state, and hence the nature of excitons or their binding energies are not obtained. The failure of semi-local functionals in predicting solid-state absorption spectra has been traced back to an incorrect description of the long-range electron-electron and electron-hole interaction, manifested by the absence of a $1/q^2$ contribution [@Gonze1995; @*Ghosez1997; @Kim2002] to the interaction, where $q$ is a wavevector in the periodic system. Several ingenious schemes for overcoming this deficiency have been suggested, including the use of an exchange-correlation kernel of the form $f_{xc}(r,r')=-\alpha/(4\pi |r-r'|)$, where $\alpha$ is a system-dependent empirical parameter [@Reining2002; @Botti2004]; a static approximation to the exchange-correlation kernel based on a jellium-with-gap model [@Trevisanutto2013]; a “bootstrap” parameter-free kernel, achieved using self-consistent iterations of the random phase approximation (RPA) dielectric function [@Sharma2011; @Sharma2014]; a related “guided iteration” RPA-bootstrap kernel [@Rigamonti2015]; and the Nanoquanta kernel [@Onida2002; @Reining2002; @Sottile2003; @Marini2003; @*Adragna2003], derived by constructing the exchange-correlation kernel from an approximate solution to the BSE. Each correction provides a major step forward. However, none is a fully DFT-based solution, as single quasiparticle excitations are obtained from GW, RPA, a DFT+U approach, or a scissors-shift correction. A different path for enabling TDDFT calculations in the solid state is the use of (global or range-separated) hybrid functionals. These are still well within density functional theory, using the generalized Kohn-Sham (GKS) framework [@Seidl1996; @Kummel2008; @Kronik2012], and their non-local Fock-like exchange component assists in the inclusion of long-range contributions. Although the time-dependent GKS equations have yet to be formally derived, hybrid functionals are already widely used for calculating optical properties. For gas-phase molecules, hybrid functionals can improve optical excitation energies, although standard hybrids still do not provide for accurate single-particle excitation energies [@Salzner1997; @Kummel2008; @Kronik2012; @Kronik2014]. TDDFT using the Heyd-Scuseria-Ernzerhof (HSE) short-range hybrid functional [@Heyd2003; @*Heyd2006], where non-local exchange is introduced only in the short-range, can improve the absorption spectra of semiconductors and insulators [@Paier2008], although some discrepancies remain. However, the HSE functional still does not provide the desired long-range non-local contribution. The B3LYP hybrid functional [@Becke1993; @*Stephens1994], in which a global 20% fraction of exact-exchange is used, was recently shown to yield TDDFT optical spectra for semiconductors in good agreement with experiment [@Bernasconi2011; @*Tomic2014], . Although in this case a non-local contribution to the kernel tail does exist, it is global and parameterized for a finite set of small organic molecules. However, global and short-range hybrid functionals were shown to be insufficient predictors of band-structures in solid-state systems [@Jain2011], notably for molecular crystals [@Refaely-Abramson2013] where excitonic effects are strong. Recently, Yang et al. [@Yang2015] suggested a screened exact-exchange (SXX) approach, in which the local part of the hybrid calculation is set to zero and the time-dependent Hartree-Fock exchange is scaled down non-empirically per system by using the inverse of the dielectric constant, based on a ground state obtained from a scissor-corrected local density approximation (LDA) calculation. Again, this led to improved performance for more strongly bound excitons. ![(Color online) (a) Band-structure (left) and density of states (right) of the pentacene solid, calculated using LDA (gray, dashed lines), G$_0$W$_0$@LDA (red, dashed lines), and OT-SRSH (black, solid lines). For all methods, the middle of the bandgap is shifted to zero. (b) The imaginary part of the dielectric function of the pantacene solid, with incident light polarization averaged over the $a$, $b$, and $c$ main unit-cell axes, calculated using TDLDA (gray, dashed lines), G$_0$W$_0$/BSE (red, dashed lines), and TD-OT-SRSH (black, solid lines). For visualization purposes, the leading absorption feature (between 0.5 to 2.5 eV) was multiplied by a factor of 10 with all computational methods used. The OT-SRSH and TD-OT-SRSH results were obtained using the parameters $\gamma=0.16$ Bohr$^{-1}$, $\alpha=0.2$, and $\varepsilon=3.6$. For computational details and convergence information, see the SI.[]{data-label="fig_pen"}](pen_DOS_abs_2h-averageLS-wan.pdf) Ideally, we seek a DFT-based method where accurate one- and two-particle excitations can be read directly off of the eigenvalues of the time-independent (G)KS and the linear-response time-dependent (G)KS equations, respectively, using a single exchange-correlation functional, from which a consistent exchange-correlation potential and kernel are derived. This challenge is not met by any of the above-surveyed methods. Recently, it was met for gas-phase systems, using the optimally-tuned range separated hybrid functional (OT-RSH) approach [@Baer2010; @Stein2010; @Kronik2012], where the long- and short- range fraction of Fock-exchange is tuned non-empirically so as to obey rigorous physical constraints. This approach, elaborated further below, was shown to yield excellent fundamental and optical gaps for molecules [@Refaely-Abramson2011; @RSHpapers]. More recently, it has been generalized so as to provide accurate single-particle excitations for both molecules [@Refaely-Abramson2012; @Egger2014; @Nguyen2015] and molecular crystals [@Refaely-Abramson2013; @Lueftner2014], and it was shown to capture gap renormalization in molecular solids [@Refaely-Abramson2013]. Can this approach, then, resolve the long-standing challenge of providing an accurate one- and two-particle excitation spectrum in solid-state systems fully within the framework of (TD)DFT? In this Letter, we present a solid-state OT-RSH approach that achieves just that. . We non-empirical calculations for pentacene, a prototypical molecular crystal provides excellent agreement with GW-BSE calculations. Furthermore, with one empirical parameter - set to reproduce the known fundamental gap - we again achieve results that are comparable with both GW-BSE and experiment for bulk silicon and LiF. The approach therefore emerges as promising for photoelectron and optical properties accurately and efficiently for a broad range of extended systems. In the range-separated hybrid approach, the Coulomb interaction is range-partitioned [@Leininger1997], e.g., via: [@Yanai2004] $$\frac{1}{r}=\frac{\alpha+\beta\mathrm{erf}(\gamma r)}{r}+\frac{1-[\alpha+\beta\mathrm{erf}(\gamma r)]}{r}, \label{alphabetagamma}$$ where $\alpha$, $\beta$ and $\gamma$ are parameters, and $r$ is the inter-electron coordinate. The exchange expression corresponding to the first term on the right-hand side of Eq. (\[alphabetagamma\]) is then treated as in Hartree-Fock theory; the exchange expression corresponding to the second term is treated within the Kohn-Sham framework, typically using the LDA or the generalized gradient approximation (GGA). In Eq. (\[alphabetagamma\]), $\gamma$ is the range-separation parameter, i.e., it controls the range at which each of the terms dominates; $\alpha$ and $\beta$ dictate the limiting behavior of the Fock-like exchange, which tends to $\alpha/r$ for $r \rightarrow 0$ and to $(\alpha + \beta )/r$ for $r \rightarrow \infty$. The resulting exchange-correlation energy is of the form: $$\begin{split} E_{xc}^{RSH}=(1-\alpha)E_{\mathrm{KS}x}^{SR}+\alpha E_{xx}^{SR}+ (1-(\alpha+\beta))E_{\mathrm{KS}x}^{LR}+\\ (\alpha+\beta)E_{xx}^{LR}+E_{\mathrm{KS}c}, \label{SRSH} \end{split}$$ where $\mathrm{KS}x$ and $\mathrm{KS}c$ denote (semi-)local KS exchange and correlation respectively, and $xx$ is a Fock-like exchange. SR and LR label short- and long-range terms, in which the Coulomb interaction is scaled using error functions. Within the GKS framework, the potential corresponding to the (semi-)local energy components is then obtained as a functional derivative, whereas the potential corresponding to the $xx$ energy components is obtained as a non-local Fock-like operator. Here we generalize this approach to the time-dependent case, within the usual linear-response formalism of Casida [@Casida1995; @Tretiak2003] and the Tamm-Dancoff approximation. This is achieved by coupling GKS electron-hole pairs via an exchange-correlation kernel. This leads to an eigenvalue equation in which the eigenvalues are related to optical excitation energies and the eigenvectors can be used to compute oscillator strengths, so that the complete optical absorption spectrum can be computed. The part of the TDDFT kernel originating from the Hartree and (semi-)local Kohn-Sham potential in the ground-state DFT can be expressed as: $${\langle ai|}\left[\frac{1}{|r-r'|}+(1-\alpha)f_{xc}^{SR}+(1-\alpha-\beta)f_{xc}^{LR}\right]{|bj\rangle} \label{ker-l}$$ where $f_{xc}^{SR}=\frac{\delta V_{xc}^{SR}}{\delta n(r)}\delta(r-r')$, $f_{xc}^{LR}=\frac{\delta V_{xc}^{LR}}{\delta n(r)}\delta(r-r')$, and $V_{xc}^{SR},V_{xc}^{LR}$ are the short- and long-range contributions of the (semi-)local KS exchange-correlation potential; and where $a,b$ and $i,j$ denote occupied and unoccupied states, respectively. The non-local exchange potential in the ground-state GKS leads to an additional term in the kernel, of the form $$-{\langle ab|}\left[ \alpha\frac{\mathrm{erfc(\gamma(|r-r'|)}}{|r-r'|}+(\alpha+\beta)\frac{\mathrm{erf(\gamma(|r-r'|)}}{|r-r'|} \right] {|ij\rangle} \label{ker-nl}$$ (see supplementary information (SI) for additional formal details). As discussed above, optimal-tuning of the RSH parameters was shown to be crucial for achieving accurate description of molecular single-particle and optical excitations. This tuning procedure is challenging in the solid-state, as it involves a calculation of the system’s ionization potential and electron affinity from total energy differences, a problematic procedure for periodic systems (see [@Vlcek2015], and references therein). For the special case of molecular crystals, however, it was shown [@Refaely-Abramson2013] that predictive bandstructures can be achieved if $\gamma$ and $\alpha$ are optimally-tuned so as to obey the ionization potential theorem for an isolated molecule, with $\beta$ chosen such that $\alpha + \beta = 1/\varepsilon_0$, where $\varepsilon_0$ is the scalar dielectric constant, itself computed from first principles. We refer to this procedure as the optimally-tuned [*screened*]{}-RSH (OT-SRSH) approach. To test the efficacy of this approach for optical properties, we examine pentacene, a molecular semiconducting crystal of extreme interest in organic electronics and photovoltaics. For pentacene, the optimal tuning parameters are found to be $\alpha=0.2$, $\beta=0.08$ (corresponding to $\varepsilon_0=3.6$), and $\gamma=0.16$ Bohr$^{-1}$ [@Refaely-Abramson2013]. With these parameters we construct the appropriate time-independent (Eq. (\[SRSH\])) and time-dependent (Eqs. (\[ker-l\]),(\[ker-nl\])) equations. These are then solved with the PARATEC [@Ihm1979] and BerkeleyGW [@Deslippe2012] codes, which we modified to handle range-separated hybrids (see SI for a complete discussion).  Figure \[fig\_pen\] (a) shows the resulting band structure and density of states (DOS) calculated from LDA, OT-SRSH, and GW eigenvalues. The band gap predicted by OT-SRSH is 1.9 eV, in good agreement with the GW gap of 2.1 eV. As expected, both improve drastically on the LDA gap, which is 0.7 eV. Save for the slight difference in gap values, the OT-SRSH and GW band-structure and DOS are remarkably similar, demonstrating for this system that the eigenvalues of the OT-SRSH method are quantitatively useful approximations to single-quasiparticle excitation energies, as they are close to GW quasiparticle energies for several eV away from the band edges. Figure \[fig\_pen\] (b) presents the resulting imaginary part of the dielectric function, Im($\varepsilon$), calculated using TDLDA, TD-OT-SRSH and BSE, with the response averaged over incident light polarization along the $a$, $b$ and $c$ directions of the pentacene lattice (see SI for additional details). Importantly, our TD-OT-SRSH results do as well as GW-BSE in providing the expected picture of excitonic binding, i.e., comparison of the results with and without electron-hole interactions suggests an exciton binding energy of 0.45 eV from both methods. Furthermore, the BSE result agrees well with previously reported ones, e.g. those of Refs. [@Tiago2003; @Cudazzo2012; @Sharifzadeh2013; @*Sharifzadeh2012-2]. The first singlet excitation is predicted to be at 1.46 eV in TD-OT-SRSH and 1.64 eV in BSE. This small quantitative difference is essentially within the expected accuracy of either calculation. Moreover, this difference is consistent with that computed at the single-particle excitation level, and likely is inherited from it. In a similar manner, all presented optical excitation energies resulting from TD-OT-SRSH and BSE are very similar, with remaining differences within the desired accuracy. However, there are differences in the oscillator strength in part of the spectrum, notably at the first peak between 1.5 eV and 2.5 eV. These differences are primarily associated with the $a$-axis direction of incident light polarization. However, the scale of these differences is small when compared to the entire spectrum. (Note that this spectral range is enhanced by a factor of 10 in Fig.\[fig\_pen\](b) to be visible.) Overall, then, the TD-OT-SRSH spectrum, while not identical to that of BSE, is in very good agreement with it. We now turn to two prototypical covalent and ionic bulk solids: Si and LiF, respectively. For these systems, we cannot use the original tuning procedure, because it was designed for systems where inter-molecular hybridization is small. However, it still possible to set $\alpha=0.2$ as a universally useful amount of short-range exact exchange [@Rohrdanz2009; @Egger2014] and demand $\alpha+\beta=1/\varepsilon_0$, as before. We are then left with only one parameter, the range-separation parameter, $\gamma$. Here, we simply choose $\gamma$ so as to obtain the fundamental gap. . As shown in the top panel of Fig \[fig\_si\](a), the SRSH Si bandstructure is fully comparable to the canonical empirical pseudopotential work of Chelikowsky and Cohen [@CC1976], as well as to GW results, whereas with LDA the gap, as expected, is too small. From the bottom panel we see that, in agreement with previous work [@Reining2002; @Botti2004], the TDLDA spectrum does not reproduce experiment satisfactorily. However, once again the TDSRSH result does almost as well as GW-BSE in predicting the experimental results, without any scissor shift or other correction operators at the single-quasiparticle level. Qualitatively, the TDSRSH lineshape is indistinguishable from that of GW-BSE. Quantitatively, the peak positions are similar to within 0.05 eV. A similar picture emerges for LiF, as both single-particle and two-particle excitations are very close to those of GW/BSE. Specifically, both TDSRSH and BSE predict the first and largest excitation at 12.6 eV, very close to the experimental value at 12.75 eV, and the overall spectral shape is satisfying. Here too, then, SRSH and TDSRSH are shown to capture excitonic effects, even though the Si excitons are much more delocalized and weakly bound compared to the case of pentacene, and the LiF excitons are strongly-bound and known to be highly challenging for many of the TDDFT methods discussed in the introduction [@Rigamonti2015; @Sharma2014; @Yang2015; @Trevisanutto2013]. What are the physical origins of this success of the TDSRSH approach in the solid state? First, the ground-state calculation, being a [*generalized*]{} Kohn-Sham one, is capable in principle of describing quasi-particle excitations owing to the non-local potential operator. Range-separation combined with long-range dielectric screening allows us to fulfill asymptotic potential constraints while retaining the crucial balance of short-range exchange and correlation components, thereby making such prediction sufficiently accurate also in practice. Second, the asymptotic form of the TDSRSH kernel generates the desired non-local $1/q^2$ contribution by construction. Furthermore, it is already scaled correctly via the non-empirical $1/\varepsilon_0$ parameter. Third, the non-locality of the TDSRSH kernel also alleviates the need for frequency-dependence, which would be necessary for bound exciton prediction with (semi-)local exchange-correlation kernels [@Sharma2014]. For these reasons, highly accurate single-quasiparticle and two-quasiparticle properties of extended systems can be obtained in a predictive manner within the unifying framework of DFT at computationally-modest cost. Last but not least, Onida [*et al.*]{} [@Onida2002] have already noted some time ago that “both the Green‘s functions and the TDDFT approaches profit from mutual insight.’’ Here, we believe we achieve an important milestone towards that vision: on the one hand, the above work can be justified entirely from (TD)DFT reasoning. On the other hand, it is clear that the work has been motivated by the need to achieve the elegant quasi-particle picture obtained so naturally within many-body perturbation theory and that by achieving this goal we have created an effective simplified framework mimicking this picture. In conclusion, we have presented a new approach for quantitative determination of single- and two-particle excitations in solids, based on range-separated hybrid density functional theory. The approach that are in quantitative agreement with those of many-body perturbation theory in the GW-BSE approximation, for three prototypical systems: Si, a covalent solid, LiF, an ionic solid, and pentacene, a molecular solid. In particular, it fully captures excitonic interactions for both strongly and weakly bound excitons. We envision that it could emerge as a useful low-cost substitute to many-body perturbation theory, as well as provoke further developments within density functional theory. This work was supported by the European Research Council, the Israel Science Foundation, the United States-Israel Binational Science Foundation, the Helmsley Foundation, and the Wolfson Foundation. S.R.A. is supported by an Adams fellowship of the Israel Academy of Sciences and Humanities. S.S. was partially supported by the Scientific Discovery through Advanced Computing (SciDAC) Partnership program funded by US Department of Energy, Office of Science, Advanced Scientific Computing Research and Basic Energy Sciences. J.B.N. was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering (Theory FWP) under Contract No. DE-AC02- 05CH11231. The work performed at the Molecular Foundry was also supported by the Office of Science, Office of Basic Energy Sciences, of the US Department of Energy. We thank the National Energy Research Scientific Computing center for computational resources.

ArXiv

--- author: - 'D. Streich' - 'R. S. de Jong' - 'J. Bailin' - 'P. Goudfrooij' - 'D. Radburn-Smith' - 'M. Vlajic' bibliography: - '../../bibliography.bib' date: 'Received .../ Accepted ...' title: 'On the relation between metallicity and RGB color in HST/ACS data' --- Introduction ============ Measuring the metallicity and its gradients in galaxies is a key issue for understanding galaxy formation and evolution. Outside the Local Group, spectroscopic metallicity determination of (resolved) stars is not feasible at the moment, except for the few very bright supergiants [@kudritzki12]. At the same time, the number of available color magnitude diagrams (CMDs) of nearby galaxies is increasing rapidly, e.g. from the GHOSTS [@radburn11] and ANGST [@dalcanton09] surveys. These CMDs can be used to derive metallicities. The color of the red giant branch (RGB) has long been known to depend on the metallicity [@hoyle55; @sandage66; @demarque82]. This has been extensively used to measure metallicities of old populations. There are many ways to convert the color to a metallicity: some authors define indices of an observed population, e.g. the color of the RGB at a given magnitude or the slope of the RGB, [as defined for example in @dacosta90; @lee93; @saviane00; @valenti04], while others measured metallicities on a star by star basis by interpolating between either globular cluster fiducial lines [e.g. @tanaka10; @tiede04] or analytic RGB functions [calibrated with globular cluster data, e.g. @zoccali03; @gullieuszik07; @held10] or stellar evolution models [e.g. @richardson09; @babusiaux05], and therefore generating metallicity distribution functions for a population. Uncertainties that arise from a specific calibration or a given isochrone or cluster template set are typically not well studied. Furthermore, an observational relation for the widely used HST ACS filters F606W-F814W is still missing in the literature[^1]. Here we aim to address these shortcomings. This paper is organized as follows: After an introduction to the data and isochrones we use in chapter \[data\], an observational color metallicity relation is derived in chapter \[results\]. A discussion and summary follow in chapters \[discussion\] and \[summary\]. Data and Isochrones {#data} =================== In this work we use the data of 71 globular clusters observed as part of the ACS Globular Cluster Survey [ACSGCS; @sarajedini07] and its extension [@dotter11]. These data contain photometry in the F606W and F814W filters and is publicly available at the homepage of the ACSGCS team[^2]. For the determination of photometric uncertainties and completeness, the results from artificial star tests are also available. A detailed description of the data reduction is given in @anderson08. To compare the different clusters, it is necessary to transform the apparent magnitudes into absolute, reddening-free magnitudes. For this purpose, we use the distance modulus and color excess from the GC database of W. Harris [@harris96; @harris10] and the extinction ratios for the ACS filters given by @sirianni05 [Table 14]. Metallicities, metallicity uncertainties and $\alpha$-abundances are taken from @carretta09 [@carretta10], if not stated otherwise. In order to measure the color of the clusters RGBs they must have a sufficient number of stars in the RGB region. We selected therefore only those clusters for our study, which have more than five stars brighter than $M_{F814W}=-2$ and least one star brighter than $M_{F814W}=-3$. A list of the clusters used is given in Appendix \[app:properties\] (Table \[GC\_literature\]). For comparison with theoretical models, we use four sets of isochrones: the new PARSEC isochrones [@bressan12] and their predecessors, the (old) Padua isochrones[^3] [@girardi10; @marigo08 and references therein], the BaSTI isochrones[^4] [@pietrinferni06; @pietrinferni04] and the Dartmouth isochrones[^5] [@dotter07]. Results ======= Color measurement {#sec:colormeasurement} ----------------- We use two indices to define the color of the RGB: $C_{-3.0}=(F606W-F814W)_{M=-3.0}$ and $C_{-3.5}=(F606W-F814W)_{M=-3.5}$, i.e. the color of the RGB at an absolute F814W magnitude of -3.0 and -3.5, respectively (see Fig. \[CMD\_diverse\] for some typical CMDs). Equivalent indices for the Johnson-Cousins filter system were already used by @dacosta90, @lee93 and also by @saviane00. These indices have the advantage of only depending on relatively bright stars and can therefore be measured in distant galaxies, as well. We use also a third index, the S-index, which is the slope of the RGB [@saviane00; @hartwick68]. This slope is measured between two points of the RGB, one at the level of the horizontal branch and the other two magnitudes brighter. While this index needs deeper data, and therefore its usage in extragalactic systems is limited, it has the advantage of being independent of extinction and distance errors. In order to provide a robust measurement of the color at a given magnitude we interpolated the RGB with a hyperbola of the form: $$M=a+b\cdot \textrm{color}+c/(\textrm{color}+d)$$ Such a function was already used by @saviane00 to find a one-parameter representation of the RGB; they defined the parameters $a$, $b$, $c$, and $d$ as a quadratic function of metallicity. Here, we are only interested in a good interpolation in sparse parts of the RGB and can therefore use $a$, $b$, $c$, and $d$ as free parameters for each cluster. In order to reduce problems due to contamination, we define a region of probable RGB stars, which also excludes the horizontal branch/red clump part of the CMD. Note that we fit the curve directly to the color/magnitude points of the stars and not to the ridge line of the RGB [in contrast to @saviane00]. More details of the fitting process and some example plots with the exclusion region are shown in the Appendix. To calculate the S-index, we first determined the horizontal branch magnitude of each system by visual inspection of the associated CMDs. This was typically F606W$\approx$0.40mag, with a 1-sigma variation of 0.10mag. We measured the color at this magnitude (and at 2 magnitudes brighter) from the fitted RGB used previously, and calculated the S-index as the slope between these points Metallicity determination ------------------------- The iron abundance \[Fe/H\] is often used synonymously with metallicity. However, from the theoretical point of view of stellar evolution, all elements are important in determining the properties of stellar atmospheres. Therefore the color of red giants is expected to depend on the overall metallicity \[M/H\] rather than on \[Fe/H\]. Unfortunately, there are very few measurements of the abundances of other elements in globular clusters. We use here the abundances given in @carretta10, who have measured \[Fe/H\] for all GCs in our sample and have compiled \[$\alpha$/Fe\] values for many of them. According to @salaris93, these two measurements can be combined to get the overall metallicity with the formula $$[M/H] = [Fe/H] + \log_{10}(0.638*10^{[\alpha/Fe]}+0.362) .$$ For clusters that have no individual $\alpha$ measurement, we had to estimate its $\alpha$ abundance. Since the spread of \[$\alpha$/Fe\] among globular clusters is rather small, such an estimate will only introduce small errors. In Fig. \[alphas\], \[$\alpha$/Fe\] is plotted against \[Fe/H\], where we have assumed an uncertainty of $0.05$ in the $\alpha$ abundance. The straight line is a linear regression, which we use for the estimation of \[$\alpha$/Fe\], where it is not available. The scatter around this regression line is 0.1dex, which we adopt as the individual uncertainty in the estimated \[$\alpha$/Fe\]. ![Alpha abundance as a function of \[Fe/H\] for all clusters in @carretta10. The text in the lower left corner gives the formula of the regression line and the scatter around this line. We used these for estimating the \[$\alpha$/Fe\] and its uncertainty for clusters without individual alpha measurement.[]{data-label="alphas"}](alpha-FeH_carretta){width="0.99\columnwidth"} Uncertainties ------------- To determine the uncertainties of our color measurements, we performed a bootstrap analysis. The uncertainty in the fit is derived by fitting the RGB of 500 samples that are drawn randomly from the original data. Each re-sample has the same number of stars as the original sample, but may contain some stars multiple times while others are absent. We also incorporated in the bootstraps a shift due to the uncertainties in extinction and distance. According to @harris10, the uncertainty in extinction is of the order 10% in E(B-V), but is at least 0.01mag, while the uncertainty in distance modulus is 0.1mag. The uncertainty in distance is important because we measure the color at a given absolute magnitude. This is particularly significant for the metal-rich clusters where the color of the RGB is strongly dependent on magnitude, as opposed to metal-poor clusters where the RGB is nearly vertical on a CMD. The resulting uncertainty in $C_{-3.5}$ ranges from approximately 0.01mag at \[M/H\]=-2 to approximately 0.1mag at \[M/H\]=-0.2. The uncertainties in the metallicity are the sum of the uncertainties in \[Fe/H\] @carretta09, and in \[$\alpha$/Fe\], which we adopt as 0.05dex for clusters with individual alpha-abundance measurements and 0.1dex for clusters with estimated values. Color metallicity relation -------------------------- Using the colors and metallicities described above, we can now look at the color-metallicity relations. The results are shown in Fig. \[metalfitindices\]. There is a clear relation between RGB color and spectroscopic metallicity. This relation can be parametrized with the function $F606W-F814W=a_0\exp(\mathrm{[M/H]}/a_1)+a_2.$ Using the orthogonal distance regression (ODR) algorithm [@boggs87; @boggs92][^6], we determined the three parameters, that are shown in Table \[fitparams\]. The ODR uses the uncertainties on both variables to determine the best fit. Hence, both the uncertainties in color and metallicity, as described above, are considered during the fit and their effects are included in the final uncertainties of the resulting fit parameters. The residual varinaces for both relations are $\sigma_{res}^2<1$, so the adopted uncertainties can explain the observed scatter in the relations. $a_0$ $a_1$ $a_2$ ------------ ----------------- ----------------- ----------------- $C_{-3.5}$ $0.95\pm0.11$ $0.602\pm0.069$ $0.920\pm0.015$ $C_{-3.0}$ $0.567\pm0.056$ $0.75\pm0.12$ $0.845\pm0.018$ S-index $3.67\pm0.76$ $-9.3\pm1.2$ $-2.08\pm0.44$ : Fit parameters of the color–metallicity relations. For $C_{-3.5}$ and $C_{-3.0}$ the relation is exponential: $C_{i}=a_0\exp(\mathrm{[M/H]}/a_1)+a_2$, for the S-index it is linear $S=a_0+a_1\mathrm{[M/H]}$. []{data-label="fitparams"} {width="99.00000%"} The S-index ----------- The slope of the RGB as a function of metallicity can be seen in Fig. \[s\_index\]. ![*top panel:* The RGB slope as a function of metallicity. The solid black line is the best-fit quadratic function, as given in the equation in the bottom left. Grey lines give the $1\sigma$, $2\sigma$ and $3\sigma$ confidence ranges of the the best fit relation. *Bottom panel:* weighted orthogonal residuals, i.e. the orthogonal distances to the best fit line divided by the respective uncertainties. []{data-label="s_index"}](s_index){width="0.99\columnwidth"} The reported uncertainties of the S-index are a combination of the uncertainties of the RGB fit (determined through a bootstrap analysis as described above) and the uncertainty in the determination of the HB level, which we set here to $\sigma_{HBmag}=0.1$mag. As expected, the slope of the RGB gets smaller with increasing metallicity, while at the low-metallicity end the RGB slope is insensitive to metallicity. We have fitted a quadratic function to the data, which is shown in Fig. \[s\_index\] together with the associated best-fit parameters. The choice of a quadratic function for the fit proves to be appropriate as no trends are seen in the residuals. Moreover, the variance of the residuals is only $\sigma_{res}^2=1.17$, i.e. the residuals are only slightly larger than expected from the individual measurement uncertainties. The maximum of the parabola is at \[M/H\]=-2.14, which is beyond the metallicity range of the observed clusters. Discussion ========== Analyzing residuals ------------------- We examine the residuals to look for a possible second parameter that influences the color or slope of the RGB and could produce some scatter in a simple color-metallicity relation. Figures \[residuals1\] and \[residuals2\] show the residuals of the fit of the color-metallicity relation, that is shown in Fig. \[metalfitindices\], and Figures \[S\_residuals1\] and \[S\_residuals2\] the residuals of the fit to the slope metallicity relation, that is shown in Fig. \[s\_index\]. ![Residuals of the fit of the color-metallicity relation as function of metallicity (upper panel), iron abundance (middle panel) and alpha enhancement (lower panel). Symbols and colors are as in Fig. \[metalfitindices\].[]{data-label="residuals1"}](residuals_metals){width="0.99\columnwidth"} ![Residuals of the fit of the color-metallicity relation as function of age [upper panel, @marinfranch09], extinction (middle panel) and galactic latitude (lower panel). Symbols and colors are as in Fig. \[metalfitindices\].[]{data-label="residuals2"}](residuals_age_pos){width="0.99\columnwidth"} ![Residuals of the fit of the slope-metallicity relation as function of metallicity (upper panel), iron abundance (middle panel) and alpha enhancement (lower panel).[]{data-label="S_residuals1"}](S_residuals_metals){width="0.99\columnwidth"} ![Residuals of the fit of the slope-metallicity relation as function of age [upper panel, @marinfranch09], extinction (middle panel) and galactic latitude (lower panel).[]{data-label="S_residuals2"}](S_residuals_agepos){width="0.99\columnwidth"} The residuals as a function of metallicity, \[Fe/H\] and \[$\alpha$/Fe\], are shown in Fig. \[residuals1\] and Fig. \[S\_residuals1\]. There is no trend with any of these parameters, neither in the color- nor slope-metallicity relations. From theoretical studies, the age is known to have an effect on the color of the RGB. In fact, a weak trend of the residuals with age can be seen in Figure \[residuals2\] (upper panel), with older clusters being slightly redder than younger clusters. The slope of the regression lines shown there is $1.94\pm2.13$ for $C_{-3.5}$ and $3.00\pm1.85$ for $C_{-3.0}$, which makes the trend significant for the $C_{-3.0}$ index. In order to quantify the effects of age on the CMD, we analyse the residuals in color space[^7] in Figure \[color\_residuals\]. ![Color residuals of the fit of the color-metallicity relation as function of age. The text gives the regression line formulas for both indices.[]{data-label="color_residuals"}](color_residuals){width="0.99\columnwidth"} Assuming a typical age for globular clusters of 12.8Gyr (as @marinfranch09 do using the isochrones of @dotter07) we can transform the slopes of the regression lines in Figure \[color\_residuals\] to actual color changes. These are $0.0062\pm0.0041$mag/Gyr for $C_{-3.5}$ and $0.0078\pm0.0026$mag/Gyr for $C_{-3.0}$. While this is a very small effect for the age range observed in our globular clusters (10Gyr to 14Gyr), it can make significant differences when extrapolated to younger populations; e.g. an 8Gyr old population would be bluer than predicted by our relation by about 0.03mag. Note also that the S index does not show any systematic trends with age. To test for problems with the extinction values, we looked for trends with E(B-V) and galactic latitude (Fig. \[residuals2\] and Fig. \[S\_residuals2\], middle and lower panel). We do not find any systematics here. Comparison ---------- We can compare our relations to those derived from stellar evolution models, and to relations from ground-based data transformed to the HST/ACS filter systems. For comparison with theoretical relations, we use the isochrone set from the Padua, Dartmouth and BaSTI groups. For all isochrone sets we used ages of 8Gyr, 10Gyr and 13Gyr. For Dartmouth, we use $\alpha$-enhancements of $[\alpha/Fe]=\{0.0,0.2,0.4\}$ and for BaSTI models $[\alpha/Fe]=\{0.0,0.4\}$. The PARSEC[^8] and Padua isochrones are available only with solar scaled abundances. {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} All these isochrone sets show a qualitatively similar behavior. The RGB gets redder (Fig. \[relations\]) and shallower (Fig. \[relations\_sindex\]) with increasing metallicity. A higher age also leads to a redder RGB, but this effect is relatively small. An age difference of 5Gyr causes the same color difference as a metallicity difference of only 0.1dex (see Fig. \[relations\], top row). At a given total metallicity \[M/H\], the $\alpha$-abundance has almost no effect on the RGB color (Fig. \[relations\], middle row). This supports the assumption that the color of the RGB is mainly influenced by \[M/H\] and not \[Fe/H\]. The increasing curvature of the RGB with increasing metallicity prevents the RGB of some metal rich clusters from reaching F814W$=-3.5$mag, but bend down at fainter magnitudes. In the Dartmouth models this applies for isochrones with \[M/H\]$>-0.4$, in Padua models isochrones with \[M/H\]$>-0.3$. However, the BaSTI RGB isochrones all reach F814W$=-3.5$mag, even at super-solar metallicities. Among our clusters, NGC6838 (\[M/H\]$=-0.53$; it also has very few stars in the RGB) and NGC6441 (\[M/H\]$=-0.29$) are affected by this. In order to quantify the agreement between our relations and other relations, we have performed a Monte Carlo resampling of our relations by drawing random parameter sets $a_i$ from a multivariate Gaussian distribution with the mean and covariance matrix as given by the best fit. The 68.3%, 95.5%, and 99.7% confidence interval are shown as contours in Figures \[metalfitindices\], \[s\_index\], \[relations\], and \[relations\_sindex\]. ![Comparison of the observed S-index metallicity relation with isochrones of varying age. The gray contours show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ confidence levels of the fit.[]{data-label="relations_sindex"}](isochrone-age_comparison_s-index){width="0.99\columnwidth"} As can be seen in Figures \[relations\] and \[relations\_sindex\], BaSTI isochrones show good agreement with our observational result. At most metallicities the $\alpha$-enhanced BaSTI isochrone falls within the the 1$\sigma$ confidence range of our observational relation; only for \[M/H\]$>$-0.4 are the isochrones significantly redder ($>3\sigma$) and shallower than our relation. The Dartmouth isochrones agree well at very low metallicities, but tend to predict slightly redder colors and shallower slopes at intermediate and higher metallicities. In contrast, results from the Padua isochrones are bluer by almost 0.15mag and much steeper at lower metallicities, and redder and shallower at the high metallicity end. Determining the reason for this offset is beyond the scope of this paper, but this problem has been known to lead to higher metallicity estimates, when Padua isochrones are used [@lejeune99]. #### Existing color-metallicity relations are given in the standard Johnson-Cousin filters. Thus to compare these with our analysis we use the transformations to the HST/ACS filter set described in @sirianni05. Two such transformations are provided, one observationally based and the other synthetic. The former uses observations of horizontal branch and RGB stars in the metal-poor (\[Fe/H\]=$-2.15$) globular cluster NGC2419. This cluster does not contain stars with $(V-I) > 1.3$, hence the transformation at these redder colors are extrapolated and should be used with caution. The transformation is: $$F606W - F814W = -0.055 + 0.762(V - I)$$ For the synthetic transformation, stellar models with $(V-I) < 1.8$ were used. Hence, for redder colors, the extrapolation should again be treated with caution. The transformation is given as: $$F606W - F814W = 0.062 + 0.646(V - I) + 0.053(V - I)^2$$ We use both these transformations on the color-metallicity relations of @saviane00, who determined relations for the indices $(V - I)_{-3.0}$ and $(V - I)_{-3.5}$, and for @dacosta90 [for $(V - I)_{-3.0}$] and @lee93 [for $(V - I)_{-3.5}$]. To shift these transformations, which are defined for \[Fe/H\], to the \[M/H\] scale, we used the same \[Fe/H\]-\[$\alpha$/Fe\] relation as for the data. Note that these two transformations have a relative offset of of about 0.05mag, which can be seen in Fig. \[relations\] as the two almost parallel lines in the lower panel.[^9] From Fig. \[relations\] it can be seen that these transformed relations are always bluer at the low metallicity end and have a steeper slope than our relations.[^10] Part of this discrepancy can be explained by the different metallicity scales used for the various relations. While we use the metallicity scale of @carretta09 [,C+09], earlier relations were determined either in the Zinn & West scale [@zinn84 ZW84] or the Carretta & Gratton scale [@carretta97 CG97]. The adopted C+09 scale is comparable to the ZW84 scale; however, the CG97 scale yields higher metallicities for \[Fe/H\]$\lesssim-1$ and lower metallicities for \[Fe/H\]$\gtrsim-1$ (see \[fehscales\]). ![Comparison of different metallicity scales. On the x-axis the C+09 scale, that is adopted in this paper, is shown. Blue crosses show the clusters from @carretta97, red crosses from @zinn84. The plus symbols show the metallicities determined in @rutledge97 [RHS] based on the Ca triplet, calibrated to both scales.[]{data-label="fehscales"}](FeH_scales){width="0.99\columnwidth"} Hence, using the CG97 scale will lead to a steeper color-metallicty relation than found from our measurements (see the bottom panel of Fig. \[relations\]). Inverting the relation ---------------------- The main purpose of the color metallicity relation is to estimate metallicities of old stellar population. The uncertainties arising from the inverted relation are highly nonlinear. In Fig. \[inverted\_residuals\] we plot the difference between the spectroscopic metallicities and the metallicities derived with our relation. It is apparent that for bluer colors (i.e. lower metallicities) the difference can be very large. If the color is near the pole of the metallicity-color function, the formal uncertainties can be infinite. Then only an upper limit on the metallicity can be derived. For all clusters with $C_{-3.5}<1.2$ (or $C_{-3.0}<1.0$) the scatter in the metallicity differences is about 0.3dex. We suggest using this as a minimum uncertainty for metallicities derived from our relation in that color range. For redder colors, the uncertainty drops in half. ![Error distribution of the metallicity determination using the inverted color metallicity relations. Lines with errorbars are the running mean and standard deviation which are computed using a bin width of 0.3mag for $C_{-3.5}$ and 0.15mag for $C_{-3.0}$. Symbols and colors are as in Fig. \[metalfitindices\]. Note the different scales on the x-axis for the two distributions.[]{data-label="inverted_residuals"}](Z_residuals){width="0.99\columnwidth"} Conclusions and summary {#summary} ======================= In this paper, we derived relations between the colors and the slope of the RGB and metallicity using data from globular clusters. The details of the relations are summarized in Table \[fitparams\]. When using these relations for determining metallicities of old resolved stellar populations, the following points should be kept in mind: - The color changes very little with metallicity for $\mbox{[M/H]}\lesssim -1.0$, the slope changes little below $[M/H]\lesssim-1.5$. Therefore, inverting the relation in this regime introduces large uncertainties. This makes a photometric metallicity determination rather inaccurate in this metallicity range. - Our relation agrees well with the prediction from BaSTI isochrones. Dartmouth isochrones are slightly redder, Padua isochrones bluer than our data. Thus, for the purpose of determining metallicities of old populations we recommend the use of BaSTI isochrones. - A comparison with other color-metallicity-relations from the literature, both empirical and theoretical, shows some scatter between these relations. Therefore a comparison of metallicities derived from different methods/relations will introduce systematic offsets. This should be kept in mind whenever the use of a homogenous method is not possible. We thank our referee, Ivo Saviane, for the careful reading of our manuscript, the detailed look into the data, and the comments that helped to improve this paper. We also thank Benne Holwerda and Antonela Monachesi for their comments on an earlier version of this paper that improved its final quality. DS gratefully acknowledges the support from DLR via grant 50OR1012 and a scholarship from the Cusanuswerk. Description of the fit of the RGB ================================= We parametrized the RGB with the function $$M=a+b\cdot \mbox{color}+c/(\mbox{color}+d) ,$$ as given in @saviane00. Since the data do not only contain RGB stars, but also the horizontal branch, blue stragglers and foreground stars, we have defined a region to guarantee a high fraction of RGB stars in our fit sample. The extent of this region can be seen as the red frame in Fig. \[CMD\_diverse\]. Note that this region excludes also the red clump. In some clusters, there is a distinct asymptotic giant branch (AGB) visible, which lies on the blue side of the RGB (it is mostly seen at F814W magnitudes between -1 and -2). As in @saviane00, we have removed these AGB stars by excluding all detections that lie blue wards of a reference line with the same slope for all clusters (denoted in Figures \[CMD\_diverse\] through \[CMDs\_diverse\_last\] by a dashed red line). The horizontal position of the reference line was set to be 0.05mag blue wards (at F814W=-0.5) of a first fit of all stars in the RGB region and then excluding. The fit including all stars is shown in the CMDs as black dashed line, while the final fit after the AGB removal is shown as black solid line. For the actual fit we used the python package scipy.odr. This routine performs an orthogonal distance regression, i.e. it minimizes the orthogonal distance between the curve and the data points. The distance of each data point is weighted with its measurement uncertainty. This method is a variation of the typical $\chi^2$ minimization, now generalized for data with uncertainties on both variables. The ACSGCS team reports photometric uncertainties for each individual star, which are typically quite small; the median uncertainty in F814W is only 0.003mag. This is much smaller than both the observed scatter in the RGB and the errors that are found in the artificial star test at a level of F814W$\approx0$. (There are no artificial star tests at brighter magnitudes.) The mean measurement error estimated from the difference of the input and recovered magnitudes in the artificial star test are 0.06mag in F814W and 0.03mag in color. These estimated are added in quadrature to the reported uncertainties of each star. The smaller error in color is due to the fact that errors in both bands are correlated. Thus the uncertainty of the difference of both bands is smaller than the uncertainty in each band. Finally, we visually inspected each CMD with its fit, to check for any residual problems of our clusters. After this inspection we excluded four more clusters from the sample: NGC6838 and NGC6441, because their RGB fits do not reach the $M_I=-3.5$ level; NGC6388 (and again NGC6441), because their red clumps seem to be to faint [@bellini13 also found problems with differential reddening and multiple populations in these two clusters]; and NGC6715, because it has a clear and strong second RGB. {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} {width="99.00000%"} Properties of the Globular clusters {#app:properties} =================================== The properties of the globular clusters, from the literature and determined in this work, are summarized in Table \[GC\_literature\] and Table \[GC\_results\] respectively. ---------- ------------ ------------ ----------- --------- ------------------ ------------------- ------------------- ----------------- ------ name RA DEC (m-M$)_V$ E(B-V) \[Fe/H\]$_{H10}$ \[Fe/H\]$_{C+10}$ $\sigma_{[Fe/H]}$ \[$\alpha$/Fe\] age \[$\deg$\] \[$\deg$\] \[mag\] \[mag\] \[dex\] \[dex\] \[dex\] \[dex\] (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) Arp 2 292.1838 -29.6444 17.59 0.10 -1.75 -1.74 0.08 0.34 0.85 IC4499 225.0769 -81.7863 17.08 0.23 -1.53 -1.62 0.09 ... ... Lynga 7 242.7652 -54.6822 16.78 0.73 -1.01 -0.68 0.06 ... 1.13 NGC104 6.0236 -71.9187 13.37 0.04 -0.72 -0.76 0.02 0.42 1.02 NGC362 15.8094 -69.1512 14.83 0.05 -1.26 -1.30 0.04 0.30 0.81 NGC1261 48.0675 -54.7838 16.09 0.01 -1.27 -1.27 0.08 ... 0.80 NGC1851 78.5282 -39.9534 15.47 0.02 -1.18 -1.18 0.08 0.38 0.78 NGC2298 102.2475 -35.9947 15.60 0.14 -1.92 -1.96 0.04 0.50 0.99 NGC2808 138.0129 -63.1365 15.59 0.22 -1.14 -1.18 0.04 0.33 0.85 NGC3201 154.4034 -45.5875 14.20 0.24 -1.59 -1.51 0.02 0.33 0.80 NGC4590 189.8666 -25.2559 15.21 0.05 -2.23 -2.27 0.04 0.35 0.90 NGC4833 194.8913 -69.1235 15.08 0.32 -1.85 -1.89 0.05 ... 0.98 NGC5024 198.2302 18.1682 16.32 0.02 -2.10 -2.06 0.09 ... 0.99 NGC5139 201.6968 -46.5204 13.94 0.12 -1.53 -1.64 0.09 ... 0.90 NGC5272 205.5484 28.3773 15.07 0.01 -1.50 -1.50 0.05 0.34 0.89 NGC5286 206.6117 -50.6257 16.08 0.24 -1.69 -1.70 0.07 ... 0.98 NGC5904 229.6384 2.0810 14.46 0.03 -1.29 -1.33 0.02 0.38 0.83 NGC5927 232.0029 -49.3270 15.82 0.45 -0.49 -0.29 0.07 ... 0.99 NGC5986 236.5125 -36.2136 15.96 0.28 -1.59 -1.63 0.08 ... 0.95 NGC6093 244.2600 -21.0239 15.56 0.18 -1.75 -1.75 0.08 0.24 0.98 NGC6101 246.4505 -71.7978 16.10 0.05 -1.98 -1.98 0.07 ... 0.98 NGC6121 245.8968 -25.4743 12.82 0.35 -1.16 -1.18 0.02 0.51 0.98 NGC6144 246.8077 -25.9765 15.86 0.36 -1.76 -1.82 0.05 ... 1.08 NGC6171 248.1328 -12.9462 15.05 0.33 -1.02 -1.03 0.02 0.49 1.09 NGC6205 250.4218 36.4599 14.33 0.02 -1.53 -1.58 0.04 0.31 0.91 NGC6218 251.8091 -0.0515 14.01 0.19 -1.37 -1.43 0.02 0.41 0.99 NGC6254 254.2877 -3.8997 14.08 0.28 -1.56 -1.57 0.02 0.37 0.89 NGC6304 258.6344 -28.5380 15.52 0.54 -0.45 -0.37 0.07 ... 1.06 NGC6341 259.2808 43.1359 14.65 0.02 -2.31 -2.35 0.05 0.46 1.03 NGC6362 262.9791 -66.9517 14.68 0.09 -0.99 -1.07 0.05 ... 1.06 NGC6426 266.2277 3.1701 17.68 0.36 -2.15 -2.36 0.06 ... ... NGC6496 269.7653 -43.7341 15.74 0.15 -0.46 -0.46 0.07 ... 0.97 NGC6541 272.0098 -42.2851 14.82 0.14 -1.81 -1.82 0.08 0.43 1.01 NGC6584 274.6567 -51.7842 15.96 0.10 -1.50 -1.50 0.09 ... 0.88 NGC6624 275.9188 -29.6390 15.36 0.28 -0.44 -0.42 0.07 ... 0.98 NGC6637 277.8462 -31.6519 15.28 0.18 -0.64 -0.59 0.07 0.31 1.02 NGC6652 278.9401 -31.0093 15.28 0.09 -0.81 -0.76 0.14 ... 1.01 NGC6656 279.0998 -22.0953 13.60 0.34 -1.70 -1.70 0.08 0.38 0.99 NGC6681 280.8032 -31.7079 14.99 0.07 -1.62 -1.62 0.08 ... 1.00 NGC6717 283.7752 -21.2985 14.94 0.22 -1.26 -1.26 0.07 ... 1.03 NGC6723 284.8881 -35.3678 14.84 0.05 -1.10 -1.10 0.07 0.50 1.02 NGC6752 287.7171 -58.0154 13.13 0.04 -1.54 -1.55 0.01 0.43 0.92 NGC6779 289.1482 30.1835 15.68 0.26 -1.98 -2.00 0.09 ... 1.07 NGC6809 294.9988 -29.0353 13.89 0.08 -1.94 -1.93 0.02 0.42 0.96 NGC6934 308.5474 7.4045 16.28 0.10 -1.47 -1.56 0.09 ... 0.87 NGC6981 313.3654 -11.4627 16.31 0.05 -1.42 -1.48 0.07 ... 0.85 NGC7006 315.3724 16.1873 18.23 0.05 -1.52 -1.46 0.06 0.28 ... NGC7078 322.4930 12.1670 15.39 0.10 -2.37 -2.33 0.02 0.40 1.01 NGC7089 323.3626 0.8233 15.50 0.06 -1.65 -1.66 0.07 0.41 0.92 NGC7099 325.0922 -22.8201 14.64 0.03 -2.27 -2.33 0.02 0.37 1.01 Pal 2 71.5246 31.3815 21.01 1.24 -1.42 -1.29 0.09 ... ... Rup 106 189.6675 -50.8497 17.25 0.20 -1.68 -1.78 0.08 -0.03 ... Terzan 8 295.4350 -32.0005 17.47 0.12 -2.16 -2.02 0.06 0.45 0.95 ---------- ------------ ------------ ----------- --------- ------------------ ------------------- ------------------- ----------------- ------ \[GC\_literature\] ---------- ---------------- ----------------- ---------------- ----------------- -------- ------------ ------------- name (V-I$)_{-3.5}$ $\sigma_{-3.5}$ (V-I$)_{-3.0}$ $\sigma_{-3.0}$ $S$ $\sigma_S$ $M_{V}(HB)$ \[mag\] \[mag\] \[mag\] \[mag\] \[mag\] (1) (2) (3) (4) (5) (6) (7) (8) Arp 2 0.999 0.027 0.920 0.011 13.415 0.826 0.45 IC4499 0.976 0.016 0.902 0.007 13.460 0.620 0.30 Lynga 7 1.321 0.042 1.158 0.016 7.829 0.587 0.40 NGC104 1.397 0.017 1.162 0.006 6.860 0.540 0.44 NGC362 1.069 0.009 0.953 0.005 11.578 0.568 0.52 NGC1261 1.092 0.024 0.977 0.007 11.190 0.622 0.53 NGC1851 1.110 0.007 0.999 0.005 11.059 0.545 0.53 NGC2298 1.054 0.019 0.969 0.010 13.736 0.762 0.35 NGC2808 1.077 0.008 0.967 0.004 11.215 0.461 0.56 NGC3201 1.048 0.012 0.973 0.005 11.307 0.512 0.30 NGC4590 0.950 0.012 0.896 0.006 13.559 0.505 0.30 NGC4833 0.976 0.004 0.916 0.003 13.888 0.476 0.30 NGC5024 0.975 0.002 0.897 0.002 12.324 0.525 0.20 NGC5139 1.044 0.011 0.960 0.006 11.553 0.521 0.40 NGC5272 1.018 0.003 0.929 0.002 12.281 0.528 0.40 NGC5286 0.985 0.005 0.913 0.003 14.017 0.554 0.40 NGC5904 1.057 0.005 0.962 0.003 11.818 0.523 0.50 NGC5927 1.883 0.091 1.353 0.025 4.849 0.680 0.51 NGC5986 1.019 0.006 0.939 0.004 12.747 0.553 0.40 NGC6093 1.053 0.006 0.968 0.006 12.131 0.589 0.40 NGC6101 1.024 0.018 0.956 0.007 13.104 0.698 0.30 NGC6121 1.179 0.028 1.068 0.015 10.401 0.849 0.30 NGC6144 1.028 0.014 0.961 0.009 14.346 0.962 0.25 NGC6171 1.231 0.017 1.108 0.017 9.498 0.622 0.42 NGC6205 1.038 0.008 0.948 0.003 11.467 0.509 0.40 NGC6218 1.119 0.015 1.003 0.007 9.887 0.583 0.30 NGC6254 1.061 0.021 0.972 0.009 11.391 0.524 0.50 NGC6304 2.051 0.176 1.392 0.043 4.257 0.848 0.49 NGC6341 0.956 0.003 0.889 0.003 13.381 0.491 0.30 NGC6362 1.186 0.023 1.034 0.009 9.554 0.571 0.54 NGC6426 1.008 0.027 0.946 0.026 13.152 1.967 0.40 NGC6496 1.587 0.078 1.308 0.036 5.976 0.772 0.53 NGC6541 0.954 0.004 0.883 0.003 12.937 0.523 0.30 NGC6584 1.015 0.005 0.925 0.003 12.120 0.557 0.40 NGC6624 1.514 0.096 1.215 0.040 6.412 0.742 0.47 NGC6637 1.427 0.023 1.159 0.009 7.243 0.675 0.46 NGC6652 1.321 0.021 1.147 0.014 7.829 0.564 0.47 NGC6656 1.014 0.012 0.949 0.005 13.284 0.523 0.50 NGC6681 1.087 0.010 0.983 0.005 12.476 0.579 0.60 NGC6717 1.091 0.062 0.995 0.041 11.462 1.647 0.60 NGC6723 1.149 0.022 1.033 0.010 10.487 0.574 0.49 NGC6752 1.075 0.025 0.980 0.010 10.063 0.564 0.25 NGC6779 0.960 0.005 0.888 0.002 13.579 0.519 0.40 NGC6809 1.011 0.007 0.938 0.006 13.041 0.638 0.40 NGC6934 1.050 0.018 0.962 0.015 10.874 1.048 0.40 NGC6981 1.015 0.009 0.933 0.004 12.173 0.559 0.40 NGC7006 1.035 0.014 0.955 0.006 12.705 0.575 0.40 NGC7078 0.927 0.002 0.861 0.002 13.877 0.520 0.30 NGC7089 0.996 0.005 0.909 0.006 12.324 0.615 0.30 NGC7099 0.978 0.005 0.908 0.003 13.211 0.475 0.40 Pal 2 1.019 0.015 0.962 0.010 11.582 0.558 0.30 Rup 106 0.965 0.014 0.890 0.009 13.509 0.873 0.30 Terzan 8 0.990 0.023 0.905 0.011 13.440 0.872 0.30 ---------- ---------------- ----------------- ---------------- ----------------- -------- ------------ ------------- \[GC\_results\] [^1]: @mormany05 actually have found such a relation, but they used only three clusters and did not publish the details. [^2]: <http://www.astro.ufl.edu/~ata/public_hstgc/> [^3]: <http://stev.oapd.inaf.it/cgi-bin/cmd> [^4]: <http://albione.oa-teramo.inaf.it/> [^5]: <http://stellar.dartmouth.edu/~models/index.html> [^6]: We used the Python implementation of this algorithm that is part of the Scipy library: <http://docs.scipy.org/doc/scipy/reference/odr.html> [^7]: i.e. we ignore uncertainties in metallicity and only look at the color offset between the data and the best fit relation [^8]: Actually, @bressan12 write about $\alpha$-enhanced PARSEC isochrones, but these are not (yet) publicly available. [^9]: The offset can already be seen in @sirianni05 [Fig. 21] as an offset in plot of (V-I) versus V-F606W. [^10]: Strictly speaking, we compare slightly different things here: The transformed relations measure the color at constant I-band magnitude, while in this work we have measured the color at constant F814W magnitude. We can ignore this difference here because the difference between I-band and F814W is small. According to the transformations given above, the differences between F814W and I are always smaller than 0.05mag and the resulting error in the color measurement of the RGB is always smaller than 0.01mag (except for the two reddest clusters, for which it can reach 0.06 mag). Therefore the effect on the total color-metallicity relation is negligible.

ArXiv

--- abstract: 'Current steps attributed to resonant tunneling through individual InAs quantum dots embedded in a GaAs-AlAs-GaAs tunneling device are investigated experimentally in magnetic fields up to 28 T. The steps evolve into strongly enhanced current peaks in high fields. This can be understood as a field-induced Fermi-edge singularity due to the Coulomb interaction between the tunneling electron on the quantum dot and the partly spin polarized Fermi sea in the Landau quantized three-dimensional emitter.' address: | $^1$Institut für Festkörperphysik, Universität Hannover, Appelstra[ß]{}e 2, D-30167 Hannover, Germany\ $^2$Institut für Theoretische Physik, Universität Hannover, Appelstra[ß]{}e 2, D-30167 Hannover, Germany\ $^3$Grenoble High Magnetic Field Laboratory, MPIF-CNRS, B.P. 166, F-38042 Grenoble Cedex 09, France\ $^4$Physikalisch-Technische Bundesanstalt Braunschweig, Bundesallee 100, D-38116 Braunschweig, Germany author: - 'I. Hapke-Wurst,$^1$ U. Zeitler,$^1$ H. Frahm,$^2$ A. G. M. Jansen,$^3$ R. J. Haug,$^1$ and K. Pierz$^4$' title: ' Magnetic-field-induced singularities in spin dependent tunneling through InAs quantum dots' --- \#1[[$\backslash$\#1]{}]{} The interaction of the Fermi sea of a metallic system with a local potential can lead to strong singularities close to the Fermi edge. Such effects have been predicted more than thirty years ago for the X-ray absorption and emission of metals[@theoXray] and observed subsequently[@expXray]. Similar singularities as a consequence of many body effects are also known from the luminescence of quantum wells[@Lum]. Matveev and Larkin were the first to predict interaction-induced singularities in the tunneling current via a localized state[@Matveev:1992] which were measured experimentally in several resonant tunneling experiments[@Geim:1994; @Cobden:1995; @Benedict:1998] from [*two-dimensional*]{} electrodes through a zero-dimensional system. Here we report on singularities observed in the resonant tunneling from highly doped [*three-dimensional*]{} (3D) GaAs electrodes through an InAs quantum dot (QD) embedded in an AlAs barrier. These Fermi-edge singularities (FES) show a considerable magnetic field dependence and a strong enhancement in high magnetic fields where the 3D electrons occupy the lowest Landau level in the emitter. We observe an asymmetry in the enhancement for electrons of different spins with an extremely strong FES for electrons carrying the majority spin of the emitter. The experimental observations are explained by a theoretical model taking into account the electrostatic potential experienced by the conduction electrons in the emitter due to the charged QD. We will show that the partial spin polarisation of the emitter causes extreme values of the edge exponent $\gamma$ not observed until present and going beyond the standard theory valid for $\gamma \ll 1$ [@Matveev:1992]. The active part of our samples are self-organized InAs QDs with 3-4 nm height and 10-15 nm diameter embedded in the middle of a 10 nm-thick AlAs barrier and sandwiched between two 3D electrodes. They consist of a 15 nm undoped GaAs spacer layer and a GaAs-buffer with graded doping. A typical InAs dot is sketched in inset (a) of Fig. \[steps\], the vertical band structure across a dot is schematically shown in inset (b). Current voltage ($I$-$V$) characteristics were measured in large area vertical diodes ($40\times 40~\mu$m$^2$) patterned on the wafer. In Fig. \[steps\] we show a part of a typical $I$-$V$-curve with several discrete steps. We have demonstrated previously that such steps can be assigned to single electron tunneling from 3D electrodes through individual InAs QDs [@Hapke:1999] consistent with other resonant tunneling experiments through self-organized InAs QDs [@tunnel]. For the positive bias voltages shown in Fig. \[steps\] the electrons tunnel from the bottom electrode into the base of an InAs QD and leave the dot via the top. The tunneling current is mainly determined by the tunneling rate through the effectively thicker barrier below the dot (single electron tunneling regime). A step in the current occurs at bias voltages where the energy level of a dot, $E_D$, coincides with the Fermi level of the emitter, $E_F$. In the following we will concentrate on the step labeled (\*) in Fig. \[steps\]. Other steps in the same structure as well as steps observed in the $I$-$V$-characteristics of other structures show a very similar behavior. After the step edge a slight overshoot in the tunneling current occurs consistent with other tunneling experiments through a localized impurity [@Geim:1994] or through InAs dots [@Benedict:1998]. This effect is caused by the Coulomb interaction between a localized electron on the dot and the electrons at the Fermi edge of the emitter. The decrease of the current $I(V)$ towards higher voltages $V >V_0$ follows a power law $I \propto (V-V_0)^{-\gamma}$ [@Geim:1994] ($V_0$ is the voltage at the step edge) with an edge exponent $\gamma = 0.02 \pm 0.01$. The evolution of step (\*) in a magnetic field applied parallel to the current direction is shown in Fig. \[Babh\]a. The step develops into two separate peaks with onset voltages marked as $V_\downarrow$ and $V_\uparrow$. The Landau quantization of the emitter leads to an oscillation of $V_\downarrow$ and $V_\uparrow$ and a shift to smaller voltages as a function of magnetic field, see Fig. \[Babh\]b. This reflects the magneto-quantum-oscillation of the Fermi energy in the emitter [@Bumbel; @Main:2000]. From the period and the amplitude of the oscillation we can extract a Fermi energy (at $B = 0$) $E_0 = 13.6~$meV and a Landau level broadening $\Gamma = 1.3~$meV in the 3D emitter. The measured $E_0 = 13.6~$meV agrees well with the expected electron concentration at the barrier derived from the doping profile in the electrodes. For $B > 6$ T only the lowest Landau level remains occupied. With a level broadening $\Gamma = 1.3~$meV the Fermi level $E_F$ for 15 T $<$ B $<$ 30 T is within less than $2~$meV pinned to the bottom of the lowest Landau band, $E_L = \hbar \omega_c/2$. As a consequence the onset voltage shifts to lower values as $\alpha e\Delta V \approx -\hbar \omega_c/2$ with $\alpha = 0.34$. The diamagnetic shift of the energy level in the dot can be neglected compared to this shift of the Fermi energy in the emitter. For the dot investigated in [@Hapke:1999] with $r_0 = 3.7$ nm the diamagnetic shift at 30 T is $\Delta E_D = 3.5$ meV negligible compared to $E_L = 26~$meV. The two distinct steps with onset voltages $V_\downarrow$ and $V_\uparrow$ originate from the spin-splitting of the energy level $E_D$ in the dot. Their distance $\Delta V_p$ is given by the Zeeman splitting $\Delta E_z = g_D \mu_B B = \alpha e \Delta V_p$ with an energy-to-voltage conversion factor $\alpha = 0.34$ [@explain-alpha]. As shown in Fig. \[Babh\]c $\Delta V_p$ is indeed linear in B, with a Landé factor $g_D = 0.8$ in agreement with other experiments on InAs dots [@Thornton:1998]. For low magnetic fields ($B \le 9~$T in our case, see graph for $B = 9~T$ in Fig. \[Babh\]a) the size of the steps is very similar for both spins and about half of the size at zero field. Also the slight overshoot in the current as the signature of a Fermi edge singularity is similar for both spin orientations and comparable to the zero field case with an edge exponent $\gamma < 0.05$ for all magnetic fields $B < 10~$T. The form of the current steps changes drastically in high magnetic fields where only the lowest Landau level of the emitter remains occupied. In particular, the second current step at higher voltage evolves into a strongly enhanced peak with a peak current of one order of magnitude higher compared to the zero-field case. Following [@Thornton:1998] we assume that $g_D$ is positive whereas the Landé factor in bulk GaAs is negative. This assumption is verified by the fact that the energetically lower lying state (first peak in Fig. \[Tabh\]) is thermally occupied at higher temperatures and can therefore be identified with the minority spin in the emitter. The strongly enhanced current peak at higher energies is due to tunneling through the spin state corresponding to the majority spin (spin up) in the emitter. The resulting spin configuration is scetched in the inset of Fig. \[Babh\]a and will also be confirmed below by our theoretical results. The shape of this current peak can be described by a steep ascent and a more moderate decrease of the current towards higher voltages. Down to temperatures $T<100$ mK the steepness of the ascent is only limited by thermal broadening. The decrease of the current for $V >V_0$ is again described with the characteristic behavior for a Fermi-edge singularity, $I \propto (V-V_0)^{-\gamma}$, where $V_0$ here is the voltage at the maximum peak current. However, along with the drastic increase of the peak current the edge exponent $\gamma$ increases dramatically reaching a value $\gamma > 0.5$ for the highest fields. A different way to visualize the signature of a FES is a temperature dependent experiment. As an example we have plotted the $I$-$V$-curve at $B=22$ T for different temperatures in Fig. \[Tabh\]. As shown in the inset the peak maximum $I_0$ for the spin-up electrons decreases according to a power law $I_0 \propto T^{-\gamma}$ with an edge exponent $\gamma = 0.43 \pm 0.05$. Such a strong temperature dependence is characteristic for a FES and allows us to exclude that pure density of states effects in the 3D emitter are responsible for the current peaks in high magnetic fields. As shown in Fig. \[Tabh\] an edge exponent $\gamma = 0.43$ also fits within experimental accuracy the observed decrease of the current for $V>V_0$. It is not possible to extract the edge exponent for the minority spin directly from temperature dependent experiments. At high magnetic fields the observed increase of the current with increasing temperature is mainly caused by an additional thermal population of the minority spin in the emitter. The general form of the curve is merely affected by temperature. Therefore, the edge exponent can only be gained from fitting the shape of the current peaks. A compilation of the edge exponents $\gamma$ for various magnetic fields and both spin orientations is shown in Fig. \[gammas\]. For the data related to the majority spin two independent methods were used to extract $\gamma$. For the minority spin only fitting of the shape of the $I$-$V$-curves was used. For a theoretical description of these effects we consider a 3D electron gas in the half space $z<0$. In a sufficiently strong magnetic field $B||\hat{z}$ all electrons are in the lowest Landau level. This defines a set of one-dimensional channels with momentum $\hbar k$ perpendicular to the boundary. This situation is different from the cases considered for scattering off point defects as in Refs. [@theoXray; @Matveev:1992] or for a 2D electron gas where the current is carried by edge states [@BaMa95]. The single particle wave functions in channel $m\ge0$ are $\psi_m(\rho,\phi) \sin kz$ with $\psi_m(\rho,\phi) \propto \rho^m \exp(-im\phi-\rho^2/4\ell_0^2)$. In the experiments the magnetic length $\ell_0=\sqrt{\hbar/eB}$ ($\ell_0 = 5.6$ nm at 20 T) is comparable to the lateral size of the QD $2 r_0 \approx 7$ nm. Hence the effect of the electrostatic potential of a charged dot on the electrons in a given channel of the emitter decreases rapidly with $m$, and the observed FES are mainly due to tunneling of electrons from the $m=0$ channel into the dot. Following [@theoXray; @Matveev:1992] tunneling processes of spin $\sigma$ electrons from the $m=0$ state in the emitter give rise to a FES with edge exponent $$\gamma_\sigma = -\frac{2}{\pi}\delta_0(k_{F\sigma}) - \frac{1}{\pi^2}\sum_{m}\sum_{\tau=\uparrow,\downarrow} \left(\delta_m(k_{F\tau})\right)^2 \label{expo}$$ where $\delta_m(k)$ is the Fermi phase shift experienced by the electrons in the $m$-th channel due to the potential of the quantum dot [@contact]. From (\[expo\]) the observed field dependence of the edge exponents is a consequence of the variation of the Fermi momenta for spin-$\sigma$ electrons with magnetic field *and* the field dependence of the effective potential in the one-dimensional channels. The former can be computed from the one-dimensional density of states (DOS) of the lowest Landau band $$D(E,B) = \frac{e\sqrt{m^{*}}}{(2\pi\hbar)^2}\,B \left( d(\epsilon_\uparrow) +d(\epsilon_\downarrow) \right)\ .$$ Here $\epsilon_\sigma = E-(\hbar\omega_c\pm g^*\mu_BB)/2$ is the energy of electrons with spin-$\sigma$ measured from the bottom of the Landau band. $g^* \approx -0.33$ [@Pfeffer:1985] is the effective Landé factor of the electrons in the emitter. The DOS for the spin-subbands is $d(\epsilon) = \sqrt{2} {\mathrm{Re}} (\epsilon+ i\Gamma)^{-{1/2}}$. Without broadening, $\Gamma=0$, one has $k_{F\sigma} = \pi^2n \ell_0^2 (1\pm b^3)$ where $n$ is the 3D density of electrons and $b$ is the magnetic field measured in units of the field necessary for complete spin polarization of the 3D emitter. Using a Fermi energy $E_0=13.6$ meV and neglecting level broadening we find that only the lowest Landau level (*both spin states!*) is occupied for $B_1 >5.2$ T. Including level broadening changes $B_1$ to a slightly higher value. With the known field dependence of the Fermi energy in the quantum limit we can calculate the field for total spin polarisation $$B_{pol} = \left( \frac{16}{9 \xi}\right) ^{1/3} \frac{m^* E_0}{\hbar e} \simeq 43~\mbox{T}$$ with $g^*= -0.33$ [@Pfeffer:1985] and $m^*=0.067\,m_0$. $\xi = \frac{1}{2} |g^*| m^*/m_0$ is the ratio between spin splitting and Landau level splitting. To make contact to the experimental observations we have to specify the interaction of the screened charge on the QD and the conduction band electrons. A Thomas-Fermi calculation gives $U(\rho,z) = (2e^2 \exp(\kappa z)/\kappa) (d/(\rho^2 +d^2)^{(3/2)})$ [@Matveev:1992]. Here $d=5\,\mathrm{nm}$ is the width of the insulating layer and $\kappa^{-1}=7\,\mathrm{nm}$ is the Debye radius. The effective potential seen by electrons in channel $m$ is $V_m\exp(\kappa z)/\kappa$ with $V_m = 2e^2d \int d\rho^2 |\psi_m(\rho,\phi)|^2/(\rho^2 +d^2)^{(3/2)}$. For large $\kappa$ we obtain for the phase shift in the $m=0$ channel $\delta_0(k) \approx -v_0f(B)k/\kappa$ where $$f(B) = \left(\frac{d}{\ell_0}\right)^2 \left\{ 1 - \sqrt{\pi\over2}\, {d\over\ell_0} {\rm e}^{d^2\over2\ell_0^2} \mathrm{erfc}\left({d\over\sqrt{2}\ell_0}\right) \right\}\$$ and $v_0 \sim (m^*e^2/\hbar^2\kappa) (\kappa d)^{-2}$ up to a numerical factor. Similarly we obtain the integrated effect of the channels $m>0$ in (\[expo\]). In Fig. 4 the resulting exponents $\gamma_\sigma$ obtained for $\sigma=\uparrow,\downarrow$ are shown for $v_0= 6.75$ and a broadening $\Gamma= 0$ and $\Gamma = 1.3$ meV, respectively. The value used for $\Gamma$ reflects its realistic experimental value. $v_0$ is the only fit parameter. Already the simple model with no level broadening ($\Gamma = 0$) is in good agreement with the experimentally measured edge exponents for both spin directions, especially in high magnetic fields where possible admixtures of higher Landau levels play a minor rule. Including level broadening leads to a less dramatic spin polarisation in the emitter and as a consequence smears out the field dependence of $\gamma$ for the minority spin. The basic features, however, remain unchanged. In particular, the edge exponent for the minority spin retains moderate values for high magnetic fields, whereas the edge exponent related to the majority spin shows a strong field dependence with very high values in high magnetic fields. In conclusion we have evaluated experimental data concerning magnetic-field-induced FES in resonant tunneling experiments through InAs QDs. We have shown that the interaction between a localized charge and the electrons in the Landau quantized emitter leads to dramatic Fermi phase shifts if only the lowest Landau level in the 3D emitter is occupied. This results in edge exponents $\gamma > 0.5$ which were measured and described theoretically. We would like to thank H. Marx for sample growing, P. König for experimental support and F. J. Ahlers for valuable discussions. Part of this work has been supported by the TMR Programme of the European Union under contract no. ERBFMGECT950077. We acknowledge partial support from the Deutsche Forschungsgemeinschaft under Grants HA 1826/5-1 and Fr 737/3. G. D. Mahan, Phys. Rev. [**163**]{}, 612 (1967); G. D. Mahan, [*Many-Particle Physics*]{} (Plenum, New York, 1981); P. Nozières and C. T. De Dominicis, Phys. Rev. [**178**]{}, 1097 (1969); K. D. Schotte and U. Schotte, Phys. Rev. [**182**]{}, 479 (1969). see, for example, T. A. Callcott, E. T. Arakawa and D. L. Ederer, Phys. Rev. B [**18**]{}, 6622 (1978) and references therein. M. S. Skolnik, J. M. Rorison, K. J. Nash, D. J. Mowbray, P. R. Tapster, S. J. Bass and A. D. Pitt, Phys. Rev. Lett. [**58**]{}, 2130 (1987); J. A. Brum and P. Hawrylak, Comments Cond. Mat. Phys. [**18**]{}, 135 (1997). K. A. Matveev and A. I. Larkin, Phys. Rev. B [**46**]{}, 15337 (1992). A. K. Geim, P. C. Main, N. La Scala, Jr., L. Eaves, T. J. Foster, P. H. Beton, J. W. Sakai, F. W. Sheard, M. Henini, G. Hill and M. A. Pate, Phys. Rev. Lett. [**72**]{}, 2061 (1994). D. H. Cobden and B. A. Muzykantskii, Phys. Rev. Lett. [**75**]{}, 4274 (1995). K. A. Benedict, A. S. G. Thornton, T. Ihn, P. C. Main, L. Eaves and M. Henini, Physica B 256-258, 519 (1998). I. Hapke-Wurst, U. Zeitler, H. W. Schumacher, R. J. Haug, K. Pierz and F. J. Ahlers, Semicond. Sci. Technol. [**14**]{}, L41 (1999). I. E. Itskevich, T. Ihn, A. Thornton, M. Henini, T. J. Foster, P. Moriarty, A. Nogaret, P. H. Beton, L. Eaves and P. C. Main, Phys. Rev. B [**54**]{}, 16401 (1996); T. Suzuki, K. Nomoto, K. Taira and I. Hase, Jpn. J. Appl. Phys. [**36**]{}, 1917 (1997); M. Narihiro, G. Yusa, Y. Nakamura, T. Noda and H. Sakaki, Appl. Phys. Lett. [**70**]{}, 105 (1997). The energy-to-voltage conversion factor $\alpha$ is derived from the temperature dependence of the width $\delta V$ of a current step caused by the thermal smearing of the Fermi-edge in the emitter. Only a part $\alpha$ of the total voltage applied drops between the emitter and the dot, the rest of the voltage drop occurs inside the electrodes and between dot and collector. A. S. G. Thornton, T. Ihn, P. C. Main, L. Eaves and M. Henini, Appl. Phys. Lett. [**73**]{}, 354 (1998). T. Schmidt, R. J. Haug, V. I. Fal’ko, K. v. Klitzing, A. Förster and H. Lüth, Phys. Rev. Lett. [**78**]{}, 1540 (1997). P. C. Main, A. S. G. Thornton, R. J. A. Hill, S. T. Stoddart, T. Ihn, L. Eaves, K. A. Benedict and M. Henini , Phys. Rev. Lett. [**84**]{}, 729 (2000). A. V. Balatsky and S. I. Matveenko, Phys. Rev. B [**52**]{}, 8676 (1995). In fact, one has to consider the change of this phase shift due to the increase in charge on the QD here. For the contact interaction considered here this is not essential. P. Pfeffer and W. Zawadzki, Phys. Rev. B [**41**]{}, 1561 (1990).\ The electronic Landé factor in the lowest Landau level in GaAs increases linearly with magnetic field from $g^*=-0.44$ at $B=0$ to $g^*=-0.29$ at $B=30$ T. Our theory uses a constant $g^*$, for the best possible connection with the experiment we have chosen to use its value at $B=22$ T.

ArXiv

‘=11 makefntext\#1[ to 3.2pt [-.9pt $^{{\ninerm\@thefnmark}}$]{}\#1]{} makefnmark[to 0pt[$^{\@thefnmark}$]{}]{} PS. @myheadings[mkbothgobbletwo oddhead[ ]{} oddfootevenheadevenfoot \#\#1\#\#1]{} 6.0in 8.5in -0.25truein 0.30truein 0.30truein \[appendixc\] \[subappendixc\] \#1 =1.5pc citex\[\#1\]\#2[@fileswauxout citeacite[forciteb:=\#2]{}[\#1]{}]{} @cghi cite\#1\#2[[$\null^{#1}$@tempswa ]{}]{} \#1[$^{\mbox{\scriptsize #1}}$]{} \#1\#2 =cmbx10 scaled1 =cmr10 scaled1 =cmti10 scaled1 =cmbxti10 scaled=cmbx10 scaled=cmr10 scaled=cmti10 scaled=cmbxti10 =cmbx10 =cmr10 =cmti10 =cmbx9 =cmr9 =cmti9 =cmbx8 =cmr8 =cmti8 Energy of a spherically symmetric charged dilaton black hole A. CHAMORRO and K. S. VIRBHADRA Departamento de Física Teórica, Universidad del País Vasco Apartado 644, 48080 Bilbao, Spain In recent years there is considerable interest in obtaining charged dilaton black hole solutions and investigating their properties $[1-6]$. Garfinkle, Horowitz, and Strominger (GHS) $[1]$ considered the action S = d\^4 x    and obtained a nice form of static and spherically symmetric charged dilaton black hole solution $[1,2]$, given by the line element ds\^2 = B dt\^2 - B\^[-1]{} dr\^2 - D r\^2 (d\^2 + \^2d\^2), the dilaton field $\Phi$, where e\^[2]{} = \^[(1-)/]{}, and the component of the electromagnetic field tensor F\_[tr]{} = , where B = (1-) (1-)\^, D = (1-)\^[1-]{}, and  =    . $r_{+}$ and $r_{-}$ are related through & &2M = r\_[+]{} +  r\_[-]{} ,\ & &Q\^2 (1+\^2) = r\_[+]{} r\_[-]{}  . $M$ and $Q$ stand for mass and charge parameters, respectively. The surface $r = r_{+}$ is the event horizon. $\beta$ is a dimensionless free parameter which controls the coupling between the dilaton and the Maxwell fields. A change in the sign of $\beta$ is the same as a change in the sign of the dilaton field. Therefore, it is sufficient to discuss only nonnegative values of $\beta$. $\beta = 0$ in GHS solution gives the well known Reissner-Nordström (RN) solution. It is known that several properties of charged dilaton black holes depend crucially on the coupling parameter $\beta$ $[2-6]$. Recently one of the present authors and Parikh $[7]$ obtained the energy of a static and spherically symmetric charged dilaton black hole for $\beta = 1$. They found that, similar to the case of the Schwarzschild black hole and unlike the RN black hole, the entire energy is confined to the interior of the black hole. It is of interest to investigate the energy associated with charged dilaton black holes for arbitrary value of $\beta$ to see what the energy distribution is for $\beta < 1$ as well as $\beta >1$ and whether or not the energy is confined to the black hole interior for any other value of $\beta$. The well known energy-momentum pseudotensor of Einstein is $[8]$ \_i\^[ k]{} = H\^[ kl]{}\_[i,  l]{}   , where H\_i\^[ kl]{} = \_[,m]{}  . Latin indices run from $0$ to $3$. $x^0$ is the time coordinate. The energy and momentum components are P\_i = \_[,]{}  dx\^1 dx\^2 dx\^3  , where the Greek index $\alpha$ takes values from $1$ to $3$. $P_0$ and $P_{\alpha}$ stand, respectively, for the energy (say $E$) and momentum components. It is known that the energy-momentum pseudotensors, for obtaining the energy and momentum associated with asymptotically flat spacetimes, give the correct result if calculations are carried out in quasi-cartesian coordinates ( those coordinates in which the metric $g_{ik}$ approaches the Minkowski metric $\eta_{ik}$ at large distance ) $[8-9]$. Transforming the line element $(2)$ to quasi-cartesian coordinates $t,x,y,z ( x = r \sin\th \cos\ph, y = r \sin\th \sin\ph, z = r \cos\th ) $ one gets ds\^2 = B dt\^2 - D (dx\^2+dy\^2+dz\^2) - (x dx+y dy +z dz)\^2. To obtain the energy the required components of $ H_i^{\ kl}$ are H\_0\^[ 01]{} &=& ,\ H\_0\^[ 02]{} &=& ,\ H\_0\^[ 03]{} &=& . By using $(13)$ with $(8)$ in $(11)$, applying the Gauss theorem, and then evaluating the integral over the surface of a sphere of radius $r$, one gets E(r) = M -   ( 1 - \^2 ). Thus one finds that the energy distribution depends on the value of the coupling parameter $\beta$. The energy is confined to its interior [*[only for]{}*]{} $\beta = 1$ and for all other values of $\beta$ the energy is shared by the interior and exterior of the black hole. $\beta=0$ in $(14)$ gives the energy distribution in the RN field (see also ref. $[9]$). $E(r)$ increases with radial distance for $\beta = 0$ (RN spacetime) as well as $\beta<1$, decreases for $\beta>1$, and remains constant for $\beta = 1$. However, the total energy ($r$ approaching infinity in $(14)$) is independent of $\beta$ and is given by the mass parameter of the black hole. The details of the present contribution will be published elsewhere. [**Acknowledgements**]{} This work has been partially supported by the Universidad del Pais Vasco under contract UPV 172.310 - EA062/93 (A.C.) and by a Basque Government post-doctoral fellowship (K.S.V.). We thank A. Achúcarro, J. M. Aguirregabiria, and I. Egusquiza for discussions. [**References**]{}\ $[1]$ D. Garfinkle, G. T. Horowitz and A. Strominger, Phys. Rev. D43\      (1991) 3140; Erratum : Phys. Rev. D45 (1992) 3888.\ $[2]$ J. H. Horne and G. T. Horowitz, Phys. Rev. D46 (1992) 1340.\ $[3]$ K. Shiraishi, Phys. Lett. A166 (1992) 298.\ $[4]$ J. A. Harvey and A. Strominger, Quantum aspects of black holes,\      preprint EFI-92-41, hep-th/9209055 .\ $[5]$ T. Maki and K. Shiraishi, Class. Quant. Grav. 11 (1994) 227.\ $[6]$ C. F. E. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447.\ $[7]$ K. S. Virbhadra and J. C. Parikh, Phys. Lett. B317 (1993) 312.\ $[8]$ C. Møller, Ann. Phys. (NY) 4 (1958) 347.\ $[9]$ K. P. Tod. Proc. Roy. Soc. Lond. A388 (1983) 467;\       K. S. Virbhadra, Phys. Rev. D41 (1990) 1086; Phys. Rev. D42 (1990) 2919;\       F. I. Cooperstock and S. A. Richardson, in Proc. 4th Canadian Conf. on General\        Relativity and Relativistic Astrophysics ( World Scientific, Singapore, 1991 );\      A. Chamorro and K. S. Virbhadra, hep-th/9406148.\

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--- author: - 'Samaneh Abbasi-Sureshjani' - Jiong Zhang - Remco Duits - Bart ter Haar Romeny bibliography: - 'manuscript.bib' date: 'Received: date / Accepted: date' title: 'Retrieving challenging vessel connections in retinal images by line co-occurrence statistics' ---

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--- abstract: 'For $R=Q/J$ with $Q$ a commutative graded algebra over a field and $J\ne0$, we relate the slopes of the minimal resolutions of $R$ over $Q$ and of $k=R/R_{+}$ over $R$. When $Q$ and $R$ are Koszul and $J_1=0$ we prove ${\operatorname{Tor}_{i}^{Q}(R,k){}}_j=0$ for $j>2i\ge0$, and also for $j=2i$ when $i>\dim Q-\dim R$ and ${\operatorname{pd}}_QR$ is finite.' address: - 'Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.' - 'Dipartimento di Matematica, Universit«a di Genova, Via Dodecaneso 35, I-16146 Genova, Italy' - 'Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.' author: - 'Luchezar L. Avramov' - Aldo Conca - 'Srikanth B. Iyengar' date: - - title: | Free resolutions over commutative\ Koszul algebras --- [^1] Let $K$ be a field and $Q$ a commutative ${{\mathbb N}}$-graded $K$-algebra with $Q_0=K$. Each graded $Q$-module $M$ with $M_j=0$ for $j\ll0$ has a unique up to isomorphism minimal graded free resolution, $F^M$. The module $F^M_i$ has a basis element in degree $j$ if and only if ${\operatorname{Tor}_{i}^{Q}(k,M){}}_j\ne0$ holds, where $k=Q/Q_{{\!\scriptscriptstyle{+}}}$ for $Q_{{\!\scriptscriptstyle{+}}}=\bigoplus_{j{{{\scriptstyle}\geqslant}}1}Q_j$. Important structural information on $F^M$ is encoded in the sequence of numbers $${t_{i}^{Q}(M){}}=\sup\{j\in{{\mathbb Z}}\mid{\operatorname{Tor}_{i}^{Q}(k,M){}}_j\ne0\}\,.$$ It is distilled through the notion of *Castelnuovo-Mumford regularity*, defined by $${\operatorname{reg}}_QM=\sup_{i{{{\scriptstyle}\geqslant}}0}\{{t_{i}^{Q}(M){}}-i\}\,.$$ One has ${\operatorname{reg}}_Qk\ge0$, and equality means that $Q$ is *Koszul*; see, for instance, [@PP]. When the $K$-algebra $Q$ is finitely generated, every finitely genetrated graded $Q$-module $M$ has finite regularity if and only if $Q$ is a polynomial ring over some Koszul algebra, see [@AP]; by contrast, the *slope* of $M$ over $Q$, defined to be the real number $${\operatorname{slope}}_{Q}M=\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{Q}(M){}}-{t_{0}^{Q}(M){}}}{i}\right\}\,,$$ is always finite; see Corollary \[cor:rate\]. Following Backelin [@Ba], we set ${\operatorname{Rate}}Q={\operatorname{slope}}_QQ_{{\!\scriptscriptstyle{+}}}$ and note that one has ${\operatorname{Rate}}Q\geq 1$, with equality if and only if $Q$ is Koszul. \[thm:main\] If $Q$ is a finitely generated commutative Koszul $K$-algebra and $J$ a homogeneous ideal with $0\ne J\subseteq(Q_{{\!\scriptscriptstyle{+}}})^2$, then for $R=Q/J$ and $c={\operatorname{Rate}}R$ one has 1. $\max\{c,2\}\le{\operatorname{slope}}_QR\le c+1$, with $c<{\operatorname{slope}}_QR$ when ${\operatorname{pd}}_QR$ is finite. 2. ${t_{i}^{Q}(R){}}=(c+1)\cdot i$ for some $i\ge1$ implies the following conditions: ${t_{h}^{Q}(R){}}=(c+1)\cdot h$ for $1\le h\le i$ and $i\le{\operatorname{rank}}_k(J/Q_{{\!\scriptscriptstyle{+}}}J)_{c+1}$. 3. ${t_{i}^{Q}(R){}}<(c+1)\cdot i$ holds for all $i>\dim Q-\dim R$ when ${\operatorname{pd}}_QR$ is finite. 4. ${\operatorname{reg}}_QR\le c\cdot{\operatorname{pd}}_QR$; when $Q$ is a standard graded polynomial ring, equality holds if and only if $J$ is generated by a $Q$-regular sequence of forms of degree $c+1$. The result is new even in the case of a polynomial ring $Q$, where a related statement was initially proved by using Gröbner bases; see \[rem:taylor\]. The theorem is proved in Section \[sec:koszul\]. Its assertions have very different underpinnings: The inequalities in (1) come from results in homological algebra, established in Section \[sec:rate\] with no finiteness or hypotheses on $Q$. The remaining statements are deduced from results about small homomorphism $Q\to R$, proved in Section \[sec:small\] by using delicate properties of commutative noetherian rings. Much of the discussion in the body of the paper concerns the general problem of relating properties of the numbers ${\operatorname{slope}}_QM$, ${\operatorname{slope}}_QR$, and ${\operatorname{slope}}_RM$, when $Q\to R$ is a homomorphism of graded $K$-algebras and $M$ is a graded module defined over $R$. The essence of our results is a comparison of two types of degrees, ones arising from homological considerations, the others induced by internal gradings of the objects under study. In constructions involving two or more gradings the index referring to an internal degree always appears last. When $y$ is a homogeneous element of a bigraded object, $|y|$ denotes the *homological degree* and $\deg(y)$ the *internal degree*. The proofs presented below involve various homological constructions that are well documented in the case of commutative local rings and their local homomorphisms, but for which graded analogs may be difficult to find in the literature. When explicit information on the behavior of internal degrees is needed, we give the statements in the graded context with references to sources dealing with the local situation. We have verified—and invite readers to follow suit—that in these instances an internal degree can be factored in all the arguments involved. Slopes of graded modules {#sec:rate} ======================== In this section ${{\varphi}}{\colon}Q\to R$ is a surjective homomorphism of graded $K$-algebras, and $M$ is a graded $R$-module with $M_j=0$ for all $j\ll0$; we set $J={\operatorname{Ker}}{{\varphi}}$. We recall a classical change-of-rings spectral sequence of Cartan and Eilenberg. \[ce\] By [@CE Ch.XVI, §5], there exists a spectral sequence of trigraded $k$-vector spaces $$\label{eq:cesequence} {{}^{r}\!\operatorname{E}_{p,q,j}}\underset{p}{\implies}{\operatorname{Tor}_{p+q}^{Q}(k,M){}}_j \quad\text{for}\quad r\ge2\,,$$ with differentials acting according to the pattern $$\label{eq:cedifferential} {{}^{r}\!\operatorname{d}_{p,q,j}}{\colon}{{}^{r}\!\operatorname{E}_{p,q,j}}\to{{}^{r}\!\operatorname{E}_{p-r,q+r-1,j}} \quad\text{for}\quad r\ge2\,,$$ with second page of the form $$\label{eq:ceE2} {{}^{2}\!\operatorname{E}_{p,q,j}}\cong \bigoplus_{j_1+j_2=j}{\operatorname{Tor}_{p}^{R}(k,M){}}_{j_1}\otimes_{k}{\operatorname{Tor}_{q}^{Q}(k,R){}}_{j_2}\,,$$ and with edge homomorphisms $$\label{eq:ceedge} {\operatorname{Tor}_{i}^{Q}(k,M){}}_{j}{\twoheadrightarrow}{{}^{\infty}\!\operatorname{E}_{i,0,j}}={{}^{i+1}\!\operatorname{E}_{i,0,j}}{\hookrightarrow}{{}^{2}\!\operatorname{E}_{i,0,j}}\cong{\operatorname{Tor}_{i}^{R}(k,M){}}_{j}$$ equal to the canonical homomorphisms of $k$-vector spaces $$\label{eq:cechange} {\operatorname{Tor}_{i}^{{{\varphi}}}(k,M){}}_j{\colon}{\operatorname{Tor}_{i}^{Q}(k,M){}}_j\to{\operatorname{Tor}_{i}^{R}(k,M){}}_j\,.$$ For all $r$, $p$, and $q$ we set $\sup{{}^{r}\!\operatorname{E}_{p,q,*}}=\sup\{j\in{{\mathbb Z}}\mid {{}^{r}\!\operatorname{E}_{p,q,j}}\ne0\}$. The proof of the next result is based on an analysis of the convergence of the preceding change-of-rings spectral sequence on the line $q=0$. \[thm:ceub\] When $J\ne QJ_1$ holds there are inequalities $${\operatorname{slope}}_{R}M \leq \max\left\{{\operatorname{slope}}_{Q}M\,, \sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{Q}(R){}}-1}{i}\right\}\right\} \leq\max\{{\operatorname{slope}}_{Q}M,{\operatorname{slope}}_QR\}\,.$$ If ${t_{i}^{Q}(R){}}$ or ${t_{i}^{Q}(M){}}$ is infinite for some $i\ge0$, then so are both maxima above, hence there is nothing to prove. Thus, we may assume that ${t_{i}^{Q}(R){}}$ and ${t_{i}^{Q}(M){}}$ are finite for every $i\ge0$; in this case the second inequality is clear. Let $m$ denote the middle term in the inequalities above. Using the equality ${t_{0}^{Q}(M){}}={t_{0}^{R}(M){}}$, we get $$\begin{aligned} \tag*{(\ref{thm:ceub}.1)${}_{i}$} {t_{i}^{Q}(M){}}&\leq mi+{t_{0}^{R}(M){}}\,; \\ \tag*{(\ref{thm:ceub}.2)${}_{i}$} {t_{i}^{Q}(R){}}&\leq mi+1\,. \end{aligned}$$ For $i\ge0$ and $r\geq 2$, from fomulas and one gets exact sequences $$\label{eq:celimit} 0{\longrightarrow}{{}^{r+1}\!\operatorname{E}_{i,0,j}}{\longrightarrow}{{}^{r}\!\operatorname{E}_{i,0,j}}{\xrightarrow}{\ {{}^{r}\!\operatorname{d}_{i,0,j}} \ }{{}^{r}\!\operatorname{E}_{i-r,r-1,j}} \,. \tag{\ref{thm:ceub}.3}$$ We set up a primary induction on $i$ and a secondary, descending one, on $r$ to prove $$\begin{aligned} \tag*{(\ref{thm:ceub}.4)${}_{i,r}$} \sup {{}^{r}\!\operatorname{E}_{i,0,*}} &\le mi + {t_{0}^{R}(M){}} \quad\text{and}\quad i+1\ge r\ge2\,. \end{aligned}$$ In view of , the validity of [(\[thm:ceub\].4)${}_{i,2}$]{} is the assertion of the proposition. The basis of the primary induction, for $i=1$, comes from and (\[thm:ceub\].1)${}_{1}$. Fix an integer $i\ge2$ and assume that (\[thm:ceub\].4)${}_{i',r}$ holds for $i'<i$. Formulas and (\[thm:ceub\].1)${}_{i}$ imply (\[thm:ceub\].4)${}_{i,i+1}$. Fix $r\in[2,i]$ and assume that (\[thm:ceub\].4)${}_{i,r'}$ holds for $i+1\ge r'>r$. The first relation in the following chain $$\begin{aligned} \sup {{}^{r}\!\operatorname{E}_{i,0,*}} &\leq \max\{\sup {{}^{r+1}\!\operatorname{E}_{i,0,*}} \,, \sup {{}^{r}\!\operatorname{E}_{i-r,r-1,*}} \}\\ &\leq \max\{mi+{t_{0}^{R}(M){}} \,, \sup {{}^{r}\!\operatorname{E}_{i-r,r-1,*}} \}\\ &\leq \max\{mi+{t_{0}^{R}(M){}} \,, \sup {{}^{2}\!\operatorname{E}_{i-r,r-1,*}} \}\\ &= \max\{mi+{t_{0}^{R}(M){}} \,, {t_{i-r}^{R}(M){}}+{t_{r-1}^{Q}(R){}} \}\\ &\leq \max\{mi+{t_{0}^{R}(M){}} \,,(m(i-r)+{t_{0}^{R}(M){}}) + (m(r-1)+1)\}\\ & = \max\{mi+{t_{0}^{R}(M){}} \,, mi+{t_{0}^{R}(M){}}-(m-1)\}\\ &\leq mi + {t_{0}^{R}(M){}} \end{aligned}$$ comes from the exact sequence . The second one holds by (\[thm:ceub\].4)${}_{i,r+1}$, the third because ${{}^{r}\!\operatorname{E}_{i-r,r-1,*}}$ is a subfactor of ${{}^{2}\!\operatorname{E}_{i-r,r-1,*}}$, the fourth by , the fifth by (\[thm:ceub\].4)${}_{i-r,2}$ and (\[thm:ceub\].2)${}_{r-1}$, and the last one because $J\ne QJ_1$ implies $m\geq1$. This completes the inductive proof of the inequality (\[thm:ceub\].4)${}_{i,r}$. Variants of the proposition have been known for some time, at least when $M$ is finitely generated and $R$ is *standard graded*; that is, $R=K[R_1]$ with ${\operatorname{rank}}_KR_1$ finite. Thus, Aramova, Bărcănescu, and Herzog in [@ABH 1.3] established the corresponding result for a related invariant, ${\operatorname{rate}}_RM=\sup_{i{{{\scriptstyle}\geqslant}}1}\{{t_{i}^{Q}(M){}}/i\}$. They used the same spectral sequence, extending an argument of Avramov for $M=k$, see [@Ba p. 97]; in the latter case, the corollary below was first proved by Anick in [@An 4.2]. \[cor:rate\] If $R$ is finitely generated over $K$, then for every finitely generated $R$-module $M$ one has ${\operatorname{slope}}_RM<\infty$. One may choose $Q$ to be a polynomial ring in finitely many indeterminates over $K$. In this case ${\operatorname{Tor}_{i}^{Q}(k,R){}}_*$ and ${\operatorname{Tor}_{i}^{Q}(k,M){}}_*$ are finitely generated over $k$ for each $i\ge0$ and are zero for almost all $i$, so ${\operatorname{slope}}_QR$ and ${\operatorname{slope}}_QM$ are finite. In the proof of the next result we again use the spectral sequence in \[ce\], this time analyzing its convergence on the line $p=0$. The hypothesis includes a condition on the maps ${\operatorname{Tor}_{i}^{{{\varphi}}}(k,M){}}_j$; see \[ch:small\] and Proposition \[prop:koszul\_small\] for situations where it is met. \[thm:celb\] If $M\ne0$ and ${\operatorname{Tor}_{i}^{{{\varphi}}}(k,M){}}$ is injective for each $i$, then one has $$\begin{aligned} {\operatorname{slope}}_{Q}R &\leq 1+ s \quad\text{where}\quad s=\sup_{i{{{\scriptstyle}\geqslant}}2}\left\{\frac{{t_{i}^{R}(M){}} - {t_{0}^{R}(M){}}-1}{i-1}\right\}\,. \end{aligned}$$ The hypothesis implies ${t_{0}^{R}(M){}}>-\infty$. There is nothing to prove if ${t_{i}^{Q}(M){}}=\infty$ for some $i$, so we assume that ${t_{i}^{Q}(M){}}$ is finite for all $i\ge0$. By the definition of the number $s$, the following inequalities then hold: $$\label{eq:sup} \tag*{(\ref{thm:celb}.1)${}_{i}$} {t_{i}^{R}(M){}} \leq s(i-1)+1+{t_{0}^{R}(M){}}\quad \text{for all}\quad i\ge 2\,.$$ It follows from and that for $r\geq 2$ there exist exact sequences $$\label{eq:cecolimit} {{}^{r}\!\operatorname{E}_{r,i-r+1,j}}{\xrightarrow}{\ {{}^{r}\!\operatorname{d}_{r,i-r+1,j}} \ }{{}^{r}\!\operatorname{E}_{0,i,j}}{\longrightarrow}{{}^{r+1}\!\operatorname{E}_{0,i,j}}{\longrightarrow}0 \tag*{(\ref{thm:celb}.2)}$$ By primary induction on $i$ and secondary, descending induction on $r$, we prove $$\begin{aligned} \tag*{(\ref{thm:celb}.3)${}_{i,r}$} \sup {{}^{r}\!\operatorname{E}_{0,i,*}}& \leq (s+1)i + {t_{0}^{R}(M){}} \quad\text{for}\quad i+2\ge r\ge2\,. \end{aligned}$$ In view of , the validity of [(\[thm:celb\].3)${}_{i,2}$]{} yields the assertion of the proposition. The injectivity of ${\operatorname{Tor}_{}^{{{\varphi}}}(k,M){}}$ and imply ${{}^{\infty}\!\operatorname{E}_{p,q,*}}=0$ for $q\ge 1$ and all $p$. It follows from and that ${{}^{n+2}\!\operatorname{E}_{0,i,*}}$ is isomorphic to ${\operatorname{Tor}_{0}^{R}(k,M){}}_*$ for $i=0$ and to $0$ for $i\ge1$, so [(\[thm:celb\].3)${}_{i,i+2}$]{} holds for all $i\ge0$. This gives the basis of the primary induction for $i=0$ and that of the secondary induction for all $i\ge1$. Fix an integer $i\ge1$ and assume that (\[thm:celb\].3)${}_{i',r'}$ holds for all pairs $(i',r')$ with $i'<i$ and $i+2\ge r'>r$. One then has a chain of relations $$\begin{aligned} \sup {{}^{r}\!\operatorname{E}_{r,i-r+1,*}} &\leq \sup {{}^{2}\!\operatorname{E}_{r,i-r+1,*}} \\ & = {t_{r}^{R}(M){}} + {t_{i-r+1}^{Q}(R){}}\\ &\leq {t_{r}^{R}(M){}} + (s+1)(i-r+1)\\ &\leq s(r-1)+1 + {t_{0}^{R}(M){}} + (s+1)(i-r+1) \\ &= (s+1)i + (2-r) + {t_{0}^{R}(M){}}\\ &\leq (s+1)i + {t_{0}^{R}(M){}}\,,\end{aligned}$$ where the first one holds because ${{}^{r}\!\operatorname{E}_{r,i-r+1,*}} $ is a subfactor of ${{}^{2}\!\operatorname{E}_{r,i-r+1,*}}$, the second by formula , the third by (\[thm:celb\].3)${}_{i-r+2,2}$ and , and the fourth by (\[thm:celb\].1)${}_{r}$. The exact sequence \[eq:cecolimit\], the preceding inequalities, and (\[thm:celb\].3)${}_{i,r+1}$ give $$\begin{aligned} \sup {{}^{r}\!\operatorname{E}_{0,i,*}} &\leq\max\{\sup{{}^{r+1}\!\operatorname{E}_{0,i,*}}\,,\sup{{}^{r}\!\operatorname{E}_{r,i-r+1,*}}\} \\ &\leq (s+1)i + {t_{0}^{R}(M){}}\,. \end{aligned}$$ Hereby, the inductive proof of the inequality (\[thm:celb\].3)${}_{i,r}$ is complete. Regular elements {#sec:reg} ================ Not surprisingly, the bounds obtained in the preceding section can be sharpened in cases when the minimal free resolution of $R$ or of $M$ over $Q$ is particularly simple. In this section we discuss a classical avatar of this phenomenon. \[thm:reg\] If $R=Q/(f)$ for a non-zero divisor $f\in Q_{{\!\scriptscriptstyle{+}}}$, then one has: $$\begin{aligned} {3} \tag{1} {\operatorname{slope}}_QM&\le\max\{{\operatorname{slope}}_RM,\deg(f)\} &\quad&\text{with equality for } \quad &f&\notin (Q_{{\!\scriptscriptstyle{+}}})^2\,. \\ \tag{2} {\operatorname{slope}}_RM&\le\max\{{\operatorname{slope}}_QM,\deg(f)/2\} &\quad&\text{with equality for} \quad &f&\in Q_{{\!\scriptscriptstyle{+}}}{\operatorname{Ann}}_QM\,. \end{aligned}$$ We start by noting an elementary inequality that will be invoked a couple of times: All pairs of real numbers $(a_1,a_2)$ and $(b_1,b_2)$ with positive $b_1$ and $b_2$ satisfy $$\label{eq:short} \frac {a_1+a_2}{b_1+b_2} \leq \max\left\{\frac{a_1}{b_1}\,,\,\frac{a_2}{b_2}\right\}\,.$$ Set $d=\deg(f)$. The minimal graded free resolution of $R$ over $Q$ is $$0{\longrightarrow}Q(-d) {\xrightarrow}{\ f \ } Q{\longrightarrow}0$$ so ${\operatorname{Tor}_{q}^{Q}(R,k){}}$ vanishes for $q\ne0,1$, is isomorphic to $k$ for $q=0$, and to $k(-d)$ for $q=1$, so for each pair $(i,j)$ the spectral sequence \[ce\] yields an exact sequence $$\label{eq:long} \begin{gathered} \xymatrixcolsep{1.3pc} \xymatrixrowsep{.3pc} \xymatrix { &&{\operatorname{Tor}_{i+1}^{R}(k,M){}}_{j}\ar@{->}[rr]^-{\delta_{i+1,j}} &&{\operatorname{Tor}_{i-1}^{R}(k,M){}}_{j-d} \\ \ar@{->}[r] &{\operatorname{Tor}_{i}^{Q}(k,M){}}_{j}\ar@{->}[r] &{\operatorname{Tor}_{i}^{R}(k,M){}}_{j}\ar@{->}[rr]^-{\delta_{i,j}} &&{\operatorname{Tor}_{i-2}^{R}(k,M){}}_{j-d} } \end{gathered}$$ The one for $i=0$ gives the following equality: $$\label{eq:zero} {t_{0}^{Q}(M){}}={t_{0}^{R}(M){}}\,.$$ \(1) For $i\ge1$ the middle three terms of the exact sequences yield $$\label{eq:long1} \begin{aligned} {t_{i}^{Q}(M){}} &\le\max\{{t_{i}^{R}(M){}},({t_{i-1}^{R}(M){}}+d)\} \end{aligned}$$ [From]{} , , and we obtain the inequalities below: $$\begin{aligned} {\operatorname{slope}}_QM &=\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{Q}(M){}}-{t_{0}^{Q}(M){}}}{i}\right\} \\ &\le \sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\max\left\{\frac{{t_{i}^{R}(M){}}-{t_{0}^{R}(M){}}}i\,,\, \frac{({t_{i-1}^{R}(M){}}-{t_{0}^{R}(M){}})+d}{(i-1)+1}\right\}\right\} \\ &\leq\sup_{i{{{\scriptstyle}\geqslant}}2}\left\{\max\left\{\frac{{t_{i}^{R}(M){}}-{t_{0}^{R}(M){}}}{i}\,, \frac{{t_{i-1}^{R}(M){}}-{t_{0}^{R}(M){}}}{i-1},d\right\}\right\} \\ &=\max\left\{\sup_{i{{{\scriptstyle}\geqslant}}1} \left\{\frac{{t_{i}^{R}(M){}}-{t_{0}^{R}(M){}}}{i}\right\},d\right\} \\ &=\max\left\{{\operatorname{slope}}_RM,d\right\}\,. \end{aligned}$$ When $f\notin(Q_{{\!\scriptscriptstyle{+}}})^2$ holds, the proof in [@Av:barca 3.3.3(1)] of a result of Nagata, implies $\delta_{i,j}=0$ in , so equalities hold in . This and give $$\begin{aligned} {t_{1}^{Q}(M){}}-{t_{0}^{Q}(M){}}&=\max\{{t_{1}^{R}(M){}}-{t_{0}^{R}(M){}},d\}\,, \\ {t_{i}^{Q}(M){}}-{t_{0}^{Q}(M){}}&\ge{t_{i}^{R}(M){}}-{t_{0}^{R}(M){}} \quad\text{for}\quad i\ge2\,. \end{aligned}$$ The preceding relations clearly imply ${\operatorname{slope}}_QM\ge\max\{{\operatorname{slope}}_RM,d\}$. \(2) For $i\ge1$ the last three terms of the exact sequences yield $$\label{eq:long2} \begin{aligned} {t_{i}^{R}(M){}} &\le\max\{{t_{i}^{Q}(M){}},({t_{i-2}^{R}(M){}}+d)\}\\ &\le\max\{{t_{i}^{Q}(M){}},({t_{i-2}^{Q}(M){}}+d),({t_{i-4}^{R}(M){}}+2d)\} \le\cdots\\ &\le\max_{0{{{\scriptstyle}\leqslant}}2h{{{\scriptstyle}\leqslant}}i}\{{t_{i-2h}^{Q}(M){}}+hd\}\,. \end{aligned}$$ [From]{} , , and we obtain the inequalities below: $$\begin{aligned} {\operatorname{slope}}_RM &=\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{R}(M){}}-{t_{0}^{R}(M){}}}{i}\right\} \\ &\le\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\max_{0{{{\scriptstyle}\leqslant}}2h{{{\scriptstyle}\leqslant}}i} \left\{\frac{{t_{i-2h}^{Q}(M){}}-{t_{0}^{Q}(M){}}+hd}{(i-2h)+(2h)}\right\}\right\} \\ &\leq\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\max_{0{{{\scriptstyle}\leqslant}}2h< i} \left\{\frac{{t_{i-2h}^{Q}(M){}}-{t_{0}^{Q}(M){}}}{i-2h}\,,\,\frac{d}{2}\right\}\right\} \\ &=\max\left\{\sup_{i{{{\scriptstyle}\geqslant}}1} \left\{\frac{{t_{i}^{Q}(M){}}-{t_{0}^{Q}(M){}}}{i}\right\}\,,\,\frac{d}{2}\right\} \\ &=\max\left\{{\operatorname{slope}}_QM\,,\,\frac{d}{2}\right\}\,. \end{aligned}$$ For $f\in Q_{{\!\scriptscriptstyle{+}}}{\operatorname{Ann}}_QM$, the proof in [@Av:barca 3.3.3(2)] of a result of Shamash shows that $\delta_{i,*}$ in is surjective, so equalities hold in ; in view of one gets $$\begin{aligned} {t_{1}^{R}(M){}}-{t_{0}^{R}(M){}}&={t_{1}^{Q}(M){}}-{t_{0}^{Q}(M){}}\,, \\ {t_{2}^{R}(M){}}-{t_{0}^{R}(M){}}&=\max\{{t_{2}^{Q}(M){}}-{t_{0}^{Q}(M){}},d\}\,, \\ {t_{i}^{R}(M){}}-{t_{0}^{R}(M){}}&\ge{t_{i}^{Q}(M){}}-{t_{0}^{Q}(M){}} \quad\text{for}\quad i\ge3\,. \end{aligned}$$ These relations clearly imply an inequality ${\operatorname{slope}}_RM\ge\max\{{\operatorname{slope}}_QM,d/2\}$. Small homomorphisms of graded algebras {#sec:small} ====================================== A homomorphism ${{\varphi}}{\colon}Q\to R$ of graded $K$-algebras is called *small* if the map $${\operatorname{Tor}_{i}^{{{\varphi}}}(k,k){}}_j{\colon}{\operatorname{Tor}_{i}^{Q}(k,k){}}_j \to {\operatorname{Tor}_{i}^{R}(k,k){}}_j$$ is injective for each pair $(i,j)\in{{\mathbb N}}\times{{\mathbb Z}}$; see \[ch:small\] for examples. Recall that *homological products* turn ${\operatorname{Tor}_{}^{Q}(k,R){}}$ into a bigraded algebra; see [@CE Ch.XI, §4]. \[thm:small\] Let $Q$ be a standard graded $K$-algebra, ${{\varphi}}{\colon}Q\to R$ a surjective small homomorphism of graded $K$-algebras with ${\operatorname{Ker}}{{\varphi}}\ne0$, and set $c={\operatorname{Rate}}R$. For every integer $i\ge1$ there are then inequalities $${{t_{i}^{Q}(R){}}} \le {\operatorname{slope}}_QR\cdot i \le (c+1)\cdot i\,,$$ and the following conditions are equivalent: 1. ${t_{i}^{Q}(R){}}=(c+1)\cdot i$. 2. ${t_{h}^{Q}(R){}}=(c+1)\cdot h$ for $1\le h\le i$. 3. ${t_{1}^{Q}(R){}}=c+1$ and ${\operatorname{Tor}_{i}^{Q}(k,R){}}_{i(c+1)}=({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^i\ne0$. Before starting on the proof of the theorem we present an application, followed by a couple of easily verifiable sufficient conditions for the smallness of ${{\varphi}}$. \[cor:small\] With $J={\operatorname{Ker}}{{\varphi}}$, the following assertions hold: 1. ${t_{i}^{Q}(R){}}=(c+1)\cdot i$ for some $i\ge1$ implies the conditions ${t_{h}^{Q}(R){}}=(c+1)\cdot h$ for $1\le h\le i$ and $i\le{\operatorname{rank}}_k(J/Q_{{\!\scriptscriptstyle{+}}}J)_{c+1}$. 2. ${t_{i}^{Q}(R){}}<(c+1)\cdot i$ holds for all $i>\dim Q-\dim R$ when ${\operatorname{pd}}_QR$ is finite. 3. ${\operatorname{reg}}_QR\le c\cdot{\operatorname{pd}}_QR$. Homological products are strictly skew-commutative for the homological degree, see [@CE Ch.XI, §4], so $({\operatorname{Tor}_{1}^{Q}(k,R){}}_*){}^i$ is the image of a canonical $k$-linear map $$\lambda_{i,*}{\colon}\textstyle{\bigwedge}^i_k(J/Q_{{\!\scriptscriptstyle{+}}}J)_*\cong \textstyle{\bigwedge}^i_k {\operatorname{Tor}_{1}^{Q}(k,R){}}_* \to {\operatorname{Tor}_{i}^{Q}(k,R){}}_*\,.$$ \(1) This follows from the map above and the implication (i)$\implies$(ii) and (iii). \(2) When ${\operatorname{pd}}_QR$ is finite one has ${\operatorname{grade}}_QR=\dim Q-\dim R$ by a theorem of Peskine and Szpiro [@PS], and $\lambda_{i,*}=0$ for $i>{\operatorname{grade}}_QR$ from a theorem of Bruns [@Br]. Thus, Theorem \[thm:small\] implies ${{\operatorname{Tor}_{i}^{Q}(k,R){}}}_j=0$ for $j\ge(c+1)i$. \(3) The theorem gives ${t_{i}^{Q}(R){}}-i\le ci$ for each $i$, hence ${\operatorname{reg}}_QR\le c\cdot{\operatorname{pd}}_QR$. A bit of notation comes in handy at this point. \[canonical\] A standard graded $K$-algebra $R$ has a *canonical presentation* $R={\widetilde}R/I_R$ with ${\widetilde}R$ the symmetric $K$ algebra on $R_1$ and $I_R\subseteq({\widetilde}R_{{\!\scriptscriptstyle{+}}})^2$, obtained from the epimorphism of $K$-algebras ${\widetilde}R\to R$ extending the identity map on $R_1$. If $Q$ is standard graded $K$-algebra and ${{\varphi}}{\colon}Q\to R$ is a surjective homomorphism with ${\operatorname{Ker}}{{\varphi}}\subseteq(Q_{{\!\scriptscriptstyle{+}}})^2$, then ${\widetilde}R\to R$ factors as ${{\widetilde}R}\cong{\widetilde}Q\to Q{\xrightarrow}{{{\varphi}}}R$. \[ch:small\] A homomorphism ${{\varphi}}$ as on \[canonical\] is small if $J={\operatorname{Ker}}{{\varphi}}$ satisfies one of the conditions: 1. $J\subseteq(f_1,\dots,f_a)$, where $f_1,\dots,f_a$ is some $Q$-regular sequence in $Q_{{\!\scriptscriptstyle{+}}}$. 2. $J_j=0$ for $j\le{\operatorname{reg}}_{{\widetilde}Q}Q$, where $Q={{\widetilde}Q}/I_Q$ is the canonical presentation. Indeed, see [@Av:small 4.3] for (a), and Şega [@Se 5.1, 9.2(2)] for (b). *The hypothesis of Theorem *\[thm:small\]* are in force for the rest of this section.* The proof of the theorem utilizes free resolutions with additional structure. A *model* of ${{\varphi}}$ is a differential bigraded $Q$-algebra $Q[X]$ with the following properties: For $n\ge1$ here exist linearly independent over $K$ homogeneous subsets $X_n=\{x\in X\mid |x|=n\}$, such that the underlying bigraded algebra is isomorphic to $Q\otimes_K\bigotimes_{n=1}^{\infty}K[X_n]$, where $K[X_n]$ is the exterior algebra of the graded $K$-vector space $KX_n$ when $n$ is odd, and the symmetric algebra of that space when $n$ is even. The differential satisfies $\deg({\partial}(y))=\deg(y)$ for every element $y\in Q[X]$, and the following sequence of homomorphisms of free graded $Q$-modules is resolution of $R$: $$\cdots {\longrightarrow}Q[X]_{n,*}{\xrightarrow}{\,{\partial}\,} Q[X]_{n-1,*}\cdots {\longrightarrow}\cdots {\longrightarrow}Q[X]_{0,*}{\longrightarrow}0$$ A $Q$-basis of $Q[X]$ is provided by the set consisting of $1$ and all the monomials $x_{1}^{d_1}\cdots x_{s}^{d_s}$ with $x_r\in X$, and with $d_r=1$ when $|x_r|$ is odd, respectively, $d_r\ge1$ when $|x_r|$ is even. The model $Q[X]$ is said to be *minimal* if for each $x\in X$, the coefficient of every $x_i\in X$ in the expansion of ${\partial}(x)$ is contained in $Q_{{\!\scriptscriptstyle{+}}}$. We summarize the properties of minimal models used in our arguments. \[model:exist\] A minimal model $Q[X]$ of ${{\varphi}}$ always exists, and is unique up to non-canonical isomorphism of differential bigraded $Q$-algebras; see [@Av:barca 7.2.4]. In such a model ${\partial}(X_1)$ is a minimal set of homogeneous generators of the ${\operatorname{Ker}}{{\varphi}}$ and $Q[X_1]$ is the Koszul complex on that set, with its standard bigrading, differential and multiplication. \[model:omega\] Let ${{\widetilde}R}[Z]$ be a minimal model for the canonical presentation ${{\widetilde}R}\to R$, see \[canonical\]. Let $Z_0$ be a $K$-basis of ${{\widetilde}R}_1$, and choose a $k$–linearly independent set $$Z'=\{z'\mid |z'|=|z|+1\text{ and }\deg(z')=\deg(z)\}_{z\in Z_0\sqcup Z}\,.$$ By [@Av:barca 7.2.6], there exists an isomorphism of bigraded $k$-vector spaces $${\operatorname{Tor}_{}^{R}(k,k){}}\cong\bigotimes_{n=1}^{\infty}k\langle Z'_n\rangle\,,$$ where $k\langle Z'_n\rangle$ denotes the exterior algebra of the graded $k$-vector space $kZ'_n$ when $n$ is odd, and the divided powers algebra of that space when $n$ is even. \[model:small\] Let $Q[X]$ be a minimal model for ${{\varphi}}$, and let ${{\widetilde}R}{\xrightarrow}{\psi}Q{\xrightarrow}{{{\varphi}}}R$ be a factorization of the canonical presentation ${\widetilde}R\to R$ as in \[canonical\]. If ${{\widetilde}R}[Y]$ is a minimal model for $\psi$, then there is a minimal model ${{\widetilde}R}[Z]$ of ${\widetilde}R\to R$ with $Z=Y\sqcup X$; see [@AI 4.11]. For every integer $i\ge2$ the following equality holds: $$\label{eq:c} {t_{i-1}^{R}(R_{{\!\scriptscriptstyle{+}}}){}}-{t_{0}^{R}(R_{{\!\scriptscriptstyle{+}}}){}}={t_{i}^{R}(k){}}-1\,.$$ Thus, for $i\ge1$ the definition of slope and Proposition \[thm:celb\] applied with $M=k$ give $$\label{eq:t} {{t_{i}^{Q}(R){}}} /i\le{\operatorname{slope}}_QR\le c+1\,.$$ It remains to establish the equivalence of the conditions in the theorem. (iii)$\implies$(ii). The condition $({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^i\ne0$ forces $({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^h\ne0$ for $h=1,\dots,i$. As ${\operatorname{Tor}_{}^{Q}(k,R){}}$ is a bigraded algebra, one gets $${\operatorname{Tor}_{h}^{Q}(k,R){}}_{(c+1)h}\supseteq({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^h\ne0\,.$$ This implies ${t_{h}^{Q}(R){}}\ge(c+1)h$, and provides the converse inequality. (ii)$\implies$(i). This implication is a tautology. (i)$\implies$(iii). The hypothesis means ${\operatorname{Tor}_{i}^{Q}(k,R){}}_{i(c+1)}\ne0$, so we have to prove $$\label{eq:kx0} {\operatorname{Tor}_{i}^{Q}(k,R){}}_{i(c+1)}=({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^i\,.$$ Let $Q[X]\to R$ be a minimal model and set $k[X]=k\otimes_QQ[X]$. The bigraded $k$-algebras ${\operatorname{H}(k[X])}$ and ${\operatorname{Tor}_{}^{Q}(k,R){}}$ are isomorphic, with $${\operatorname{Tor}_{i}^{Q}(k,R){}}_j\cong{\operatorname{H}_{i}(k[X])}_j\,. \label{eq:hkx}$$ In view of \[model:small\] each $x\in X_n$ can be viewed as an indeterminate of a minimal model of ${{\widetilde}R}\to R$, and so by \[model:omega\] it defines an element $x'$ in ${\operatorname{Tor}_{n+1}^{R}(k,k){}}$ with $\deg(x) =\deg(x')$. [From]{} this equality and we obtain $$\label{eq:x} \deg(x) =\deg(x') \le{t_{n+1}^{R}(k){}} \le cn+1=c|x|+1\,.$$ The $k$-vector space $k[X]_{i,(c+1)i}$ has a basis of monomials $x_1^{d_1}\cdots x_s^{d_s}$ with $x_r\in X$ and $d_r\ge1$. The following relations hold, with the inequality coming from : $$\begin{aligned} \sum_{r=1}^sd_r|x_r| &=\big|x_1^{d_1}\cdots x_s^{d_s}\big|=i=(c+1)i-ci\\ &=\deg\big(x_1^{d_1}\cdots x_s^{d_s}\big)-c\big|x_1^{d_1}\cdots x_s^{d_s}\big| \\ &=\sum_{r=1}^sd_r(\deg(x_r)-c|x_r|) \\ &\le\sum_{r=1}^sd_r\,. \end{aligned}$$ All $d_r$ and $|x_r|$ are positive integers, so for $1\le r\le s$ we get first $|x_r|=1$, then $|x_r|=\deg(x_r)-c|x_r|$; thats is, $\deg(x_r)=c+1$. We have now proved $$k[X]_{i,(c+1)i}=k[X_1]_{i,(c+1)i}=(kX_{1,c+1})^i\,.$$ The isomorphism maps ${\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1}$ to $kX_{1,c+1}$ and ${\operatorname{Tor}_{i}^{Q}(k,R){}}_{(c+1)i}$ to a quotient of $k[X]_{i,(c+1)i}$, so the equalities above establish . Koszul agebras {#sec:koszul} ============== In this section we prove and discuss the theorem stated in the introduction. Here $Q$ is a standard graded $K$-algebra, ${{\varphi}}{\colon}Q\to R$ a surjective homomorphism of graded $K$-algebras, and $M$ a graded $R$-module. As in [@PP], we say that $M$ is *Koszul* over $Q$ if ${\operatorname{Tor}_{i}^{Q}(k,M){}}_j=0$ unless $i=j$. In the following proposition the Koszul hypotheses are related to the injectivity of ${\operatorname{Tor}_{}^{{{\varphi}}}(k,M){}}$ through the following lemma. \[prop:koszul\_small\] Assume that $J$ is contained in $(Q_{{\!\scriptscriptstyle{+}}})^2$. 1. If $Q$ is Koszul, then ${{\varphi}}$ is small. 2. If ${{\varphi}}$ is small and $M$ is Koszul over $Q$, then ${\operatorname{Tor}_{}^{{{\varphi}}}(k,M){}}$ is injective. Forming vector space duals, one sees that the injectivity of ${\operatorname{Tor}_{}^{{{\varphi}}}(k,M){}}$ is equivalent to surjectivity of the homomorphism of bigraded $k$-vector spaces $${\operatorname{Ext}^{}_{{{\varphi}}}(M,k){}}{\colon}{\operatorname{Ext}^{}_{R}(M,k){}}\to{\operatorname{Ext}^{}_{Q}(M,k){}}\,.$$ \(1) For $M=k$ the map above is a homomorphism of $K$-algebras, with multiplication given by Yoneda products. The map ${\operatorname{Ext}^{1}_{{{\varphi}}}(k,k){}}_*$ is isomorphic to $${\operatorname{Hom}_{R}({{\varphi}}_1,k)}_*{\colon}{\operatorname{Hom}_{R}(R_1,k)}_*\to{\operatorname{Hom}_{Q}(Q_1,k)}_*\,,$$ which is bijective as $J\subseteq(Q_{{\!\scriptscriptstyle{+}}})^2$ holds. As $Q$ is Koszul, the $k$ -algebra ${\operatorname{Ext}^{}_{Q}(k,k){}}$ is generated by ${\operatorname{Ext}^{1}_{Q}(k,k){}}$, see [@PP Ch.2, §1, Def.1], so ${\operatorname{Ext}^{}_{{{\varphi}}}(k,k){}}$ is surjective. \(2) Yoneda products turn ${\operatorname{Ext}^{}_{{{\varphi}}}(M,k){}}$ into a homomorphism of bigraded left modules over ${\operatorname{Ext}^{}_{R}(k,k){}}$, with this algebra acting on ${\operatorname{Ext}^{0}_{Q}(M,k){}}$ through ${\operatorname{Ext}^{}_{{{\varphi}}}(k,k){}}$. The bigraded module ${\operatorname{Ext}^{}_{Q}(M,k){}}$ is generated over ${\operatorname{Ext}^{}_{Q}(k,k){}}$ by ${\operatorname{Ext}^{0}_{Q}(M,k){}}$, because $M$ is Koszul over $Q$; see [@PP Ch.2, §1, Def.2]. Since ${{\varphi}}$ is small, ${\operatorname{Ext}^{0}_{{{\varphi}}}(k,k){}}_*$ is surjective, and hence ${\operatorname{Ext}^{0}_{Q}(M,k){}}$ generates ${\operatorname{Ext}^{}_{Q}(M,k){}}$ as an ${\operatorname{Ext}^{}_{R}(k,k){}}$-module as well. The map ${\operatorname{Ext}^{0}_{{{\varphi}}}(M,k){}}_*$ is surjective, because it is canonically isomorphic to the identity map of ${\operatorname{Hom}_{k}(M_0,k)}_*$. It follows that ${\operatorname{Ext}^{}_{{{\varphi}}}(M,k){}}$ is surjective. Recall that $Q$ is Koszul, $J$ is a non-zero ideal of $Q$ with $J_1=0$, and $c={\operatorname{slope}}_R{R_{{\!\scriptscriptstyle{+}}}}$. Note that ${{\varphi}}$ is small by Proposition \[prop:koszul\_small\](1). \(1) The inequality ${\operatorname{slope}}_QR\le c+1$ was proved as part of Theorem \[thm:small\]. One has ${t_{i}^{Q}(k){}}=i$ for $1\le i<{\operatorname{pd}}_Qk+1$ by the Koszul hypothesis on $Q$, and ${t_{i}^{Q}(R){}}\ge i+1$ for $1\le i<{\operatorname{pd}}_QR+1$ by the conditions $J_1=0$. The exact sequence $${\operatorname{Tor}_{i+1}^{Q}(k,k){}}\to{\operatorname{Tor}_{i}^{Q}(k,R_{{\!\scriptscriptstyle{+}}}){}}\to{\operatorname{Tor}_{i}^{Q}(k,R){}}\,.$$ of graded vector spaces, which holds for every $i\ge1$, therefore implies $${t_{i}^{Q}(R_{{\!\scriptscriptstyle{+}}}){}}\le\max\{{t_{i+1}^{Q}(k){}},{t_{i}^{Q}(R){}}\}={t_{i}^{Q}(R){}}\,,$$ and hence ${\operatorname{slope}}_QR_{{\!\scriptscriptstyle{+}}}\le\sup_{i{{{\scriptstyle}\geqslant}}1}\{({t_{i}^{Q}(R){}}-1)/i\}$. Now Proposition \[thm:ceub\] gives $$c \le\max\left\{{\operatorname{slope}}_Q{R_{{\!\scriptscriptstyle{+}}}},\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{Q}(R){}}-1}i\right\}\right\} \le\sup_{i{{{\scriptstyle}\geqslant}}1}\left\{\frac{{t_{i}^{Q}(R){}}-1}i\right\} \le{\operatorname{slope}}_QR\,.$$ When ${\operatorname{pd}}_QR$ is finite the last inequality is strict, so one has $c<{\operatorname{slope}}_QR$. The inequalities in (2), (3), and (4) were proved as part of Corollary \[cor:small\]. Finally, assume that $Q$ is a standard graded polynomial ring and ${\operatorname{reg}}_QR=cp$ holds with $p={\operatorname{pd}}_QR$. Theorem \[thm:small\] then shows that $({\operatorname{Tor}_{1}^{Q}(k,R){}}_{c+1})^p$ is not zero, and so ${\operatorname{Ker}}{{\varphi}}$ needs at least $p$ minimal generators of degree $c+1$. As a bigraded $k$-algebra, ${\operatorname{Tor}_{}^{Q}(k,R){}}$ is isomorphic to the homology of the Koszul $E$ complex on some $K$-basis of $Q_1$, so one also has $({\operatorname{H}_{1}(E)})^p\ne0$. Now a theorem of Wiebe, see [@BH 2.3.15], implies that ${\operatorname{Ker}}{{\varphi}}$ is generated by a $Q$-regular sequence of $p$ elements. \[prop:canonical\] For a Koszul $K$-algebra $Q$ and $R=Q/J$ with $J\subseteq(Q_{{\!\scriptscriptstyle{+}}})^2$ one has $$2\le{\operatorname{slope}}_QR\le{\operatorname{slope}}_{{\widetilde}R}R\,,$$ where $R={\widetilde}R/I_R$ is the canonical presentation. Equalities hold when $R$ is Koszul. The canonical presentation factors as ${{\widetilde}R}\to Q{\xrightarrow}{{{\varphi}}}R$; see \[canonical\]. Part (1) of the main theorem, applied to the homomorphism ${{\widetilde}R}\to Q$ and the $Q$-module $R$, gives inequalities $2\le{\operatorname{slope}}_{{\widetilde}R}Q\le{\operatorname{Rate}}Q+1=2$, so Proposition \[thm:ceub\] yields $${\operatorname{slope}}_QR\le\max\{{\operatorname{slope}}_{{\widetilde}R}R,{\operatorname{slope}}_{{\widetilde}R}Q\} =\max\{{\operatorname{slope}}_{{\widetilde}R}R,2\}={\operatorname{slope}}_{{\widetilde}R}R\,.$$ When $R$ is Koszul, the computation above gives $2\le{\operatorname{slope}}_{{\widetilde}R}R\le{\operatorname{Rate}}R+1=2$. The last assertion of Proposition \[prop:canonical\] does not admit a converse. To demonstrate this we appeal to a family of graded algebras constructed by Roos [@Ro]. Recall that the formal power series $H_{M}(s)=\sum_{j\in{{\mathbb N}}}{\operatorname{rank}}_KM_js^j$ in ${{\mathbb Z}}[\![s]\!]$ is called the *Hilbert series* of $M$, and the formal Laurent series $P^{R}_k(s,t)=\sum_{i\in{{\mathbb N}},j\in{{\mathbb Z}}}\beta_{i,j}^R(M)\,s^jt^i $ in ${{\mathbb Z}}[s^{\pm1}][\![t]\!]$, where $\beta_{i,j}^R(M)={\operatorname{rank}}_k{\operatorname{Tor}_{i}^{R}(k,M){}}_j$, is known as its *graded Poincaré series*. \[ch:roos\] Let $P=K[x_1,x_2,x_3,x_4,x_5,x_6]$ be a polynomial ring. For each integer $a\ge2$ set $R{(a)}=P/I{(a)}$, where $I{(a)}$ is the ideal $$\big(\{x_i^2\}_{1\le i\le 6}\,,\,\{x_{i}x_{i+1}\}_{1\le i\le 5}\,,\, x_1x_3+ax_3x_6-x_4x_6\,,\,x_1x_4+x_3x_6+(a-2)x_4x_6\big)\,.$$ When the characteristic of $K$ is zero, Roos [@Ro Thm.1$'$] proves the equalities $$H_{R(a)}(s)=1+6s+8s^2 \quad\text{and}\quad P^{R{(a)}}_k(s,t)=\frac1{H_{R(a)}(-st)-(st)^{a+1}(s+st)}\,.$$ For each $a\ge2$ the graded $K$-algebra $R{(a)}$ from \[ch:roos\] satisfies $${\operatorname{slope}}_{P}R{(a)}-1=1<1+(1/a)\le{\operatorname{Rate}}R{(a)}\le1+(2/a)\,.$$ Indeed, one has ${t_{1}^{P}(R(n)){}}=2$ because $I(a)$ is generated by quadrics. The isomorphism ${\operatorname{Tor}_{i}^{P}(k,R(a)){}}_j\simeq{\operatorname{H}_{i}(E\otimes_P{R(a)})}_j$, where $E$ denotes the Koszul complex on some basis of ${P}_1$, and the equalities ${R(a)}_j=0$ for $j\ge3$ imply ${t_{i}^{P}(R(a)){}}\le i+2$ for $2\le i\le 6$. Comparing the numbers ${t_{i}^{P}(R(a)){}}/i$, one gets ${\operatorname{slope}}_{P}R{(a)}=2$. Following [@ABH], for each $f(s,t)=\sum_{i,j{{{\scriptstyle}\geqslant}}0}b_{i,j}s^jt^i\in{{\mathbb R}}[s][\![t]\!]$ we set $${\operatorname{rate}}(f(s,t))=\sup_{i,j}\{j/i\mid i\ge1\text{ and }b_{i,j}\ne0\}\,.$$ Writing $h(s,t)=6-8st+s^{a+1}t^{a}+s^{a+1}t^{a+1}$, we obtain the expression $$P^{R(a)}_{R(a)_{{\!\scriptscriptstyle{+}}}}(s,t)=\frac{P^{R(a)}_k(s,t)-1}{t} =\frac{sh(s,t)}{1-(st)h(s,t)} =\sum_{i{{{\scriptstyle}\geqslant}}1}s^it^{i-1}h(s,t)^i\,.$$ The momomial $s^jt^i$ with least $i\ge1$ and largest $j$, which appears with a non-zero coefficient in the sum on the right, is $s^{a+2}t^{a}$. This gives the first inequality below: $$\begin{aligned} \frac{a+1}a &\le{\operatorname{slope}}_{R(a)}(R(a)_{{\!\scriptscriptstyle{+}}}) ={\operatorname{rate}}\left(\frac{s\cdot h(s,t)}{1-(st)h(s,t)}\right)\\ &\le\max\big\{{\operatorname{rate}}(s\cdot h(s,t))\,,{\operatorname{rate}}(1-(st)h(s,t))\big\}\\ &=\max\left\{\frac{a+2}a\,,\frac{a+2}{a+1}\right\} =\frac{a+2}a\,. \end{aligned}$$ The second inequality comes from [@ABH 1.1]. The desired inequalities follow. Slopes and Gröbner bases ======================== Let $R$ be a standard graded $K$-algebra and $R={{\widetilde}R}/I_R$ its canonical presentation. Let $T(R)$ denote the set of all term orders on all $K$-bases of ${{\widetilde}R}_1$. Letting ${\operatorname{in}}_\tau(I_R)$ denote the initial ideal corresponding to $\tau\in T$, Eisenbud, Reeves, and Totaro [@ERT] set $$\Delta(R)=\inf_{\tau\in T(R)}\{ t^{{\widetilde}R}_1({{\widetilde}R}/{\operatorname{in}}_\tau(I_R)) \}\,.$$ In words: $\Delta(R)$ is the smallest number $a$ such that $I_R$ has a Gröbner basis of elements of degree $\leq a$ with respect to a term order on some coordinate system. Now we set $$\Delta^{\ell}(R)=\inf\{ \Delta(Q) \}\,,$$ where $Q$ ranges over the set of all graded $K$-algebras satisfying $Q/L\simeq R$ for some ideal $L$ generated by a $Q$-regular sequence of elements of degree $1$. \[prop:a1\] When $R$ is not a polynomial ring the following inequalities hold: $$2\le{\operatorname{Rate}}R+1\le\Delta^{\ell}(R)\,. \qedhere$$ For $R\cong Q/(l)$ with $l$ a non-zero-divisor in $Q_1$, one has a chain $${\operatorname{Rate}}R={\operatorname{slope}}_{R}R_{{\!\scriptscriptstyle{+}}}={\operatorname{slope}}_{Q}R_{{\!\scriptscriptstyle{+}}} ={\operatorname{slope}}_{Q}Q_{{\!\scriptscriptstyle{+}}}={\operatorname{Rate}}Q\le\Delta(Q)-1\,.$$ where the first and third equalities hold by definition, the second one by Proposition \[thm:reg\](1), and the last one from the exact sequence $0\to Q(-1)\to Q_{{\!\scriptscriptstyle{+}}}\to R_{{\!\scriptscriptstyle{+}}}\to 0$; the inequality, announced without proof by Backelin [@Ba Claim, p.98], is established in [@ERT Prop. 3]. The second inequality in the proposition follows. Combining the main theorem and the preceding proposition, one obtains: \[cor:taylor\] The following inequalities hold. 1. ${\operatorname{slope}}_{{\widetilde}R}R\le\Delta^{\ell}(R)$. 2. $t_i^{{\widetilde}R}(R)< \Delta^{\ell}(R)\cdot i$ for all $i>({\operatorname{rank}}_KR_1-\dim R)$. 3. ${\operatorname{reg}}_{{\widetilde}R} R\leq(\Delta^{\ell}(R)-1)\cdot({\operatorname{rank}}_KR_1-{\operatorname{depth}}R)$. The research reported in this paper was prompted by the inequalities above, which were initially obtained by a very different argument; we proceed to sketch it. \[rem:taylor\] For any isomorphism $R\simeq Q/L$, with $L$ generated by a regular sequence of linear forms, and for each $\tau\in T(Q)$ and every pair of integers $(i,j)$ one has: $$\label{eq:betti} \beta_{i,j}^{{\widetilde}R}(R)=\beta_{i,j}^{{\widetilde}Q}(Q)\le \beta_{i,j}^{{\widetilde}Q}({\widetilde}Q/{\operatorname{in}}_\tau(I_Q))\,;$$ see, for instance, [@BC 3.13]. The Taylor resolution of the monomial ideal ${\operatorname{in}}_\tau(I_Q)$, see [@Fr §5], yields inequalities ${t_{i}^{{\widetilde}Q}({{\widetilde}Q}/{\operatorname{in}}_\tau(I_Q)){}}\leq {t_{1}^{{\widetilde}Q}({{\widetilde}Q}/{\operatorname{in}}_\tau(I_Q)){}}\cdot i$, which are strict for $i>{\operatorname{rank}}_KQ_1-\dim Q$. From these observations one obtains: $${\operatorname{slope}}_{{\widetilde}R}R={\operatorname{slope}}_{{\widetilde}Q}{Q}=\sup_{i{{{\scriptstyle}\geqslant}}1}\{{t_{i}^{{\widetilde}Q}(Q){}}/i\} \le\inf_{\tau\in T(Q)}\{{t_{1}^{{\widetilde}Q}({{\widetilde}Q}/{\operatorname{in}}_\tau(I_Q)){}}\}=\Delta(Q)\,.$$ These inequalities imply part (1) of Corollary \[cor:taylor\]; part (3) is a formal consequence. In [@Co], algebras $R$ satisfying $\Delta(R)=2$ are called *G-quadratic*, and those with $\Delta^{\ell}(R)=2$ are called *LG-quadratic*. A G-quadratic algebra is LG-quadratic by definition, and an LG-quadratic one is Koszul, see Proposition \[prop:a1\]. The first one of the preceding implications is not invertible: By an observation of Caviglia, see [@Co 1.4], complete intersections of quadrics are LG-quadratic, while it is known that not all of them are G-quadratic, see [@ERT]. Which leaves us with: Is every Koszul algebra LG-quadratic? The *Betti numbers* $\beta^{{\widetilde}R}_i(R)=\sum_{j\in{{\mathbb Z}}}{\operatorname{rank}}_k{\operatorname{Tor}_{i}^{{\widetilde}R}(k,R){}}_j$ might help separate the two notions. Indeed, when $R$ is LG-quadratic one has $R\cong Q/L$ and $Q={\widetilde}Q/I_Q$, where $Q$ is a standard graded $K$-algebra, $L$ is an ideal generated by a $Q$-regular sequence of linear forms, and the initial ideal ${\operatorname{in}}_\tau(I_Q)$ for some $\tau\in T(Q)$ is generated by quadrics. As a consequence, one has $\beta_1^{{\widetilde}Q}(Q)=\beta_1^{{\widetilde}Q}({\widetilde}Q/{\operatorname{in}}_\tau(I_Q))$, so we obtain $$\begin{aligned} \beta_i^{{\widetilde}R}(R) \le\beta_i^{{\widetilde}Q}({\widetilde}Q/{\operatorname{in}}_\tau(I_Q)) \le\binom{\beta^{{\widetilde}Q}_1({\widetilde}Q/{\operatorname{in}}_\tau(I_Q))}i =\binom{\beta^{{\widetilde}R}_1(R)}i\,, \end{aligned}$$ with inequalities coming from and the Taylor resolution. Thus, we ask: If $R$ is a Koszul algebra, does $\beta^{{\widetilde}R}_i(R)\le\displaystyle\binom{\beta^{{\widetilde}R}_1(R)}i$ hold for every $i$? [20]{} D. J. Anick, *On the homology of associative algebras*, Trans. Amer. Math. Soc. **296** (1986), 641-659. A. Aramova, J. Herzog, Ş. Bărcănescu, *On the rate of relative Veronese submodules*, Rev. Roumaine Math. Pures Appl. **40** (1995), 243–251. L. L. Avramov, *Small homomorphisms of local rings*, J. Algebra **50** (1978), 400–453. L. L. Avramov, *Infinite free resolutions*, Six lectures on commutative algebra (Bellaterra, 1996), Progr. Math. **166**, BirkhŠuser, Basel, 1998; 1–118. L. L. Avramov, S. Iyengar, *André-Quillen homology of algebra retracts*, Ann. Sci. École Norm. Sup. (4) **36** (2003), 431–462. L. L. Avramov, I. Peeva, *Finite regularity and Koszul algebras*, Amer. J. Math. **123** (2001), 275–281. J. Backelin, *On the rates of growth of the homologies of Veronese subrings*, Algebra, algebraic topology, and their interactions (Stockholm, 1983), Lecture Notes in Math. **1183**, Springer, Berlin, 1986; 79–100. W. Bruns, *On the Koszul algebra of a local ring*, Illinois J. Math. **37** (1993), 278–283. W. Bruns, A. Conca, *Gröbner bases and determinantal ideals*, Commutative algebra, singularities and computer algebra (Sinaia, 2002), NATO Sci. Ser. II Math. Phys. Chem., **115**, Kluwer Acad. Publ., Dordrecht, 2003; 9–66. W. Bruns, A. Conca, T. Römer, *Koszul homology and syzygies of Veronese subalgebras*, `arXiv:0902.2431`. W. Bruns, J. Herzog, *Cohen-Macaulay rings*, Revised edition, Cambridge Studies Adv. Math. **39**, University Press, Cambridge, 1998. H. Cartan, S. Eilenberg, *Homological algebra*, Princeton Univ. Press, Princeton, NJ, 1956. A. Conca, *Koszul algebras and Gröbner bases of quadrics*, Proceedings of the 29th Symposium on Commutative Algebra in Japan, Nagoya, Japan, 2007; 127-133; `arXiv:0903.2397`. D. Eisenbud, A. Reeves, B. Totaro, *Initial ideals, Veronese subrings, and rates of algebras*, Adv. Math. 109 (1994), no. 2, 168–187. R. Fröberg, *Some complex constructions with applications to Poincaré series*, Séminaire d’Algèbre Paul Dubreil, (Paris, 1977–1978), Lecture Notes in Math., **740**, Springer, Berlin, 1979; 272–284. C. Peskine, L. Szpiro, *Syzygies et multiplicités*, C. R. Acad. Sci. Paris Sér. A **278** (1974), 1421–1424. A. Polishchuk, L. Positselski, *Quadratic algebras*, Univ. Lecture Ser. **37**, Amer. Math. Soc., Providence, RI, 2005. J.-E. Roos, *Commutative non Koszul algebras having a linear resolution of arbitrarily high order. Applications to torsion in loop space homology*, C. R. Acad. Sci. Paris **316** (1993), 1123–1128. L. M. Şega, *Homological properties of powers of the maximal ideal of a local ring*, J. Algebra **241** (2001), 827-858. [^1]: Research partly supported by NSF grants DMS 0803082 (LLA) and 0602498 (SBI)

ArXiv

--- abstract: 'In this paper, we consider finitely many interval maps simultaneously acting on the unit interval $I = [0,\, 1]$ in the real line $\mathbb{R}$; each with utmost finitely many jump discontinuities and study certain important statistical properties. Even though we use the symbolic space on $N$ letters to reduce the case of simultaneous dynamics to maps on an appropriate space, our aim in this paper remains to resolve ergodicity, rates of recurrence, decay of correlations and invariance principles leading upto the central limit theorem for the dynamics that evolves through simultaneous action. In order to achieve our ends, we define various Ruelle operators, normalise them by various means and exploit their spectra.' author: - | Aswin Gopakumar\ [[email protected]]{}\ Kirthana Rajasekar\ [[email protected]]{}\ Shrihari Sridharan\ [[email protected]]{}\ [*Indian Institute of Science Education and Research*]{}\ [*Thiruvananthapuram (IISER-TVM), India.*]{} title: '<span style="font-variant:small-caps;">Simultaneous Action of Finitely Many Interval Maps: Some Dynamical and Statistical Properties</span>' --- --------------------- --- ----------------------------------------------------- **Keywords** : Growth of typical trajectories; Invariance principles; Ruelle operator and the pressure function; Simultaneous action of finitely many interval maps. **AMS Subject** : 37E05, 37C35, 37D35, 37B10. **Classifications** --------------------- --- ----------------------------------------------------- Introduction ============ Various dynamical properties and statistical properties help us understand the behaviour of dynamics caused by the action of a transformation $T$ on some phase space $X$. Important among such properties include the Birkhoff’s pointwise ergodic theorem, asymptotic estimates on rates of recurrence of typical orbits, decay of correlations, invariance principles, central limit theorem, law of iterated logarithms [*etc*]{}. Each of these theorems provide us a deeper glimpse into the structure, the dynamics builds in its phase space of action or an invariant subset, thereof. Birkhoff’s pointwise ergodic theorem observes a considered dynamical system through a real-valued continuous function and states that the sequence of local time averages along the orbit of any typical point converges to the global space average, whenever the transformation $T$ acting on $X$ is ergodic. Though the result is extremely strong, it does allow some points (though negligible, meaning with collective measure zero) to fluctuate from this mean behaviour. Thus, an interesting study in the dynamics of such ergodic systems is to obtain a good understanding of the set of points that violate the Birkhoff’s ergodic conditions. An easy way to approach this subject locally is to work out the ergodic sums of the observables and consider the cardinality of the set of points whose ergodic sum calculated at various times remain inside some chosen interval $[a,\, b] \subset \mathbb{R}$. However, on a global scale, an alternate way to understand the deviation from the average behaviour of typical orbits is achieved by formalising the central limit theorem. The central limit theorem is an important tool in mathematics that distributes the random variables along a bell-curve (normal distribution), as more and more independent random variables are appropriately included under the ambit of study. This is a central object of investigation in understanding deterministic dynamical systems, owing to its natural appeal, when we consider the various orbits in the phase space. However, as important as the central limit theorem is, we see that they are subsumed by more general invariance principles. A sequence of random variables $\big\{ X_{n} \big\}_{n\, \ge\, 1}$ is said to satisfy an almost sure invariance principle if the sequence can be approximated almost surely by another sequence, preferably with certain desired properties and with a relatively small margin of error. Several mathematicians have studied these properties in many deterministic dynamical systems, where the phase space is a compact interval of the real line, [@md:86; @ci:96; @lsv:99; @ps:02; @cr:07; @cm:15], the Julia set of some rational map that occurs as a compact subset of the Riemann sphere, [@du:91; @dpu:96; @ss:07; @ss:09], [*etc.*]{}, however, with a single transformation acting on the appropriate space, based on which the dynamics evolves. Examples of continuous time dynamical systems that has remained in the focus of the dynamics community include expanding flows restricted on a compact subset of the Riemannian manifold, [@dp:84; @spl:89; @mp:91] or some mixing Axiom $A$ diffeomorphism restricted on a basic set, [@mr:73; @ks:90; @rs:93; @na:00; @ps:01]. A particularly desirable feature of all the above-mentioned maps restricted on their respective sets is that they can be studied through an associated symbolic model [@rb:73; @mr:73]. There are also various studies carried out by several mathematicians that analyse statistical results in various settings of dynamical systems. Prominent among them include [@ps:75; @hh:80; @cp:90; @pp:90; @ps:94; @lsy:99; @si:99; @fmt:03; @mtk:05; @mn:05; @hntv:17]. What we intend to investigate in this paper is slightly richer in dynamics, than what is explained so far. In this paper, we consider the compact unit interval $[0,\, 1] \subset \mathbb{R}$; however with finitely many maps acting on the space simultaneously. Thus, the dynamics evolves along the multiple branches provided by each of these maps. In fact, we work with finitely many interval maps defined on $[0,\, 1]$; each of which has a discrete set of utmost finitely many discontinuities. As an expert reader may realise, these results are readily transferable to various settings including the simultaneous action of finitely many rational maps restricted on the appropriate Julia set, as defined in [@hs:00] or to the action of a holomorphic correspondence restricted on the support of its Dinh-Sibony measure, as defined in [@bs:16]. We shall explain our claim of transferability of the main theorems of this paper, as written above, in the final section, . This paper is structured as follows: In the next two introductory sections and , we narrate the basic settings of this paper and develop certain notations and dynamical notions, however only as skeletal to enable us to state the main theorems of this paper in section . The main results of this paper describe the ergodicity of the system in theorems and , the rates of recurrence in theorems and , the exponential decay of correlations in theorems and , almost sure invariance principles in theorems and and a few more statistical properties such as the central limit theorems and the laws of iterated logarithms in theorems and . The reason why each theorem appears twice in the list above will be clear, by the time we reach section . In section , we recall the setting of symbolic dynamics that comes in handy as a book-keeping mechanism in our study. In sections , and , we define three kinds of Ruelle operators on the appropriate Banach space of continuous functions and Hölder continuous functions defined on the phase spaces that interest us, compare their spectra and normalise them in different ways in order that they help us in proving our main theorems. Having achieved these, we embark on writing the proofs of the main theorems in sections , , , and . We conclude the paper with a few remarks in section . Preliminaries and the pressure function {#prelims} ======================================= In this section, we explain the setting of our paper and define certain basic terminologies that help us in constructing the necessary notions to state our main results. Let $I$ denote the unit interval on the real line, [*i.e.*]{}, $I = [0,\, 1]$. We are interested in studying the dynamics of finitely many interval maps acting simultaneously on $I$, *i.e.*, given $N \in \mathbb{N}$ and $1 \leq d \leq N$, we consider the interval maps $T_{d} : I \longrightarrow I$ of degree $(d + 1)$ given by $$T_{d}\, (x)\ \ :=\ \ (d + 1)\; x \pmod 1.$$ The simultaneous action is explained as follows: For any $x_{0} \in I$, its forward orbit at times $t = 0,\, 1,\, 2,\, \cdots,\, n,\, \cdots$ is defined as $$\label{1storbit} \left\{ x_{0},\, x_{1} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{0}),\, x_{2} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{1}),\, \cdots,\, x_{n} \in \bigcup_{d\, =\, 1}^{N} T_{d} (x_{n - 1}), \cdots \right\}.$$ Thus, at every stage, we have $N$ many maps to choose from to move forward and the totality of all these branches describe the forward orbit. Observe that the dynamics that arises out of such a process can also be described by the action of a semigroup generated by the same interval maps, $\mathscr{S} = \big\langle T_{1},\, T_{2},\, \cdots,\, T_{N} \big\rangle$, or as a correspondence on $I \times I$, appropriately defined. Suppose $\mathcal{C} (V, \mathbb{F})$ denotes the space of all continuous functions defined on the space $V$ that takes values on the field $\mathbb{F}$. Then, for any $f \in \mathcal{C}(I, \mathbb{C})$, the set of all complex-valued continuous functions defined on $I$, we define a *composition operator*, $\mathscr{O}$ by $$\mathscr{O} (f) \in \bigcup\limits_{d\, =\, 1}^{N} \big( f \circ T_{d} \big).$$ Along every orbit of the point $x_{0}$, as described in , this composition operator chooses the map $T_{d}$ every time in such a fashion that $T_{d} (x_{k - 1})\, =\, x_{k}$. Hence, even though $\mathscr{O} (f)$ is not single-valued, we have by definition that $\left( \mathscr{O} (f) \right) (x_{0})$ to be single-valued along every chosen orbit of $x_{0}$. We further describe this idea in section . To assist us in this study, we will make use of the space consisting of infinitely long words on $N$ symbols, [*i.e.*]{}, suppose $S = \big\{ 1,\, 2,\, \cdots,\, N \big\}$, we consider $$\Sigma_{N}^{+}\ \ :=\ \ S^{\mathbb{Z}_{+}}\ \ =\ \ \Big\{ w = \left( w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots \right) : w_{i} \in S \Big\}.$$ As we shall explain in section , $\Sigma_{N}^{+}$ is a compact measurable metric space equipped with the Bernoulli measure, where the shift map $\sigma$ defined by $\left( \sigma (w) \right)_{n} = w_{n\, +\, 1}$ is continuous and non-invertible, but a local homeomorphism. From now on, we denote by $X$ the product phase space given by $X := \Sigma_{N}^{+} \times I$, where we define a skew-product map $T$ as $$\label{spm} T (w,\, x)\ \ :=\ \ ( \sigma w,\, T_{w_{1}} x),\ \ \ \ \text{where}\ \ w = (w_{1}\, w_{2}\, \cdots).$$ By a standard argument due to Tychonoff, as may be found in [@mj:00], we consider the natural product topology on $X$ that gives rise to the metric $d_{X} (\cdot,\, \cdot)$ on $X$. Further, we also have the product sigma-algebra and the product measure defined on $X$. Let $\mu$ denote some $T$-invariant measure supported on $X$. For example, the appropriate product measure of the Bernoulli measure on cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on open intervals of $I$ is a $T$-invariant probability measure on $X$. The definition of the skew-product map $T$ entails that the forward orbit of $(w, x)$ at times $t = 0,\, 1,\, 2,\, \cdots,\, n,\, \cdots$ under $T$ is given by $$\Big\{ (w,\, x),\; (\sigma w,\, T_{w_{1}} x),\; \left( \sigma^{2} w,\, \left( T_{w_{2}} \circ T_{w_{1}} \right) x \right),\; \cdots,\; \left( \sigma^{n} w,\, \left( T_{w_{n}} \circ T_{w_{n - 1}} \cdots \circ T_{w_{1}} \right) x \right),\; \cdots \Big\}.$$ Thus, for a chosen $w$ in $\Sigma_{N}^{+}$, the natural projection on the second co-ordinate $\mathbf{Proj}_{2} : X \longrightarrow I$ captures the sectional idea behind the orbit of $x \in I$ as described in . Taking the union over all possible $w \in \Sigma_{N}^{+}$ captures the idea in its entirety. For ease of notations, we fix little letters like $f,\, g,\, h,$ [*etc*]{}. to represent functions defined on the interval, $I$ and use big letters like $F,\, G,\, H$ [*etc*]{}. to represent functions defined on the product space $X = \Sigma_{N}^{+} \times I$. Although one may think of $f$ as being a restriction of $F$ on the interval, [*i.e.*]{}, $F = f \circ \mathbf{Proj}_{2}$ that yields $F((w,\, x)) = f(x)$, it need not be the case always. The space $X$ now facilitates us to redefine the composition operator, in this setting denoted by $\mathscr{Q}$ defined on $\mathcal{C} (X, \mathbb{C})$ given by $$\mathscr{Q} (F)\ \ :=\ \ F \circ T.$$ Let $\mathscr{F}_{\alpha}(X, \mathbb{C})$ denote the space of all complex-valued $\alpha$-Hölder continuous functions defined on $X$. For $F \in \mathscr{F}_{\alpha}(X, \mathbb{C})$, we define the following norm, $$\big\Vert F \big\Vert_{\alpha}\ \ :=\ \ \big\vert F \big\vert_{\alpha} + \big\Vert F \big\Vert_{\infty},$$ where $$\big\vert F \big\vert_{\alpha}\ \ :=\ \ \sup\limits_{(w,\, x)\; \neq\; (v,\, y)} \left\{ \frac{\big\vert F((w,\, x))\; -\; F((v,\, y)) \big\vert}{\big( d_{X}((w,\, x),\; (v,\, y)) \big)^{\alpha}}\ :\ (w,\, x),\; (v,\, y) \in X \right\}$$ denotes the $\alpha$-Hölder semi-norm and $\big\Vert F \big\Vert_{\infty}$ denotes the usual supremum norm. Then, $\mathscr{F}_{\alpha}(X, \mathbb{C})$ is a Banach space under the norm $\big\Vert \cdot \big\Vert_{\alpha}$. Given any function $F \in \mathcal{C}(X, \mathbb{R})$, its *pressure* is defined as $$\label{pressure} \mathfrak{P} (F)\ \ :=\ \ \sup \left\{ h_{\mu} (T) + \int\! F d \mu \right\},$$ where the supremum is taken over all $T$-invariant probability measures supported on $X$. Further, $h_{\mu} (T)$ is the measure theoretic entropy of $T$ with respect to $\mu$. An *equilibrium measure* for the function $F$ denoted by $\mu_{F}$ is defined as that measure for which the supremum is attained in the definition of pressure, as stated in . The unique existence of $\mu_{F}$ for every $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ is assured by Denker and Urbanski in [@du:91] and Sumi and Urbanski in [@su:09], for an analogous setting. The pressure function and the equilibrium measure have respective analogues for dynamics under simultaneous actions of the interval maps. We will establish the same later in section . However, for the sake of stating the results, we mention the following properties. Given $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$, its *pressure* under simultaneous dynamics that we denote by $\mathbb{P}(f)$ coincides with the quantity $\mathfrak{P}(f \circ \mathbf{Proj}_{2})$. Similarly, the measure $\mathfrak{m}_{f}$ on $I$, whose relation with $\mu_{f \circ \mathbf{Proj}_{2}}$, which will be an easy observation once defined, is given by, $$\int\! g\, d \mathfrak{m}_{f}\ \ =\ \ \int\! ( g \circ \mathbf{Proj}_{2})\, d \mu_{f\, \circ\, \mathbf{Proj}_{2}},\ \ \ \forall g \in \mathscr{F}_{\alpha}(I, \mathbb{R}).$$ Periodic points and other dynamical notions {#peri} =========================================== A point $(w,\, x) \in X$ is said to be a *periodic point* of period $p$ for $T$ iff $\sigma^{p} w\; =\; w$ and $\left( T_{w_{p}} \circ T_{w_{p - 1}} \circ \cdots \circ T_{w_{1}} \right)\, x\; =\; x$. We denote the set of all $p$-periodic points by ${\rm Fix}_{p} (T)$. Once we determine the periodic points of $T$, it becomes easier for us to identify the periodic orbits of simultaneous action of the $N$ many interval maps. We first introduce a few notations here. For any $x_{1} \in I$, let $\mathscr{R}_{n} (x_{1})$ denote the set of all rays starting from $x_{1}$ that describe the initial $n$-long itinerary of the trajectory of $x_{1}$ in the order that the point visits, [*i.e.*]{}, $$\mathscr{R}_{n} (x_{1})\ \ :=\ \ \Big\{ \left( x_{1},\, x_{2},\, \cdots,\, x_{n} \right) \in I^{n}\ :\ \forall 2 \le k \le n,\ \exists 1 \le d \le N\ \text{such that}\ T_{d} (x_{k - 1}) = x_{k} \Big\}.$$ Further, we denote by $\mathscr{R} (x_{1}) = \mathscr{R}_{\infty} (x_{1})$ the set of all infinite rays starting from $x_{1}$, where we produce each point in $\mathscr{R}_{n} (x_{1})$ to an infinitely long sequence, as allowed by the dynamics, [*i.e.*]{}, $$\mathscr{R} (x_{1})\ \ :=\ \ \Big\{ \left( x_{1},\, x_{2},\, \cdots\, \right) \in I^{\mathbb{Z}_{+}}\ :\ \forall k \ge 2,\ \exists 1 \le d \le N\ \text{for which}\ T_{d} (x_{k - 1}) = x_{k} \Big\}.$$ For any $m \le n < \infty$, we define the following projection operators on the set $\mathscr{R}_{n} (x_{1})$ as follows: $$\begin{array}{c c c c l c r c l} \Pi_{m} & : & \mathscr{R}_{n} (x_{1}) & \longrightarrow & \mathscr{R}_{m} (x_{1}) & \text{defined by} & \Pi_{m} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{n} ) \right) & = & \left( x_{1},\, x_{2},\, \cdots,\, x_{m} \right); \\ \pi_{m} & : & \mathscr{R}_{n} (x_{1}) & \longrightarrow & I & \text{defined by} & \pi_{m} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{n} ) \right) & = & x_{m}. \end{array}$$ Allowing a slight abuse of notations, for any $m < \infty$, analogous definitions can be written for the projection operators $\Pi_{m}$ and $\pi_{m}$ defined on $\mathscr{R} (x_{1})$. $$\begin{array}{c c c c l c r c l} \Pi_{m} & : & \mathscr{R} (x_{1}) & \longrightarrow & \mathscr{R}_{m} (x_{1}) & \text{defined by} & \Pi_{m} \left( ( x_{1},\, x_{2},\, \cdots ) \right) & = & \left( x_{1},\, x_{2},\, \cdots,\, x_{m} \right); \\ \pi_{m} & : & \mathscr{R} (x_{1}) & \longrightarrow & I & \text{defined by} & \pi_{m} \left( ( x_{1},\, x_{2},\, \cdots ) \right) & = & x_{m}. \end{array}$$ We say $x_{1} \in I$ is a *periodic point* of period $p$ with *periodic orbit* $(x_{1},\, x_{2},\, \cdots,\, x_{p}) \in \mathscr{R}_{p} (x_{1})$ pertaining to the combinatorial data given by some $p$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{p})$ on $N$ letters (the length of $w$ denoted by $|w| = p$), if 1. $\pi_{1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ =\ \ \pi_{p + 1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)$; 2. $p$ is the least such positive integer for which the first condition is true, [*i.e.*]{},\ $\pi_{1} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ \ne\ \ \pi_{q} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \forall q \le p$; and 3. there does not exist any distinct $1 \le q, r \le p$ for which\ $\pi_{q} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{p} ) \right)\ \ =\ \ \pi_{r} \left( ( x_{1},\, x_{2},\, \cdots,\, x_{p} ) \right)$. We identify such a periodic point $x_{1} \in I$ with period $p$ and periodic orbit $(x_{1},\, x_{2},\, \cdots,\, x_{p})$ by looking for periodic blocks of points in $\mathscr{R} (x_{1})$ that satisfy, $$\Pi_{p} \left( ( x_{1},\, x_{2},\, \cdots ) \right)\ \ =\ \ \Pi_{p} \left( ( x_{mp\, +\, 1},\, x_{mp\, +\, 2},\, \cdots ) \right)\ \ \ \forall m \in \mathbb{Z}_{+}.$$ It is a simple observation that corresponding to any $p$-periodic point $x \in I$, there exists a $p$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{p})$ on $N$ letters such that $$T_{w} x\ \ :=\ \ \big( T_{w_{p}} \circ \cdots \circ T_{w_{1}} \big) x\ \ =\ \ x.$$ For any $n$-lettered word $w = ( w_{1}\, w_{2}\, \cdots\, w_{n} )$, we collect the points satisfying $T_{w} x = x$ in the set ${\rm Fix} (T_{w})$. For any $T$-invariant probability measure $\mu$ supported on $X$, let $F, G$ be any two complex-valued integrable functions defined on $X$, the appropriate space denoted by $L^{1} (\mu)$. We say the functions $F$ and $G$ are *cohomologous* to each other if there exists a function $H \in L^{1} (\mu)$ such that $F\, -\, G\ =\ \mathscr{Q} (H)\, -\, H$. If $F$ is cohomologous to the constant function $\mathbf{0}$, then $F$ is called a *coboundary*. For any $F \in \mathcal{C} (X, \mathbb{C})$, we denote and define its $n$-th ergodic sum by $$\label{nthergodicsumF} F^{n}\ \ :=\ \ F + \mathscr{Q} (F) + \mathscr{Q}^{2} (F) + \cdots + \mathscr{Q}^{n - 1} (F).$$ Hence, for any two cohomologous functions $F$ and $G$, it is obvious that their $n$-th ergodic sums evaluated at a periodic point of period $n$ must be the same, [*i.e.*]{}, $$F^{n}((w,\, x))\ \ =\ \ G^{n}((w,\, x))\ \ \forall (w,\, x) \in {\rm Fix}_{n} (T).$$ Let $F,\, G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ with $F$ not being cohomologous to any constant function. Then, by a result due to Ruelle in [@dr:78], the function $t \longmapsto \mathfrak{P}(F + t G)$, where $t \in \mathbb{R}$, is convex and real analytic. Further, from [@cp:90] and [@pp:90], we have $$\begin{aligned} \label{firstderivative} \left. \frac{d}{dt} \Big( \mathfrak{P}(F + t G) \Big) \right|_{t\, =\, 0} & = & \int\! G\, d \mu_{F} \\ \label{secondderivative} \left. \frac{d^{2}}{d t^{2}} \Big( \mathfrak{P}(F + t G) \Big) \right|_{t\, =\, 0} & = & \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \left( G^{n} - n \int\! G\, d \mu_{F} \right)^{2}\, d \mu_{F}\ >\ 0.\end{aligned}$$ For simultaneous action of $N$ interval maps at a point $x \in I$, the $n$-th order ergodic sum of any $f \in \mathcal{C} (I, \mathbb{C})$ must be calculated over its appropriate orbit, [*i.e.*]{}, given a $n$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n})$ on $N$ letters, we define the composition operator $\mathscr{O}_{w}$ by $$\mathscr{O}_{w} (f)\ \ :=\ \ f \circ T_{w},\ \ \ \ \text{where}\ \ T_{w}\ :=\ T_{w_{n}} \circ \cdots \circ T_{w_{1}}.$$ Then, the $n$-th order ergodic sum of the function $f$ with respect to the given $n$-lettered word $w$, or (by a slight abuse of notations) the $n$-th order ergodic sum of the function $f$ with respect to any given infinite-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots)$ on $N$ letters that agree with our $n$-lettered word on the initial $n$ positions is given by $$\begin{aligned} f^{n}_{w} (x) & := & \left( f + \mathscr{O}_{(w_{1})} (f) + \cdots + \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n - 1})} (f) \right) (x) \\ & = & \left( f + f ( T_{w_{1}} ) + \cdots + f \left( T_{w_{n - 1}} \circ \cdots \circ T_{w_{2}} \circ T_{w_{1}} \right) \right) (x). \end{aligned}$$ For a given $n$-lettered word $w = (w_{1}\, w_{2}\, \cdots\, w_{n})$ on $N$ letters, we say that two Lebesgue integrable functions $f, g \in L^{1} (\lambda)$ defined on $I$ are *$w$-cohomologous* to each other if there exists an integrable function $h \in L^{1} (\lambda)$, also defined on $I$ such that $f\, -\, g\ =\ \mathscr{O}_{w} h\, -\, h$. Hence, for any two $w$-cohomologous functions $f$ and $g$, we observe that the values of the function evaluated at a periodic point $x_{1}$ of period $n$ with periodic orbit $(x_{1},\, x_{2},\, \cdots,\, x_{n})$ pertaining to the combinatorial data given by the $n$-lettered word $w$, necessarily agree; and so do their $n$-th order ergodic sums. Further, if $f$ is $w$-cohomologous to the constant function $\mathbf{0}$, then $f$ is called a *$w$-coboundary*. Statements of results {#main} ===================== In this section, we state the main theorems of this paper. The first five results concern the setting of the dynamics of the skew-product map $T$ defined on $X = \Sigma_{N}^{+} \times I$, while the next five results concern the setting of simultaneous dynamics of the concerned interval maps on $I$; thus generalising the situation to maps that evolve with multiple branches. \[erg1\] The action of $T$ on the product space $X$ is necessarily ergodic with respect to the product measure $\mu$. In other words, the measure of any subset $B \subseteq X$, in the product sigma-algebra of $X$ that satisfies $T^{-1} B = B$, is necessarily $0$ or $1$. \[ror1\] Consider $F \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ that satisfies the approximability condition, [*i.e.*]{}, there exists distinct points $(w_{1},\, x_{1}) \in {\rm Fix}_{p_{1}} (T),\ (w_{2},\, x_{2}) \in {\rm Fix}_{p_{2}} (T)$ and $(w_{3},\, x_{3}) \in {\rm Fix}_{p_{3}} (T)$ with $p_{i} \neq p_{j}$ for $i \neq j$ for which $$\label{dioF} \frac{F^{p_{2}}((w_{2},\, x_{2}))\; -\; F^{p_{1}}((w_{1},\, x_{1}))}{F^{p_{3}}((w_{3},\, x_{3}))\; -\; F^{p_{1}}((w_{1},\, x_{1}))}\ \ =:\ \ \mathfrak{d}_{1}$$ is a Diophantine number, [*i.e.*]{}, there exists $l > 2$ and $m > 0$ such that we have $$\label{dioph} \left| \mathfrak{d}_{1} - \frac{p}{q} \right|\ \ \geq\ \ \frac{m}{q^{l}},\ \ \forall p, q \in \mathbb{Z}_{+}.$$ Further, suppose that there exists a unique real number $\kappa$ such that $$\int\! F d \mu_{\kappa F}\ \ =\ \ 0.$$ Then, for every $n \in \mathbb{Z}_{+},\ a, b \in \mathbb{R}$ with $a < b$, there exists a positive real constant $C_{1} > 0$ such that $$\label{ror1eq} \# \Big\{ (w,\, x) \in {\rm Fix}_{n} (T) : a \leq F^{n}((w,\, x)) \leq b \Big\}\ \ \sim\ \ C_{1}\ \frac{e^{n \mathfrak{P} (\kappa F)}}{\sqrt{n}}\ \int_{a}^{b}\!\! e^{- \kappa t}\, dt.$$ Here, in equation , by $a_{n} \sim b_{n}$, we mean that $\displaystyle{\lim\limits_{n\, \to\, \infty} \frac{a_{n}}{b_{n}} = 1}$. \[doc1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, there exists a constant $\vartheta \in (0, 1)$ such that for all $G,\ H \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, we have $C_{2} > 0$ (depending on $G$ and $H$) that satisfies $$\label{doce1} \left\vert \int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\; -\; \int\! G\, d \mu_{F}\, \int\! H\, d \mu_{F} \right\vert\ \ \le\ \ C_{2} \vartheta^{n};\ \ \forall n \geq 1.$$ The preceding theorem defines the exponential decay of correlations of the distributions $\mathscr{Q}^{n} (G)$ and $H$ with respect to the measure $\mu_{F}$ as $n \rightarrow \infty$. The exponential nature of the decay is evident in the statement of the theorem. The next theorem relates the ergodic sum of $G \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ to what is known as the Brownian motion on some richer probability space. \[asip1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, consider $G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ satisfying $$\int G\, d \mu_{F}\ \ =\ \ 0.$$ Suppose the variance of $H \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ is defined as $$\left( \varsigma (H) \right)^{2}\ \ :=\ \ \lim_{n \rightarrow \infty} \frac{1}{n} \int\! \left( H^{n}\; -\; n\, \int\! H\, d \mu_{F} \right)^{2}\, d \mu_{F}.$$ Then, there exists a Hölder continuous function $\Phi \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ cohomologous to $G$, a one-dimensional Brownian motion $\Big\{ \mathfrak{B}(t) \Big\}_{t\, \ge\, 0}$ with variance $t \left( \varsigma (G) \right)^{2}$ and a sequence of random variables $\big\{ \mathfrak{Y}_{n} : \Omega \longrightarrow \mathbb{R} \big\}_{n\, \ge\, 0}$ such that $\big\{ \mathfrak{Y}_{n} \big\}_{n\, \ge\, 0}$ and $\big\{ \Phi^{n} \big\}_{n\, \ge\, 0}$ are equal in distribution and given any $\delta > 0$, $$\mathfrak{Y}_{\lfloor t \rfloor}(\omega)\ \ =\ \ \mathfrak{B} (t) (\omega)\; +\; O(t^{\frac{1}{4}\, +\, \delta}),\ \ \forall t \geq 0,\ \ \mu_{F} \text{-a.e.},$$ provided $G$ is not a coboundary. The almost sure invariance principle leads to a few important corollaries such as the central limit theorem and the law of iterated logarithms. \[osp1\] For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ whose equilibrium measure is denoted by $\mu_{F}$, consider $G \in \mathscr{F}_{\alpha}(X, \mathbb{R})$ satisfying $$\int\! G\, d \mu_{F}\ \ =\ \ 0.$$ Suppose $G$ is not a coboundary. Then, 1. $G$ satisfies the central limit theorem, [*i.e.*]{}, $\frac{1}{\sqrt{n}} G^{n}$ converges in distribution to a normal distribution with mean zero and variance $\left( \varsigma (G) \right)^{2}$ as $n \rightarrow \infty$. 2. $G$ satisfies the law of iterated logarithms, [*i.e.*]{}, $$\limsup_{n\, \rightarrow\, \infty} \frac{G^{n} ((w,\, x))}{\varsigma (G) \sqrt{2n \log \log n}}\ \ =\ \ 1\ \ \mu_{F}\text{-a.e.}$$ In the next five theorems, we state theorems captioned under the same titles, however, by suppressing the first co-ordinate of $X$ and looking at a genuine simultaneous action of the finitely many interval maps under consideration. \[erg2\] Consider the interval maps $T_{1},\, T_{2},\, \cdots,\, T_{N}$ that act simultaneously on the interval $I$. Let $\lambda$ denote the Lebesgue measure on $\mathbb{R}$. Then, for any real-valued Lebesgue integrable function $f \in L^{1} (\lambda)$, for $\lambda$-a.e. $x \in I$, we have $$\lim_{n\, \to\, \infty}\, \frac{1}{n}\, \frac{1}{N^{n}}\, \sum_{w\; :\; |w|\, =\, n} \Big[ f\, +\, \mathscr{O}_{(w_{1})} (f)\, +\, \cdots\, +\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n - 1})} (f) \Big] (x)\ \ =\ \ \int_{0}^{1}\! f\, d \lambda.$$ \[ror2\] Consider $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$ that satisfies the approximability condition, [*i.e.*]{}, there exists distinct periodic points $x,\ y$ and $z$ in $I$ with distinct periods $p_{x},\ p_{y}$ and $p_{z}$, pertaining to the combinatorial data given by $w_{x},\ w_{y}$ and $w_{z}$ such that $$\label{dioph2} \frac{f^{p_{y}}_{w_{y}} (y)\ -\ f^{p_{x}}_{w_{x}} (x)}{f^{p_{z}}_{w_{z}} (z)\ -\ f^{p_{x}}_{w_{x}} (x)}\ \ =\ \ \mathfrak{d}_{2}$$ is a Diophantine number. Further, suppose there exists unique $\kappa > 0$ such that $$\int\! f d\mathfrak{m}_{\kappa f}\ \ =\ \ 0.$$ Then, for every $n \in \mathbb{Z}_{+},\ a, b \in \mathbb{R}$ with $a < b$, there exists a positive real constant $C_{3} > 0$ such that $$\sum\limits_{w\; :\; |w|\, =\, n} \# \big\{ x \in {\rm Fix} (T_{w})\ :\ a \leq f^{n}_{w}(x) \leq b \big\}\ \ \sim\ \ C_{3} \frac{e^{n \mathbb{P}(\kappa f)}}{\sqrt{n}}\ \int_{a}^{b}\! e^{-\kappa t}\, d t.$$ \[doc2\] Let $\lambda$ denote the Lebesgue measure on $I$. Then, there exist a constant $\vartheta \in (0, 1)$ such that for all $\alpha$-Hölder continuous functions $g,\ h \in \mathscr{F}_{\alpha}(I, \mathbb{R})$, we have $C_{4} > 0$ (depending on $g,\ h$ and some $n$-lettered word $w$) that satisfies $$\label{doce2} \left\vert \int\! \mathscr{O}_{w} (g) h\, d \lambda\; -\; \int\! g\, d \lambda \, \int\! h\, d \lambda \right\vert\ \ \le\ \ C_{4} \vartheta^{n};\ \ \forall n \geq 1.$$ \[asip2\] Let $\lambda$ denote the Lebesgue measure on $I$. For any $g \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ with $$\int\! g\, d \lambda\ \ =\ \ 0,$$ and $w = (w_{1}\, w_{2}\, \cdots) \in \Sigma_{N}^{+} $, assume that the variance of $g$ with respect to the word $w$ denoted by $\big( \varsigma_{w} (g) \big)^{2}$ and defined by $$\big( \varsigma_{w} (g) \big)^{2}\ \ :=\ \ \lim\limits_{n\, \to\, \infty} \frac{1}{n} \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda\ \ >\ \ 0.$$ Then, there exists a probability space $( \Omega,\; \mathscr{A},\; \nu )$, a sequence of random variables $\big\{ Y_{w}^{n} \big\}_{n\, \ge\, 0}$ and a standard Brownian motion $\big\{ \mathfrak{B}^{*} (t) \big\}_{t\, \ge\, 0}$ such that $g_{w}^{n}$ and $Y_{w}^{n}$ are equal in distribution and given any $\delta > 0$, $$\label{varsigmawng} Y_{w}^{n} (\omega)\ -\ \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right) (\omega)\ \ =\ \ O \big( n^{\frac{1}{4}\, +\, \delta} \big),\ \ \ \ \nu\text{-a.e.}\ \ \ \text{where}\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\ \ =\ \ \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda,$$ provided $g_{\Pi_{n} (w)}$ is not a $\Pi_{n} (w)$-coboundary for any $n \ge 1$. \[osp2\] For a function $g \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ that satisfies $$\int\! g\, d \lambda\ \ =\ \ 0,$$ suppose $w \in \Sigma_{N}^{+} $ is such that the variance $\big( \varsigma_{w} (g) \big)^{2} > 0$ and $g_{\Pi_{n} (w)}$ is not a $\Pi_{n} (w)$-coboundary for any $n \ge 1$. Then, 1. $g$ satisfies the central limit theorem *i.e.,* $\frac{1}{\sqrt{n}} g_{w}^{n} $ converges in distribution to a normal distribution with mean zero and variance $\big( \varsigma_{w} (g) \big)^{2}$ as $n \to \infty$. 2. $g$ satisfies the law of iterated logarithms, [*i.e.*]{}, $$\limsup_{n\, \rightarrow\, \infty} \frac{g_{w}^{n} (x)}{\varsigma_{w} (g) \sqrt{2n \log \log n}}\ \ =\ \ 1\ \ \lambda\text{-a.e.}$$ A book-keeping mechanism, $\Sigma_{N}^{+}$ {#SigmaNplus} ========================================== In this section, we define the space $\Sigma_{N}^{+}$ and discuss certain properties that will be useful in the sequel. Interested readers may refer to [@bk:98], for more details on this space. Recall the definition of $\Sigma _{N}^{+}$ from section , $$\Sigma_{N}^{+}\ \ :=\ \ S^{\mathbb{Z}_{+}}\ \ =\ \ \big\{ 1,\, 2,\, \cdots,\, N \big\}^{\mathbb{Z}_{+}}\ \ =\ \ \Big\{ w = \left( w_{1}\, w_{2}\, \cdots\, w_{n}\, \cdots \right)\ :\ w_{i} \in \left\{ 1,\, 2,\, \cdots,\, N \right\} \Big\}.$$ Observe that one can define the maps $\Pi_{m}$ and $\pi_{m}$ on the symbolic space $S^{n}$ as well as $\Sigma_{N}^{+}$, analogous to its definitions on $I^{n}$ and $I^{\mathbb{Z}_{+}}$. We make use of the same to define a metric between any words $v, w \in \Sigma_{N}^{+}$. Fix any $\theta \in (0, 1)$, and define $$d_{\Sigma_{N}^{+}} (v,\, w)\ \ :=\ \ \theta^{n (v,\, w)},\ \ \text{where}\ \ n (v,\, w)\ :=\ \sup \Big\{ k \in \mathbb{Z}_{+}\ :\ \Pi_{k} (v)\ =\ \Pi_{k} (w) \Big\}.$$ Here, we define $n ( v, v ) := \infty$, thereby $d_{\Sigma_{N}^{+}} ( v, v ) = 0$. Thus, it is clear that we have a family of metrics on the space $\Sigma_{N}^{+}$. The discrete topology that separates any two distinct symbols on the set $\{ 1, 2, \cdots, N \}$ accords a product topology on $\Sigma_{N}^{+}$ with which the above described family of metrics is compatible. We shall fix a value of $\theta$, according to our need in a later section. In this topology, the cylinder sets given by fixing a finite set of co-ordinates, are the sets that are both closed and open. For ease of explanations, we shall always consider cylinder sets whose co-ordinates are fixed from the first co-ordinate onwards, for example, a cylinder set of length $m$ looks like $$\big[ v_{1}\, v_{2}\, \cdots\, v_{m} \big]\ \ =\ \ \Big\{ w \in \Sigma_{N}^{+}\ :\ \Pi_{m} (w)\ =\ (v_{1}\, v_{2}\, \cdots\, v_{m}) \Big\}.$$ These cylinder sets form a basis for the $\sigma$-algebra on $\Sigma_{N}^{+}$ on which one could define a measure for $\Sigma_{N}^{+}$. An easily describable measure on the space, $\Sigma_{N}^{+}$ is the Bernoulli measure defined thus. For any fixed probability vector $p = \big( p_{1},\, p_{2},\, \cdots,\, p_{N} \big)$, the measure is defined as $$\mu \Big( \big[ v_{1}\, v_{2}\, \cdots\, v_{m} \big] \Big)\ \ =\ \ p_{v_{1}}\; p_{v_{2}}\; \cdots\; p_{v_{m}}.$$ Observe that the shift map $\sigma$ defined on $\Sigma_{N}^{+}$ satisfies the properties asked for with respect to the topology defined on $\Sigma_{N}^{+}$. Further, $\Sigma_{N}^{+}$ is a compact metric space with topological dimension $0$. We now appeal to Tychonoff and accord some structure on $X$. For any two points $(w,\, x)$ and $(v,\, y)$ in $X$, we define a metric $$d_{X} ((w,\, x),\, (v,\, y))\ \ :=\ \ \max \Big\{ d_{\Sigma_{N}^{+}} (w,\, v),\; |x - y| \Big\}.$$ Thus, we work with the appropriate product topology and the product $\sigma$-algebra and the product measure, while we work with $X$. Various Ruelle operators {#vro} ======================== Making use of the Ruelle operator, as given in [@pp:90] for every $1 \le d \le N$, we define a Ruelle operator for the skew-product map in this section. Later, we consider each of these Ruelle operators to define a collective Ruelle operator for the case of simultaneous action of all these maps. For every $1 \le d \le N$, fix $f \in \mathcal{C} (I, \mathbb{C})$, consider $\mathcal{L}_{f}^{(d)} : \mathcal{C} (I, \mathbb{C}) \longrightarrow \mathcal{C} (I, \mathbb{C})$ given by $$\label{ruelled} \left( \mathcal{L}_{f}^{(d)} g \right) (x)\ \ :=\ \ \sum_{T_{d} y\, =\, x} e^{f(y)} g(y).$$ Observe that this definition entails the following iterative formula given by, $$\label{iterue} \left( \left( \mathcal{L}_{f}^{(d)} \right)^{\!\circ n} g \right) (x)\ \ :=\ \ \sum_{T_{d}^{n} y\, =\, x} e^{f^{n}_{(d\, d\, \cdots\, d)} (y)} g(y).$$ This is the usual Ruelle operator, as defined in [@pp:90]. For such an operator $\mathcal{L}_{f}^{(d)}$, we have the Ruelle operator theorem, as stated in [@pp:90]. \[rotd\] Suppose $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$. Then, the Ruelle operator $\mathcal{L}_{f}^{(d)}$ has a simple maximal positive eigenvalue, $\rho^{(d)}$. The remainder of the spectrum lies in a disc of radius strictly smaller than $\rho^{(d)}$. The eigenfunction $\phi_{d}$ corresponding to the maximal eigenvalue is strictly positive. Further, there exists an eigenmeasure corresponding to the maximal eigenvalue, in the space of all $T_{d}$-invariant probability measures supported on $I$, for the dual operator $\left( \mathcal{L}_{f}^{(d)} \right)^{\!*}$. Appealing to variational principles studied by several authors including Bowen in [@rb:73], Ruelle in [@dr:78] and Parry and Pollicott in [@pp:90], we know that the maximal eigenvalue for the Ruelle operator $\mathcal{L}_{f}^{(d)}$, for $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ can also be described using the $d$-pressure function, [*i.e.*]{}, $\rho^{(d)} = e^{\mathcal{P}^{(d)} (f)}$, where $$\mathcal{P}^{(d)} (f)\ \ :=\ \ \sup \left\{ h_{m^{(d)}} (T_{d}) + \int\! f\, d m^{(d)} \right\}.$$ The supremum in the above definition is taken over all $T_{d}$-invariant probability measures supported on $I$ and $h_{m^{(d)}} (T_{d})$ is the measure theoretic entropy of $T_{d}$ with respect to the measure $m^{(d)}$. Then, the existence of the unique equilibrium measure, denoted by $m_{f}^{(d)}$, that realises the supremum in the definition of $d$-pressure is assured by Denker and Urbanskii in [@du:91]. Further, the variational principle states that this unique equilibrium measure, $m_{f}^{(d)}$ is equivalent to the eigenmeasure corresponding to the maximal eigenvalue $\rho^{(d)}$ for the dual operator $\left( \mathcal{L}_{f}^{(d)} \right)^{\!*}$, as given in theorem . Taking cue from definition , we define the Ruelle operator for the skew-product setting as follows: Fix $F \in \mathcal{C} (X, \mathbb{C})$ and consider $\mathfrak{L}_{F} : \mathcal{C} (X, \mathbb{C}) \longrightarrow \mathcal{C} (X, \mathbb{C})$ given by $$\label{ruelleT} \left( \mathfrak{L}_{F} G \right) ((w,\, x))\ \ :=\ \ \sum_{T ((v,\, y))\, =\, (w,\, x)} e^{F((v,\, y))} G((v,\, y)).$$ It is a simple observation that the iterates of the Ruelle operator $\mathfrak{L}_{F}$ agrees with the appropriate iterative formula given in . We merely state the same here. $$\left( \left( \mathfrak{L}_{F} \right)^{\!\circ n} G \right) ((w,\, x))\ \ :=\ \ \sum_{T^{n} ((v,\, y))\, =\, (w,\, x)} e^{F^{n} ((v,\, y))} G((v,\, y)).$$ Further, the operator $\mathfrak{L}_{F}$ satisfies the properties mentioned in the Ruelle operator theorem, as mentioned in theorem , whenever $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$. We include the statement of the theorem, in the context of the skew-product map, for readers’ convenience. \[rotskew\] Suppose $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$. Then, the Ruelle operator $\mathfrak{L}_{F}$ has a simple maximal eigenvalue at $\varrho = e^{\mathfrak{P} (F)}$. The remainder of the spectrum lies in a disc of radius strictly smaller than $e^{\mathfrak{P} (F)}$. The eigenfunction $\Phi$ corresponding to the maximal eigenvalue is strictly positive. Further, the eigenmeasure corresponding to the maximal eigenvalue for the dual operator $\left( \mathfrak{L}_{F} \right)^{\!*}$ is equivalent to the equilibrium measure $\mu_{F}$, that realises the supremum in the definition of pressure, as stated in equation . Fixing $f \in \mathcal{C} (I, \mathbb{C})$, we now define a third Ruelle operator, denoted by $$\mathbb{L}_{f}\ :\ \mathcal{C} (I, \mathbb{C})\ \longrightarrow\ \mathcal{C} (I, \mathbb{C}),$$ that captures the idea of simultaneous action of interval maps. Making sense of the $n$-th iterate of the Ruelle operators $\mathcal{L}_{f}^{(d)}$ and $\mathfrak{L}_{F}$ that captures the set of all $n$-th order pre-images of the point where the operator acts and taking the $n$-th ergodic sum over each of those orbits, we are inclined to define the $n$-th iterate of the Ruelle operator $\mathbb{L}_{f}$ acting on a point $x$ by considering those points that would reach $x$ in $n$ steps, by the action of a combination of $n$ many maps from the collection $\{ T_{1},\, T_{2},\, \cdots,\, T_{N} \}$ and taking the $n$-th ergodic sum dictated by all such $n$-lettered words. The same can be expressed as $$\begin{aligned} \left( \left( \mathbb{L}_{f} \right)^{\!\circ n} g \right) (x) & = & \sum_{w\, :\, |w|\, =\, n} \sum_{\left( T_{w_{n}} \circ T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} \right) y\, =\, x} e^{f^{n}_{w} (y)} g(y) \\ & = & \sum_{w_{n}\, =\, 1}^{N} \cdots \sum_{w_{1}\, =\, 1}^{N} \sum_{\left( T_{w_{n}} \circ T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} \right) y\, =\, x} e^{f (y)\, +\, f (T_{w_{1}} y)\, +\, \cdots\, +\, f ( T_{w_{n - 1}} \circ \cdots \circ T_{w_{1}} y)} g(y). \end{aligned}$$ This understanding paves the way for us to define the Ruelle operator, in this case as $$\label{ruellesimul} \mathbb{L}_{f} g (x)\ \ :=\ \ \sum_{d\, =\, 1}^{N} \sum_{T_{d} y\, =\, x} e^{f(y)} g(y)\ \ =\ \ \sum_{d\, =\, 1}^{N} \mathcal{L}_{f}^{(d)} g (x).$$ Spectrum of the operators $\mathfrak{L}_{F}$ and $\mathbb{L}_{f}$ {#spectrum} ================================================================= In this section, we establish a relationship between the Ruelle operators $\mathfrak{L}_{F}$ and $\mathbb{L}_{f}$, as defined in equations and . Let $$\label{qoff} Q\ :\ \mathscr{F}_{\alpha} (I, \mathbb{C}) \longrightarrow \mathscr{F}_{\alpha} (X, \mathbb{C})\ \ \text{be defined as}\ \ \left( Q(f) \right) (w,\, x)\ \ :=\ \ f (x).$$ Then, for any Hölder continuous function $f \in \mathscr{F}_\alpha (I, \mathbb{C})$, $$\begin{aligned} \big| Q(f) ((w,\, x))\ -\ Q(f)((v,\, y)) \big| & = & \big| f(x) - f(y) \big| \\ & \le & M_{f} \left| x - y \right|^{\alpha} \\ & \le & M_{f}\, \big[ d_{X} ((w,\, x),\ (v,\, y)) \big]^{\alpha}, \end{aligned}$$ for some $M_{f} > 0$ and for any $0 < \theta < 1$, on which the metric on $\Sigma_{N}^{+}$ depends. Thus, the map $Q$ is well defined. Further, the above inequality also proves that the Hölder constant $M_{f}$ remains unperturbed for the function $Q(f)$ in the product space as well, [*i.e.*]{}, $M_{f} \equiv M_{Q(f)}$. Moreover, it is clear from the definition of the various Ruelle operators that $$Q \left( \mathbb{L}_{f} g \right)\ \ =\ \ \mathfrak{L}_{Q(f)} Q(g),\ \ \forall f,\, g \in \mathscr{F}_{\alpha} (I, \mathbb{C}).$$ Thus, the action of $\mathbb{L}_{f}$ is similar to that of the action of $\mathfrak{L}_{Q(f)}$ restricted to the subspace, ${\rm Image} (Q) \subseteq \mathscr{F}_{\alpha} (X, \mathbb{C})$. As a consequence, we can relate the spectrum of $\mathbb{L}_{f}$ and $\mathfrak{L}_{Q(f)}$. The following lemma narrates the same. \[rotsa\] For some fixed $f \in \mathscr{F}_{\alpha}(I, \mathbb{C})$, let $\Phi \in \mathscr{F}_{\alpha} (X, \mathbb{C})$ be an eigenfunction of $\mathfrak{L}_{Q(f)}$ with corresponding eigenvalue $\varrho$, [*i.e.*]{}, $\mathfrak{L}_{Q(f)} \Phi = \varrho\, \Phi$. Then, $\Phi \in {\rm Image}(Q) \subset \mathscr{F}_{\alpha} (X, \mathbb{C})$. In order to prove this lemma, it is sufficient to prove that the eigenfunction $\Phi$ is independent of the first co-ordinate, [*i.e.*]{}, $\Phi ((v,\, x)) = \Phi ((w,\, x)),\ \forall v,\, w \in \Sigma_{N}^{+}$ and $x \in [0,\, 1]$. In fact, we prove that given any $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{Z}_{+}$ such that $$\Big\vert \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x))\ -\ \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big\vert\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon}.$$ A mere application of the eigenfunction equation to the above inequality, then yields $$\varrho^{n} \Big| \Phi ((v,\, x))\ -\ \Phi ((w,\, x)) \Big|\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon},$$ implying $$\Big| \Phi ((v,\, x))\ -\ \Phi ((w,\, x)) \Big|\ \ \le\ \ \epsilon.$$ The proof is then complete, appealing to the arbitrary choice that we can make for $\epsilon$. Using the definition of the Ruelle operator, we have $$\begin{aligned} & & \Big| \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x)) - \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big| \\ & = & \left| \sum_{u\; :\; |u|\, =\, n} \left( \sum_{T^{n} ((u v,\, y))\, =\, (v,\, x)} e^{(Q(f))^{n} (u v,\, y)} \Phi ((u v,\, y)) \right. \right. \\ & & \hspace{+2in} \left. \left. -\ \sum_{T^{n} ((u w,\, y))\, =\, (w,\, x)} e^{(Q(f))^{n} (u w,\, y)} \Phi ((u w,\, y)) \right) \right| \\ & \le & \sum_{u\; :\; |u|\, =\, n} \sum_{T_{u_{n}} \circ \cdots \circ T_{u_{1}} y\, =\, x} \Big| e^{f_{u}^{n} (y)} \Big| \Big| \Phi ((u v,\, y)) - \Phi ((u w,\, y)) \Big| \\ & \leq & e^{n \| f \|_{\infty}}\, M_{\Phi}\, N^{n}\, \theta^{n \alpha} \\ & = & \varrho^{n}\, M_{\Phi}\, \left( \frac{e^{\| f \|_{\infty}}\, N\, \theta^{\alpha}}{\varrho} \right)^{n}. \end{aligned}$$ Once we choose the functions $f$ and $\Phi$, observe that the quantities $\| f \|_{\infty}$ and $M_{\Phi}$ are determined. Thus, we can only rely on our choice of $\theta$ to make the above estimate, as small as necessary. We therefore fix $\theta$ in such fashion that $$\frac{e^{\| f \|_{\infty}}\, N\, \theta^{\alpha}}{\varrho}\ \ <\ \ 1.$$ Further, we remark that we shall use the same $\theta$, so fixed to suit our purpose in the above inequality, in the remainder of this paper. Now, it is clear that we have some threshold, say $M_{\epsilon}$ such that $$\Big| \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((v,\, x))\ -\ \left( \mathfrak{L}_{Q(f)} \right)^{\!\circ n} \Phi ((w,\, x)) \Big|\ \ \le\ \ \varrho^{n} \epsilon,\ \ \forall n \ge M_{\epsilon}.$$ A careful reader may observe that lemma is merely the statement of the Ruelle operator theorem stated for the simultaneous dynamics of finitely many interval maps. In particular, $\Phi \in {\rm Image}(Q)$ implies that there exists $\phi \in \mathscr{F}_{\alpha} (I, \mathbb{C})$ such that $Q(\phi) = \Phi$ and $\mathbb{L}_{f} \phi = \varrho \phi$. Thus, the set of eigenvalues of $\mathfrak{L}_{Q(f)}$ and $\mathbb{L}_{f}$ remains equal. In particular, the simple maximal eigenvalue $\varrho$ of $\mathfrak{L}_{Q(f)}$ is also the simple maximal eigenvalue of $\mathbb{L}_{f}$. Normalising the operators $\mathcal{L}_{f}^{(d)},\ \mathfrak{L}_{F}$ and $\mathbb{L}_{f}$ {#Normalising} ========================================================================================= For technical convenience, we normalise the Ruelle operators $\mathcal{L}_{f}^{(d)},\ \mathfrak{L}_{Q(f)}$ and $\mathbb{L}_{f}$, in different ways that would suit our purposes to prove the main theorems. For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, we define the normalised Ruelle operator by considering the Ruelle operator, as defined in , corresponding to the function $$\widetilde{f}_{d}\ \ :=\ \ f\; +\; \log \phi_{d}\; -\; \log \phi_{d} \circ T_{d}\; -\; \mathcal{P}^{(d)} ( f ),$$ where $\phi_{d}$ is the eigenfunction corresponding to the maximal eigenvalue of the operator $\mathcal{L}_{f}^{(d)}$. Thus, $$\widetilde{\mathcal{L}}_{f}^{(d)}\, \mathbf{1} (x)\ \ :=\ \ \mathcal{L}_{\widetilde{f}_{d}}^{(d)}\, \mathbf{1} (x)\ \ =\ \ \mathbf{1} (x).$$ For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, we define the normalised Ruelle operator in the skew-product setting by considering the Ruelle operator, as defined in , corresponding to the function $$\widetilde{Q (f)}\ \ :=\ \ Q(f)\; +\; \log \Phi\; -\; \log \Phi \circ T\, ,$$ where $\Phi$ is the eigenfunction corresponding to the maximal eigenvalue $\varrho = e^{\mathfrak{P} ( Q(f))}$ of the operator $\mathfrak{L}_{Q(f)}$. Thus, this normalisation effects the eigenfunction of the normalised operator to be equal to the constant function $\mathbf{1}$, however, leaves the the maximal eigenvalue $\varrho$ unchanged, [*i.e.*]{}, $$\widetilde{\mathfrak{L}}_{Q(f)}\, \mathbf{1} ((w,\, x))\ \ :=\ \ \mathfrak{L}_{\widetilde{Q(f)}}\, \mathbf{1} ((w,\, x))\ \ =\ \ \varrho\, \mathbf{1} ((w,\, x)).$$ In particular, the operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$ has $1$ as its maximal eigenvalue with corresponding eigenfunction $\mathbf{1}$. Further, since by definition, $Q(f)$ and $\widetilde{Q(f)}$ are cohomologous to each other, their ergodic sums are preserved. Further, once we normalise the operators $\mathcal{L}_{f}^{(d)}$ and $\mathfrak{L}_{Q(f)}$ as prescribed, we also observe that the equilibrium measure with respect to the Hölder continuous functions $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ for the map $T_{d}$ and $Q(f) \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ for the skew-product map $T$, namely $m_{f}^{(d)}$ and $\mu_{Q(f)}$ are nothing but the eigenmeasures corresponding to the maximal eigenvalues of the normalised operators $\widetilde{\mathcal{L}}_{f}^{(d)}$ and $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$ respectively. Thus, we use the notations $m_{f}^{(d)}$ and $\mu_{Q(f)}$ for the equilibrium measure, as well as the eigenmeasure corresponding to the maximal eigenvalue of the normalised operators $\widetilde{\mathcal{L}}_{f}^{(d)}$ as well as $\varrho^{-1} \widetilde{\mathfrak{L}}_{Q(f)}$, respectively. Moreover, we define the normalised operator in the setting of simultaneous dynamics thus: For any $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, let $$\widetilde{\mathbb{L}}_{f} g (x)\ \ :=\ \ \frac{1}{\phi (x)}\, \mathbb{L}_{f + \log \phi} g (x)\ \ =\ \ \frac{1}{\phi (x)}\, \sum_{d\, =\, 1}^{N} \sum_{T_{d} y\, =\, x} e^{f(y)\, +\, \log \phi (y)} g(y),$$ where $\phi$ is the strictly positive eigenfunction corresponding to the maximal eigenvalue of the operator $\mathbb{L}_{f}$, that we denote by $e^{\mathbb{P} (f)}$, where $\mathbb{P} (f)$ is the *pressure* of the function $f$, in this setting. It must be noted that this normalisation does not change the maximal eigenvalue, but has an effect on the corresponding eigenfunction, making it to be $\mathbf{1}$. Further, this normalisation also takes a toll in this setting; $\widetilde{\mathbb{L}}_{f}$ is no longer a Ruelle operator, but merely a bounded linear operator. The operator $Q : \mathcal{C} (I, \mathbb{C}) \longrightarrow \mathcal{C} (X, \mathbb{C})$, as defined in , has a natural transpose $$Q^{*}\ :\ \mathcal{C}^{*} (X, \mathbb{C}) \longrightarrow \mathcal{C}^{*} (I, \mathbb{C}).$$ On a restricted space, this transpose gives us the map, $$Q^{*}\ :\ \big\{ \mu\ :\ \mu\ \text{is a probability measure on}\ X \big\} \longrightarrow \big\{ \mathfrak{m}\ :\ \mathfrak{m}\ \text{is a probability measure on}\ I \big\},$$ defined by $$\int\! f\, d \left( Q^{*} \mu \right)\ \ :=\ \ \int\! Q(f)\, d \mu,\ \ \forall f \in \mathcal{C} (I, \mathbb{C}).$$ For $f \in \mathscr{F}_{\alpha}(X, \mathbb{R})$, we define $\mathfrak{m}_{f} := Q^{*}(\mu_{Q(f)})$, thereby, $$\int_{I}\! g\, d \mathfrak{m}_{f}\ \ =\ \ \int_{X}\! Q(g)\, d \mu_{Q(f)},\ \ \forall g \in \mathscr{F}_{\alpha} (I, \mathbb{R}).$$ Thus, it is an easy observation that as a consequence of the definitions of $\mathfrak{m}_{f}$ and $\widetilde{\mathbb{L}}_{f},\ \mathfrak{m}_{f}$ coincides with the eigenmeasure corresponding to maximal eigenvalue of the operator $e^{-\mathbb{P}(f)} \left( \widetilde{\mathbb{L}}_{f} \right)^{*}$. In the sequel, we will be interested in a complex perturbation of the operators $\widetilde{\mathfrak{L}}_{Q(f)}$ and $\widetilde{\mathbb{L}}_{f}$ for $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, [*i.e.*]{}, for $\zeta = \kappa + i \xi \in \mathbb{C}$, we consider the operators $\mathfrak{L}_{\kappa Q(f)}$ and $\mathbb{L}_{\kappa f}$, normalise the same as explained above respectively and then perturb the operator. We denote them respectively by $\widetilde{\mathfrak{L}}_{\zeta Q(f)}$ and $\widetilde{\mathbb{L}}_{\zeta f}$. We conclude this section with a few lemmas that provide a bound on the operator norm for the iterates of the normalised, yet perturbed operators. For any $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, there exists a positive constant $C_{5} > 0$, such that for every $n \ge 0$, and any $G \in \mathscr{F}_{\alpha} (X, \mathbb{C})$, we have $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\!\circ n}\, G \bigg\Vert_{\alpha}\ \ \ge\ \ C_{5}\, | \xi |\, \big\Vert G \big\Vert_{\infty}\; +\; \alpha^{n}\, \big\vert G \big\vert_{\alpha}.$$ \[zeropointtwo\] Let $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfy the approximability condition, as mentioned in equations and of theorem . Then, there exists positive constants $C_{6},\ C_{7}$ and $C_{8}$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\! \circ (2 n R)} \bigg\Vert\ \ \le\ \ C_{6}\, | \xi |\, \left( 1 - \frac{1}{| \xi |^{C_{7}}} \right)^{n - 1},\ \ \ \forall n \ge 1,$$ where $\left| \xi \right|$ is sufficiently large and $R$ is the greatest integer contained in $C_{8} \log \left| \xi \right|$. We conclude this section with a lemma that provides a bound on the operator norm for the iterates of the normalised, but perturbed operator $\widetilde{\mathbb{L}}_{\zeta f}$, for some $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$. We also include a short proof of the same, for readers’ convenience. \[eightpointthree\] Let $f \in \mathscr{F}_{\alpha}(I, \mathbb{R})$ satisfy the approximability condition, as mentioned in equation of theorem . Let $\zeta = \kappa + i \xi \in \mathbb{C}$. Then, there exists positive constants $C_{9},\ C_{10}$ and $C_{11}$ such that $$\bigg\Vert \left( e^{-\mathbb{P}(\kappa f)} \widetilde{\mathbb{L}}_{\zeta f} \right)^{\!\circ (2 n R)} \bigg\Vert\ \ \le\ \ C_{9}\ \left\vert \xi \right\vert\ \Bigg( 1 - \frac{1}{\left\vert \xi \right\vert^{C_{10}}} \Bigg)^{n - 1},\ \ \ \forall n \ge 1,$$ where $\left\vert \xi \right\vert$ is sufficiently large and $R$ is the greatest integer contained in $C_{11} \log \left\vert \xi \right\vert$. Given $f \in \mathscr{F}_{\alpha} (I, \mathbb{R})$ that satisfies the approximability condition, it is an easy observation that $Q(f) \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfies the appropriate approximability condition. Thus, lemma comes to our rescue. Further, from the definitions of the operators as stated in and , we obtain that $$\label{landl} \bigg\Vert \left( \widetilde{\mathbb{L}}_{\zeta f} \right)^{\!\circ n} \bigg\Vert\ \ \le\ \ \bigg\Vert \left( \widetilde{\mathfrak{L}}_{\zeta Q(f)} \right)^{\!\circ n} \bigg\Vert\ \ \ \ \forall n \ge 1.$$ Proof of the ergodicity theorems {#ergproof} ================================ In this section, we prove the ergodicity theorems as stated in for the skew-product setting and for the simultaneous dynamics setting. Let $B \subseteq X$ be a completely $T$-invariant set in the $\sigma$-algebra of $X$ with strictly positive measure, [*i.e.*]{}, $\mu (B) > 0$. As said earlier in section , a natural candidate for $\mu$ is the product measure of the Bernoulli measure on cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on open intervals of $I$. Since we have assumed that the set $B$ is completely $T$-invariant, [*i.e.*]{}, $T^{-1} B\, =\, B$ with strictly positive measure, it is sufficient for us to show that $\mu (B) = 1$, to prove the theorem. It is easy to observe that, in its most general form, the set $B$ can be expressed as $$B\ \ =\ \ \bigcup_{j\, \ge\, 1} \big( U_{j} \times V_{j} \big),$$ where $U_{j}$’s are cylinder sets in $\Sigma_{N}^{+}$ and $V_{j}$’s are open subsets of $I$. Let $U \times V \subseteq B$ be a product set in this collection where $U = \big[ v_{1}\, v_{2}\, \cdots\, v_{n} \big]$ and $V \subseteq I$ is an open set. Since $B$ is completely $T$-invariant, we have $U \times V \subseteq T^{-1} B$. Thus, there exists $B'$ in the $\sigma$-algebra of $X$ such that $B' \subseteq B$ and $U \times V \subseteq T^{-1} (B')$. For the smallest such subset $B'$, the countably many possibilities for the form of $B'$ are given by 1. $\big[ v_{2}\, v_{3}\, \cdots\, v_{n} \big] \times V'$; 2. $\bigcup\limits_{d\, =\, 1}^{N} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d \big] \times V_{d}' \Big)$; 3. $\bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2} \big] \times V_{(d_{1}\, d_{2})}' \Big)$; 4.   $\cdots$. It is clear that in the above enumeration of possibilities, the cylinder sets in all cases starting from (1) onwards are subsumed by the cylinder set in case (0). Suppose we also prove that the appropriate open subsets of $I$ in each of the possibilities starting from case (1) are subsumed by the open subset of $I$ in case (0), it is sufficient to merely work with case (0). For the same purpose, we consider the general case (m), as listed in the above possibilities, namely, 1. $\bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big)$. Observe that $$\begin{aligned} \label{UtimesV} U \times V & \subseteq & T^{-1} B' \nonumber \\ & = & T^{-1} \Bigg( \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg) \nonumber \\ & = & \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N}\Bigg( T^{-1} \Big( \big[ v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg) \nonumber \\ & = & \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N} \Bigg( \bigcup_{d\, =\, 1}^{N} \Big( \big[ d\, v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times T_{d}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big) \Bigg). \nonumber \\ & & \end{aligned}$$ However, since $U = \big[ v_{1}\, v_{2}\, \cdots\, v_{n} \big]$, the only possibility in the right hand side of the countable union in equation where $U \times V$ can be a subset reduces to $$U \times V\ \ \subseteq\ \ \bigcup\limits_{d_{1}\, =\, 1}^{N} \bigcup\limits_{d_{2}\, =\, 1}^{N} \cdots \bigcup\limits_{d_{m}\, =\, 1}^{N} \Big( \big[ v_{1}\, v_{2}\, v_{3}\, \cdots\, v_{n}\, d_{1}\, d_{2}\, \cdots\, d_{m} \big] \times T_{v_{1}}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}' \Big).$$ The above union is over sets that are disjoint with respect to their first component and thus $V \subseteq T_{v_{1}}^{-1} V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}'$, for any $m$-lettered word $(d_{1}\, d_{2}\, \cdots\, d_{m})$. Now using minimality of the set $B'$, we observe that the set $V_{(d_{1}\, d_{2}\, \cdots\, d_{m})}'$ remains independent of the $m$-lettered word $(d_{1}\, d_{2}\, \cdots\, d_{m})$. Hence, $B' = \big[ v_{2}\, v_{3}\, \cdots\, v_{n} \big] \times V'$, as mentioned in case (0). Based on our observation that for every $U \times V = U_{0} \times V_{0} \subseteq B$, there exists a $B_{1} = U_{1} \times V_{1} \subseteq B$ that satisfies $U_{0} \times V_{0} \subseteq T^{-1} \left( U_{1} \times V_{1} \right)$, we obtain a sequence of sets $\left( U_{k} \times V_{k} \right)$ that satisfy, $\left( U_{k - 1} \times V_{k - 1} \right) \subseteq T^{-1} \left( U_{k} \times V_{k} \right)$. However, this process must end in a finite number of steps, precisely $n$, since $U_{0}$ is a cylinder set that fixes only $n$ many positions. Thus, after those finitely many steps, we obtain $\Sigma_{N}^{+} \times V_{n} \subseteq B$ for some open subset $V_{n} \subseteq I$ with strictly positive Lebesgue measure, [*i.e.*]{}, $\lambda (V_{n}) > 0$. Choose the maximal subset $V \subseteq I$ such that $\Sigma_{N}^{+} \times V \subseteq B$, [*i.e.*]{}, suppose there exists a subset $V' \subseteq I$ such that $\Sigma_{N}^{+} \times V' \subseteq B$ then, $V' \subseteq V$. For such a maximal subset $V \subseteq I$, consider $\big[ w \big] \times V$ for some choice of $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. It is obvious that $\big[ w \big] \times V \subseteq \Sigma_{N}^{+} \times V$. Then, there exists a $V' \subseteq I$ such that $$\label{wtimesV} \big[ w \big] \times V\ \ \subseteq\ \ T^{-1} \big( \Sigma_{N}^{+} \times V' \big)\ \ =\ \ \bigcup_{d\, =\, 1}^{N} \Big( \big[ d \big] \times T_{d}^{-1} V' \Big).$$ Arguing as earlier, we reduce the countable union in the right hand side of equation to $$\big[ w \big] \times V\ \ \subseteq\ \ \big[ w \big] \times T_{w}^{-1} V'.$$ However, owing to the maximality of $V$, we have $V' \subseteq V$ that implies $T_{w}^{-1} V' \subseteq T_{w}^{-1} V$. Thus, $$\big[ w \big] \times V\ \ \subseteq\ \ \big[ w \big] \times T_{w}^{-1} V'\ \ \subseteq \big[ w \big] \times T_{w}^{-1} V.$$ This implies $V \subseteq T_{w}^{-1} V$ for any choice of $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. However, each of these interval maps preserves the Lebesgue measure, [*i.e.*]{}, $\lambda (V) = \lambda (T_{w}^{-1} V)$ for all $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. Thus, by eliminating an appropriate set of Lebesgue measure zero from $V$, we obtain $\widetilde{V}$ that satisfies $\widetilde{V} = T_{w}^{-1} \widetilde{V}$ for all $w \in \big\{ 1,\, 2,\, \cdots,\, N \big\}$. Finally, we appeal to the ergodicity of each of these interval maps $T_{w}$ to conclude that the completely $T_{w}$-invariant set $\widetilde{V}$ for all $w$ of strictly positive measure must be of Lebesgue measure $1$. Thus, the completely $T$-invariant set $B$ that satisfies $\mu (B) > 0$ has measure $1$, thereby completing the proof of theorem . \[erg1rem\] The above proof, in fact provides a stronger result than is stated in theorem , namely, for any set $B$ in the $\sigma$-algebra of $X$ that is completely $T$-invariant, meaning $T^{-1} B = B$, we have either $\mu (B) = 0$ or $B$ can be expressed as $\Sigma_{N}^{+} \times \widetilde{V}$ where $\widetilde{V}$ is a set of full Lebesgue measure in $I$, [*i.e.*]{}, $\lambda \left( \widetilde{V} \right) = 1$. We now prove the ergodic theorem for simultaneous action of finitely many interval maps, as stated in theorem . For a Lebesgue integrable real-valued function $f$ defined on $I$, define $F := Q(f) \in L^{1} (\mu)$, where $\mu$ is the product measure of the Bernoulli measure on the cylinder sets of $\Sigma_{N}^{+}$ and the Lebesgue measure on the open intervals of $I$. Define $$E\ :=\ \Bigg\{ (w,\, x) \in X\; :\; \liminf_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ =\ \limsup_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ =\ \int\! F\, d \mu \Bigg\}.$$ One can easily observe that $E$ is a $T$-invariant subset of $X$. Thus, $\mu (E)$ is either zero or one, by theorem . The set of points collected in $E$ is the set of all points in $X$ that satisfies the Birkhoff’s pointwise ergodic theorem to the dynamical system $T$ acting on $X$. Hence, $\mu (E) = 1$. Further, from remark , we have that $E = \Sigma_{N}^{+} \times E'$, where $\lambda(E') = 1$. Hence, we now have for $\lambda$-almost every $x \in I$ and for every $w \in \Sigma_{N}^{+}$, $$\lim_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x)\ \ =\ \ \int\! F\, d \mu.$$ \[thereshold\] For a fixed $x_{0} \in E'$, given $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{N}$, independent of $w$, such that for all $n \geq M_{\epsilon}$, we have $$\left| \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \int\! F\, d \mu \right|\ \ <\ \ \epsilon,\ \ \ \forall w \in \Sigma_{N}^{+}.$$ We initially prove the theorem, assuming claim to be true. We shall prove the claim immediately thereafter. From the definition of the composition operators, we have for any arbitrary $\epsilon > 0$, there exists $M_{\epsilon} \in \mathbb{N}$ such that for every $n \geq M_{\epsilon}$, $$\left| \frac{1}{n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right|\ \ \le\ \ \epsilon,\ \ \text{for}\ \lambda\text{-a.e.}\ x \in I\ \text{and}\ \forall w = (w_{1}\, w_{2}\, \cdots\, w_{n}).$$ This implies $$\left| \sum\limits_{w\ :\ |w|\, =\, n} \left( \frac{1}{n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right) \right|\ \ \le\ \ N^{n} \epsilon.$$ Thus, $$\left| \frac{1}{n} \frac{1}{N^{n}} \sum\limits_{w\ :\ |w|\, =\, n} f_{w}^{n} (x)\ -\ \int_{0}^{1}\! f\, d \lambda \right|\ \ \le\ \ \epsilon,\ \ \text{for}\ \ \lambda\text{-a.e.}\ x \in I,$$ proving the theorem. We now complete this section, by proving the claim in . For a fixed $x_{0} \in E'$, consider the sequence $\displaystyle{\left\{ \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0}) \right\}_{n\, \ge\, 1}}$ of functions defined on the compact space $\Sigma_{N}^{+}$, converging to the constant $\int\! F\, d \mu$. This convergence is uniform over $w \in \Sigma_{N}^{+}$, [*i.e.*]{}, given any $\epsilon > 0$, there exists $M_{1} = M_{1} (\epsilon)$ such that for all $n > M_{1}$, we have $$\label{kirone} \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0}) \right|\ \ \le\ \ \frac{\epsilon}{2},\ \ \forall w, v \in \Sigma_{N}^{+}.$$ We already know that for some fixed $v \in \Sigma_{N}^{+}$, there exists $M_{2} = M_{2}(\epsilon, v)$ such that for all $n > M_{2}$, we have $$\label{kirtwo} \left| \frac{1}{n} \sum\limits_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0})\ -\ \int\! F\, d \mu \right|\ \ \le\ \ \frac{\epsilon}{2}.$$ Taking $M_{3} := \max\{ M_{1},\, M_{2} \}$ where $M_{1}$ and $M_{2}$ are the quantities prescribed by equations and , we get for all $n > M_{3}$ and $w \in \Sigma_{N}^{+}$, $$\begin{aligned} & & \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \int\! F\, d \mu \right| \\ & \le & \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (w,\, x_{0})\ -\ \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0}) \right|\ +\ \left| \frac{1}{n} \sum_{j\, =\, 0}^{n - 1} \mathscr{Q}^{j} (F) (v,\, x_{0})\ -\ \int\! F\, d \mu \right| \\ & \le & \epsilon.\end{aligned}$$ It is clear from the definition that $M_{3}$ is independent of the word $w$ and only dependent on $\epsilon$ and $x_{0}$. Rates of recurrence {#rorproof} =================== In this section, we write the proofs of theorems and . Throughout this section, we work with a fixed $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$, along with the corresponding normalised operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ with equilibrium measure $\mu_{F}$. We know that the iterates of this operator obey the result given in lemma . Further, for some $\zeta = \kappa + i \xi \in \mathbb{C}$, we note that the pressure function associated with the operator $\mathfrak{L}_{\zeta F}$ and its normalised version $\widetilde{\mathfrak{L}}_{\zeta F}$ are one and the same, owing to the definition of pressure and the method of normalisation. The following lemma gives an approximation for the eigenvalue of the normalised yet perturbed operator, $\widetilde{\mathfrak{L}}_{\zeta F}$. We urge the reader to observe that the statement of the lemma and hence, its proof, are merely mentioned for a change of variables along the imaginary variable, even though more is true, as one may obtain from, say [@ps:94]. \[morse\] For $\zeta = \kappa + i \xi$, there exists a change of variables $\Upsilon = \Upsilon(\xi)$ such that for $| \xi | < \delta$, we can expand $$e^{\mathfrak{P} \left( \zeta F \right)}\ \ =\ \ e^{\mathfrak{P} \left( \kappa F \right)}\; \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big),$$ where $\Theta$ is a real-valued function that satisfies $\Theta (\Upsilon) = O ( | \Upsilon |^{3} )$. By perturbation theory, we know that there exists $\delta > 0$ such that, for $G \in \mathscr{F}_{\alpha}(X, \mathbb{C})$ satisfying $\| G - \kappa F \|_{\alpha} < \delta$, $$\label{analytic} G\ \ \longmapsto\ \ \mathfrak{P}(G)\ \ \ \ \text{is an analytic map}.$$ The analyticity of the above map from the Banach space $\mathscr{F}_{\alpha} (X, \mathbb{C})$ to $\mathbb{C}$, in a neighbourhood of $\kappa F$ assures the existence of a linear map, say $\mathfrak{D} : \mathscr{F}_{\alpha} (X, \mathbb{C}) \longrightarrow \mathbb{C}$ such that $$\lim_{G\, \rightarrow\, \kappa F} \frac{\mathfrak{P} \big( G \big)\ -\ \mathfrak{P} \big( \kappa F \big)\ -\ \mathfrak{D} \big( G - \kappa F \big)}{\| G\ -\ \kappa F \|_{\alpha}}\ \ =\ \ 0,$$ where $\mathfrak{D}$ is the differential of $\mathfrak{P}$ at $\kappa F$. From equation , we have for $\xi \in \mathbb{R}$, $$\lim_{\xi \rightarrow 0} \frac{\mathfrak{P} \big( ( \kappa + \xi) F \big)\ -\ \mathfrak{P} \big( \kappa F \big)\ -\ \mathfrak{D} \big( \xi F \big)}{\| \xi F \|_{\alpha}}\ \ =\ \ 0,$$ for the choice $$\frac{\mathfrak{D} \big( F \big)}{\| F \|_{\alpha}}\ \ =\ \ \left. \frac{d}{d\xi} \mathfrak{P} \big( ( \kappa + \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \int\! F\, d \mu_{\kappa F}.$$ Since $\mathfrak{D}$ is linear, we get $$\left. \frac{d}{d \xi} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{\xi\, \rightarrow\, 0} \frac{\mathfrak{D} \big( i \xi F \big)}{\| i \xi F \|_{\alpha}}\ \ =\ \ i \lim_{\xi\, \rightarrow\, 0} \frac{\mathfrak{D} \big( \xi F \big)}{\| \xi F \|_{\alpha}}\ \ =\ \ \int\! i F\, d \mu_{\kappa F}\ \ =\ \ 0.$$ Similarly from equation we have for $\xi \in \mathbb{R}$, $$\left. \frac{d^{2}}{d \xi^{2}} \mathfrak{P} \big( ( \kappa + \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \Big( F^{n} ((w,\, x)) \Big)^{2}\, d \mu_{\kappa F}.$$ This implies $$\left. \frac{d^{2}}{d \xi^{2}} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{1}{n} \int\! \Big( i F^{n} \Big)^{2}\, d \mu_{\kappa F}\ \ =\ \ \lim_{n\, \rightarrow\, \infty} \frac{-1}{n} \int\! \Big( F^{n} \Big)^{2}\, d \mu_{\kappa F}\ \ <\ \ 0.$$ Since $G \longmapsto \mathfrak{P} (G)$ is an analytic map, as mentioned in , we have the map $\zeta \longmapsto \mathfrak{P} ( \zeta F )$ to be analytic too in a neighbourhood of $\kappa$, where $\zeta = \kappa + i \xi \in \mathbb{C}$. This implies that ${\rm Im} \big( \mathfrak{P} ( \zeta F ) \big)$ is a harmonic function around $\kappa$, [*i.e.*]{}, for $\xi \in \mathbb{R}$, $$\left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ \left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + i \xi ) F ) \big) \right|_{\xi\, =\, 0}.$$ We know that for $\xi \in \mathbb{R},\ \mathfrak{P} \big( ( \kappa + \xi ) F \big) \in \mathbb{R}$, [*i.e.*]{}, $$\left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ 0\ \ \ \ \Longrightarrow\ \ \ \ \left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \big( \mathfrak{P} ( ( \kappa + i \xi ) F ) \big) \right|_{\xi\, =\, 0}\ \ =\ \ 0.$$ Thus we have $$\begin{aligned} \left. \frac{d}{d \xi} \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right|_{\xi\, =\, 0} & = & 0; \\ \left. \frac{d^{2}}{d \xi^{2}} {\rm Re} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right) \right|_{\xi\, =\, 0} & < & 0; \\ \left. \frac{d^{2}}{d \xi^{2}} {\rm Im} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right) \right|_{\xi\, =\, 0} & = & 0. \end{aligned}$$ Thus, the map $\xi \longmapsto {\rm Re} \left( \mathfrak{P} \big( ( \kappa + i \xi ) F \big) \right)$ satisfies the hypothesis of Morse lemma for non degenerate critical points. Thus, similar to lemma (4) in [@ps:94] that has been derived in an analogous context as lemma (6) in [@ss:07], we obtain a change of variables $\Upsilon = \Upsilon(\xi)$ for $- \delta < \xi < \delta$ such that $$e^{\mathfrak{P} \big( ( \kappa + i \xi ) F \big)}\ \ =\ \ e^{\mathfrak{P} ( \kappa F )} \big( 1\ -\ \Upsilon^{2}\ +\ i \Theta \left( \Upsilon \right) \big),$$ where $\Theta \left( \Upsilon ( \xi ) \right) = e^{- \mathfrak{P} ( \kappa F )} {\rm Im} \big( e^{\mathfrak{P} \big( ( \kappa + i \xi ) F \big)} \big)$ and $\Theta \left( \Upsilon \right) = O \big( \left| \Upsilon \right|^{3} \big)$, thus proving the lemma. We now start by considering the left hand side of equation , as mentioned in theorem , that measures the cardinality of the set $$\Big\{ (w,\, x)\, \in\, {\rm Fix}_{n} (T)\ :\ a\, \leq\, F^{n}((w,\, x))\, \leq\, b \Big\}.$$ It is clear that it can be expressed in terms of the indicator function, $\chi_{[a,\, b]}$, [*i.e.*]{}, $$\# \Big\{ (w,\, x)\, \in\, {\rm Fix}_{n} (T)\ :\ a\, \leq\, F^{n}((w,\, x))\, \leq\, b \Big\}\ \ =\ \ \sum_{(w,\, x)\, \in\, {\rm Fix}_{n} (T)} \chi_{[a,\, b]} \Big( F^{n} ((w,\, x)) \Big).$$ Since we know that any characteristic function can be approximated by a sequence of smooth functions with compact support under the integral norm, we initially prove a slightly modified result, as stated in proposition , where the indicator function $\chi_{[a,\, b]}$ is replaced by some smooth function with compact support, say $\tau : \mathbb{R} \longrightarrow \mathbb{R} $. \[prop 1\] Suppose $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ satisfies the hypothesis in theorem , [*i.e.*]{}, $F$ satisfies the approximability condition and there exists a unique real number $\kappa$ such that $\int\! F\, d \mu_{\kappa F} = 0$. Then, there exists a positive constant $C_{12} > 0$ such that $$\digamma_{\!\! \tau} (n)\ \ :=\ \ \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \tau \Big( F^{n} ((w,\, x)) \Big)\ \ \sim\ \ C_{12}\; \frac{e^{n\, \mathfrak{P}(\kappa F)}}{\sqrt{n}}\; \int_{\mathbb{R}}\! \tau (t)\, e^{- \kappa t}\; d t.$$ For $y \in \mathbb{R}$ and $\kappa$ as in the hypothesis, define $$\label{tausubkappa} \tau_{\kappa} (y)\ \ :=\ \ \tau (y) e^{-\, \kappa y},$$ in order that $\digamma_{\!\! \tau} (n)$ can now be expressed as $$\label{taukappa} \digamma_{\!\! \tau} (n)\ \ =\ \ \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \tau_{\kappa}\; \Big( F^{n} ((w,\, x)) \Big)\; e^{\kappa F^{n} ((w,\, x))}.$$ Using inverse Fourier transform and Fubini’s theorem, we rewrite equation as $$\begin{aligned} \digamma_{\!\! \tau} (n) & = & \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi)\; e^{(\kappa\, +\, i \xi)\; F^{n} ((w,\, x))}\; d \xi \\ & = & \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \sum_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi)\; F^{n} ((w,\, x))}\; d \xi. \end{aligned}$$ By definition, $\tau_{\kappa}$ lives inside a compact support. Hence, by an application of the Paley-Wiener theorem, we deduce that it is sufficient to estimate $$\sum\limits_{(w,\, x)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi)\; F^{n}((w,\, x))},\ \ \text{for some}\ \xi\; \in\; \mathbb{R}.$$ The following lemma, from [@dr:73] helps us approximate this sum in terms of the iterates of the appropriate normalised operator such that $$\widetilde{\mathfrak{L}}_{\kappa F} \mathbf{1}\ \ =\ \ \varrho\, \mathbf{1}\ \ \ \ \text{where we recall}\ \ \varrho\ \ =\ \ e^{\mathfrak{P}(\kappa F)}.$$ \[ror1lemma1\] Let $\kappa \in \mathbb{R}$ be the unique real number that satisfies $\int\! F\, d \mu_{\kappa F} = 0$. Then, there exists $0 < \eta < \varrho^{-1}$ such that for every point $(w, x)\; \in\; {\rm Fix}_{n} (T)$, we have $$\sum_{(v,\, y)\; \in\; {\rm Fix}_{n} (T)} e^{(\kappa\, +\, i \xi) F^{n}((v,\, y))}\ \ =\ \ \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; \Big( 1\; +\; O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big).$$ Using the above lemma, we write $\digamma_{\!\! \tau} (n)$ as $$\label{split1} \digamma_{\!\! \tau} (n)\ \ =\ \ \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\ \Big( 1\, +\, O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big)\; d \xi.$$ The second term in the above equation is dominated by $(\varrho \eta)^{n}$, which converges to zero faster than any polynomial of $n$. In the remainder of the proof, we estimate the first term of equation . If $\Xi_{\zeta F} : \mathscr{F}_{\alpha}(X, \mathbb{C}) \longrightarrow \mathscr{F}_{\alpha}(X, \mathbb{C})$ is the one-dimensional eigenprojection associated with $\widetilde{\mathfrak{L}}_{\zeta F}$, for $- \delta < \xi < \delta$, we know by perturbation theory that $\Xi_{\zeta F} (\mathbf{1}) = \mathbf{1} + O \big( \left| \Upsilon \right| \big)$. Thus, by perturbation theory and lemma , for $- \delta < \xi < \delta$ and some $0 < \vartheta < 1$, we obtain $$\begin{aligned} \left( \widetilde{\mathfrak{L}}_{\zeta F} \right)^{\!\circ n} \mathbf{1} & = & e^{n \mathfrak{P} ( \zeta F )} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ) \\ & = & e^{n \mathfrak{P} \big( \kappa F \big)} \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ). \end{aligned}$$ The above equation facilitates the splitting of the integral in equation into two integrals given by, $$\begin{aligned} \label{split2} \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi & = & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi \nonumber \\ & & + \int_{| \xi |\, \ge\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\; d \xi. \nonumber \\ & & \end{aligned}$$ We first estimate the first integral in equation , using the change in variables from lemma . $$\begin{aligned} \label{split3} & & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathfrak{L}}_{(\kappa\, +\, i \xi) F} \right)^{\!\circ n} \mathbf{1} \right) ((w,\, x))\, d \xi \nonumber \\ & = & \int_{| \xi |\, <\, \delta} e^{n \mathfrak{P} ( \kappa F )} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big) \Big( \widehat{\tau_{\kappa}} \big( \xi ( \Upsilon ) \big) \Big) \frac{d \xi}{d \Upsilon} d \Upsilon + O ( \vartheta^{n} ) \nonumber \\ & = & C_{13}\; \widehat{\tau_{\kappa}} (0)\; e^{n \mathfrak{P} ( \kappa F )} \int_{| \xi |\, <\, \delta} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big)\; d \Upsilon\ +\ O(n^{-1})\ +\ O( \vartheta^{n} ), \nonumber \\ & & \end{aligned}$$ where $C_{13} > 0$ is a constant dependent on $\tau_{\kappa}$ and the Jacobian of change of variables. We now apply binomial expansion to the expression in the integral of equation , thus splitting it into three parts and define them as follows: $$\begin{aligned} I_{0} (n) & := & \int_{- \delta}^{\delta}\! \big( 1\ -\ \Upsilon^{2} \big)^{n}\, d \Upsilon; \\ \sum_{j\, =\, 1}^{n} I_{j} (n) & := & \left| \sum_{j\, =\, 1}^{n} \binom{n}{j} \int_{- \delta}^{\delta} \big( 1\ -\ \Upsilon^{2} \big)^{n - j}\; \big( i \Theta (\Upsilon) \big)^{j}\; \left( 1\ +\ O \big( |\Upsilon| \big) \right)\, d \Upsilon \right|; \\ J(n) & := & \int_{- \delta}^{\delta}\! \big( 1\ -\ \Upsilon^{2} \big)^{n}\; O \big( |\Upsilon| \big)\, d \Upsilon. \end{aligned}$$ Using techniques from [@ps:94], we estimate the above integrals to get the following inequalities. $$\begin{aligned} I_{0} (n) & = & C_{14} \frac{\Gamma \left( n + 1 \right)}{\Gamma \left( n + 1 + \frac{1}{2} \right)} + O ( ( 1 - \delta^{2} )^{n} ) \\ \sum_{j\, =\, 1}^{n} I_{j} (n) & \le & C_{15} \frac{\Gamma \left( n + 1 \right)}{\Gamma \left( n + 1 + \frac{1}{2} \right)}\;\frac{1}{\sqrt{n}} \\ | J(n) | & \le & C_{16} \frac{\Gamma(n + 1)}{\Gamma(n + 2)}, \end{aligned}$$ for some positive constants $C_{14},\ C_{15}$ and $C_{16}$. We now use the identity, $$\lim_{n\, \rightarrow\, \infty} \frac{\Gamma(n + \beta)}{\Gamma(n) n^{\beta}}\ \ =\ \ 1,$$ to conclude that $I_{0} (n)\ \sim\ \frac{1}{\sqrt{n}} C_{14}$ and that the rest of the terms inside the integral in equation converge to zero faster than $\frac{1}{\sqrt{n}}$. What remains now is the integral in equation over $| \xi | \geq \delta$. For that, we make use of the following lemma, which is a standard result in the theory of Fourier transforms. \[lemmafourier\] Let $\chi : \mathbb{R} \longrightarrow \mathbb{R}$ be a compactly supported $\mathcal{C}^{r}$ function. Then the Fourier transform $\widehat{\chi} (\xi) = O ( | \xi |^{- r} )$ as $\xi \rightarrow \infty$. Since $\tau_{\kappa}$ is smooth, we may suppose that $\tau_{\kappa}$ is a compactly supported $\mathcal{C}^{r}$ function, for any arbitrary $r \in \mathbb{N}$. Now using lemma and lemma , we obtain the following expression. $$\int_{| \xi |\, \ge\, \delta} \widehat{\tau_{\kappa}} ( \xi )\; \Big( \left( \widetilde{\mathfrak{L}}_{(\kappa + i \xi ) F} \right)^{\!\circ n} \mathbf{1} \Big)((w, x))\, d \xi\ \ =\ \ O \left( e^{n \mathfrak{P} \big( \kappa F \big)} \int_\delta^{\infty}\! \left( 1 - \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi \right),$$ where $R = [C_{8} \log \xi]$. We now prove that the integral in the right hand side of the above quantity approaches zero faster than $\frac{1}{\sqrt{n}}$ as $n$ goes to $\infty$, using techniques from [@ps:94]. In order to achieve the same, we further split the integral into two parts, as $$\int_{\delta}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi\ \ =\ \ \int_{\delta}^{n^{\delta'}} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi\ +\ \int_{n^{\delta'}}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi,$$ where $\delta < \delta' < \frac{1}{C_{7}}$. Thus, by our choice of $\delta'$, we have $C_{7} \delta' < 1$. Hence, we get the following estimates. The convergence rate of the first term to zero is faster than any polynomial while the second part converges to zero at a polynomial rate dependent on $r$. $$\begin{aligned} \label{split5.1} \int_{\delta}^{n^{\delta'}} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi & = & O \left( n^{\delta'} \left( 1\ -\ \frac{1}{n^{C_{7} \delta'}} \right)^{\frac{n}{2 \delta' \beta \log n}} \right) \\ \label{split5.2} \int_{n^{\delta '}}^{\infty} \left( 1\ -\ \frac{1}{\xi^{C_{7}}} \right)^{\frac{n}{2R}} \xi^{- r}\, d \xi & = & O \left( n^{( 1 - r ) \delta'} \right) \end{aligned}$$ Using the bounds in equations and , we now estimate the remaining part of equation to be the following. $$\int_{| \xi |\, \ge\, \delta} \widehat{\tau_{\kappa}} (\xi) \Big( \widetilde{\mathfrak{L}}_{(\kappa + i \xi) F} \Big)^{\! \circ n}\mathbf{1} ((w, x))\, d \xi\ \ =\ \ O \Big( e^{n \mathfrak{P} ( \kappa F )} n^{(1 - r) \delta'} \Big).$$ Since $\widehat{\tau_{\kappa}}$ is smooth, we prove that the rate of growth of the second term in equation is smaller than $\frac{1}{\sqrt{n}} e^{n \mathfrak{P} ( \kappa F )}$, by considering $r > 1 + \frac{1}{2 \delta'}$. Thus, we obtain the following asymptotic relation. $$\digamma_{\!\! \tau}(n)\ \ \sim\ \ C_{12} \frac{e^{n\mathfrak{P} (\kappa F)}}{\sqrt{n}} \widehat{\tau_{\kappa}} (0).$$ The proof of theorem now follows from proposition where we replace the function $F$ by the characteristic function $\chi_{[a,\, b]}$. We now proceed to prove theorem , using techniques similar to those used in the above proof. Hence, we only highlight the important steps to complete this proof. As earlier, we begin with a generalisation of the left hand side of the assertion of theorem for a compactly supported function and define $\widetilde{\digamma}_{\!\! \tau} (n)$ as $$\widetilde{\digamma}_{\!\! \tau} (n)\ \ :=\ \ \sum\limits_{w\; :\, |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} T_{w}} \tau ( f^{n}_{w} (x) ).$$ Clearly, $\widetilde{\digamma}_{\chi_{[a,\, b]}} (n)$ coincides with the expression we need to estimate. By replacing $\tau$ by $\tau_{\kappa}$, as defined in equation and applying Fourier transforms we get the following: $$\widetilde{\digamma}_{\tau} (n)\ \ =\ \ \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \sum\limits_{w\; :\; |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} (T_{w})} e^{((\kappa + i \xi ) f^{n}_{w} (x))}\, d \xi.$$ Thus, in order to estimate $\widetilde{\digamma}_{\tau} (n)$, we first estimate the sum inside the integral, namely $$\sum\limits_{w\; :\; |w|\, =\, n} \sum\limits_{x\, \in\, {\rm Fix} (T_{w})} e^{((\kappa + i \xi) f^{n}_{w} (x))}\ \ \text{for}\ \ \xi \in \mathbb{R}.$$ The following lemma gives a relation between the sum to be estimated and the iterates of the normalised Ruelle operator, $\widetilde{\mathbb{L}}_{(\kappa + i \xi) f}$. By observing the relation between $(\widetilde{\mathbb{L}}_{(\kappa + i \xi) f})^{\circ n}$ and $(\widetilde{\mathfrak{L}}_{(\kappa + i \xi) Q(f)})^{\circ n}$, as prescribed in equation , we write the following lemma which is nothing but an easy corollary of lemma . Let $\kappa \in \mathbb{R}$ be the unique real number that satisfies $\int\! f\, d \mathfrak{m}_{\kappa f} = 0$. Then, there exists $0 < \eta < e^{- \mathbb{P}(\kappa f)}$ such that for every point $x$ that satisfies $T_{w}^{n} x = x$ for some $n$-lettered word $w$, we have $$\sum_{v\; :\; |v|\, =\, n} \sum\limits_{y\, \in\, {\rm Fix} (T_{v})} e^{(\kappa\, +\, i \xi) f^{n}_{v} (y)}\ \ =\ \ \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; \Big( 1\; +\; O \left( n \eta^{n}\, \max\big\{ 1,\; |\xi| \big\} \right) \Big).$$ Using the above lemma, we rewrite $\widetilde{\digamma}_{\tau}(n)$ as, $$\label{integralform} \widetilde{\digamma}_{\tau}(n) = \int_{\mathbb{R}} \widehat{\tau}_{\kappa}(\xi) \left( \left((\widetilde{\mathbb{L}}_{(\kappa+i\xi)f})^{\circ n} \mathbf{1} \right)(x)\ \left(1 + O(n \eta^{n} \max \{1, |\xi|\})\right) \right) d\xi.$$ The second term in the above integral is dominated by $(\eta e^{\mathbb{P} (\kappa f)})^{n}$ and thus goes to zero, faster than polynomial of $n$, as $n$ tends to infinity. Using the definition of $\mathbb{P} ( ( \kappa + i \xi ) f )$, theorem and lemma , we conclude that the pressure functions $\mathbb{P} ( ( \kappa + i \xi ) f )$ and $\mathfrak{P} ( ( \kappa + i \xi ) Q(f) )$ coincide. Further, since $\int\! Q(f)\, d \mu_{\kappa Q(f)} = \int\! f\, d \mathfrak{m}_{\kappa f} = 0$, we obtain the following lemma as a corollary of lemma . For $\zeta = \kappa + i \xi$, there exists a change of variables $\Upsilon = \Upsilon(\xi)$ such that for $| \xi | < \delta$, we can expand $$e^{\mathbb{P} \left( \zeta f \right)}\ \ =\ \ e^{\mathbb{P} \left( \kappa f \right)}\; \Big( 1\; -\; \Upsilon^{2}\; +\; i\, \Theta \left( \Upsilon \right) \Big),$$ where $\Theta$ is a real-valued function that satisfies $\Theta (\Upsilon) = O ( | \Upsilon |^{3} )$. By perturbation theory, the one dimensional eigenprojection associated with $\widetilde{\mathbb{L}}_{( \kappa + i \xi ) f}$ is of the form $$\Xi_{( \kappa + i \xi ) f} \mathbf{1}\ \ =\ \ \mathbf{1} + O ( | \Upsilon | ).$$ Thus for $- \delta < \xi < \delta$ and for some $0 < \vartheta < 1$, we have $$\left( \widetilde{\mathbb{L}}_{( \kappa + i \xi ) f} \right)^{\!\circ n} \mathbf{1}\ \ =\ \ e^{n \mathbb{P} \big( ( \kappa + i \xi ) f \big)} \Big( \mathbf{1} + O \left( \big| \Upsilon \big| \right) \Big) + O ( \vartheta^{n} ).$$ The integral in the equation can now be split accordingly to get $$\begin{aligned} \label{split integral} \int_{\mathbb{R}}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi & = & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi \nonumber \\ & & + \int_{| \xi |\, \ge\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\; d \xi. \nonumber \\ & & \end{aligned}$$ We then apply change of variables to the first integral in the above equation to get $$\begin{aligned} & & \int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\, d \xi \\ & = & C_{17}\; \widehat{\tau_{\kappa}} (0)\; e^{n \mathfrak{P} ( \kappa f )} \int_{| \xi |\, <\, \delta} \Big( 1 - \Upsilon^{2} + i \Theta \left( \Upsilon \right) \Big)^{n} \Big( 1 + O \left( \big| \Upsilon \big| \right) \Big)\; d \Upsilon\ +\ O(n^{-1})\ +\ O( \vartheta^{n} ), \\ \end{aligned}$$ where the expression inside the integral is the same as in equation . Thus, the techniques used in the proof of theorem can be used again to obtain $$\int_{| \xi |\, <\, \delta}\! \widehat{\tau_{\kappa}} (\xi) \left( \left( \widetilde{\mathbb{L}}_{(\kappa\, +\, i \xi) f} \right)^{\!\circ n} \mathbf{1} \right) (x)\, d \xi\ \sim\ \frac{1}{\sqrt{n}} C_{18} e^{n \mathbb{P} ( \kappa f)} \widehat{\tau_{\kappa}}(0).$$ Now what remains is to prove that the second integral in equation grows at a rate strictly smaller than $\frac{1}{\sqrt{n}} e^{n \mathbb{P} ( \kappa f )}$, as $n \to \infty$, to complete the proof. This can be achieved again using lemma , as earlier. Decay of correlations {#docsec} ===================== In this section, we prove theorems pertaining to the decay of correlations in the skew-product setting and in the setting of simultaneous action of finitely many interval maps, namely theorems and . Fix $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ and consider the normalised Ruelle operator $\widetilde{\mathfrak{L}}_{F}$ along with its corresponding equilibrium measure $\mu_{F}$. Denote by $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$, the set of all $\alpha$-Hölder continuous functions whose integral with respect to $\mu_{F}$ is zero, [*i.e.*]{}, $$\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})\ \ :=\ \ \Bigg\{ G \in \mathscr{F}_{\alpha} (X, \mathbb{R})\ :\ \int\! G\, d \mu_{F} = 0 \Bigg\}.$$ It is easily verifiable that $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ is a subspace of $\mathscr{F}_{\alpha} (X, \mathbb{R})$. Further, the space $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ is preserved by the action of the operator $\widetilde{\mathfrak{L}}_{F}$, [*i.e.*]{}, $\widetilde{\mathfrak{L}}_{F} : \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}) \longrightarrow \mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$. The Ruelle operator theorem states that the action of $\widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$ has a spectral radius strictly smaller than $\varrho = e^{\mathfrak{P} (F)}$. Equivalently one may consider the operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$ that has a spectral radius, say $\varrho_{F} < 1$. We first state and prove a lemma, that will be useful to prove our main results in this section. \[lem4.1\] For any $\vartheta \in (\varrho_{F},\, 1)$, there exists a positive constant $C_{19} > 0$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ C_{19}\, \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha}\ \ \forall n \ge 0\ \ \text{and}\ \ \forall G \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}).$$ Fix a number $\vartheta \in (\varrho_{F},\, 1)$. Choose $\epsilon > 0$ such that $\varrho_{F} + \epsilon < \vartheta$. For this $\epsilon$, there exists $M_{\epsilon} \in \mathbb{Z}_{+}$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \bigg\Vert^{\frac{1}{n}}\ \ <\ \ \varrho_{F} + \epsilon,\ \ \forall n \ge M_{\epsilon},\ \ \ \text{since}\ \ \varrho_{F}\ \ = \inf\limits_{n\, \ge\, 1} \bigg\Vert \left( \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \bigg\Vert^{\frac{1}{n}},$$ where we only consider the action of $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ on $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$. This implies $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ <\ \ \left( \varrho_{F} + \epsilon \right)^{n}\, \big\Vert G \big\Vert_{\alpha}\ \ <\ \ \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha},\ \ \forall n \ge M_{\epsilon},\ \ \text{and}\ \ \forall G \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R}).$$ Further, since $\widetilde{\mathfrak{L}}_{F}$ is a bounded operator, there exists a positive constant $D_{0} > 0$ such that $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ D_{0}\, \big\Vert G \big\Vert_{\alpha},\ \ \forall n \ge 1.$$ Now, an application of the Archimedean property of $\mathbb{R}$ results in finitely many finite constants $D_{1},\, D_{2},\, \cdots,\, D_{M_{\epsilon}}$ that satisfy $D_{n} \vartheta^{n} > D_{0}$ for $1 \le n \le M_{\epsilon}$. Hence, $$\bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G \bigg\Vert_{\alpha}\ \ \le\ \ D_{n}\, \vartheta^{n}\, \big\Vert G \big\Vert_{\alpha},\ \ \text{for}\ 1 \le n \le M_{\epsilon}.$$ The result now follows by taking $C_{19} = \max \big\{ 1,\, D_{1},\, D_{2},\, \cdots,\, D_{M_{\epsilon}} \big\}$. We are now ready to prove theorem that states the decay of correlations result for the skew-product map $T$. For any $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, the left hand side of equation in theorem can be written as $$\begin{aligned} \int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\ -\ \int\! G\, d \mu_{F} \int\! H\, d \mu_{F} & = & \int\! \mathscr{Q}^{n} (G) H\, d \mu_{F}\ -\ \int\! \mathscr{Q}^{n} (G)\, d \mu_{F} \int\! H\, d \mu_{F} \\ & = & \int\! \mathscr{Q}^{n} (G) \left( H - \int\! H\, d \mu_{F} \right)\, d \mu_{F}. \end{aligned}$$ Suppose we denote $\displaystyle{\widetilde{H} := \left( H - \int\! H\, d \mu_{F} \right)}$, then it is easy to see that $\widetilde{H} \in \mathscr{F}^{F}_{\alpha} (X, \mathbb{R})$. Further, on the space of $\mu_{F}$-square integrable real-valued functions defined on $X$ denoted by $L^{2} (\mu_{F})$, the operator $\widetilde{\mathfrak{L}}_{F}$ has a natural extension, with its adjoint given by the operator $\mathscr{Q}$, [*i.e.*]{}, $$\big\langle \mathscr{Q} \Phi,\, \Psi \big\rangle\ \ =\ \ \big\langle \Phi,\, \widetilde{\mathfrak{L}}_{F} \Psi \big\rangle,\ \ \forall \Phi, \Psi \in L^{2} (\mu_{F}).$$ Hence, $$\int\! \mathscr{Q}^{n} (G) \widetilde{H}\, d \mu_{F}\ \ =\ \ \int\! G \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H}\, d \mu_{F}.$$ Therefore, $$\left\vert \int\! G \left( \varrho^{-1} \widetilde{ \mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H}\, d \mu_{F} \right\vert\ \ \le\ \ \int\! \left\vert G \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \right\vert\, d \mu_{F}\ \ \le\ \ \left\Vert G \right\Vert_{2}\, \left\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \right\Vert_{2}.$$ Further, $$\begin{aligned} \bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \bigg\Vert_{2} & \le & \bigg\Vert \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \widetilde{H} \bigg\Vert_{\alpha} \\ & \le & C_{19}\, \vartheta^{n}\, \left\Vert \widetilde{H} \right\Vert_{\alpha}\ \hspace{+6cm} (\text{using lemma \eqref{lem4.1}}) \\ & \le & C_{19}\, \vartheta^{n}\, \left( \big\Vert H \big\Vert_{\alpha}\ +\ \left\vert \int H\, d \mu_{F} \right\vert \right)\ \hspace{+2.7cm} (\text{using definition of}\ \widetilde{H}) \\ & \le & 2 C_{19}\, \vartheta^{n} \big\Vert H \big\Vert_{\alpha}. \end{aligned}$$ Thus, we obtain the result with the constant $C_{2}\; =\; 2 C_{19}\, \left\Vert G \right\Vert_{2}\, \left\Vert H \right\Vert_{\alpha}$ to complete the proof of theorem . To prove theorem , we start by considering the functions $f_{d} = - \log | T_{d}' | \in \mathscr{F}_{\alpha} (I, \mathbb{R})$, for $1 \le d \le N$. By the definition of the Ruelle operator $\mathcal{L}_{f_{d}}^{(d)}$, as stated in equation , we know that $$\Big( \mathcal{L}_{f_{d}}^{(d)} g \Big) (x)\ \ =\ \ \sum_{T_{d} y\, =\, x} \frac{g(y)}{| T_{d}' (y) |}.$$ Observe that $\mathcal{L}_{f_{d}}^{(d)} = \widetilde{\mathcal{L}}_{f_{d}}^{(d)}$, [*i.e.*]{}, the operator $\mathcal{L}_{f_{d}}^{(d)}$ has eigenvalue $1$, with corresponding eigenfunction $\mathbf{1}$. Further, it is evident from Boyarsky and Góra ([@bg:97], section (4.3)) that the dual operator $\left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!*}$ fixes the Lebesgue measure $\lambda$ [*i.e.*]{}, $\left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!*} \lambda = \lambda$. Moreover, for every $1 \le d \le N$, the operator $\mathscr{O}_{d}$ defined by $\mathscr{O}_{d} g = g \circ T_{d}$ satisfies $\mathcal{L}_{f_{d}}^{(d)} \mathscr{O}_{d} = {\rm id}$, the identity operator in $\mathcal{C} (I, \mathbb{R})$. Denoting by $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$, the set of all real-valued $\alpha$-Hölder continuous functions on $I$ whose Lebesgue integral is equal to $0$, [*i.e.*]{}, $$\label{Falphalambda} \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})\ \ :=\ \ \left\{ f \in \mathscr{F}_{\alpha} (I, \mathbb{R})\ :\ \int\! f \, d \lambda = 0 \right\},$$ and observing that $\mathcal{L}_{f_{d}}^{(d)}$ preserves the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ for all $1 \le d \le N$, we now state a lemma whose proof runs [*mutatis mutandis*]{} as the proof of lemma . We know that the action of $\mathcal{L}_{f_{d}}^{(d)}$ on $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ has a spectral radius, say $\rho_{\lambda}^{(d)} < 1$, owing to theorem . \[elevenpointtwo\] For any $\vartheta^{(d)} \in (\rho_{\lambda}^{(d)},\, 1)$, there exists a constant $C_{20}^{(d)} > 0$ such that $$\bigg\Vert \left( \mathcal{L}_{f_{d}}^{(d)} \right)^{\!\circ n} g \bigg\Vert_{\alpha}\ \ \le\ \ C_{20}^{(d)}\, \left( \vartheta^{(d)} \right)^{n}\, \big\Vert g \big\Vert_{\alpha}\ \ \forall n \ge 1\ \ \text{and}\ \ \forall g \in \mathscr{F}^{\lambda}_{\alpha} (I, \mathbb{R}),\ \ \text{for}\ \ 1 \le d \le N.$$ We are now thoroughly equipped to prove theorem . For any $g \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$, the left hand side of equation in theorem can be written as $$\begin{aligned} \int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! g\, d \lambda \int\! h\, d \lambda & = & \int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! \mathscr{O}_{w} (g)\, d \lambda \int\! h\, d \lambda \\ & = & \int\! \mathscr{O}_{w} (g) \left( h - \int\! h\, d \lambda \right)\, d \lambda. \end{aligned}$$ Let $\displaystyle{\widetilde{h} = h - \int\! h\, d \lambda} \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$. Then, since $\mathcal{L}_{f_{d}}^{(d)}$ is the adjoint of $\mathscr{O}_{d}$ in the space of Lebesgue square integrable real-valued functions defined on $I,\ L^2(I,\mathbb{R})$, we have $$\begin{aligned} \left\vert \int\! \left( \mathscr{O}_{w} g \right) \widetilde{h}\, d \lambda \right\vert & = & \left\vert \int\! g \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h}\, d \lambda \right\vert \\ & \le & \left\Vert g \right\Vert_{2} \left\Vert \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \widetilde{h} \right\Vert_{2}. \end{aligned}$$ Now making use of the inequality $$\left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{2}\ \ \le\ \ \left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{\alpha},$$ and redistributing the operators for $1 \le d \le N$, we obtain $$\begin{aligned} \left\Vert \left( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \right) \widetilde{h} \right\Vert_{\alpha} & = & \left\Vert \left( \left( \mathcal{L}_{f_{1}}^{(1)} \right)^{\gamma_{1}} \left( \mathcal{L}_{f_{2}}^{(2)} \right)^{\gamma_{2}} \cdots \left( \mathcal{L}_{f_{N}}^{(N)} \right)^{\gamma_{N}} \right) \widetilde{h} \right\Vert_{\alpha} \\ & \le & C_{20}^{(1)}\, C_{20}^{(2)}\, \cdots\, C_{20}^{(N)}\, \left( \vartheta^{(1)} \right)^{\gamma_{1}} \left( \vartheta^{(2)} \right)^{\gamma_{2}} \cdots \left( \vartheta^{(N)} \right)^{\gamma_{N}} \left\Vert \widetilde{h} \right\Vert_{\alpha}, \end{aligned}$$ appealing to lemma . Finally, defining $C_{20} := C_{20}^{(1)}\, C_{20}^{(2)}\, \cdots\, C_{20}^{(N)}$ and $\vartheta := \max \left\{ \vartheta^{(1)},\, \vartheta^{(2)},\, \cdots,\, \vartheta^{(N)} \right\}$, we obtain $$\begin{aligned} \left\vert \int\! \mathscr{O}_{w} (g) h\, d \lambda\ -\ \int\! g\, d \lambda \int\! h\, d \lambda \right\vert & \le & C_{20} \vartheta^{n} \left\Vert g \right\Vert_{2} \left\Vert \widetilde{h} \right\Vert_{\alpha} \\ & \le & 2 C_{20} \vartheta^{n} \left\Vert g \right\Vert_{2} \left\Vert h \right\Vert_{\alpha}, \end{aligned}$$ thus completing the proof. Almost sure invariance principles {#asipsec} ================================= In this section, we prove the almost sure invariance principles as stated in theorems and for both the settings, that we focus in this paper. As in section , we begin by fixing a real-valued Hölder continuous function $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ and considering the corresponding normalised Ruelle operator $\varrho^{-1} \widetilde{\mathfrak{L}}_{F}$ along with its equilibrium measure $\mu_{F}$ and the subspace $\mathscr{F}_{\alpha}^{F} (X, \mathbb{R}) \subseteq \mathscr{F}_{\alpha} (X, \mathbb{R})$. The proof closely follows the method of proof given by Pollicott and Sharp in [@ps:02] and Sridharan in [@ss:09]. For any function $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, define $$H\ \ :=\ \ \sum\limits_{n\, \ge \, 1} \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} G.$$ Observe that the infinite series that defines $H$ converges, owing to lemma . Then, $$\widetilde{\mathfrak{L}}_{F} \Big( G\, +\, H\, -\, \mathscr{Q} (H) \Big)\ \ =\ \ \widetilde{\mathfrak{L}}_{F} G\, +\, \widetilde{\mathfrak{L}}_{F} H\, -\, \varrho H\ \ =\ \ \mathbf{0}.$$ Thus, defining $\Phi\; :=\; G + H - \mathscr{Q} (H)$, we observe that $$\begin{aligned} \Big\vert G^{n} ((w,\, x))\; -\; \Phi^{n} ((w,\, x)) \Big\vert & = & \Big\vert \mathscr{Q}^{n} H ((w,\, x))\; -\; H ((w,\, x)) \Big\vert \\ & \le & \Big\vert \mathscr{Q}^{n} H ((w,\, x)) \Big\vert\; +\; \Big\vert H ((w,\, x)) \Big\vert \\ & \le & 2 \big\Vert H \big\Vert_{\alpha}. \end{aligned}$$ Thus, we have proved: (c.f.[@ps:02], Lemma 2) For any function $G \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$, there exists a function $H \in \mathscr{F}_{\alpha}^{F} (X, \mathbb{R})$ such that $\Phi = G + \big( H - \mathscr{Q} (H) \big)$ satisfies $$\widetilde{\mathfrak{L}}_{F} \Phi\ \ =\ \ 0\ \ \ \ \text{and}\ \ \ \ G^{n} ((w,\, x))\ \ =\ \ \Phi^{n} ((w,\, x)) + O(1).$$ Given that $\Phi$ and $G$ are cohomologous to each other, we have $$\label{eqnvar} \varsigma(G)^{2}\ \ =\ \ \int\! \Phi ((w,\, x))^{2}\, d \mu_{F}\; +\; 2 \sum_{n\, \ge\, 0} \int\! \Phi ((w,\, x)) \Phi (T^{n} ((w,\, x)))\, d \mu_{F}.$$ Since $\widetilde{\mathfrak{L}}_{f} \Phi = 0$, we obtain $$\begin{aligned} \int\! \Phi((w,\, x)) \Phi(T^{n} ((w,\, x)))\, d \mu_{F} & = & \int\! \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F} \right)^{\!\circ n} \left( \Phi ((w,\, x)) \Phi(T^{n} ((w,\, x))) \right)\, d \mu_{F} \\ & = & \int\! \left( \varrho^{-1} \widetilde{\mathfrak{L}}_{F}\right)^{\!\circ n} \Phi((w,\, x)) \Phi((w,\, x))\, d \mu_{F} \\ & = & 0. \end{aligned}$$ Therefore, equation becomes $$\varsigma(G)^{2}\ \ =\ \ \int\! \Phi ((w,\, x))^{2}\, d \mu_{F}.$$ Let $$\widehat{X}\ \ :=\ \ \Big\{ (w_{n},\, x_{n})_{n\, \le\, 0} \in X^{- \mathbb{N}}\ :\ T((w_{n - 1},\, x_{n - 1})) = (w_{n},\, x_{n}) \Big\}.$$ For the purposes of proofs in this section, we fix the following notations. Elements in $\widehat{X}$ will be represented as $\overline{(w,\, x)} = (w_{n},\, x_{n})_{n\, \le\, 0}$. Making use of the canonical projection $${\rm Pr} : \widehat{X} \longrightarrow X\ \ \ \text{defined by}\ \ \ {\rm Pr} \left( \overline{(w,\, x)} \right)\ =\ (w_{0},\, x_{0}),$$ we denote and define the natural extension of the map $T : X \longrightarrow X$ on $\widehat{X}$ by $$\widehat{T}\ :\ \widehat{X} \longrightarrow \widehat{X}\ \ \ \text{such that}\ \ \ {\rm Pr} \left( \widehat{T} \left( \overline{(w,\, x)} \right) \right)\ =\ T((w_{0},\, x_{0})).$$ Given a function $\Phi \in \mathcal{C} (X, \mathbb{R})$, let $\widehat{\Phi}$ be its natural extension on $\widehat{X}$ given by $$\label{widehatPhi} \widehat{\Phi} \left( \overline{(w,\, x)} \right)\ \ =\ \ \Phi \left( {\rm Pr} \left (\overline{(w,\, x)} \right) \right)\ \ =\ \ \Phi( (w_{0},\, x_{0}) ).$$ Thus, the function $F \in \mathscr{F}_{\alpha} (X, \mathbb{R})$ that we fixed in the beginning of this section along with its equilibrium measure $\mu_{F}$ are written as $\widehat{F}$ and $\widehat{\mu_{F}}$ on the space $\widehat{X}$. Since $\mu_{F}$ is a $T$-invariant probability measure on $X$, it is clear that $\widehat{\mu_{F}}$ is a $\widehat{T}$-invariant probability measure on $\widehat{X}$. Suppose $\mathscr{B}$ is a $\sigma$-algebra on $X$, define a sequence of $\sigma$-algebras on $\widehat{X}$ by $$\mathscr{B}_{0}\ \ :=\ \ {\rm Pr}^{- 1} \mathscr{B}\ \ \ \ \text{and}\ \ \ \ \mathscr{B}_{n}\ \ :=\ \ \left( \widehat{T} \right)^{n} \left( \mathscr{B}_{0} \right)\ \ \text{for}\ \ n \in \mathbb{N}.$$ On a probability space $(\Omega, \nu)$, let $\big\{ \mathscr{B}_{n} \big\}_{n\, \ge\, 0}$ be an increasing sequence of $\sigma$-algebras and $\big\{ \Psi_{n} : \Omega \longrightarrow \mathbb{R} \big\}_{n\, \ge\, 0}$ be a collection of functions. Then, $\big\{ \Psi_{n}, \mathscr{B}_{n} \big\}_{n\, \ge\, 0}$ is called an *increasing martingale* if $\Psi_{n}$ is $\mathscr{B}_{n}$-measurable and ${\rm E} \left[ \Psi_{n + 1} \mid \mathscr{B}_{n} \right] = \Psi_{n}$ for $n\geq 0$. Thus, defining $$\begin{aligned} \left( \widehat{\Phi} \right)^{n} \left( \overline{(w,\, x)} \right) & := & \widehat{\Phi} \left( \left( \widehat{T} \right)^{- 1} \left( \overline{(w,\, x)} \right) \right)\; +\; \widehat{\Phi} \left( \left( \widehat{T} \right)^{- 2} \left( \overline{(w,\, x)} \right) \right) \\ & & \hspace{+4cm} +\; \cdots +\; \widehat{\Phi} \left( \left( \widehat{T} \right)^{- n} \left( \overline{(w,\, x)} \right) \right), \end{aligned}$$ that captures the $n$-th ergodic sum $\Phi^{n}((w,\, x))$, as defined in equation , on the base space, helps us form an increasing martingale on $\widehat{X}$, related to $\Phi^{n}$. [@ps:02] The sequence $\left\{ \left( \widehat{\Phi} \right)^{n}, \mathscr{B}_{n} \right\}_{n\, \ge\, 1}$ forms an increasing martingale on $\widehat{X}$. Before we embark on the proof of theorem , we state the Skorokhod embedding theorem, as in Appendix I of [@hh:80]. The statement of this theorem will come in handy, in writing the proof. Let $\left\{ \widehat{\Psi}_{n}, \mathscr{B}_{n} \right\}_{n\, \ge\, 0}$ be a zero mean and square integrable martingale on $\widehat{X}$. Then, there exists a probability space $( \Omega, \mathcal{A}, \nu)$ that supports a Brownian motion $\mathfrak{B}$ such that $\mathfrak{B}(t)$ has variance $t$, an increasing sequence of $\sigma$-algebras $\big\{ \mathcal{F}_{n} \big\}_{n\, \ge\, 0}$ and a sequence of non negative random variables $\big\{ \mathfrak{X}_{n} \big\}_{n\, \ge\, 1}$ such that if $\mathcal{S}_{0} = 0$ and $\mathcal{S}_{n} = \sum\limits_{j\, =\, 1}^{n} \mathfrak{X}_{j}$ for $n \geq 1$, then 1. $\mathfrak{Y}_{n}\ \ :=\ \ \mathfrak{B} \left( \mathcal{S}_{n} \right)\ \ \stackrel{{\rm d}}{=}\ \ \widehat{\Psi}_{n}$,\ where $\stackrel{{\rm d}}{=}$ represents equality in distribution, [*i.e.*]{}, for any Borel measurable set $V$ in $\mathbb{R}$, $$\widehat{\mu_{F}} \left( \left\{ \overline{(w,\, x)} \in \widehat{X}\ :\ \widehat{\Psi}_{n} \left( \overline{(w,\, x)} \right) \in V \right\} \right)\ \ =\ \ \nu \left( \big\{ \omega \in \Omega\ :\ \mathfrak{Y}_{n} (\omega) \in V \big\} \right);$$ 2. $\mathfrak{Y}_{n}$ and $\mathcal{S}_{n}$ are $\mathcal{F}_{n}$-measurable; 3. ${\rm E}\left[ \mathfrak{X}_{n} \mid \mathcal{F}_{n - 1}) \right]\ \ =\ \ {\rm E}\left[ \left( \mathfrak{Y}_{n} - \mathfrak{Y}_{n - 1} \right)^{2} \mid \mathcal{F}_{n - 1} \right],\ \nu$-a.e. for $n \ge 1$. We now make use of the Skorokhod embedding theorem and prove theorem . Since $\left( \widehat{\Phi} \right)^{n}$ is a square integrable function with mean zero, we can apply the Skorokhod embedding thoerem. Further, making use of the definition of $\widehat{\Phi}$, as given in equation , we obtain $$\mathfrak{Y}_{n}\ \ \stackrel{{\rm d}}{=}\ \ \left( \widehat{\Phi} \right)^{n}\ \ \stackrel{{\rm d}}{=}\ \ \Phi^{n}.$$ Thus, in order to complete the proof of theorem , we make the following claim. \[claim2\] Given any $\delta > 0$, $$\mathfrak{Y}_{n} (\omega)\ \ =\ \ \mathfrak{B} (n) (\omega)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right)\ \ \forall n \ge 0,\ \ \ \nu\text{-a.e.}$$ Pending proof of claim , it follows from the properties of Brownian motion that $$\mathfrak{Y}_{\lfloor t \rfloor}\ \ =\ \ \mathfrak{B} (t)\; +\; O \left( t^{\frac{1}{4}\, +\, \delta} \right)\ \ \forall t \ge 0,\ \ \ \nu\text{-a.e.}$$ This proves the theorem. We now prove our claim . Since $\mathfrak{Y}_{n} = \mathfrak{B} ( \mathcal{S}_{n} )$, we approximate $\mathcal{S}_{n}$ by $n \varsigma(G)^{2}$, as follows. $$\begin{aligned} \label{eqn12} \mathcal{S}_{n}\; -\; n \varsigma(G)^{2} & = & \sum_{j\, =\, 1}^{n} \Big( \mathfrak{X}_{j}\, -\, {\rm E} \big[ \mathfrak{X}_{j} \mid \mathcal{F}_{j - 1} \big] \Big) \nonumber \\ & & +\; \sum_{j\, =1\, }^{n} \Big( {\rm E} \big[ \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \mid \mathcal{F}_{j - 1} \big]\; -\; \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \Big) \nonumber \\ & & +\; \sum_{j\, =\, 1}^{n} \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2}. \end{aligned}$$ Given any sequence $\left\{ \widehat{\Psi}_{n} \right\}_{n\, \ge\, 0}$ of functions and an increasing sequence of $\sigma$-algebras $\big\{ \mathcal{F}_{n} \big\}_{n\, \ge\, 0}$ such that $\widehat{\Psi}_{n}$ is $\mathcal{F}_{n}$ measurable for all $n \ge 0$, the sequence defined by $$\left\{ \mathbf{\widehat{\Psi}}_{n}\ \ :=\ \ \sum\limits_{j\, =\, 1}^{n} \left( \widehat{\Psi}_{j}\, -\, {\rm E} \big[ \widehat{\Psi}_{j} \mid \mathcal{F}_{j - 1} \big] \right),\; \mathcal{F}_{n} \right\}_{n\, \ge\, 1}$$ forms a martingale. Hence, the first and the second terms on the right hand side of equation are martingales. By the strong law of large numbers for martingales, as can be found in [@wf:71], we can see that for every $\delta > 0$ $$\begin{aligned} \sum_{j\, =\, 1}^{n} \Big( \mathfrak{X}_{j}\, -\, {\rm E} \big[ \mathfrak{X}_{j} \mid \mathcal{F}_{j - 1} \big] \Big) & = & O \left( n^{\frac{1}{2}\, +\, \delta} \right); \\ \sum_{j\, =1\, }^{n} \Big( {\rm E} \big[ \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \mid \mathcal{F}_{j - 1} \big]\; -\; \left( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \right)^{2} \Big) & = & O \left( n^{\frac{1}{2}\, +\, \delta} \right). \end{aligned}$$ We can therefore write equation as $$\label{eqn13} \mathcal{S}_{n}\; -\; n \varsigma(G)^{2}\ \ =\ \ \sum_{j\, =\, 1}^{n} \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2}\; +\; O \left( n^{\frac{1}{2}\, +\, \delta} \right).$$ We estimate the sum on the right hand side of equation with the help of the following series. $$\sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right].$$ The following integrals are equal. $$\begin{aligned} \mathfrak{I} (\delta) & := & \int\! \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right] \right)^{2}\, d \nu \\ & = & \int\! \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Phi \left( T^{j} ((w, x)) \right)^{2}\, -\, \int\! \Phi^{2}\, d \mu_{F} \right] \right)^{2}\, d \mu_{F}. \end{aligned}$$ We already have that $\mathfrak{Y}_{n} \stackrel{{\rm d}}{=} \Phi^{n}$. Thus, from a proposition of Brieman, L. as in [@lb:68], we deduce that for any measurable function $\Theta : \mathbb{R}^{\mathbb{N}} \longrightarrow \mathbb{R}$, $$\int\! \Theta \Big( \big( \mathfrak{Y}_{j} (\omega) \big)_{j\, =\, 0}^{\infty} \Big)\, d \nu\ \ =\ \ \int\! \Theta \Big( \big( \Phi^{j} ((w, x)) \big)_{j\, =\, 0}^{\infty} \Big)\, d \mu_{F}.$$ The result follows from an appropriate choice of the function $\Theta$, say $$\Theta \Big( \big( y_{j} \big)_{j\, \ge\, 0} \Big)\ \ =\ \ \left( \sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \left[y_{j + 1}\, -\, y_{j} \right]^{2}\; -\; \int\! \Phi^{2}\, d \mu \right] \right)^{2}.$$ A simple calculation now yields that for any $\delta > 0,\ \mathfrak{I} (\delta) < \infty$. Hence, $$\sum_{j\, \ge\, 1} \frac{1}{j^{\frac{1}{2}\, +\, \delta}} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; \varsigma(G)^{2} \right]\ \ <\ \ \infty,\ \ \nu\text{-a.e.}$$ Applying the Kronecker lemma as in [@hh:80], we deduce that $$\label{eqn18} \sum_{j\, =\, 1}^{n} \left[ \Big( \mathfrak{Y}_{j}\, -\, \mathfrak{Y}_{j - 1} \Big)^{2}\; -\; n \varsigma(G)^{2} \right]\ \ =\ \ O \left( n^{\frac{1}{2}\, +\, \delta} \right).$$ Thus, from equations ) and , we have $\mathcal{S}_{n}\; -\; n \varsigma(G)^{2}\ =\ O \left( n^{\frac{1}{2}\, +\, \delta} \right),\ \nu$-a.e. Finally, defining $\widetilde{\mathfrak{B}} (t) := \mathfrak{B} ( t \varsigma(G)^{2} )$, we have for $n \ge 0$, $$\mathfrak{B} ( \mathcal{S}_{n} )\ \ =\ \ \mathfrak{B} \left( n \varsigma(G)^{2} \right)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right)\ \ =\ \ \widetilde{\mathfrak{B}} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu\text{-a.e.}$$ This proves the equation in claim , namely, $$\mathfrak{Y}_{n}\ \ =\ \ \widetilde{\mathfrak{B}} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu\text{-a.e.}$$ We now proceed to prove the next theorem on almost sure invariance principles for simultaneous action of the interval maps, as stated in theorem . We draw motivation from the proof of a similar result in an article by Haydn, Nicol, Török and Vaienti in [@hntv:17] and achieve a better bound. We first state a theorem due to Cuny and Merlévede as in [@cm:15], that would be helpful in our proof. [@cm:15] \[CM 2.3\] Let $\big\{ U_{n} \big\}_{n\, \ge\, 0}$ be a sequence of square integrable random variables adapted to some non-increasing sequence of $\sigma$-algebras $\big\{ \mathscr{A}_{n} \big\}_{n\, \ge\, 0}$ on $\mathbb{R}$. Assume that $${\rm E} \big[ U_{n} \mid \mathscr{A}_{n + 1} \big]\ \ =\ \ 0\ \text{a.s.};\ \ \ \ \varsigma_{n}^{2}\ \ =\ \ \sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ U_{k}^{2} \big]\ \ \to\ \ \infty;\ \ \ \ \sup\limits_{n\, \ge\, 0} {\rm E} \big[ U_{n}^{2} \big]\ \ <\ \ \infty.$$ Let $\big\{ a_{n} \big\}_{n\, \ge\, 0}$ be a non-decreasing sequence of positive numbers such that $$\left\{ \frac{a_{n}}{\varsigma_{n}} \right\}_{n\, \ge\, 0}\ \ \text{is non-decreasing}\ \ \ \text{and}\ \ \ \left\{ \frac{a_{n}}{\varsigma_{n}^{2}} \right\}_{n\, \ge\, 0}\ \ \text{is non-increasing}.$$ Further, assume that 1. $\sum\limits_{k\, =\, 0}^{n - 1} \Big( {\rm E} \big[ U_{k}^{2} \mid \mathscr{A}_{k + 1} \big]\; -\; {\rm E} \big[ U_{k}^{2} \big] \Big)\ \ =\ \ o(a_{n}),\ \lambda$-a.s.; 2. $\sum\limits_{n\, \ge\, 0} a_{n}^{- r} {\rm E} \big[ |U_{n}|^{2r} \big]\ \ <\ \ \infty$ for some $1 \le r \le 2$. Then enlarging our probability space, if necessary, it is possible to find a sequence $\big\{ \mathcal{U}_{n} \big\}_{n\, \ge\, 0}$ of independent centered Gaussian variables with ${\rm E} \big[ \mathcal{U}_{n}^{2} \big] = {\rm E} \big[ U_{n}^{2} \big]$ such that $$\sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} U_{j}\; -\; \sum_{j\, =\, 0}^{k} \mathcal{U}_{j} \right|\ \ =\ \ o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right),\ \ \ \lambda\text{-a.s.}$$ Note that the assertion of theorem can be rewritten by considering another probability space $(\Omega, \mathscr{A}, \nu)$ and a sequence of random variables, say $\big\{ \mathcal{V}_{n} \big\}_{n\, \ge\, 0}$ such that $U_{n} \stackrel{{\rm d}}{=} \mathcal{V}_{n}$. Then, $$\sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} \mathcal{V}_{j}\; -\; \sum_{j\, =\, 0}^{k} \mathcal{U}_{j} \right|\ \ =\ \ o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right),\ \ \ \nu\text{-a.s.}$$ We will now prove theorem . Recall the definition of the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ from equation , $$\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})\ \ :=\ \ \left\{ f \in \mathscr{F}_{\alpha} (I, \mathbb{R})\ :\ \int\! f \, d \lambda = 0 \right\},$$ and the property that for $f_{d} = - \log | T_{d}' |$, the operator $\mathcal{L}_{f_{d}}^{(d)}$ preserves the space $\mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ for all $1 \le d \le N$. Let $g \in \mathscr{F}_{\alpha}^{\lambda} (I, \mathbb{R})$ and $w \in \Sigma_{N}^{+}$. Then, define a sequence of $\sigma$-algebras $$\mathscr{B}_{w}^{n}\ \ :=\ \ \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \right)^{- 1} \mathscr{B}\ \ \ \ \text{for}\ \ n \ge 0,$$ where $\mathscr{B}$ is the Borel $\sigma$-algebra on $I$. Suppose for all $n \ge 1$, we denote by $\mathfrak{g}_{w}^{n}$, the sum $$\mathfrak{g}_{w}^{n}\ \ :=\ \ \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \Big) g\; +\; \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \Big) g\; +\; \cdots\; +\; \Big( \mathcal{L}_{f_{w_{n}}}^{(w_{n})} \mathcal{L}_{f_{w_{n - 1}}}^{(w_{n - 1})} \cdots \mathcal{L}_{f_{w_{1}}}^{(w_{1})} \Big) g$$ and $\mathfrak{g}_{w}^{0} = 0$ for all $w \in \Sigma_{N}^{+}$. It is easy to see that $$\mathcal{L}_{f_{w_{n + 1}}}^{(w_{n + 1})} \mathbb{g}_{w}^{n}\ \ =\ \ 0,\ \ \ \ \text{where}\ \ \ \ \mathbb{g}_{w}^{n}\ \ =\ \ g\; +\; \mathfrak{g}_{w}^{n}\; -\; T_{w_{n + 1}} \mathfrak{g}_{w}^{n + 1}.$$ Defining $\mathbb{h}_{w}^{n} = \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \left( \mathbb{g}_{w}^{n} \right)$, one can observe that $\mathbb{h}_{w}^{n}$ agrees with the definition of a reverse martingale difference sequence for the sequence of $\sigma$-algebras $\mathscr{B}_{w}^{n}$, as defined in Conze and Raugi [@cr:07], as given below. Given a sequence of random variables $\big\{ X_{n} \big\}_{n\, \in\, \mathbb{N}}$ adapted to a non-increasing sequence of $\sigma$- algebras $\big\{ \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}},\ \big\{ X_{n},\, \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}}$ is a *reverse martingale* or equivalently, $\big\{ X_{n} \big\}_{n\, \in\, \mathbb{N}}$ is a reverse martingale adapted to $\big\{ \mathscr{A}_{n} \big\}_{n\, \in\, \mathbb{N}}$ if $\bigg\{ \widetilde{X}_{n},\, \widetilde{\mathscr{A}}_{n} \bigg\}_{n\, \le\, -1}$ forms a martingale, where $\widetilde{X}_{n} = X _{- n}$ and $\widetilde{\mathscr{A}}_{n} = \mathscr{A}_{- n}$ for each $n \in - \mathbb{N}$. Now, $$\begin{aligned} \sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} & = & \sum_{k\, =\, 0}^{n - 1} \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g\, +\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} \mathfrak{g}_{w}^{k}\, -\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k + 1})} \mathfrak{g}_{w}^{k + 1} \right) \\ & = & \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g\, -\, \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n}. \end{aligned}$$ Further, $\left\Vert \mathfrak{g}_{w}^{n} \right\Vert_{\alpha}$ is uniformly bounded. Hence, $$\begin{aligned} {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} \right)^{2} \right] & = & {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g \right)^{2} \right]\; +\; {\rm E} \big[ \left( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n} \right)^{2} \big] \\ & & \hspace{+2cm} -\; 2 {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{k})} g \right) \big( \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \mathfrak{g}_{w}^{n} \big) \right] \\ & = & \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( \left( \varsigma_{w}^{(n)} (g) \right) \right), \end{aligned}$$ where we recall the definition of $\varsigma_{w}^{(n)} (g)$ from equation as $$\left( \varsigma_{w}^{(n)} (g) \right)^{2}\ \ =\ \ \int\! \left( g_{w}^{n} \right)^{2}\, d \lambda.$$ Haydn, Nicol, Török and Vaienti in [@hntv:17] show us that ${\rm E} \big[ \mathbb{h}_{w}^{j} \mathbb{h}_{w}^{k} \big] = 0$, for $j \ne k$ and therefore $$\sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ =\ \ {\rm E} \left[ \left( \sum_{k\, =\, 0}^{n - 1} \mathbb{h}_{w}^{k} \right)^{2} \right]\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( \left( \varsigma_{w}^{(n)} (g) \right) \right),$$ which implies, $\sum\limits_{k\, =\, 0}^{n - 1} {\rm E} \big[\left( \mathbb{h}_{w}^{k} \right)^{2} \big] \to \infty$. Thus, we have constructed a sequence of square integrable random variables $\left\{ \mathbb{h}_{w}^{n} \right\}_{n\, \ge\, 0}$ adapted to a non-increasing sequence of $\sigma$-algebras $\big\{ \mathscr{B}_{w}^{n} \big\}_{n\, \ge\, 0}$ that satisfies $${\rm E} \big[ \mathbb{h}_{w}^{n} \mid \mathscr{B}_{w}^{n + 1} \big]\ \ =\ \ 0\ \text{a.s.};\ \ \ \ \sum_{k\, =\, 0}^{n - 1} {\rm E} \big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ \to\ \ \infty;\ \ \ \ \sup\limits_{n\, \ge\, 0} {\rm E} \big[ \left( \mathbb{h}_{w}^{n} \right)^{2} \big]\ \ <\ \ \infty.$$ Further, defining a sequence $\left\{ a_{n} := \left( \varsigma_{w}^{(n)} (g) \right)^{1\, +\, \epsilon} \right\}_{n\, \ge\, 0}$ for some sufficiently small $\epsilon > 0$, we observe that the sequences satisfy $$\left\{ \frac{a_{n}}{\left( \varsigma_{w}^{(n)} (g) \right)} \right\}_{n\, \ge\, 0}\ \ \text{is non-decreasing}\ \ \ \text{and}\ \ \ \left\{ \frac{a_{n}}{\left( \varsigma_{w}^{(n)} (g) \right)^{2}} \right\}_{n\, \ge\, 0}\ \ \text{is non-increasing}.$$ Thus, in order to appeal to theorem and exploit the assertions there, we only need to verify the two enumerated assumptions in the statement. We will, for now take the relevant assumptions to be true and proceed to complete the proof of theorem . Once the proof is complete, we will complete the verifications of the enumerated statements. By theorem , we have that there exist sequences $\big\{ \mathcal{Y}_{w}^{n} \big\}_{n\, \ge\, 0}$ and $\big\{ Z_{w}^{n} \big\}_{n\, \ge\, 0}$ such that $$\begin{aligned} \sup_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} \mathcal{Y}_{w}^{j}\; -\; \sum_{j\, =\, 0}^{k} Z_{w}^{j} \right| & = & o \left( \left[ a_{n} \left( \left| \log \left( \frac{\varsigma_{n}^{2}}{a_{n}} \right) \right|\; +\; \log \log a_{n} \right) \right]^{\frac{1}{2}} \right) \\ & = & o \left( \left( n^{\frac{1}{2}\, +\, \epsilon} \left( \left| \log \left( n^{\frac{1}{2}\, -\, \epsilon} \right) \right|\; +\; \log \log n^{\frac{1}{2}\, +\, \epsilon} \right) \right)^{\frac{1}{2}} \right) \\ & = & O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \ \ \nu\text{-a.s., for some}\ \delta > 0. \end{aligned}$$ Further, by the result due to Cuny and Merlevede [@cm:15], we know that $$\sum_{j\, =\, 0}^{n - 1} {\rm E} \big[ \left( Z_{w}^{j} \right)^{2} \big]\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; O \left( \varsigma_{w}^{(n)} (g) \right)\ \ =\ \ \left( \varsigma_{w}^{(n)} (g) \right)^{2}\; +\; o \left( n^{\frac{1}{2}\; +\; \delta'} \right),$$ for some $\delta' > 0$. Hence, we can replace the random variables with a standard Brownian motion $\big\{ \mathfrak{B}^{*} (t) \big\}_{t\, \ge\, 0}$ such that $$\sup\limits_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} Z_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(k)} (g) \right)^{2}\; +\; o \left( k^{\frac{1}{2}\, +\, \delta'} \right) \right) \right|\ \ =\ \ 0,\ \ \ \nu\text{-a.s.}$$ which implies that $$\sup\limits_{0\, \le\, k\, \le\, n - 1} \left| \sum_{j\, =\, 0}^{k} Z_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(k)} (g) \right)^{2} \right) \right|\ \ =\ \ o \left( n^{\frac{1}{4}\, +\, \delta'} \right),\ \ \ \nu\text{-a.s.}$$ Therefore, if we replace the independent centered Gaussian variables with the standard Brownian motion, we get $$\sum_{j\, =\, 0}^{n - 1} \mathcal{Y}_{w}^{j}\; -\; \mathfrak{B}^{*} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right)\ \ =\ \ O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \ \ \nu\text{-a.s.}$$ Further, it is easy to see that there exist a sequence of random variables $\big\{ Y_{w}^{n} \big\}_{n\, \ge\, 0}$ such that $$\left\vert Y_{w}^{n}\; -\; \sum_{j\, =\, 0}^{n} \mathcal{Y}_{w}^{j} \right\vert\ \ =\ \ O(1),$$ and $g_{w}^{n}$ and $Y_{w}^{n}$ are equal in distribution, thus proving theorem . We now complete the verifications of the enumerated conditions in theorem . \[cond1\] The first of the enumerated condition in theorem looks like $$\sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \right] \right)\ \ =\ \ o(a_{n}).$$ \[cond2\] The second of the enumerated conditions in theorem looks like $$\sum\limits_{n\, \ge\, 0} a_{n}^{- r} {\rm E} \big[ \left| \mathbb{h}_{w}^{n} \right|^{2r} \big]\ \ <\ \ \infty,\ \ \text{for some}\ 1 \le r \le 2.$$ From Conze and Raugi [@cr:07], we get that $${\rm E} \big[ \left( \mathbb{h}_{w}^{n} \right)^{2} \mid \mathscr{B}_{w}^{n + 1} \big]\ \ =\ \ \mathscr{O}_{(w_{1}\, w_{2}\, \cdots\, w_{n})} \left( \mathcal{L}_{f_{w_{n + 1}}}^{(w_{n + 1})} \left( \mathbb{g}_{w}^{n} \right)^{2} \right)\ \ \text{and}$$ $$\int\! \left| \sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \right] \right) \right|^{2}\, d \lambda\ \ \le\ \ C_{21} \sum_{k\, =\, 0}^{n - 1} {\rm E}\big[ \left( \mathbb{h}_{w}^{k} \right)^{2} \big]\ \ \le\ \ C_{22} \left( \varsigma_{w}^{(n)} (g) \right)^{2},$$ for some positive constants $C_{21}$ and $C_{22}$. Hence, by Gal Koksma Theorem as in [@zl:14; @sw:60], we have $$\begin{aligned} \sum_{k\, =\, 0}^{n - 1} \left( {\rm E} \left[ \left( \mathbb{h}_{w}^{k} \right)^{2} \mid \mathscr{B}_{w}^{k + 1} \right]\; -\; {\rm E} \left[ \left( \mathbb{h}_{w}^{k}\right)^{2} \right] \right) & = & O \left( \left( \varsigma_{w}^{(n)} (g) \right)\; +\; \log^{\frac{3}{2}\, +\, \epsilon} \left( \left( \varsigma_{w}^{(n)} (g) \right)^{2} \right) \right) \\ & = & o \left( \left( \varsigma_{w}^{(n)} (g) \right)^{1\, +\, \epsilon'} \right) \\ & = & o(a_{n}), \end{aligned}$$ where $\epsilon'$ is some small positive quantity, possibly less than or equal to $\epsilon$. Here, we begin with an easy observation that $\left( \varsigma_{w}^{(n)} (g) \right)^{2} = O(n)$. Thus given $\delta > 0$, there exists a threshold $M_{\delta} \in \mathbb{N}$ and a positive real number $C_{23} > 0$ such that $$\left| \frac{\varsigma_{w}^{(n)} (g)}{\sqrt{n}}\; -\; C_{23} \right|\ \ \le\ \ \delta,\ \ \ \forall n \ge M_{\delta}.$$ Suppose $m < M_{\delta}$. Then, by the Archimedean property of the reals, we have $$\varsigma_{w}^{(m)} (g)\ \ \ge\ \ \sqrt{m} D_{m}.$$ Choosing $C_{24} = \min \big\{ C_{23} - \delta,\, D_{1},\, D_{2},\, \cdots,\, D_{N_{\delta}} \big\}$, we have $$\varsigma_{w}^{(n)} (g)\ \ \ge\ \ \sqrt{n} C_{24},\ \ \ \ \forall n \in \mathbb{N}.$$ For $r = 2$ in condition (2) in the enumerated statement of theorem , we have $$\begin{aligned} \sum_{n\, \ge\, 0} a_{n}^{- 2} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] & = & \sum_{n\, \ge\, 0} \left( \varsigma_{w}^{(n)} (g) \right)^{- (2\, +\, \epsilon)} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] \\ & \le & C_{25}\; +\; \sum_{n\, \ge\, 1} \frac{1}{C_{24}^{2\, +\, \epsilon} n^{1\, +\, 2 \epsilon}} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] \\ & \le & C_{25}\; +\; C_{26} \sum_{n\, \ge\, 1} \frac{1}{n^{1\, +\, 2 \epsilon}} \\ & < & \infty, \end{aligned}$$ since $\sup\limits_{n\, \ge\, 0} {\rm E} \left[ \left\vert \mathbb{h}_{w}^{n} \right\vert^{4} \right] < \infty$. Finally, when we replace the standard Brownian motion with a Brownian motion $\big\{ \widetilde{\mathfrak{B}^{*}} (t) \big\}_{t\, \ge\, 0}$ such that $\widetilde{\mathfrak{B}^{*}} (t)$ has variance $t \left( \varsigma_{w} (g) \right)^{2}$, we get $$Y_{w}^{n}\; -\; \widetilde{\mathfrak{B}^{*}} \left( \left( n \right) \right)\ \ =\ \ O \left( n^{\frac{1}{2}\, -\, \gamma} \right),\ \ \ \ \nu\text{-a.s., for some}\ \gamma > 0.$$ Proofs of other statistical properties {#seccor} ====================================== In this section, we write the proofs of the other statistical properties such as the central limit theorem, weak invariance principles and the law of iterated logarithms, as mentioned in theorems and . 1. [*Proof of the central limit theorem*]{}: Recall from the proof of theorem that 1. $\mathfrak{Y}_{n}\ \ \stackrel{\text{d}}{=}\ \ \Phi^{n}\ \ \forall n \ge 1$ and 2. $\mathfrak{Y}_{n}\ \ =\ \ \mathfrak{B} (n)\; +\; O \left( n^{\frac{1}{4}\, +\, \delta} \right),\ \ \nu$-a.e. Further, owing to condition (b) above, we have that for some $\epsilon > 0$, $$\frac{1}{\sqrt{n}} \mathfrak{Y}_{n}\ \ =\ \ \frac{1}{\sqrt{n}} \mathfrak{B} (n)\; +\; O\left( n^{- \epsilon} \right),\ \ \nu\text{-a.e.}$$ and therefore, $\frac{1}{\sqrt{n}} \big( \mathfrak{Y}_{n} - \mathfrak{B} (n) \big) \stackrel{{\rm p}}{\longrightarrow} 0$, [*i.e.*]{}, converges in probability to $0$ as $n \rightarrow \infty$. But $\frac{1}{\sqrt{n}} \mathfrak{B} (n)$ is a normal distribution with mean zero and variance $\varsigma(G)^{2}$ for all $n \geq 1$. Further, owing to condition (a), we have that $$\frac{1}{\sqrt{n}} \mathfrak{Y}_{n}\ \ \stackrel{\text{d}}{=}\ \ \frac{1}{\sqrt{n}} \Phi^{n}\ \ \forall n \ge 1.$$ Making use of both the conditions, we have $$\frac{1}{\sqrt{n}} \Phi^{n}\ \ \stackrel{\text{d}}{\longrightarrow}\ \ \mathcal{N} \left( 0,\, \varsigma(G)^{2} \right),$$ where $\mathcal{N} \left( 0,\, \varsigma^{2} \right)$ denotes the normal distribution with mean $0$ and variance $\varsigma^{2}$. The result follows since $G^{n}((w, x)) = \Phi^{n} ((w, x)) + O(1)$. 2. [*Proof of the law of iterated logarithms*]{}: If $\Phi$ in theorem satisfies the law of iterated logarithms, then so does $G$, since $$\limsup_{n\, \rightarrow\, \infty} \left[ \frac{G^{n} ((w, x)) - \Phi^{n} ((w, x))}{\varsigma(G) \sqrt{2n \log \log n}} \right]\ \ \stackrel{\text{p}}{\longrightarrow}\ \ 0\ \ \ \ \text{as}\ n \rightarrow \infty.$$ The following lemma is the key to proving the law of iterated logarithms for the given function $\Phi$. Any Brownian motion with variance $\varsigma^{2}$ satisfies the law of iterated logarithms [*i.e.*]{}, $$\limsup_{t\, \rightarrow\, \infty} \frac{\mathfrak{B} (t) (\omega)}{\varsigma \sqrt{2t \log \log t}}\ \ =\ \ 1,\ \ \ \nu\text{-a.e.}$$ Since we have a Brownian motion which by condition (b) in the proof of the central limit theorem is equal to $\mathfrak{Y}_{n} + O \left( n^{\frac{1}{4}\, +\, \delta} \right)$, we have by theorem that $$\limsup_{n\, \rightarrow\, \infty} \frac{\mathfrak{Y}_{n} (\omega)}{\varsigma \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \nu\text{-a.e.}$$ Since $\mathfrak{Y}_{n}$ and $\Phi^{n}$ have the same distribution with the latter having variance $\big( \varsigma (G) \big)^{2}$, we conclude that $$\limsup_{n\, \rightarrow\, \infty} \frac{\Phi^{n} ((w, x))}{\varsigma(G) \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \mu_{F}\text{-a.e.}$$ We now prove theorem to conclude the proofs of all the theorems in this paper. The proofs of both the statements run [*mutatis mutandis*]{} as the proofs of their analogous statements in theorem . Hence, we merely highlight the following for readers’ convenience. 1. $Y_{w}^{n}\ \ \stackrel{\text{d}}{=}\ \ g_{w}^{n}\ \ \ \forall n \geq 1$. 2. $Y_{w}^{n}\ \ =\ \ \widetilde{\mathfrak{B}}^{*} (n) + O \left( n^{\frac{1}{2}\, -\, \gamma} \right),\ \ \nu$-a.e. Further, $$\limsup_{n\, \rightarrow\, \infty} \frac{g_{w}^{n}(x)}{\varsigma_{w}(g) \sqrt{2n \log \log n}}\ \ =\ \ 1,\ \ \ \lambda\text{-a.e.}$$ Concluding remarks {#concl} ================== As stated in the introductory section, the theorems proved in this paper are easily transferable to several other analogous settings of dynamical systems. We conclude this paper by merely pointing to some of those. - Instead of working with the specified interval maps $T_{d} : I \longrightarrow I$ of degree $(d + 1)$ for $1 \le d \le N$ given by $T_{d} (x) = (d + 1) x \pmod 1$, one may well consider the action of any $N$ (piecewise) linear interval maps, $S_{d} : I \longrightarrow I;\ 1 \le d \le N$, each with integer degree at least $2$. The measure, in this case still remains Lebesgue. - One might as well consider the simple monomial maps $P_{d};\ 1 \le d \le N$, defined on the Riemann sphere $\overline{\mathbb{C}} = \mathbb{C} \cup \{ \infty \}$ and given by $P_{d} (z) = z^{k_{d}}$ where $k_{d} \ge 2$ for all $1 \le d \le N$. We know from, say [@afb:91], that the Julia set $\mathbb{J} (P_{d})$ of the monomial map $P_{d}$ is the unit circle $\mathbb{S}^{1}$ in $\mathbb{C}$. In such a case, the Julia set $\mathbb{J} (P)$ of the skew-product map $P$ appropriately defined analogous to equation is also the unit circle, as one may find from [@hs:00]. Owing to the Julia set $\mathbb{J} (P)$ being completely $P$-invariant, one may undertake an analogous study, as done in this paper, to the dynamics generated by the monomial maps $P_{d};\ 1 \le d \le N$ restricted on the Julia set $\mathbb{J} (P) = \mathbb{S}^{1} \subset \mathbb{C}$ and obtain analogous results, employing the Haar measure on $\mathbb{S}^{1}$. - Let $R_{d};\ 1 \le d \le N$ be a collection of rational maps acting on the Riemann sphere $\overline{\mathbb{C}}$; each with degree $k_{d} \ge 2$. We suppose that the rational maps are so chosen that the Julia set $\mathbb{J} (R_{d})$ of the map $R_{d}$ is topologically connected. Then, defining the skew-product map $R$ appropriately and restricting its action on the $R$-invariant Julia set $\mathbb{J} (R)$, as one may obtain from [@hs:00], it is possible to investigate analogous results for Boyd’s measure, as defined in [@db:99]. It must be borne in mind that the Boyd’s measure is a generalisation of the Lyubich’s measure, as in [@myl:86]. - Finally, we consider the dynamical system obtained by iterating certain relations on $\mathbb{C}$. The relation can be explained as the zero set of a polynomial, say $Q \in \mathbb{C}[\zeta,\, \omega]$ of a certain form such that: - $Q \big( \cdot,\, \omega \big)$ and $Q \big( \zeta,\, \cdot \big)$ are generically multiple-valued; - if $\mathscr{G}_{Q}$ denotes the biprojective completion of $\big\{ Q\, =\, 0 \big\}$ in $\overline{\mathbb{C}}\, \times\, \overline{\mathbb{C}}$, then no irreducible component of $\mathscr{G}_{Q}$ is of the form $\big\{ a \big\}\, \times\, \overline{\mathbb{C}}$ or $\overline{\mathbb{C}}\, \times\, \big\{ a \big\}$, where $a \in \overline{\mathbb{C}}$. Such dynamical systems have been studied by Dinh and Sibony in [@ds:06] and Bharali and Sridharan in [@bs:16]. 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--- address: 'Department of Mathematics, Hokkaido University, Sapporo, 060-0810 Japan' author: - Toshiyuki Akita title: | A formula for the Euler characteristics of\ even dimensional triangulated manifolds --- [^1] A finite simplicial complex $K$ is called an [*Eulerian manifold*]{} (or a [*semi-Eulerian complex*]{} in the literature) if all of maximal faces have the same dimension and, for every nonempty face $\sigma\in K$, $$\chi({\operatorname{Lk}}\sigma)=\chi(S^{\dim K-\dim\sigma-1})$$ holds, where ${\operatorname{Lk}}\sigma$ is the link of $\sigma$ in $K$ and $S^n$ is the $n$-dimensional sphere. Note that $K$ is not necessary connected. Any triangulation of a closed manifold is an Eulerian manifold. More generally, a triangulation of a homology manifold without boundary provides an Eulerian manifold. The purpose of this short note is to prove the following alternative formula for the Euler characteristics of even dimensional Eulerian manifolds. \[main\] Let $K$ be a $2m$-dimensional Eulerian manifold. Then $$\label{eq-main} \chi(K)=\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^i f_i(K)$$ holds, where $f_i(K)$ is the number of $i$-simplices of $K$. A finite simplicial complex $L$ is called a [*flag complex*]{} if every collection of vertices of $L$ which are pairwise adjacent spans a simplex of $L$. The formula was proved in [@akita] under the additional assumptions that $K$ is a PL-triangulation of a closed $2m$-manifold and is a flag complex. M. W. Davis pointed out that the formula follows from a result in [@davis], provided $K$ is a flag complex (see [*Note added in proof*]{} in [@akita]). Both results follow from the considerations of the Euler characteristics of Coxeter groups. In this note, we deduce the formula from the generalized Dehn-Sommerville equations proved by Klee [@klee]. Let $K$ be a finite $(d-1)$-dimensional simplicial complex and $f_i=f_i(K)$ the number of $i$-simplices of $K$ as before. The $d$-tuple $(f_0,f_1,\dots,f_{d-1})$ is called the [*$f$-vector*]{} of $K$. The [*$f$-polynomial*]{} $f_K(t)$ of $K$ is defined by $$f_K(t)=t^d+f_0t^{d-1}+\cdots+f_{d-2}t+f_{d-1}.$$ Define the [*$h$-polynomial*]{} $h_K(t)$ of $K$, $$h_K(t)=h_0t^d+h_1t^{d-1}+\cdots+h_{d-1}t+h_d,$$ by the rule $h_K(t)=f_K(t-1)$. The $(d+1)$-tuple $(h_0,h_1,\dots,h_d)$ is called the [*$h$-vector*]{} of $K$. The $h$-vector of $K$ satisfies the generalized Dehn-Sommerville equations, as stated below in Theorem \[DS\]. \[DS\] Let $K$ be a $(d-1)$-dimensional Eulerian manifold. Then $$h_{d-i}-h_i=(-1)^i\binom{d}{i}(\chi(K)-\chi(S^{d-1}))$$ holds for all $i$ $(0\leq i\leq d)$. Klee stated the generalized Dehn-Sommerville equations in terms of the $f$-vector rather than the $h$-vector. The formulae quoted in Theorem \[DS\] are equivalent to those in [@klee] and can be found in [@ed]. Theorem \[DS\] was also proved in [@panov] by a quite different method, provided that $K$ is a triangulation of a closed manifold. Now we prove Theorem \[main\]. We have $$h_K(-1)=\sum_{i=0}^{2m+1}(-1)^{2m+1-i}h_i =\sum_{i=0}^{m} (-1)^i (h_{2m+1-i}-h_i).$$ Now Theorem \[DS\] asserts that $$h_{2m+1-i}-h_i=(-1)^i\binom{2m+1}{i}(\chi(K)-2).$$ Hence we obtain $$\label{h-poly} h_K(-1)=(\chi(K)-2)\sum_{i=0}^m\binom{2m+1}{i} =2^{2m}(\chi(K)-2).$$ On the other hand, we have $$\label{f-poly} f_K(-2)=(-2)^{2m+1}+\sum_{i=0}^{2m}(-2)^{2m-i}f_i =2^{2m}\left( -2+\sum_{i=0}^{2m}\left(-\frac{1}{2}\right)^if_i \right).$$ Since $h_K(-1)=f_K(-2)$ by the definition of the $h$-polynomial $h_K(t)$, Theorem \[main\] follows from and . [1]{} T. Akita, Euler characteristics of Coxeter groups, PL-triangulations of closed manifolds, and cohomology of subgroups of Artin groups, J. London Math. Soc. (2) 61 (2000), 721–736. V. M. Buchstaber, T. E. Panov, [*Torus actions and their applications in topology and combinatorics*]{}, University Lecture Series 24, American Mathematical Society, Providence, 2002. R. Charney, M. W. Davis, Reciprocity of growth functions of Coxeter groups, Geom. Dedicata 39 (1991), 373–378. V. Klee, A combinatorial analogue of Poincaré’s duality theorem, Canad. J. Math. 16 (1964), 517–531. E. Swartz, From spheres to manifolds, preprint (2005). [^1]: Partially supported by the Grant-in-Aid for Scientific Research (C) (No.17560054) from the Japan Society for Promotion of Sciences.

ArXiv

--- abstract: 'In the present contribution, we investigate first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and non-conservative first order terms. Whereas the theory of first-order systems of conservation laws is well established and the conditions for the existence of supplementary conservation laws, and more specifically of an entropy supplementary conservation law for smooth solutions, well known, there exists so far no general extension to obtain such supplementary conservation laws when non-conservative terms are present. We propose a framework in order to extend the existing theory and show that the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to a supplementary conservation law. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms with the objective of obtaining specifically an entropy supplementary conservation law. The theory as well as developed computer algebra tool are then applied to a Baer-Nunziato two-phase flow model and to a multicomponent plasma fluid model. The first one is a first-order fluid model, with non-conservative terms impacting on the linearly degenerate field and requires a closure since there is no way to derive interfacial quantities from averaging principles and we need guidance in order to close the pressure and velocity of the interface and the thermodynamics of the mixture. The second one involves first order terms for the heavy species coupled to second order terms for the electrons, the non-conservative terms impact the genuinely nonlinear fields and the model can be rigorously derived from kinetic theory. We show how the theory allows to recover the whole spectrum of closures obtained so far in the literature for the two-phase flow system as well as conditions when one aims at extending the thermodynamics and also applies to the plasma case, where we recover the usual entropy supplementary equation, thus assessing the effectiveness and scope of the proposed theory.' author: - 'Pierre CORDESSE[^1]' - 'Marc MASSOT[^2]' bibliography: - '../biblatex/jabref\_bdd.bib' title: 'Entropy supplementary conservation law for non-linear systems of PDEs with non-conservative terms: application to the modelling and analysis of complex fluid flows using computer algebra[^3]' --- Nonlinear PDEs with non-conservative terms, supplementary conservation law, entropy, computer algebra, two-phase flow, Baer-Nunziato model, multicomponent plasma fluid model 35L60; 68W30; 76N15; 76T10; 82D10 Introduction ============ First-order nonlinear systems of partial differential equations and more specifically systems of conservation laws have been the subject of a vast literature since the second half of the twentieth century because they are ubiquitous in mathematical modelling of fluid flows and are used extensively for numerical simulation in applications and industrial context [[@Bissuel_2018; @Gaillard_2016]]{}. Such systems of equation can either be rigorously derived from kinetic theory of gases through various expansion techniques [[@Ferziger_1972; @Woods_1975]]{}, or can be derived using rational thermodynamics and fluid mechanics including stationary action principle (SAP) [[@Serrin_1959; @Landau_1976; @Truesdell_1969]]{}. As far as Euler or Navier-Stokes equations are concerned for a gaseous flow field, the outcome of both approaches are similar and the mathematical properties of these systems have been thoroughly investigated for the past decades. An interesting related problem is the quest for supplementary conservation laws. Noether’s theorem [[@Olver_1986]]{} leads, within the framework of SAP, to the derivation of supplementary conservation laws based on symmetry transformations of the variational problem under investigation[^4]. Examples of such derivations on two-phase flow modelling can be found in [[@Gavrilyuk_Saurel_2002; @Drui_JFM_2019]]{}. However, to the authors knowledge, no symmetry transformations have been identified yielding a conservative law on the entropy of the system. In fact, SAP does not allow to reach a closed system of equations, and one has to provide a closure for the entropy (see [[@Gouin_2009]]{} for example). A specific type of supplementary conservation equation for smooth solution is especially important, namely the *entropy equation*, derived through the theory developed in [[@Godunov_1961; @Friedrichs_1971]]{} for systems of conservation laws. Such systems of PDEs are hyperbolic at any point where a locally convex entropy function exists [[@Mock_1980]]{}, and when they are equipped with a strictly convex entropy, they can be symmetrized [[@Friedrichs_1971]]{} [[@Harten_Hyman_1983]]{} and thus are hyperbolic. These properties have been at the heart of the mathematical theory of existence and uniqueness of smooth solutions [[@Kawashima_1988]]{} [[@Giovangigli_1998]]{}, but they are also a corner stone for the study of weak solutions for which the work of [[@Kruzkov_1970]]{} proves the well-posedness of Cauchy problem for one-dimensional systems. Nonetheless, for a number of applications, where reduced-order fluid models have to be used for tractable mathematical modelling and numerical simulations, be it in the industry or in other disciplines, micro-macro kinetic-theory-like approaches as well as rational thermodynamics and SAP approaches often lead to system of conservation laws involving *non-conservative terms*. Among the large spectrum of applications, we focus on two types of models, which exemplify the two approaches: 1- two phase flows models which rely on a hierarchy of diffuse interface models among which stands the Baer-Nunziato [[@Baer_Nunziato_1986]]{} model used when full disequilibrium of the phases must be taken into account. Since this model is derived through rational thermodynamics, the macroscopic set of equations can not be derived from physics at small scale of interface dynamics and thus require closure of interfacial pressure and velocity, 2- multicomponent fluid modelling of plasmas flows out of thermal equilibrium, where the equations can be derived rigorously from kinetic theory using a multi-scale Chapman-Enskog expansion mixing a hyperbolic scaling for the heavy species and a parabolic scaling for the electrons [[@Graille_2007]]{}. Concerning the thermodynamics, whereas for the first model it has to be postulated and requires assumptions, it can be obtained from kinetic theory in the second model. In both cases, the models involve non-conservative terms, but these terms do not act on the same fields; linearly degenerate field is impacted for the two-phase flow model, whereas it acts on the genuinely nonlinear fields in the second [[@Wargnier_2018]]{}. Whereas hyperbolicity depends on the closure and is not guaranteed for the first class of models [[@Gallouet_2004]]{}, the second is naturally hyperbolic [[@Graille_2007]]{} and also involves second-order terms and eventually source terms [[@Magin_2009]]{}. Thus, the presence of *non-conservative terms* encompasses several situations and requires a general theoretical framework. While Noether’s theorem can still applied to obtain some supplementary conservation laws, it does not permit to exhibit all of them and especially not an entropy supplementary conservation law. A unifying theory extending the standard approach for systems of conservations laws (entropy supplementary conservation law, entropic symmetrization, Godunov-Mock theorem, hyperbolicity) is still missing for such systems even if some key advances exist. The system has been shown to be symmetrizable by [[@Coquel_2013]]{} – not in the sens of Godunov-Mock – far from the resonance condition for which hyperbolicity degenerates. In [[@Forestier_2011]]{}, the model is proved to be partially symmetrizable in the sense of Godunov-Mock. The present paper first proposes an extension of the theory of supplementary conservation laws for system of conservation laws to first-order nonlinear systems of partial differential equations which are constituted of two parts: a system of conservation laws and *non-conservative first order terms*.We emphasize how the presence of non-conservative terms somewhat complexifies the problem since numerous combinations of the conservative and non-conservative terms can lead to supplementary conservation laws. We then identify a restricted framework in order to design and analyze physical models of complex fluid flows by means of computer algebra and thus obtain the entire ensemble of possible combination of conservative and non-conservative terms to obtain an entropy supplementary conservation law. The proposed theoretical approach is then applied to the two systems identified so far for their diversity of behaviour. Even if the whole theory is valid for any supplementary conservation law, we focus on obtaining an *entropy* supplementary conservation law. For the two-phase flow model, assuming a thermodynamics of non-miscible phases, we derive conditions to obtain an entropy supplementary conservative equation together with a compatible thermodynamics and closures for the non-conservative terms. Interestingly enough, all the closures proposed so far in the literature are recovered [[@Baer_Nunziato_1986; @Kapila_1997; @Bdzil_1999; @Lochon_PhdThesis_2016; @Saurel_Gavrilyuk_2003]]{}. The strength of the formalism lies also in the capacity to derive such conditions for some level of mixing of the phases. By introducing a mixing term in the definition of the entropy, the new theory brings out constraints on the form of the added mixing term. We recover not only the closure proposed to account for a configuration energy as in the context of deflagration-to-detonation [[@Baer_Nunziato_1986]]{} or in [[@Coquel_2002]]{}, but we also rigorously find new closures leading to a conservative system of equations[^5]. We also prove that the theory encompasses the plasma case, where we recover the usual *entropy* supplementary equation assessing the effectiveness and scope of the proposed theory. The paper is organized as follows. The extension of the theory for system of conservation laws to first-order nonlinear systems of partial differential equations including non-conservative terms, as well as the framework to apply the theory by means of computer algebra are introduced in Section \[sec:theory\]. These results are then applied first to the Baer-Nunziato model in Section \[sec:BNZ\] and then to the plasma model in Section \[sec:plasma\] to obtain an entropy supplementary conservation law compatible with the model closure. **Notations:** Let ${\boldsymbol{a}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, ${\boldsymbol{b}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, $\mathcal{B} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p}$, $\mathcal{C} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p}$, ${{\boldsymbol{\mathcal{D}}}} \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p\times p \times p}$ be a $p$-component line first-order tensor, a $p$-component column first-order tensor, two $p$-square second-order tensor and a third-order tensor respectively. We introduce the following notations: - ${\boldsymbol{a}} \mathcal{B}$ is a line first-order tensor in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ whose $i$ component are defined by $$\begin{aligned} \left({\boldsymbol{a}} \mathcal{B}\right)_{i} = \sum_{j=1,p} {\boldsymbol{a}}_{j} \mathcal{B}_{j,i}, \end{aligned}$$ - $\mathcal{B} {\boldsymbol{b}}$ is a column first-order tensor in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ whose $i$ component is defined by $$\begin{aligned} \left(\mathcal{B} {\boldsymbol{b}}\right)_{i} = \sum_{j=1,p} \mathcal{B}_{i,j}{\boldsymbol{b}}_{j}, \end{aligned}$$ - $\mathcal{B} \times \mathcal{C}$ is $p$-square second-order tensor whose $(i,j)$ component is defined by $$\begin{aligned} \left(\mathcal{B} \times \mathcal{C}\right)_{i,j} = \sum_{k=1,p} \mathcal{B}_{i,k} \mathcal{C}_{k,j}, \end{aligned}$$ - ${\boldsymbol{a}} \otimes \mathbb{D}$ is a $p$-square second-order tensor whose $(i,j)$ component is defined by $$\begin{aligned} \left( {\boldsymbol{a}} \otimes \mathbb{D} \right)_{(i,j)} = \sum_{k=1,p} {\boldsymbol{a}}_{k} \times \mathbb{D}_{k,i,j}. \end{aligned}$$ Hereafter, we will name zero- first- and second-order tensors by scalar, vector and matrix respectively and for convenience we will use vector and matrix representations of functions. Moreover, given a scalar function $S$, the partial differentiation of $S$ by a column vector ${\boldsymbol{a}}$, $\partial_{{\boldsymbol{a}}}S$ is a line vector in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$. Finally, $\cdot$ denotes the Euclidean scalar product in ${ \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$. Supplementary conservation law {#sec:theory} ============================== First we recall the theory of the existence of a supplementary conservative equation for first-order nonlinear systems of conservation laws. Second, this notion is extended to systems containing first order non-conservative terms. Third, we introduce a framework to apply this new theory to design and analyze physical models using computer algebra. A one-dimensional framework is adopted from now on, $x \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, in order to simplify the derivation. Nonetheless, the results can easily be extended to the multi-dimensional approach as presented in [[@Godlewski_1996]]{} for systems of conservation laws. First-order nonlinear conservative systems. {#ssec:CS_theory} ------------------------------------------- The homogeneous form of a first-order nonlinear system of $p$ conservation laws writes $$\begin{aligned} \label{eq:cons_syst_non_linear} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \partial_{x} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}},\end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ denotes the conservative variables with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ and ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ the conservative fluxes. Focusing on smooth solution of the system , its quasi-linear form is given by $$\begin{aligned} \label{eq:cons_syst_quasi_linear} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ \[theo:cons\_syst\_SCE\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. The following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:cons_syst_entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:cons_syst_compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square symmetric matrix. The proofs of the theorem can be found in the literature. We would like to recall how the last statement is obtained. Assuming $(C_{2})$, differentiating Equation  leads to $$\begin{aligned} \label{eq:cons_syst_sym_cond} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ where $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square matrix defined as $\sum_{i} \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{i}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}_{i}({{{\boldsymbol{\mathrm{u}}}}})$ which is a linear combination of Hessian matrices and hence symmetric. Moreover, the RHS of Equation  $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. Therefore $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. Theorem \[theo:cons\_syst\_SCE\] applies for any type of supplementary conservative equations and other formulations of Theorem \[theo:cons\_syst\_SCE\] can be found in the literature [[@Harten_Hyman_1983; @Godlewski_1996; @Despres_2005]]{}. Extension to systems involving non-conservative terms. {#ssec:NC_theory} ------------------------------------------------------ Let us now consider the homogeneous form of a first-order nonlinear system of partial differential equations constituted of two parts: conservations laws and first-order non-conservative terms. Its quasi-linear form can be written as $$\begin{aligned} \label{eq:NC_syst} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}},\end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$, ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ the conservative fluxes, ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ the $p$-square matrix containing the first-order non-conservative terms. In the following we extend the theory introduced in Section \[ssec:CS\_theory\] to system . Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, multiplying system  by the line vector $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ yields $$\begin{aligned} \label{eq:NC_syst_SCE} \partial_{t} {{\mathsf{H}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}= 0.\end{aligned}$$ Compared to Equation , the presence of the non-conservative terms in Equation  complexifies the question of the existence of a supplementary conservative equation. Therefore we propose to decompose in a specific way the conservative and non-conservative terms in Definition \[def:decomposition\]. \[def:decomposition\] Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ and a first-order nonlinear non-conservative system , let us define the four $p$-square matrices, ${{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ in ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p \times p}$ such that $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) & = {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}), \\ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) & = {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})+{{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ with the condition $$\begin{aligned} \label{eq:decomposition_condition} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ In light of Definition \[def:decomposition\], Theorem \[theo:cons\_syst\_SCE\] can be extended as follows: \[theo:NC\_syst\_SCE\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear system of non-conservation laws , if we introduce the decomposition as in Definition \[def:decomposition\], then the following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:NC_syst_compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] $ is a $p$-square symmetric matrix. Rewriting Equation  using the decomposition of the conservative and non-conservative terms as $$\begin{aligned} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}= - \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}\end{aligned}$$ outlines the result. \[remark:symmetry\_condition\]Theorem \[theo:NC\_syst\_SCE\] applies for any type of supplementary conservative equations. The usual symmetry condition on which relies the existence of a supplementary conservation equation is strongly modified when non-conservation terms are present. From Theorem \[theo:cons\_syst\_SCE\] to Theorem \[theo:NC\_syst\_SCE\] the condition $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric},\end{aligned}$$ is modified into $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \text{ symmetric}.\end{aligned}$$ In the context of systems of conservation laws, an interesting algebraic approach is proposed in [[@Barros_2006]]{} based on the reinterpretation of the symmetric Condition $(C_{3})$ in Theorem \[theo:cons\_syst\_SCE\] as a Frobenuis problem. Nevertheless, when dealing with additional non-conservative terms, the above new symmetry condition prevents us from applying efficiently such an approach. In Definition \[def:decomposition\], the condition  implies that the conservative and non-conservative terms depend only on the variables ${{{\boldsymbol{\mathrm{u}}}}}$, and not on their gradient. Some authors have allowed the matrices ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ to depend also on the gradients of the variables ${{{\boldsymbol{\mathrm{u}}}}}$, then a more general condition for the decomposition can be written $$\begin{aligned} \label{eq:theo_b1b2_extended} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x}{{{\boldsymbol{\mathrm{u}}}}}\leq 0.\end{aligned}$$ In Section \[sec:BNZ\], we will see that such a condition has been chosen to close the Baer-Nunziato model [[@Saurel_Gavrilyuk_2003]]{}. However, since it changes the mathematical nature of the PDE under investigation, we will not include it in our study. From a modelling perspective, System  under consideration is not necessary closed. Therefore, the following corollary yields conditions on the model to obtain a supplementary conservative equation once we have postulated the thermodynamics. Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear system of non-conservation laws where ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ and ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ are unknown functions to be modelled. If we introduce the decomposition as in Definition \[def:decomposition\], then System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq_corro} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ a scalar function, if and only if the following conditions hold 1. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\times \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\otimes \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \left[{{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] $ is a $p$-square symmetric matrix. 2. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}$. Design or analysis of physical models using computer algebra. {#ssec:NC_theory_applied} ------------------------------------------------------------- We would like to apply the theory on first-order nonlinear non-conservative systems introduced in Section \[ssec:NC\_theory\] to physical models such as the Baer-Nunziato model and the plasma model in order to design and analyze them. We recall that our prior interest is to obtain an *entropy* supplementary conservation law. However, the difficulty is manifold: - The combination of the non-conservative terms and conservative terms proposed in Definition \[def:decomposition\] to build a supplementary conservative equation is not unique and thus many degrees of freedom exist in defining the matrices ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$. - When the model is derived trough rational thermodynamics, terms in the system of equations might need closure and the thermodynamics has to be postulated. Therefore, the matrices ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ can contain unknowns related to the system and the definition of ${{\mathsf{H}}}$. - The calculations needed to derive a supplementary conservative equation are heavy and choice-based. Any change of ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ that respects Definition \[def:decomposition\], or any new postulated thermodynamics would require to derive again all the equations, and eventually a very limited range of possibilities would be examined. These difficulties to apply the theory and examine all the possibilities makes computer algebra very appealing since it allows symbolic operations to be implemented and thus can derive equations systematically and quasi-instantaneously for any combinations of conservative and non-conservative terms as well as model closure and ${{\mathsf{H}}}$ definition. Furthermore, the generic level handled by computer algebra is not unlimited and therefore Definition \[def:decomposition\] requires further assumptions to circumscribe the number of degrees of freedom that can be accounted for. Even if the theory proposed hereinbefore is valid to obtain any kind of supplementary conservation laws, we are mainly interested in obtaining an entropy supplementary conservation law. We thus need to define the notions of *entropy* and *entropic variables* in the following two definitions. ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is said to be an *entropy* of the system  if ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ is a convex scalar function of the variables ${{{\boldsymbol{\mathrm{u}}}}}$ which fulfills Theorem \[theo:cons\_syst\_SCE\]. The supplementary conservative equation  is then named the *entropy equation* and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is the associated *entropy flux*. \[def:entropic\_variable\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Given a first-order nonlinear conservative system , let us define the *entropic variables* ${{{\boldsymbol{\mathrm{v}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ such that $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})\right)^{t}.\end{aligned}$$ The entropic variables have been studied in [[@Giovangigli_1998]]{} in order to obtain symmetric and normal forms of the system of equation and used in the framework of gaseous mixtures, where the mathematical entropy ${{\mathsf{H}}}$ is usually defined as the opposite of a physical entropy density per unit volume of the system [[@Giovangigli_1998]]{}. \[def:decomposition\_applied\] Given a scalar function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$, a first-order nonlinear non-conservative system , and the four $p$-square matrices ${{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})$ in ${ \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ defined in Definition \[def:decomposition\], we introduce the unknown line vector $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ such that $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}),\\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ The condition of Equation  rewrites into $$\begin{aligned} \label{eq:decomposition_condition_applied} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ Since Definition \[def:decomposition\_applied\] is a projection of the matrix equations of Definition \[def:decomposition\] on the vector $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$, it may be interesting to introduce an unknown matrix ${{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p \times p}$ associated to the unknown line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ such that $$\begin{aligned} \label{def:gamma_matrix} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}).\end{aligned}$$ Thus, Definition \[def:decomposition\_applied\] can be formulated as follows $$\begin{aligned} {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}), \\ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) &= {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}),\end{aligned}$$ with the condition $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - {{{\boldsymbol{\mathcal{T}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] = {{\boldsymbol{\mathrm{0}}}}.\end{aligned}$$ The unknown functional line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ represents the transfer of non-conservative terms to the conservative terms. In the degenerate case where $\transfer{v}={{\boldsymbol{\mathrm{0}}}}$, ${{{\boldsymbol{\mathcal{C}}}}_{k}}$ receives all the conservative terms and ${{{\boldsymbol{\mathcal{Z}}}}_{k}}$ all the non-conservative terms. Condition  forces all the non-conservative terms to vanish and System  is fully conservative, hence the theory of conservative system can be applied. Definition \[def:decomposition\_applied\] being more restrictive than Definition \[def:decomposition\], computer algebra is now applicable to analyze the properties of a first-order nonlinear non-conservative system leading to a reformulation of Theorem \[theo:NC\_syst\_SCE\]. \[theo:NC\_SCE\_applied\] Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. Consider a first-order nonlinear system of non-conservation laws . If we introduce the decomposition as in Definition \[def:decomposition\_applied\], then the following statements are equivalent: 1. System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:entropy_eq} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution of System  and ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function. 2. There exists a scalar function ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ such that $$\begin{aligned} \label{eq:compatibility_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}). \end{aligned}$$ 3. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is a $p$-square symmetric matrix. Injecting Definition \[def:decomposition\_applied\] into Theorem \[theo:NC\_syst\_SCE\] leads to these results. When ${{\mathsf{H}}}$ is the entropy of the system, Theorem \[theo:NC\_SCE\_applied\] provides equations that relate the thermodynamics of the model through ${{\mathsf{H}}}$, the model itself with possible terms to be closed in ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$, and the unknown line vector $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$. Combined with the Definition \[def:decomposition\_applied\], Theorem \[theo:NC\_SCE\_applied\] brings out conditions on the model to obtain a supplementary conservative equation given a postulated thermodynamics and it leads to the following corollary. \[coro:NC\_syst\_applied\_metho\] Consider a first-order nonlinear system of non-conservation laws where ${{{\boldsymbol{\mathrm{u}}}}}\in \Omega \subset { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ is a smooth solution with $\Omega$ an open convex of ${ \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{p}$ but ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ and ${{{\boldsymbol{\mathcal{N}}}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rpp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rpp}{}]}$ are unknown functions to be modelled. Let ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ be a scalar function, not necessarily convex. If we introduce the decomposition as in Definition \[def:decomposition\_applied\], then System  admits a supplementary conservative equation $$\begin{aligned} \label{eq:NC_syst_entropy_eq_corro_applied} \partial_{t} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{x} {{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}}) = 0, \end{aligned}$$ where ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is a scalar function if and only if the following conditions hold 1. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is symmetric. 2. $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}$. The previous framework can be extended to the multi-dimensional case in a straightforward manner. If the original system is isotropic, such as for the applications we have in mind, then the previous conditions will be the same in the various directions. In the framework of more general non-isotropic systems, which satisfy Galilean and rotational invariances for example, we will obtain different conditions and we have to check that the decomposition we perform in the various directions satisfies some compatibility relations so that the obtained conservation law satisfies the original invariance properties of the system. Methodology. {#ssec:NC_metho} ------------ Corollary \[coro:NC\_syst\_applied\_metho\] draws the methodology we have implemented in the Maple computer algebra software[^6] in order to obtain an *entropy* supplementary conservation law. Our methodology is the following: 1. We define the thermodynamics by postulating - if need be - an entropy function ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$. 2. We then use Condition $(C_{1})$ and $(C_{2})$ of Corollary \[coro:NC\_syst\_applied\_metho\] to ensure the existence of an entropy flux ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ and solve $$\begin{aligned} \label{eq:NC_SCE_applied_cond} \left\{ \, \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{l} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric}, \\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}. \end{IEEEeqnarraybox} \right. \end{aligned}$$ In System , $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ is systematically an unknown, ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$, ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ as well as ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ can include unknown terms for which the variable dependency is specified. Maple generates then an exhaustive solution for $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ and constraints on all the other unknown terms. 3. From that, the software derives the admissible entropy flux ${{\mathsf{G}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ which gives then the supplementary conservative equation. Application to the Baer-Nunziato model {#sec:BNZ} ====================================== Context and presentation of the model. -------------------------------------- The Baer-Nunziato model has been derived through rational thermodynamics in [[@Baer_Nunziato_1986]]{} and describes a two-phase flow out of equilibrium. Extended by the work of [[@Saurel_1999]]{} thanks to the introduction of interfacial quantities, the homogeneous form of the Baer-Nunziato model is $$\begin{aligned} \label{sys:BNZ_eq} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{c} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= {{\boldsymbol{\mathrm{0}}}},\\ \\ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \begin{pmatrix} 0 & {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} \\ {{\boldsymbol{\mathrm{0}}}} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{2}} {{{\boldsymbol{\mathrm{f}}}}}_{2}({{{\boldsymbol{\mathrm{u}}}}}_{2}) & {{\boldsymbol{\mathrm{0}}}} \\ {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}_{1}} {{{\boldsymbol{\mathrm{f}}}}}_{1}({{{\boldsymbol{\mathrm{u}}}}}_{1}) \end{pmatrix}, \ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) = \begin{pmatrix} \speed{sint} & {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}}\\ {{{\boldsymbol{\mathrm{n}}}}_{2}}& {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}}\\ {{{\boldsymbol{\mathrm{n}}}}_{1}}& {{\boldsymbol{\mathrm{0}}}} & {{\boldsymbol{\mathrm{0}}}} \end{pmatrix}, \end{IEEEeqnarraybox}\end{aligned}$$ where the column vector ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{7}$ is defined by ${{{\boldsymbol{\mathrm{u}}}}}^{{T}} = \left(\alpha_{2},\, {{{\boldsymbol{\mathrm{u}}}}}_{2}^{{T}},\, {{{\boldsymbol{\mathrm{u}}}}}_{1}^{{T}} \right)$, ${{{\boldsymbol{\mathrm{u}}}}}_{k}^{{T}} = ( \alpha_{k} \rho_{k},\, \allowbreak \allowbreak \alpha_{k} \rho_{k} \speed{sk},\, \allowbreak \alpha_{k} \rho_{k} E_{k} )$. The conservative flux ${{{\boldsymbol{\mathrm{f}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R7}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R7}{}]}$ reads ${{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = (0,\, {{{\boldsymbol{\mathrm{f}}}}}_{2}({{{\boldsymbol{\mathrm{u}}}}}_{2})^{{T}},\, {{{\boldsymbol{\mathrm{f}}}}}_{1}({{{\boldsymbol{\mathrm{u}}}}}_{1})^{{T}})$ with ${{{\boldsymbol{\mathrm{f}}}}}_{k}({{{\boldsymbol{\mathrm{u}}}}}_{k})^{{T}} = (\alpha_{k} \rho_{k} \speed{sk},\, \allowbreak \alpha_{k}(\rho_{k} \speed{sk}^{2}+p_{k}),\, \allowbreak \alpha_{k} ( \rho_{k} E_{k}+p_{k})\speed{sk} )$. ${{{\boldsymbol{\mathcal{N}}}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R77}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R77}{}]}$ is the matrix containing the non-conservative terms with ${{{\boldsymbol{\mathrm{n}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = - {{{\boldsymbol{\mathrm{n}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = (0,\, \allowbreak -{p_{I}},\, \allowbreak -{p_{I}}\speed{sint})$. Then, $\alpha_{k}$ is the volume fraction of phase $k \in \left[ 1,2 \right]$, $\rho_{k}$ the partial density, $\speed{sk}$ the phase velocity, $p_{k}$ the phase pressure, $E_{k}=\epsilon_{k} + \allowbreak \speed{sk}^{2}/2$ the total energy per unit of mass, $\epsilon_{k}$ the internal energy, $\speed{sint}$ the interfacial velocity and ${p_{I}}$ the interfacial pressure. Two levels of ingredients are still missing for this model. First, the macroscopic set of equations includes the interface dynamics through the interfacial terms $\speed{sint}$ and ${p_{I}}$ and thus needs closure on these terms. Second the thermodynamics has to be postulated. The mathematical properties of the model have been studied by [[@Embid_Baer_1992; @Coquel_2002; @Gallouet_2004]]{} among others and many closure have been proposed for the interfacial terms based on wave-type considerations and the entropy inequality. Regarding the thermodynamics, for non-miscible phases, the entropy ${{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}})$ is commonly defined by Equation  as in [[@Coquel_2002; @Lochon_PhdThesis_2016]]{}, $$\begin{aligned} \label{eq:mixture_entropy_immiscible} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) = - \sum_{k=1,2} \alpha_{k} \rho_{k} s_{k},\end{aligned}$$ with $s_{k}=s_{k}(\rho_{k},p_{k})$ the phase entropy which takes for the Ideal Gas equation of state the form $$\begin{aligned} s_{k}= c_{v,k} \text{ln}\left(\frac{p_{k}}{\rho_{k}^{\gamma_{k}}}\right),\end{aligned}$$ with $c_{v,k}$ the heat capacity, $p_{k}$ the pressure, $\rho_{k}$ the density and $\gamma_{k}$ the isentropic coefficient of phase $k$. If we were to account for partial miscibility between the two phases, we would have to add a mixing term to the definition of the non-miscible entropy. The mixing term could take the form proposed in [[@Gallouet_2004]]{}, so that the entropy rewrites $$\begin{aligned} \label{eq:mixture_entropy_extended} {{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k}(\rho_{k},p_{k}) - \psi_{k}(\alpha_{k}) \right],\end{aligned}$$ with $\psi_{k}$, $k=\left[ 1,2 \right]$, two strictly convex nonlinear arbitrary functions depending on the volume fraction. Nevertheless, so far in the literature, no explicit expressions of these functions have been proposed. In [[@Gallouet_2004]]{}, in order to obtain a supplementary conservative equation using the entropy defined in Equation , the authors show that the following condition has to be fulfilled $$\begin{aligned} \label{eq:mixing_term_condition_BNZ} \psi_{k}(\alpha_{k}) = \psi_{k^{\prime}}(\alpha_{k^{\prime}}).\end{aligned}$$ In this section, we apply to the Baer-Nunziato model the framework introduced in Section \[sec:theory\] by means of computer algebra. We will firstly assume the phases are non-miscible and derive an entropy supplementary conservative equation along with conditions on the interfacial terms. All the closures proposed in the literature will be recovered. Secondly, we will also apply the methodology in the case of a thermodynamics with partial miscibility and derive an entropy supplementary conservative equation together with conditions on both the interfacial terms and the mixing terms of the entropy. Not only all the closures proposed in the literature are recovered but also new ones and we also propose explicit formulations of the mixing terms and show that depending on their expression, the condition expressed in [[@Gallouet_2004]]{} is not necessary. Methodology and decomposition. ------------------------------ We start without any condition on $(\speed{sint}, {p_{I}})$. We need initially to fix a decomposition of $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ and ${{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}})$ including a certain degree of freedom as explained in Section \[ssec:NC\_theory\_applied\]. Given an entropy ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ of System , by expressing the entropic variables as ${{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = \left( {\mathrm{v_{\alpha}}}, {{{\boldsymbol{\mathrm{v}}}}}_{2}^{{T}}, {{{\boldsymbol{\mathrm{v}}}}}_{1}^{{T}}\right)$, we use the decomposition proposed in Definition . Since we do not want to generate other non-conservative terms, we choose to define the line vector $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{Rp}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: Rp}{}]}$ by $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = \left( \transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}}), {{\boldsymbol{\mathrm{0}}}}, {{\boldsymbol{\mathrm{0}}}} \right)$ where $\transfer{salpha}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ is the unknown scalar function a priori of all the variables ${{{\boldsymbol{\mathrm{u}}}}}$. We obtain the following decompositions [rClrClrClrCl]{}\[eq:final\_entropic\_condition\_decomp\_BNZ\] ( \_ )\^[[T]{}]{} &=& ()\ \_[2]{} \_[\_[2]{}]{}\_[2]{}(\_[2]{})\ \_[1]{} \_[\_[1]{}]{}\_[1]{}(\_[1]{}) ,\ (\_ )\^[[T]{}]{} &=& -() + + \_\_[k]{} [\_[k]{}]{}\ \ . $\transfer{salpha}$ allows fractions of the non-conservative terms to feed the matrix ${{{\boldsymbol{\mathcal{C}}}}_{k}}$. Given this decomposition, we use the methodology proposed in Section \[ssec:NC\_metho\]. ($Step$ 2) will be split here into two sub-steps. 1. Condition $(C_{1})$ on the symmetry of the matrix $\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$ ensures the existence of an entropy flux ${{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$. It will determine $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$. 2. Knowing $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})$, Condition $(C_{2})$, $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \, {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) - \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) = {{\boldsymbol{\mathrm{0}}}}$, will return an equation linking $(\speed{sint},{p_{I}})$ and also $\psi_{k}$ when miscibility is accounted for. Non-miscible phases entropy. ---------------------------- We start applying our method $(Step \ 1)$ by postulating ${{\mathsf{H}}}$ as in Equation . The thermodynamics is entirely known and we use the Ideal Gas EOS. The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are then $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} {\mathrm{v_{\alpha}}}\\ {{{\boldsymbol{\mathrm{v}}}}}_{2} \\ {{{\boldsymbol{\mathrm{v}}}}}_{1} \end{pmatrix} \text{ with } {\mathrm{v_{\alpha}}}= \frac{p_{1}}{T_{1}} - \frac{p_{2}}{T_{2}} \text{ and } {{{\boldsymbol{\mathrm{v}}}}}_{k} = \frac{1}{T_{k}}\begin{pmatrix} g_{k} - \speed{sk}^{2}/2 \\ \speed{sk} \\ -1 \end{pmatrix},\end{aligned}$$ with $g_{k}$ the Gibbs free energy, $g_{k} =\epsilon_{k} + p_{k}/\rho_{k} - T_{k}s_{k}$. We now apply the conditions to determine $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$ and derive the equation that links the interfacial quantities $\speed{sint}$ and ${p_{I}}$. \[theo:BNZ\_ab\_classic\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} {s}_{k}$ then with the decomposition proposed in Equations  $$\begin{aligned} \label{eq:classic_cond_a} \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \Leftrightarrow \transfer{salpha} ({{{\boldsymbol{\mathrm{u}}}}}) = {F}(\alpha_{2}) + \frac{p_{1}}{T_{1}}u_{1} - \frac{p_{2}}{T_{2}}u_{2},\end{aligned}$$ with ${F}$ a strictly convex arbitrary function depending on the volume fraction $\alpha_{2}$. As a consequence the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ gives $$\begin{aligned} \label{eq:classic_cond_b} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] &= {{\boldsymbol{\mathrm{0}}}} \\ \Leftrightarrow \ - {F}(\alpha_{2}) + \sum_{k=1,2} \frac{(-1)^{k}}{T_{k}} ({p_{I}}- p_{k} )( \speed{sk} - \speed{sint}) &= 0. \end{IEEEeqnarraybox}\end{aligned}$$ The function $\transfer{salpha}$ is found relying on symbolic computation and it holds as a proof. As explained in $(Step\ 2.a)$, Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}$ associated with the mixture entropy ${{\mathsf{H}}}$ chosen as in Equation  by defining the unknown function $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$. Then as described in $(Step\ 2.b)$, Equation  relates the interfacial terms $(\speed{sint}, {p_{I}})$. By choosing ${F}(\alpha_{2}) =0$, the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\times \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ writes $$\begin{aligned} \label{eq:classic_cond_b_reduced} \sum_{k = 1,2} \frac{1}{T_{k}} \left(p_{k}-{p_{I}}\right) \left(\speed{sint}-\speed{sk}\right) = 0.\end{aligned}$$ So now, to obtain a closed model along with a supplementary conservative equation, we can postulate an interfacial velocity $\speed{sint}$ and derive the corresponding ${p_{I}}$. We will limit ourselves to defining $\speed{sint}$ such that the field associated to $\speed{sint}$ is linearly degenerate. In that case, the only admissible interfacial velocities are $\speed{sint} = \beta u_{1} + (1-\beta) u_{2}$ with $\beta \in \left[ 0,1, \alpha_{1} \rho_{1}/\rho \right]$ [[@Coquel_2002]]{}, [[@Lochon_PhdThesis_2016]]{}. We will focus on the particular case where ${F}(\alpha_{2}) =0$. We obtain the following results: - If $\speed{sint} = \speed{sk}$, then Equation  returns ${p_{I}}= p_{k^{\prime}}$. $(\speed{sk},p_{k^{\prime}})$ is the closure proposed first by [[@Baer_Nunziato_1986]]{}, [[@Kapila_1997]]{}, [[@Bdzil_1999]]{}, in the context of deflagration-to-detonation. - If $\speed{sint} = \beta u_{1} + (1-\beta) u_{2}$ with $\beta = \alpha_{1} \rho_{1}/\rho$, then Equation  returns ${p_{I}}=\mu p_{1} + (1- \mu ) p_{2}$ with $\mu\left(\beta\right) =(1-\beta) T_{2} / ( \beta T_{1} + (1-\beta) T_{2})$. It is the closure found in [[@Lochon_PhdThesis_2016]]{} among others. We see that first these closures are a specific case where $F(\alpha_{2})$ is chosen to be zero in Equation . Second, one could have chosen another interfacial velocity $\speed{sint}$ and it would have led to another interfacial pressure ${p_{I}}$ compatible with an entropy pair. If we had used the extended condition expressed in Equation , then the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ would be $$\begin{aligned} \label{eq:classic_cond_b_extended} & \sum_{k = 1,2} \frac{1}{T_{k}} \left[p_{k}-{p_{I}}\left({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right) \right] \left[\speed{sint}\left({{{\boldsymbol{\mathrm{u}}}}}, \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right)-\speed{sk}\right] \partial_{x} \alpha_{k} \leq 0 \\ \Leftrightarrow \ & - \sum_{k = 1,2} \frac{1}{T_{k}} \frac{Z_{k}}{(Z_{1}+Z_{2})^{2}}\left[ p_{k^{\prime}}-p_{k} + sgn\left(\partial_{x} \alpha_{1} \right) (u_{k^{\prime}}-\speed{sk}) Z_{k^{\prime}} \right]^{2} \leq 0,\end{aligned}$$ where $Z_{k}$ is defined by $Z_{k} = \rho_{k} a_{k}$ with the phase sound speed $a^{2}_{k} = \left. \partial p_{k} / \partial \rho_{k} \right|_{\scriptstyle s_{k}}$. From Equation , one sees that the dependency on $\partial_{x} {{{\boldsymbol{\mathrm{u}}}}}$ reduces to $\partial_{x} \alpha_{2}$ otherwise some terms would not be signable. Then closures such as the one found through Discrete Element Method (DEM) [[@Saurel_Gavrilyuk_2003]]{} are obtained $$\begin{aligned} \speed{sint} &= \frac{Z_{1} u_{1} + Z_{2} u_{2}}{Z_{1} + Z_{2}} + sgn\left(\partial_{x} \alpha_{1} \right) \frac{p_{2}-p_{1}}{Z_{1}+Z_{2}}, \\ {p_{I}}&= \frac{Z_{2} p_{1} + Z_{1} p_{2}}{Z_{1} + Z_{2}} + sgn\left(\partial_{x} \alpha_{1} \right) \frac{Z_{1}Z_{2}}{Z_{1}+Z_{2}} \left(u_{2}-u_{1}\right).\end{aligned}$$ Partially miscible phases entropy. ---------------------------------- Now, let us add a degree of freedom in the thermodynamics by introducing mixing terms in the definition of the entropy ${{\mathsf{H}}}$ as in Equation  to account for partial miscibility of the phases. The added terms, $\psi_{k}$, functions of the volume fraction $\alpha_{k}$ only, are to be determined. The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are $$\begin{aligned} \label{eq:v_r_ln_alpha} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} \sum\limits_{\text{\tiny $k{=}1,2$}} (-1)^{k+1} \dfrac{p_{k}}{T_{k}} \left[ 1 - \dfrac{\alpha_{k}}{r_{k}} \psi_{k}^{\prime}(\alpha_{k}) \right]\\ {{{\boldsymbol{\mathrm{v}}}}}_{2} \\ {{{\boldsymbol{\mathrm{v}}}}}_{1} \end{pmatrix} \text{ with } {{{\boldsymbol{\mathrm{v}}}}}_{k} = \frac{1}{T_{k}}\begin{pmatrix} g_{k} - \speed{sk}^{2}/2\\ \speed{sk} \\ -1 \end{pmatrix}\end{aligned}$$ \[theo:BNZ\_ab\_mixing\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k} - \psi_{k}(\alpha_{k}) \right]$ with $\psi_{k}$, $k=\left[ 1,2 \right]$, two strictly convex arbitrary functions depending on the volume fraction, then with the decomposition proposed in Equations , we have $$\begin{aligned} \label{eq:mixture_cond_a} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}+ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v} \text{ symmetric } \\ \Leftrightarrow \, &\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}}) = {F}(\alpha_{2}) + \frac{p_{1}}{T_{1}}u_{1} \left[ 1 - \frac{\alpha_{1}}{r_{1}} \psi_{1}^{\prime}(\alpha_{1}) \right] - \frac{p_{2}}{T_{2}}u_{2} \left[ 1 - \frac{\alpha_{2}}{r_{2}} \psi_{2}^{\prime}(\alpha_{2}) \right] \end{IEEEeqnarraybox}\end{aligned}$$ with ${F}$ a strictly convex arbitrary function depending on the volume fraction. As a consequence the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ gives $$\begin{aligned} \label{eq:mixture_cond_b} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rcl} & {{\boldsymbol{\mathrm{0}}}} &= \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \\ \Leftrightarrow \ & 0 &= - {F}(\alpha_{2}) + \sum_{k=1,2} (-1)^{k+1} \alpha_{k}\rho_{k} \psi_{k}^{\prime}(\alpha_{k}) (u_{k}-\speed{sint})\\ & &+ \sum_{k=1,2} \frac{(-1)^{k}}{T_{k}} ({p_{I}}- p_{k} )( \speed{sk} - \speed{sint}) \end{IEEEeqnarraybox}\end{aligned}$$ Again, Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}({{{\boldsymbol{\mathrm{u}}}}})$ conditioning the function $\transfer{salpha}({{{\boldsymbol{\mathrm{u}}}}})$ $(Step\ 2.a)$. The interfacial quantities $(\speed{sint},{p_{I}})$ and $\psi_{k}$ are linked by Equation  $(Step \ 2.b)$. The difference with the previous case for immiscible phases is that there are two supplementary unknowns $\psi_{k}$, $k=1,2$. We thus are free to either postulate first an interfacial velocity $\speed{sint}$ and then derive the corresponding ${p_{I}}$ and $\psi_{k}$ or postulate first the functions $\psi_{k}$ and see what choices we have for the interfacial terms. In the following we investigate the two approaches. ### Interfacial closures impacting thermodynamics. Let us postulate $\speed{sint}$ and limit ourselves to the case ${F}(\alpha_{2}) =0$. We will again seek a linearly degenerate field for $\speed{sint}$. In such case, the results in Table \[table:mixing\_entropy\_cond\_case12\] are obtained. [c|c|c|c]{} & $\speed{sint}$ & ${p_{I}}$ & $\left( \psi_{k} , \psi_{k^{\prime}}\right)$\ Case 1 &$\speed{sk}$ & $p_{k^{\prime}}$ & $\left( \psi_{k}, 0 \right)$\ Case 2 & -------------------------------------- $\beta u_{1} + (1-\beta) u_{2}$ $\beta = \alpha_{1} \rho_{1} / \rho$ -------------------------------------- : Admissiblethermodynamics and model closures obtained by postulating$\speed{sint}$[]{data-label="table:mixing_entropy_cond_case12"} & --------------------------------------------------------------------------------- $\mu p_{1} + (1- \mu ) p_{2}$ $\mu\left(\beta\right) = \frac{(1-\beta) T_{2}}{\beta T_{1} + (1-\beta) T_{2}}$ --------------------------------------------------------------------------------- : Admissiblethermodynamics and model closures obtained by postulating$\speed{sint}$[]{data-label="table:mixing_entropy_cond_case12"} & $\psi_{k}(\alpha_{k})=\psi_{k^{\prime}}(\alpha_{k^{\prime}})$ In Case 1 of Table \[table:mixing\_entropy\_cond\_case12\], $\psi_{k}$ can be interpreted as a configuration energy of phase $k$ as in [[@Baer_Nunziato_1986]]{}, [[@Kapila_1997]]{} [[@Bdzil_1999]]{}, in the context of deflagration-to-detonation. It is a term defining an interaction of one phase with itself only. More importantly, Equation  shows that it is not possible to include a configuration energy for each phase when choosing the closure $(\speed{sint},{p_{I}})=(\speed{sk},p_{k^{\prime}})$. In Case 2 of Table \[table:mixing\_entropy\_cond\_case12\], the condition on the mixing term introduced in Equation  by [[@Gallouet_2004]]{} is recovered and the closures are the one stated in [[@Coquel_2002]]{}. However, the condition on the mixing terms imposes a constraint on the volume fraction and thus on the flow topology. Since mixing of the phases should be able to occur disregarding the flow topology, these terms fail to introduce free mixing among the phases. ### Thermodynamics impacting interfacial term closures. Since Case 1 and Case 2 of Table \[table:mixing\_entropy\_cond\_case12\] do not allow the phases to mix, let us choose first the thermodynamics of the system and induce the admissible interfacial terms. It has been shown that the mixing entropy of an ideal compressible binary mixture is of the form $\sum_{k=1,2} \alpha_{k} \text{ln}(\alpha_{k})$. Therefore, we choose to define the functions $\psi_{k}$ by $\psi_{k}(\alpha_{k}) =r_{k} \text{ln}(\alpha_{k})$. In this case, the entropy writes $$\begin{aligned} {{\mathsf{H}}}= - \sum_{k=1,2} \alpha_{k} \rho_{k} \left[ {s}_{k} - r_{k}\text{ln}(\alpha_{k}) \right],\end{aligned}$$ with $r_{k}$ the specific gas constant of phase $k$, we now account for quasi-miscibility between the phases. The condition on $\transfer{salpha}$ degenerates, $\transfer{salpha} = {F}(\alpha_{2})$ and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right]$ is now $$\begin{aligned} \label{eq:mixture_cond_b_rln} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{c} - {F}(\alpha_{2}) + {p_{I}}\left( \frac{u_{1}-\speed{sint}}{T_{1}}-\frac{u_{2}-\speed{sint}}{T_{2}} \right) = 0. \end{IEEEeqnarraybox}\end{aligned}$$ It is no more possible to obtain the classic definition on $\speed{sint}$ and ${p_{I}}$. In the case ${F}(\alpha_{2})=0$ two choices are possible to verify Equation  and summarized in Table \[table:mixing\_entropy\_cond\]. $\speed{sint}$ ${p_{I}}$ -------- ---------------------------------------------------------------------- --------------- Case 3 $\beta u_{1} + (1-\beta) u_{2}$ with $\beta = T_{2} / (T_{2}-T_{1})$ no constraint Case 4 no constraint 0 : Admissiblethermodynamics and model closures obtained by postulating$\psi_{k}$[]{data-label="table:mixing_entropy_cond"} Case 3 of Table \[table:mixing\_entropy\_cond\] proposes a temperature-based averaged velocity for $\speed{sint}$, which does not seem to be physically reasonable. In Case 4, the interfacial pressure must vanish for the system to admit an entropy supplementary conservation equation and the Baer-Nunziato model becomes a conservative system if one assumes the field associated to $\speed{sint}$ to be linearly degenerate. One knows how much it simplifies the problem in terms of numerical implementation. This result can be interpreted as an incompatibility between the existence of a mixing process in the thermodynamics of the mixture and an interfacial pressure, that stays meaningful as long as there is an interface between the two phases. ### Link with dispersed phase flow. When the thermodynamics accounts for mixing (Case 4 Table \[table:mixing\_entropy\_cond\]), the existence of an entropy supplementary conservative equation is incompatible with the interfacial pressure, and thus the nozzling terms ${p_{I}}\partial_{x} \volfrac{k}$ vanish. In separated two-phase flows, these terms are known to be necessary to preserve uniformity in velocity and pressure of the flow during its temporal evolution [[@Andrianov_2003]]{} and are usually compared to the terms obtained in a single gas with a variable section [[@Saurel_2001]]{}. Whereas these arguments seem valid for separated two-phase flows, one may question the role these terms play in a dispersed phase flows. Taking the particular case ${p_{I}}= 0$ and $p_{2}=0$ in the Baer-Nunziato model seems to lead to a system of equations similar to one that would describe a flow of incompressible suspended particles, where 1 would denote the carrier phase and 2 the dispersed phase. Doing so, one recovers not only the Marble model [[@Marble_1963]]{}, which proposes a pressureless gas dynamic equations for the particle phase, valid in the limit where $\alpha_{2} < 10^{-3}$, but also the model obtained by Sainsaulieu [[@Sainsaulieu_1995]]{} in the asymptotic limit where the volume fraction of the particles $\alpha_{2} \rightarrow 0$. Nevertheless, even if the partial differential equations are alike, the thermodynamics associated to Marble and Sainsaulieu models differ from the one we propose for the Baer-Nunziato model. The latter accounts for compressibility of the two phases and partial miscibility whereas the thermodynamics of the Marble model assumes incompressibility of the particles and non-miscibility between the two phases. To conclude, if one aims at unifying the description of both separated phases and dispersed flow through a unique model, the thermodynamics must be treated together with the system modelling. Application to the plasma model {#sec:plasma} =============================== The multicomponent fluid modelling of plasma flows out of thermal equilibrium has been derived rigorously from kinetic theory using a multi-scale Chapman-Enskog expansion mixing a hyperbolic scaling for the heavy species with a parabolic scaling for the electrons [[@Graille_2007]]{}. The system takes the form $$\begin{aligned} \label{sys:plasma_eq_full} \partial_{t} {{{\boldsymbol{\mathrm{u}}}}}+ \left[ \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}= \partial_{x} \left( {{{\boldsymbol{\mathcal{D}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \partial_{x} {{{\boldsymbol{\mathrm{u}}}}}\right),\end{aligned}$$ with $$\begin{aligned} \label{eq:plasma_eq} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 1 & 0 & 0 & 0 \\ (\kappa/2-1)\speed{s}^2 & (2-\kappa)\speed{s} & \kappa & 0 & 0\\ (\kappa/2 \speed{s}^2 - \frac{h^{tot}}{\rho_{h}})\speed{s} & \frac{h^{tot}}{\rho_{h}} - \kappa \speed{s}^2 & (1+\kappa)\speed{s} & 0 & 0\\ -\frac{\rho_{e}}{\rho_{h}} \speed{s} & \frac{\rho_{e}}{\rho_{h}} & 0 & \speed{s} & 0 \\ - \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \speed{s} & \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} & 0 & 0 & \speed{s} \end{pmatrix}, \\ {{{\boldsymbol{\mathcal{N}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ -\frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \kappa \speed{s} & \frac{\rho_{e} \epsilon_{e}}{\rho_{h}} \kappa & 0 & 0 & 0 \end{pmatrix}, \\ {{{\boldsymbol{\mathcal{D}}}}}({{{\boldsymbol{\mathrm{u}}}}}) &= \begin{pmatrix} 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} & \frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} + \gamma D\\ 0 & 0 & 0 & 0 & \frac{D\kappa}{T_{e}}\\ 0 & 0 & 0 & -\frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} & \frac{\lambda \kappa \epsilon_{e}}{\rho_{e}} + \gamma D \end{pmatrix},\end{aligned}$$ where the column vector ${{{\boldsymbol{\mathrm{u}}}}}\in { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}^{5}$ is defined by ${{{\boldsymbol{\mathrm{u}}}}}^{{T}} = \left( \rho_{h}, \rho_{h} \speed{s}, E, \rho_{e}, \rho_{e} \epsilon_{e}\right)$ with $\rho_{h}$ is the density of the heavy particles, $\speed{s}$ the hydrodynamic velocity, $E$ the total energy defined by $E= \rho_{h} \speed{s}^2/2 + \rho_{h} \epsilon_{h} + \rho_{e} \epsilon_{e}$, $ \epsilon_{h}$ the internal energy of the heavy particles, $\rho_{e}$ the density of the electrons, $\epsilon_{e}$ the internal energy of the electrons, $h^{tot}$ the total enthalpy defined by $h^{tot}= E + p$ with $p=p_{h}+p_{e}$, $T_{e}$ the temperature of the electrons, the constant $\kappa$ defined by $\kappa = \gamma-1$ with $\gamma$ the isentropic coefficient, $p_{h}$ is the pressure of the heavy particles and $p_{e}$ is the pressure of the electrons. In the diffusive terms, $\lambda$ is the electron thermal conductivity, D the electron diffusion coefficient. Concerning the thermodynamics, it can be obtained from kinetic theory. The electrons and the heavy particles thermodynamics are defined by an ideal gas equation of state, and they share both the same isentropic coefficient: $p_{h} = \kappa \rho_{h} \epsilon_{h}$, $p_{e} = \kappa \rho_{e} \epsilon_{e}$ where $p_{h}$ is the pressure of the heavy particles and $p_{e}$ is the pressure of the electrons, $r$ is the constant of the gas $r=c_{v} \kappa$ with $c_{v}$ the calorific heat at constant volume, the model being adimensionalized $r=c_{v}(\gamma-1)=1$. The model is naturally hyperbolic [[@Graille_2007]]{} and also involves second-order terms and eventually source terms [[@Magin_2009]]{}. Here we considered the homogeneous form. In this section, we would like to derive the usual entropy supplementary conservative equation found by [[@Graille_2007]]{} and show that it is unique, to attest the effectiveness of the theory. Decomposition. -------------- We need to proceed to the decomposition of the conservative and non conservative terms of System . We restrict ourselves again to the decomposition proposed in Definition  and we add a degree of liberty to each non-null non-conservative components by defining $\transfer{v}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R5}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R5}{}]}$ as $\transfer{v}({{{\boldsymbol{\mathrm{u}}}}})^{{T}} = ( \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}}), \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}}), 0, 0, 0)$ such that the following decompositions are obtained $$\begin{aligned} \label{eq:final_entropic_condition_decomp_plasma} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{C}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} = {{{\boldsymbol{\mathrm{v}}}}}({{{\boldsymbol{\mathrm{u}}}}}) \cdot \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \begin{pmatrix} \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}})\\ \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}})\\ 0 \\ 0 \\ 0 \end{pmatrix}, \\ \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} =\begin{pmatrix} - \transfer{s1}({{{\boldsymbol{\mathrm{u}}}}}) - \frac{\rho_{e}}{\rho_{h}} \left(1-\frac{T_{e}}{T_{h}}\right) \speed{s} \\ - \transfer{s2}({{{\boldsymbol{\mathrm{u}}}}}) + \frac{\rho_{e}}{\rho_{h}} \left(1-\frac{T_{e}}{T_{h}}\right) \\ 0 \\ 0 \\ 0 \end{pmatrix}.\end{aligned}$$ The unknown scalar functions $\transfer{v}_{k}({{{\boldsymbol{\mathrm{u}}}}})$ give the possibility to fractions of the non-conservative terms to be given to the matrix ${{{\boldsymbol{\mathcal{C}}}}_{k}}$. Ideal Gas entropy. ------------------ The entropy ${{\mathsf{H}}}: {{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ for two perfect gases is defined as $$\begin{aligned} \label{eq:mixture_entropy_plasma} {{\mathsf{H}}}= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e},\end{aligned}$$ with the partial entropies defined by $$\begin{aligned} {s}_{h}= c_{v} \, \text{ln}\left(\frac{p_{h}}{\kappa \rho_{h}^{\transfer{v}}}\right), && {s}_{e}= c_{v} \, \text{ln}\left(\frac{p_{e}}{\kappa \rho_{e}^{\transfer{v}}}\right).\end{aligned}$$ This entropy includes mixing between the electrons and the heavy particles. Thus, we start applying our method $(Step \ 1)$ by postulating ${{\mathsf{H}}}$ as in Equation . The entropic variables ${{{\boldsymbol{\mathrm{v}}}}}$ are then $$\begin{aligned} {{{\boldsymbol{\mathrm{v}}}}}= \begin{pmatrix} \dfrac{1}{T_{h}}\left(g_{h} - \speed{s}^{2}/2\right)\\ \dfrac{1}{T_{h}}\speed{s}\\ -\dfrac{1}{T_{h}}\\ \dfrac{1}{T_{e}} g_{e}\\ \dfrac{1}{T_{h}}-\dfrac{1}{T_{e}} \end{pmatrix},\end{aligned}$$ with $g_{k}$ the Gibbs free energy, $g_{k} =\epsilon_{k} + p_{k}/\rho_{k} - T_{k}s_{k}$. In the fourth component of the entropic variable, the kinetic energy of the electrons has vanished. This is due to the low-Mach assumption made for the electrons. We now apply the conditions to determine $\transfer{sk}({{{\boldsymbol{\mathrm{u}}}}})$. \[theo:plasma\_ab\_classic\] Consider System . If the mixture entropy is defined as ${{\mathsf{H}}}= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e}$, then with the decomposition proposed in Equations , we have $$\begin{aligned} \label{eq:plasma_classic_cond_a} \begin{IEEEeqnarraybox}[\IEEEeqnarraystrutmode \IEEEeqnarraystrutsizeadd{1pt}{1pt}][c]{rl} & \partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \\ \Leftrightarrow \ & \transfer{s1} ({{{\boldsymbol{\mathrm{u}}}}}) = \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\speed{s} \text{ and } \transfer{s2} ({{{\boldsymbol{\mathrm{u}}}}}) = -\frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right) \end{IEEEeqnarraybox},\end{aligned}$$ and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ is $$\begin{aligned} \label{eq:plasma_classic_cond_b} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] & = (0,\, 0,\, 0,\, 0,\, 0).\end{aligned}$$ Using Maple, we find $$\begin{aligned} &\partial_{{{{\boldsymbol{\mathrm{u}}}}}{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \times \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \partial_{{{{\boldsymbol{\mathrm{u}}}}}} \transfer{v}({{{\boldsymbol{\mathrm{u}}}}}) \text{ symmetric } \nonumber \\ \Leftrightarrow & \transfer{s1} ({{{\boldsymbol{\mathrm{u}}}}}) = \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\speed{s} + \int \left[ -\speed{s} \partial_{\speed{s}}{F}_{1}(\rho_{h},\speed{s}) +\rho_{h} \partial_{\rho_{h}}{F}_{1}(\rho_{h},\speed{s}) \right] d\speed{s} + {F}_{2}(\rho_{h}) \nonumber \\ & \text{and } \transfer{s2} ({{{\boldsymbol{\mathrm{u}}}}}) = -\frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right) + {F}_{1}(\rho_{h},\speed{s}),\end{aligned}$$ with $\mathsf{F_{1}}$, $\mathsf{F_{2}}$ two arbitrary functions and the condition on $\partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right]$ is $$\begin{aligned} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}({{{\boldsymbol{\mathrm{u}}}}}) \left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}({{{\boldsymbol{\mathrm{u}}}}}) + {{{\boldsymbol{\mathcal{Z}}}}_{2}}({{{\boldsymbol{\mathrm{u}}}}}) \right] \right)^{{T}} & = \begin{pmatrix} - \int \left[ -\speed{s} \partial_{\speed{s}}{F}_{1}(\rho_{h},\speed{s}) +\rho_{h} \partial_{\rho_{h}}{F}_{1}(\rho_{h},\speed{s}) \right] d\speed{s} - {F}_{2}(\rho_{h}) \\ -{F}_{1}(\rho_{h},\speed{s}) \\ 0\\ 0\\ 0 \end{pmatrix}\\ &= {\boldsymbol{0}}.\end{aligned}$$ One sees that the last equation imposes first ${F}_{1}=0$ and thus ${F}_{2}=0$. Reinjecting these terms into the first equation gives the result. As explained in $(Step\ 2.a)$, the Equation  guarantees the existence of an entropy flux ${{\mathsf{G}}}:{{{\boldsymbol{\mathrm{u}}}}}\in \Omega \mapsto { \IfEqCase{R}{ {M}{\mathcal{M}} {TM}{\mathcal{TM}} {TqM}{\mathcal{T}_{\pathM}\mathcal{M}} {TR}{\mathcal{TR}} {R7}{\mathbb{R}^{7}} {R77}{\mathbb{R}^{7\times7}} {R3}{\mathbb{R}^{3}} {R5}{\mathbb{R}^{5}} {R}{\mathbb{R}} {Rp}{\mathbb{R}^{p}} {Rpp}{\mathbb{R}^{p\times p}} }[\PackageError{space}{Undefined option to space: R}{}]}$ associated with the entropy ${{\mathsf{H}}}$ defined in Equation  by solving the unknown functions $\transfer{s1}({{{\boldsymbol{\mathrm{u}}}}})$ and $\transfer{s2}({{{\boldsymbol{\mathrm{u}}}}})$. Therefore, for the entropy ${{\mathsf{H}}}$ defined in Equation , there is a unique decomposition which ensures the existence of a supplementary conservative equation which is given by $$\begin{aligned} \label{eq:final_entropic_condition_decomp_plasma_determined} \left( \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{C}}}}_{1}}+ {{{\boldsymbol{\mathcal{C}}}}_{2}}\right] \right)^{{T}}= {{{\boldsymbol{\mathrm{v}}}}}^{{T}} \cdot \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{{\boldsymbol{\mathrm{f}}}}}({{{\boldsymbol{\mathrm{u}}}}}) + \begin{pmatrix} \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)v\\ \frac{\rho_{e}}{\rho_{h}}\left(1-\frac{T_{e}}{T_{h}}\right)\\ 0 \\ 0 \\ 0 \end{pmatrix},\end{aligned}$$ $$\begin{aligned} \partial_{{{{\boldsymbol{\mathrm{u}}}}}} {{\mathsf{H}}}\left[ {{{\boldsymbol{\mathcal{Z}}}}_{1}}+ {{{\boldsymbol{\mathcal{Z}}}}_{2}}\right] = {\boldsymbol{0}}.\end{aligned}$$ It leads to the following entropy flux couple $$\begin{aligned} {{\mathsf{H}}}&= - \rho_{h} {s}_{h} - \rho_{e} {s}_{e}, \\ {{\mathsf{G}}}&= - \left( \rho_{h} {s}_{h} + \rho_{e} {s}_{e} \right) \speed{s}.\end{aligned}$$ The theory recovers the supplementary conservative equation already found in the literature from the kinetic theory [[@Graille_2007]]{}. Conclusion ========== In the present contribution, we have proposed a theoretical framework for the derivation of supplementary conservation laws for systems of partial differential equation including first-order non-conservative terms - commonly encountered in modeling of complex flows - thus extending the standard approach for systems of conservation laws. Since our main objective is deriving an entropy supplementary conservation law, we have used this framework to make a first step to extend the theory of Godunov-Mock to such non-conservative systems. Given a reasonable choice in the combination of the conservative and non-conservative terms, we have been able to show how to use the theory to design or analyze systems by means of computer algebra on two applications chosen for their numerous differences in terms of model and thermodynamics closure as well as the nature of the waves impacted by the non-conservative terms. Firstly, applied to the Baer-Nunziato two-phase flow model derived from rational thermodynamics, the theory has brought about entropy supplementary conservative equations together with constraints on the interfacial quantities and the definition of the thermodynamics for non-miscible fluids and also when accounting for some level of mixing of the two phases. A new closure for the interfacial quantities has been proposed and leads to a conservative system. Secondly, for a plasma model obtained rigorously from the kinetic theory of gases, where the thermodynamics is also provided, the approach allows to recover as unique the supplementary conservation equation related to the kinetic entropy and is thus assessed. The content of the paper is a first step into studying the entropic symmetrization in the sense of Godunov-Mock and relation to source terms for two-phase flow modeling. Some partial symmetrization of the Baer-Nunziato model has been obtained in the classical framework by [@Forestier_2011]. Combining such symmetrization theory with source terms can then be envisioned such as in the case of plasma flows [@Magin_2009], even if the symmetrization is only partial in the framework of [@Graille_2007] where the electron are considered in a low-Mach limit. Nevertheless, for such a study to be complete, several other steps have to be handled first: the question of the strict convexity of the entropy for the change of variable to be admissible and its relation to thermodynamics (a difficult question [@Coquel_2002; @Gallouet_2004]); it is a part of Pierre Cordesse’s PhD Thesis [@Cordesse_PhD]. This loss of strict convexity in the framework of non-interacting thermodynamics has been investigated in [[@Cordesse_CMT_2019]]{} where a mixing thermodynamics for multi-fluids has been developed. Based on this new developments, we hope that equipping the Baer-Nunziato system with an extended thermodynamics closure will lead to a strictly convex entropy and thus allow the study of entropic full symmetrization and source terms, in the spirit of [@Giovangigli_1998; @Massot_2002; @Giovangigli_2004; @Magin_2009]. This is the subject of our current research. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to acknowledge the support of a CNES/ONERA PhD Grant for P. Cordesse and the help of M. Théron (CNES). They would like to express their special thanks to F. Coquel, S. Kokh, V. Giovangigli and A. Murrone for their invaluable help and numerous pieces of advice during the writing of the paper. We also would like to thank discussions with J.M. Hérard, which prompted this research path. Part of this work was conducted during the Summer Program 2018 at NASA Ames Research Center and the support and help of Nagi N. Mansour is also gratefully acknowledged. [^1]: ONERA, DMPE, 8 Chemin de la Hunière, 91120 Palaiseau, France, and CMAP, Ecole polytechnique, Route de Saclay 91128 Palaiseau Cedex, France, ([email protected]) [^2]: CMAP, Ecole polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, ([email protected]) [^3]: Received date, and accepted date (The correct dates will be entered by the editor). [^4]: Among the most well-known symmetry transformations, the time translation yields the conservation of the total energy of the system if the associated Lagrangian is invariant to time-shift and the space translation yields the conservation of the total momentum of the system if the Lagrangian is invariant to space-shift [^5]: Such closure is similar to the one used in [[@Powers_1988; @Powers_1990]]{} which led to a controversy [[@Drew_1983; @Bdzil_1999; @Andrianov_2003]]{} [^6]: Maple is a trademark of Waterloo Maple Inc.

ArXiv

--- abstract: 'We present preliminary results of follow-up optical observations, both photometric and spectroscopic, of stellar X-ray sources, selected from the cross-correlation of ROSAT All-Sky Survey (RASS) and TYCHO catalogues. Spectra were acquired with the E[lodie]{} spectrograph at the 193-cm telescope of the Haute Provence Observatory (OHP) and with the REOSC echelle spectrograph at the 91-cm telescope of the Catania Astrophysical Observatory (OAC), while $UBV$ photometry was made at OAC with the same telescope. In this work, we report on the discovery of six late-type binaries, for which we have obtained good radial velocity curves and solved for their orbits. Thanks to the OHP and OAC spectra, we have also made a spectral classification of single-lined binaries and we could give first estimates of the spectral types of the double-lined binaries. Filled-in or pure emission H$\alpha$ profiles, indicative of moderate or high level of chromospheric activity, have been observed. We have also detected, in near all the systems, a photometric modulation ascribable to photospheric surface inhomogeneities which is correlated with the orbital period, suggesting a synchronization between rotational and orbital periods. For some systems has been also detected a variation of H$\alpha$ line intensity, with a possible phase-dependent behavior.' author: - 'A. Frasca, P. Guillout, E. Marilli, R. Freire Ferrero, K. Biazzo' title: 'Newly discovered active binaries in the RasTyc sample of stellar X-ray sources' --- Introduction {#sec:Intro} ============ The cross-correlation between the ROSAT All-Sky Survey ($\simeq$ 150000 sources) and the TYCHO mission ($\simeq$ 1000000 stars) catalogues has selected about 14000 stellar X-ray sources (RasTyc sample, [@Guillout99]). Although most of these soft X-ray sources are expected to be the youngest stars in the solar neighborhood, neither the contamination by older RS CVn systems nor the fraction of BY Dra binaries are actually known. This information is, however, of fundamental importance for studying the recent local star formation history and, for instance, for putting constrains on the scale height of the spatial distribution of nearby young stars around the galactic plane. We thus started a spectroscopic observation campaign aimed at a deep characterisation of a representative sub-sample of the RasTyc population. In addition to derive chromospheric activity levels (from H$\alpha$ emission) and rotational velocities (from Doppler broadening), high resolution spectroscopic observations allow to infer ages (by means of Lithium abundance) and to single out spectroscopic and active binaries. In this work we present some preliminary results of follow-up observations, both photometric and spectroscopic, of some RasTyc stars performed with the 193-cm telescope of OHP and the 91-cm telescope of the Catania Astrophysical Observatory (OAC). In particular, we analyse six new late-type binaries, for which we have obtained good radial velocity curves and orbital solutions. An accurate spectral classification for the single-lined binaries has been also performed and the projected rotational velocity $v\sin i$ has been measured for all stars. The chromospheric activity level and the lithium content have been also investigated using as diagnostics the H$\alpha$ emission and the Li[i]{}$\lambda\,6708$ line, respectively. [llcccccccr]{}\ RasTyc & Name & P$_{\rm orb}$ & $\gamma$ & $k$ (P/S) & $M\sin^3i$ & $v\sin i$ (P/S) & Sp. Type & $B-V$ & W$_{\rm LiI}$\ & & (days) & (kms$^{-1}$) & (kms$^{-1}$) & $M_{\odot}$ & (kms$^{-1}$) & & & (mÅ)\ \ 193137 & HD 183957 & 7.954 & $-$29.0 & 57.5/63.1 & 0.758/0.691 & 4.0/4.4 & K0-1V/K1-2V & 0.84 & $< 10$\ 215940 & OT Peg & 1.748 & $-$27.0 & 16.6/23.2 & 0.007/0.005 & 9.2/9.4 & K0V/K3-5V & 0.79 & 50\ 221428 & BD+334462 & 10.12 & $-$20.9 & 59.2/60.4 & 0.905/0.887 & 16.1/32.6 & G2 + K & 0.70 & 15:\ 040542 & DF Cam & 12.60 & $-$19.5 & 22.8 & SB1 & 35 & K2III & 1.14 & —\ 072133 & V340 Gem & 36.20 & +37.0 & 42.1 & SB1 & 40 & G8III & 0.83 & 70\ 102623 & BD+382140 & 15.47 & +47.4 & 31.3 & SB1 & 11.5 & K1IV & 1.03 & 40\ \ \ Observations and reduction {#sec:Obs} ========================== Spectroscopy ------------ Spectroscopic observations have been obtained at the [*Observatoire de Haute Provence*]{} (OHP) and at the [*M.G. Fracastoro*]{} station (Mt. Etna, 1750 m a.s.l.) of Catania Astrophysical Observatory (OAC). At OHP we observed in 2000 and 2001 with the E[lodie]{} echelle spectrograph connected to the 193-cm telescope. The 67 orders recorded by the CCD detector cover the 3906-6818 Å  wavelength range with a resolving power of about 42000 ([@Bar96]). The E[lodie]{} spectra were automatically reduced on-line during the observations and the cross-correlation with a reference mask was produced as well. The observations carried out at Catania Observatory have been performed in 2001 and 2002 with the REOSC echelle spectrograph at the 91-cm telescope. The spectrograph is fed by the telescope through an optical fiber (UV - NIR, $200\,\mu m$ core diameter) and is placed in a stable position in the room below the dome level. Spectra were recorded on a CCD camera equipped with a thinned back-illuminated SITe CCD of 1024$\times$1024 pixels (size 24$\times$24 $\mu$m). The échelle crossed configuration yields a resolution of about 14000, as deduced from the FWHM of the lines of the Th-Ar calibration lamp. The observations have been made in the red region. The detector allows us to record five orders in each frame, spanning from about 5860 to 6700 Å. The OAC data reduction was performed by using the [echelle]{} task of IRAF[^1] package following the standard steps: background subtraction, division by a flat field spectrum given by a halogen lamp, wavelength calibration using the emission lines of a Th-Ar lamp, and normalization to the continuum through a polynomial fit. Photometry ---------- The photometric observations have been carried out in 2001 and 2002 in the standard $UBV$ system also with the 91-cm telescope of OAC and a photon-counting refrigerated photometer equipped with an EMI 9789QA photomultiplier, cooled to $-15\degr$C. The dark noise of the detector, operated at this temperature, is about $1$ photon/sec. For each field of the RasTyc sources, we have chosen two or three stars with known $UVB$ magnitudes to be used as local standards for the determination of the photometric instrumental “zero points". Additionally, several standard stars, selected from the list of Landolt ([@Lan92]), were also observed during the run in order to determine the transformation coefficients to the Johnson standard system. A typical observation consisted of several integration cycles (from 1 to 3, depending on the star brightness) of 10, 5, 5 seconds, in the $U$, $B$ and $V$ filter, respectively. A 21$\arcsec$ diaphragm was used. The data were reduced by means of the photometric data reduction package PHOT designed for photoelectric photometry of Catania Observatory ([@LoPr93]). Seasonal mean extinction coefficient for Serra La Nave Observatory were adopted for the atmospheric extinction correction. Results ======= Radial velocity and photometry {#sec:RV} ------------------------------ The radial velocity (RV) measurements for the E[lodie]{} data have been performed onto the cross-correlation functions (CCFs) produced on-line during the data acquisition. Radial velocities for OAC spectra were obtained by cross-correlation of each echelle spectral order of the RasTyc spectra with that of bright radial velocity standard stars. For this purpose the IRAF task [fxcor]{}, that computes RVs by means of the cross-correlation technique, was used. The wavelength ranges for the cross-correlation were selected to exclude the H$\alpha$ and Na[I]{} D$_2$ lines, which are contaminated by chromospheric emission and have very broad wings. The spectral regions heavily affected by telluric lines (e.g. the O$_2$ lines in the $\lambda~6276-\lambda~6315$ region) were also excluded. The observed RV curves are displayed in Fig. \[fig:RV\], where, for SB2 systems, we used dots for the RVs of primary (more massive) components and open circles for those secondary (less massive) ones. We initially searched for eccentric orbits and found in any case very low eccentricity values (e.g. $e=0.010$ for HD 183957, $e=0.030$ for 221428). Thus, following the precepts of [-@Lucy71], we adopted $e=0$. The circular solutions are also represented in Fig. \[fig:RV\] with solid and dashed lines for the primary and secondary components, respectively. The orbital parameters of the systems, orbital period ($P_{\rm orb}$), barycentric velocity ($\gamma$), RV semi-amplitudes ($k$) and masses ($M\sin^3i$), are listed in Table \[tab:param\], where P and S refer to the primary and secondary components of the SB2 systems, respectively. With the only exception of HD 183957, for which any modulation is visible neither in OAC data nor in TYCHO $V_{\rm T}$ magnitudes, all sources show a photometric modulation well correlated with the orbital period, indicating a high degree of synchronization. The low amplitude of the light curve of 215940 and the very low values of $M\sin^3i$ imply a very low inclination of orbital/rotational axis. Spectral type and $v\sin i$ determination {#sec:Spty} ----------------------------------------- For SB1 systems observed with E[lodie]{} we have determined effective temperatures and gravity (i.e. spectral classification) by means of the TGMET code, available at OHP ([@Katz98]). We have also used ROTFIT, a code written by one of us ([@Frasca03]) in IDL (Interactive Data Language, RSI), which simultaneously find the spectral type and the $v\sin i$ of the target by searching, into a library of standard star spectra, for the standard spectrum which gives the best match of the target one, after the rotational broadening. As standard star library, we used a sub-sample of the stars of the TGMET list whose spectra were retrieved from the E[lodie]{} Archive ([@Prugniel01]). The ROTFIT code was also applied to the OAC spectra, using standard star spectra acquired with the same instrument. This was especially advantageous for DF Cam, for which we have no E[lodie]{} spectrum. For SB2 systems we made a preliminary classification on the basis of a visual inspection of E[lodie]{} and OAC spectra. However, we are developing a code for spectral type determination in double-lined binaries which will allow us to improve the spectral classification. We found at least two binaries composed by main sequence stars, while the remaining systems contain an evolved (giant or sub-giant) star. Measurements of $v\sin i$ were also made using the E[lodie]{} CCFs and the calibration relation between CCF width and $v\sin i$ proposed by [-@Queloz98]. The lower rotation rate ($v\sin i \simeq$4 kms$^{-1}$) has been detected for both components of HD 183957, which display also the lowest H$\alpha$ activity among the six sources. H$\alpha$ emission and Lithium content {#sec:Halpha} -------------------------------------- The H$\alpha$ line is an important indicator of chromospheric activity. Only the very active stars show always H$\alpha$ emission above the continuum, while in less active stars only a filled-in absorption line is observed. The detection of the chromospheric emission contribution filling in the line core is hampered in double-lined systems in which both spectra are simultaneously seen and shifted at different wavelengths, according to the orbital phase. Therefore a comparison with an “inactive” template built up with two stellar spectra that mimic the two components of the system in absence of activity is needed to emphasize the H$\alpha$ chromospheric emission. The inactive templates have been built up with rotationally broadened E[lodie]{} archive spectra (HD 10476, K1V for both components of HD 183957; $\gamma$ Cep, K1IV for 102623; $\delta$ Boo, G8III for 072133; HD 17382, K1V for 215940) or with OAC spectra of $\alpha$ Ari (K2III), for DF Cam, acquired during the observing campaigns. The two components of HD 183957 show only a small filling of their H$\alpha$ profiles (Fig. \[fig:Halpha2\]), while the other RasTyc stars display H$\alpha$ emission profiles with a variety of shapes, going from a simple symmetric emission profile (102623) to a double-peaked strong emission line (215940). It has been also observed a very broad, complex feature with a filled-in core and an emission blue wing (072133). A H$\alpha$ profile similar to that displayed by the latter star has been sometimes observed in some long-period RS CVn’s, like HK Lac (e.g. Catalano & Frasca 1994). RasTyc 072133 was classified as a semi-regular variable after Hipparcos, but it displays all the characteristics of a RS CVn SB1 binary. The E[lodie]{} spectra of 221428 in the H$\alpha$ region show that the secondary (less massive) component displays a H$\alpha$ line always in emission with a stronger intensity around phase 0$\fp$7. The OAC spectra of DF Cam always display a pure H$\alpha$ emission line, whose intensity varies with the orbital/rotational phase. Similarly to 072133, DF Cam, considered as a semi-regular variable after Hipparcos photometry, is very likely an active binary of the RS CVn or BY Dra class. The equivalent width of the lithium $\lambda$6708 line, $EW_{\rm Li}$, was measured on the E[lodie]{} spectra. For the three sources for which we were able to detect and measure $EW_{\rm Li}$, we deduced lithium abundance, $\log N(Li)$, in the range 1.3–1.8, according to [-@Pav96] NLTE calculations. We are grateful to the members of the staff of OHP and OAC observatories for their support and help with the observations. This research has made use of SIMBAD and VIZIER databases, operated at CDS, Strasbourg, France. Baranne A., Queloz D., Mayor M., et al., 1996, A&AS 119, 373 Catalano S. and Frasca A. 1994, A&A 287, 575 Frasca A., Alcalà J.M., Covino E., Catalano S., Marilli E. and Paladino R. 2003, A&A 405, 149 Guillout P., Schmitt J. H. M. M., Egret D., Voges W., Motch C. and Sterzik M. F. 1999, A&A 351, 1003 Katz D., Soubiran C., Cairel R., Adda M. and Cautain R. 1998, A&A 338, 151 Landolt, A. U. 1992, AJ, 104, 340 Lo Presti, C., & Marilli, E. 1993, PHOT. Photometrical Data Reduction Package. Internal report of Catania Astrophysical Observatory N. 2/1993 Lucy, L. B. and Sweeney, M. A., 1971, AJ 76, 544 Pavlenko Y.V. & Magazzù A. 1996, A&A 311, 961 Prugniel, P. and Soubiran, C. 2001, A&A 369, 1048 Queloz D., Allain S., Mermilliod J.-C., Bouvier J. and Mayor, M. 1998, A&A 335, 183 [^1]: IRAF is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation.

ArXiv

--- abstract: | Experience collected in mesoscopic dynamic modeling of externally driven systems indicates absence of potentials that could play role of equilibrium or nonequilibrium thermodynamic potentials yet their\ thermodynamics-like modeling is often found to provide a good description, good understanding, and predictions that agree with results of experimental observations. This apparent contradiction is explained by noting that the dynamic and the thermodynamics-like investigations on a given mesoscopic level of description are not directly related. Their relation is indirect. They both represent two aspects of dynamic modeling on a more microscopic level of description. The thermodynamic analysis arises in the investigation of the way the more microscopic dynamics reduces to the mesoscopic dynamics (reducing dynamics) and the mesoscopic dynamic analysis in the investigation of the result of the reduction (reduced dynamics). author: - | Miroslav Grmela[^1]\ École Polytechnique de Montréal,\ C.P.6079 suc. Centre-ville Montréal, H3C 3A7, Québec, Canada title: ' **Externally driven macroscopic systems: Dynamics versus Thermodynamics**' --- Introduction {#Intr} ============ Boussinesq equation is a well known example of mathematical formulation of mesoscopic dynamics of externally driven macroscopic systems. The mesoscopic level on which the physics is regarded in this example is the level of fluid mechanics, the system itself is a horizontal layer of fluid heated from below (Rayleigh-Bénard system), and the external driving forces are the gravitational force and imposed temperature gradient. Analysis of solutions of the Boussinesq equations reveals properties observed in experiments (e.g. the observed passage from less organized to a more organized behavior presents itself as a bifurcation in solutions). Many other examples of this type can be found for instance in [@Halp]. One of the common features of the dynamical equations that arise in the examples (the feature that has been noted in [@Halp]) is that there does not seem to be possible, at least in general, to associate them with a potential whose landscape would provide a pertinent information about their solutions \[ *“... there is no evidence for any global minimization principles controlling the structure ...” - see the last paragraph of Conclusion in [@Halp]*\]. Since potential (or potentials) of this type are essential in any type of thermodynamics, the observed common feature seems to point to the conclusion that there is no thermodynamics of externally driven systems. On the other hand, there is a long tradition (starting with Prigogine in [@Prigogine]) of investigating externally driven systems with methods of thermodynamics. Roughly speaking, responses of macroscopic systems to external forces are seen as adaptations minimizing their resistance. The thermodynamic potentials involved in this type of considerations (i.e. potentials used to characterize the “resistance”) are usually various versions of the work done by external forces and the entropy production. There are many examples of very successful and very useful considerations of this type (see e.g. [@Umb]). In Section \[EX5\] we illustrate the thermodynamic analysis in the context of an investigation of the morphology of immiscible blends. Specifically, we show how the thermodynamic argument provides an estimate of concentrations at the point of phase inversion, i.e. at the point at which the morphology of a mixture of two immiscible fluids changes in such a way that the roles of being encircled and encircling changes (i.e. the continuous phase and the dispersed phase exchange their roles). The experience collected in investigations of externally driven systems can be thus summed up by saying that mesoscopic dynamical modeling indicates an impossibility of using thermodynamics-like arguments yet this type of arguments are often found to be very useful and pertinent. There are in fact well known examples [@Keizer] in which both dynamic and thermodynamic approaches were developed and the potentials used in the thermodynamic analysis are proven to play no significant role in the dynamic analysis. Our objective in this paper is to suggest an explanation of this apparent contradiction. We show that the dynamic and the thermodynamic analysis made on a given mesoscopic level of description are not directly related. Their relation is indirect. They are both two aspects of a single dynamic analysis made on a more microscopic (i.e. involving more details) level of description. An investigation of the way the microscopic dynamics is reducing to the mesoscopic dynamics provides the mesoscopic thermodynamics (Section \[RD\]) and the investigation of the final result of the reduction provides the mesoscopic dynamics. It is important to emphasize that we are using in this paper the term “thermodynamics” in a general sense (explained in Section \[RD\]). While the classical equilibrium thermodynamics and the Gibbs equilibrium statistical mechanics are particular examples of the general thermodynamics presented in Section \[RD\], they are not the ones that are the most pertinent for discussing externally driven systems. Multiscale Mesoscopic Models {#MMM} ============================= Given an externally driven system (or a family of such systems), how do we formulate its dynamical model? The most common way to do it (called hereafter a direct derivation) proceeds in the following three steps. First, behavior of the externally driven macroscopic systems under consideration is observed experimentally in certain types of measurements called hereafter *meso-measurements*. In the second step, the experience collected in the meso-measurements together with an insight into the physics taking place in the observed systems leads to the choice of the level of description, i.e. the choice of state variables (we shall denote them by the symbol $y$), and equations $$\label{Gdyn} \dot{y}=g(y,\zeta, \mathcal{F}^{meso})$$ governing their time evolution. By $\zeta$ we denotes the material parameters (i.e. the parameters through which the individual nature of the physical systems under consideration is expressed) and $\mathcal{F}^{meso}$ denotes the external forces. In the third step, the governing equations (\[Gdyn\]) are solved and the solutions are compared with results of observations. If the comparison is satisfactory, the model represented by (\[Gdyn\]) is called a well established mesoscopic dynamical model (e.g. the Boussinesq model is a well established model of the Rayleigh-Bénard systems). The choice of state variables $y$ in the second step is usually made by trying to formulate the simplest possible model in the sense that the chosen state variables are related as close as possible to the quantities observed in the *meso* measurements. The original derivation of the Boussinesq equations constituting the dynamic model of the Rayleigh-Bénard system provides a classical example of the direct derivation. The chosen mesoscopic level is in this example the level of fluid mechanics (the classical hydrodynamic fields serve as state variables $y$). The comparison made in the third step shows indeed agreement between predictions of the model and results of experimental observations. Hereafter, we shall refer to the collection of *meso* measurements and the mathematical model (\[Gdyn\]) as a *meso level* description. We now pick one well established mesoscopic model (e.g. the Boussinesq model). There are immediately two conclusions that we can draw. The first one is that there exist more microscopic levels (i.e. levels involving more details, we shall call them *MESO levels*) on which the physical system under investigation can be described. This is because the chosen *meso level* (e.g. the level of fluid mechanics) ignores many microscopic details that appear to be irrelevant to our interests (determined by meso-measurements and also by intended *meso* applications). We recall that there always exists at least one well established *MESO level* on which states are described by position vectors and velocities of $\sim 10^{23}$ particles composing the macoscopic systems under consideration (provided we remain in the realm of classical physics). Such ultimately microscopic model will be hereafter denoted as *MICRO* model. The second conclusion is that if we choose a *MESO level* and we found it to be well established (i.e. its predictions agree with results of more detailed *MESO* measurements), then we have to be able to see in solutions to its governing equations the following two types of dynamics: (i) reducing dynamics describing approach to the *MESO* dynamics to the *meso* dynamics, and (ii) reduced *MESO* dynamics that is the *meso* dynamics. This is because both the original *meso* model and the more microscopic *MESO* model have been found to be well established. Following further the second conclusion, we see that we have now an alternative way to derive the governing equations of our original *meso* model. In addition to its direct mesoscopic derivation described above in the first paragraph, we can derive it also by constructing first a more microscopic *MESO* model and then recognizing the *meso* model as a pattern in solutions to its governing equations. This new way of deriving the *meso* model seems to be complicated and indeed, it is rarely used. Nevertheless, it is important that this alternative way of derivation exists and that, by following it, we arrive at least at two new results: (a) the material parameters $\zeta$ through which the individual nature of macroscopic systems is expressed in the *meso* model (\[Gdyn\]) appear as functions of the material parameters playing the same role in the more microscopic *MESO* model, and (b) the reducing dynamics, giving rise to thermodynamics (as we show in Section \[RD\]). The above consideration motivates us to start our investigation of externally forced macroscopic systems with two mesoscopic models instead of with only one such model (\[Gdyn\]). The second model (*MESO* model) is formulated on a more microscopic level than the level on which the model (\[Gdyn\]) is formulated. By “a more microscopic model” we mean that more details are taken into account in the model. We write the governing equations of the second model formally as $$\label{Fdyn} \dot{x}=G(x,\varsigma, \mathcal{F}^{MESO})$$ where $x$ denotes state variables, $\varsigma$ material parameters and $\mathcal{F}^{MESO}$ the external influence. The state space used in the *meso* model (\[Gdyn\]) is denoted by the symbol $N$ (i.e. $y\in N$ ) and the state space used on the more microscopic *MESO* model (\[Fdyn\]) is denoted by the symbol $M$ (i.e. $x\in M$). We shall call hereafter the dynamics described by (\[Fdyn\]) as *MESO* dynamics and the dynamics described by (\[Gdyn\]) by *meso* dynamics. How do we formulate the *MESO* model (\[Fdyn\])? In its direct derivation we proceed in the same way as we do in the direct derivation of the [meso]{} model (\[Gdyn\]). The difference is only in that the *meso* measurements are replaced by more detailed *MESO* measurements and that the same type of physics as the one expressed in (\[Gdyn\]) is now expressed in (\[Fdyn\]) in a more detail. As an example of *meso* dynamics (\[Gdyn\]) we can take Boussinesq equations describing, on the level of fluid mechanics (i.e. the *meso level* in this example is the level of fluid mechanics), the Rayleigh-Bénard system. The corresponding to it *MESO level* could be the level of kinetic theory on which the state variable $x$ is the one particle distribution function and Eq.(\[Fdyn\]) is a kinetic equation expressing the same physics as the one expressed in the Boussinesq equations but on the level of kinetic theory. Having both *MESO* and *meso* dynamics, we are in position to provide a new derivation of the *meso* dynamics (\[Gdyn\]) and also to identify reducing $MESO\rightarrow meso$ dynamics that, as we shall see below in Section \[RDMmde\], provides us with a new *meso thermodynamics*. The process leading from *MESO level* to *meso level* is conveniently seen (see Section \[EX2\]) as a pattern recognition in the *MESO* phase portrait. By *MESO* phase portrait we mean a collection of trajectories (i.e. solutions to (\[Fdyn\]) ) passing through all $x\in M$ for a large family of the material parameters $\xi$ and external forces $\mathcal{F}^{MESO}$. The pattern that we search is the one which can be interpreted as representing the mesoscopic phase portrait corresponding to the *meso* dynamics (\[Gdyn\]). We prefer to refer to the process involved in the passage from *MESO* to *meso* dynamics as a pattern recognition process rather than the more frequently used “coarse graining” process since the latter term evokes procedures (as e.g. making pixels and averaging in them) that are manifestly coordinate dependent and thus geometrically (and consequently also physically) meaningless. Reducing Dynamics, Thermodynamics {#RD} ================================== We now proceed to investigate the pattern recognition process leading from *MESO* dynamics to *meso* dynamics. We recognize first its complexity. We recall for instance that this type of investigation constitutes in fact the famous Hilbert’s 6th problem (see [@GKHilb]). Roughly speaking, any investigation of the $MESO\,\rightarrow\,meso $ passage consists essentially in splitting the *MESO* dynamics (\[Fdyn\]) into the *meso* dynamics (\[Gdyn\]) (that we call *reduced dynamics* if we regard it in the context of $MESO\,\rightarrow\,meso$ passage) and another dynamics that makes the reduction (that we call *reducing dynamics*). While most investigations of the $MESO\,\rightarrow\,meso $ passages have focused in the past on the reduced dynamics, we show that investigations of the reducing dynamics are also interesting and bring in fact an additional important information that can be interpreted as an introduction of thermodynamics on the *meso* level. The reduced dynamics (i.e. *meso* dynamics) together with the thermodynamics implied by the reducing dynamics express then (on *meso level*) the complete physics of the macroscopic system under consideration. More details of the behavior of the macroscopic systems under consideration are seen on the *MESO* level (represented by (\[Fdyn\]) ) than on the *meso* level. Let $\mathcal{P}^{MESO}$ and $\mathcal{P}^{meso}$ be the phase portraits corresponding to the *MESO* dynamics (\[Fdyn\]) and the *meso* dynamics (\[Gdyn\]) respectively. Our problem is to recognize $\mathcal{P}^{meso}$ as a pattern inside $\mathcal{P}^{MESO}$. In the pattern recognition process we recover the less detailed viewpoint expressed in (\[Gdyn\]) (that arises in the pattern recognition process as the reduced dynamics) but in addition we also begin to see the reducing dynamics making the pattern to emerge. In this section we argue that the reducing dynamics, is in its essence thermodynamics. In order to be able to justify the use of the term “thermodynamics” we begin by recalling the standard (i.e. Gibbs) formulation of the classical thermodynamics and show subsequently that the reducing dynamics is indeed its natural extension. The level of description used in the classical equilibrium thermodynamics is called in this paper *equilibrium level*. In this section we concentrate on establishing a unified formulation of the reducing dynamics. We show that the formalism puts under a single umbrella the thermodynamics of driven systems and well established classical, microscopic, and mesoscopic equilibrium and nonequilibrium thermodynamics. The unification power of the formalism is in this section the principal argument supporting it. In the following section (Section \[EX\]) we then collect illustrative examples and applications providing additional support. Classical equilibrium thermodynamics; statics {#RDET} --------------------------------------------- The point of departure of the classical equilibrium thermodynamics is the postulate ***equilibrium Postulate 0*** of the *existence of equilibrium states*. For example, Callen formulates [@Callen] it as follows: \[*“... in all systems there is a tendency to evolve toward states in which the properties are determined by intrinsic factors and not by previously applied external influences. Such simple terminal states are, by definition, time independent. They are called equilibrium states...”*\]. The level of description on which investigations are limited only to macroscopic systems at equilibrium states will be called *equilibrium* level. No time evolution takes place on this level. The next postulate addresses the *state variables* used on *equilibrium* level to characterize the equilibrium states introduced in the previous postulate. ***equilibrium Postulate I*** *The state variables on *equilibrium* level are the state variables needed to formulate overall macroscopic mechanics (the number of moles $N$, the volume $V$, and the macroscopic mechanical kinetic energy $E_{mech}$) and in addition the internal energy $E_{int}$ that is a new, extra mechanical quantity, serving as an independent state variable. The internal energy $E_{int}$ then combines with the macroscopic mechanical $E_{mech}$ to define the overall total energy $E=E_{mech}+E_{int}$. We shall denote the state variables of the classical equilibrium thermodynamics by the symbol $\omega$ (i.e. $\omega=(E,N,V)$) an the equilibrium state space $\Omega$ (i.e. $\omega\in \Omega$)*. The third postulate addresses the way the equilibrium states are reached. ***equilibrium Postulate II*** \(i) *The fundamental thermodynamic relation consists of three potentials* $$\label{classftr} N^{(ee)}(\omega); \, E^{(ee)}(\omega);\,S^{(ee)}(\omega)$$ The two potentials, namely the number of moles $N^{(ee)}$ and the energy $N^{(ee)}$ are universal: $N^{(ee)}=N;\,\,E^{(ee)}=E$. The entropy $S^{(ee)}(\omega)$ is not universal. It is the quantity in which, on *equilibrium level*, the individual nature of the macroscopic systems under consideration are expressed. The association between $S^{(ee)}(\omega)$ and the macroscopic systems can be obtained, if we remain inside *equilibrium level*, only by experimental observations (whose results are collected in the so called thermodynamic tables). The entropy $S^{(ee)}(E,V,N)$ is required to satisfy the following three properties. (i) $S^{(ee)}(E,V,N)$ is a real valued and sufficiently regular function of $z$, (ii) $S^{(ee)}(E,V,N)$ is homogeneous of degree one (i.e. $S^{(ee)}(\lambda E,\lambda V,\lambda N)= \lambda S^{(ee)}(E,V,N)$ which means that the energy, number of moles, volume, and entropy are all extensive variables), and (iii) $S^{(ee)}(E,V,N)$ is a concave function (we exclude from our considerations in this paper critical states and phase transitions). \(ii) *Equilibrium states are defined as states at which the entropy $S^{(ee)}(\omega)$ reaches its maximum allowed by constraints (i.e. MaxEnt principle on equilibrium level)*. Since we consider in this paper thermodynamics associated with passages between two general levels, we need a clear notation. The upper index $(ee)$ in potentials introduced in (\[classftr\]) means $equilibrium \rightarrow equilibrium$, i.e. the passage in which the starting level is *equilibrium* level and the target level is also *equilibrium* level. If the passage that we investigate is *MICRO* $\rightarrow$ *equilibrium* (in Section \[RDMee\] below), we shall use $(MIe)$, if the passage is *MESO* $\rightarrow$ *equilibrium* (in Section \[RDMMee\]), we shall use $(Me)$, and in the investigation of the passage *MESO* $\rightarrow$ *meso* (in Section \[RDMmde\]), we shall use $(Mm)$. The first letter in the upper index denotes always the level on which the quantity is defined and the second letter the level to which the reduction aims or the level from which it is reduced (see (\[MIimpl\]), or (\[Meimpl\]) below). In order to write explicitly the MaxEnt principle, we introduce $$\label{Phiclass} \Phi^{(ee)}(\omega;T,\mu)=-S^{(ee)}(\omega)+E^{*}E^{(ee)}(\omega)+N^{*}N^{(ee)}(\omega)$$ called a thermodynamic potential on *equilibrium* level. By $\omega^*=(E^*,N^*,V^*)$ we denote conjugate state variables; $E^{*}$ is conjugate to $E$ (i.e. $E^*=S^{(ee)}_E$), $N^*$ is conjugate to $N$ (i.e. $N^*=S^{(ee)}_N$), and $V^*$ is conjugate to $V$ (i.e.$V^*=S^{(ee)}_V$). We use hereafter the shorthand notation $S_E=\frac{\partial S}{\partial E}$,... . In the classical equilibrium thermodynamics the conjugate variables have particular names, namely, $E^*=\frac{1}{T}, N^*=S_N=-\frac{\mu}{T}, V^*=-\frac{P}{T}$, where $T$ is the temperature, $\mu$ the chemical potential, and $P$ the pressure. Entropy $S^{(ee)}(E,V,N)$ transforms, under the Legendre transformation, into its conjugate $S^{(ee)*}(\mu,T)$, $$\label{eqimp} S^{(ee)*}(\mu,T)=[\Phi^{(ee)}(\omega;T,\mu)]_{\omega=\omega_{eq}(T,\mu)}$$ where $\omega_{eq}(T,\mu)$ is a solution of $\Phi^{(ee)}_{\omega}=0$. As a direct consequence of the homogeneity of $S^{(ee)}$, $S^{(ee)*}(T,\mu) =-\frac{P}{VT}$. We note that the MaxEnt principle in the classical equilibrium thermodynamics does not address the time evolution leading to the equilibrium states (i.e. it does not address the process of preparing macroscopic systems to equilibrium thermodynamic observations). It addresses only the question of what is the final result of such time evolution. We shall introduce such time evolution later in this paper. MICRO $\rightarrow$ equilibrium; Gibbs equilibrium statistical mechanics; statics {#RDMee} ---------------------------------------------------------------------------------- Another part of the classical equilibrium theory is the Gibbs equilibrium statistical mechanics that investigates the passage $MICRO \rightarrow equilibrium$. We shall formulate the physical basis of the Gibbs theory again in three postulates that are direct adaptations of the three postulates in Section \[RDET\] to *MICRO level*. The first postulate, Postulate 0, is the same as in the classical equilibrium thermodynamics except that we include in it the statement that *MICRO level* is also well established. The second postulate addresses the state variables ***MICRO$\rightarrow$ equilibrium Postulate I***. *State variables on MICRO level are position vectors ${{\boldmath \mbox{$r$}}}=({{\boldmath \mbox{$r$}}}_1,...,{{\boldmath \mbox{$r$}}}_N)$ and momenta ${{\boldmath \mbox{$v$}}}=({{\boldmath \mbox{$v$}}}_1,...,{{\boldmath \mbox{$v$}}}_N)$ of $N$ particles, $N\sim 10^{23}$, (or alternatively the $N$-particle distribution function $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$)*. Next, we proceed to the third postulate that addresses the time evolution. Since the reduced time evolution in the passage *MICRO* $\rightarrow$ *equilibrium* is no time evolution, the time evolution taking place on *MICRO level* is the reducing time evolution. The *MICRO level* time evolution $({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_0\mapsto({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_t$ is governed by Hamilton’s equations $\left(\begin{array}{cc}\dot{{{\boldmath \mbox{$r$}}}}\\ \dot{{{\boldmath \mbox{$v$}}}}\end{array}\right)=\left(\begin{array}{cc}0&1\\-1&0\end{array}\right)\left(\begin{array}{cc}E^{(MICRO)}_{{{\boldmath \mbox{$r$}}}}\\E^{(MICRO)}_{{{\boldmath \mbox{$v$}}}}\end{array}\right)$, where $E^{(MICRO)}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ is the microscopic energy. This microscopic time evolution induces the time evolution $f_0({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\mapsto f_t({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})=f_0(({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_{-t})$. In the Gibbs equilibrium statistical mechanics only two aspect of the *MICRO* time evolution are retained: (1) conservations of the total mass $N^{(MIe)}(f)$ and the total energy $E^{(MIe)}(f)$ defined below in (\[microftr\]), and (2) an assumption about the *MICRO* trajectories, namely an ergodic-type hypothesis. The second postulate is thus the following. ***MICRO $\rightarrow$ equilibrium Postulate II*** \(i) *The fundamental thermodynamic relation consists of three potentials $$\begin{aligned} \label{microftr} N^{(MIe)}(f)&=&\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\nonumber \\ E^{(MIe)}(f)&=&\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}E^{(MICRO)}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}}) f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\nonumber \\ S^{(MIe)}(f)&=&-k_B\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\ln f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\end{aligned}$$ where $k_B$ is the Boltzmann constant, $N^{(MIe)}(f)$ has the physical interpretation of number of moles, $E^{(MIe)}(f)$ is the energy.* The map leading from the state space of the Liouville representation of classical mechanics to the state space of the classical equilibrium thermodynamics will be denoted by the symbol $\mathfrak{P}^{(MIe)}$, i.e. $$\label{PMIe} f\mapsto \mathfrak{P}^{(MIe)}(f)=(N^{(MIe)}(f),E^{(MIe)}(f))$$ \(ii) *$N^{(MIe)}(f)$ and $E^{(MIe)}(f)$ introduced in the fundamental thermodynamic relation (\[microftr\]) are conserved during the time evolution*. \(iii) *Particle trajectories $({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_0\mapsto({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_t$ fill up the microscopic phase space $M^{(MICRO)}$ (i.e. $({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\in M^{(MICRO)}$) so that time averages can be replaced with averages (by using certain measures) in $M^{(MICRO)}$ (*the ergodic hypothesis*)*. (iv)*Equilibrium states are defined as states at which $S^{(MIe)}(f)$ reaches its maximum allowed by constraints (i.e. MaxEnt principle for the MICRO $\rightarrow$ equilibrium passage). The expression (\[microftr\]) for $S^{(MIe)}(f)$ is in the Gibbs theory universally valid for all macroscopic systems. The quantity that on MICRO level expresses the individual nature of the macroscopic systems under consideration is only the energy $E(f)$*. In order to write explicitly the MaxEnt principle on the *MICRO* level, we introduce, as we did in the previous section, the thermodynamic potential $$\label{Phi} \Phi^{(MIe)}(f;T,\mu)=-S^{(MIe)}(f)+\frac{1}{T}E^{(MIe)}(f)-\frac{\mu}{T}N^{(MIe)}(f)$$ The fundamental thermodynamic relation on *equilibrium level* implied by the fundamental thermodynamic relation (\[microftr\]) on *MICRO level* is given by $$\begin{aligned} \label{MIimpl} N^{(eMI)}(\omega)&=&[N^{(MIe)}(f)]_{f=f_{eq}}\nonumber \\ E^{(eMI)}(\omega)&=&[E^{(MIe)}(f)]_{f=f_{eq}}\nonumber \\ S^{(eMI)*}(\mu,T)&=&[\Phi^{(MIe)}(f,T.\mu)]_{f=f_{eq}}=-\frac{P}{VT}\end{aligned}$$ where $f_{eq}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}};T,\mu)$, solutions of $\Phi^{(MIe)}_{f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})}=0$, are equilibrium states. They form a manifold $\mathcal{M}_{eq}\subset M$ (i.e. $f_{eq}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}};T,\mu)\in \mathcal{M}_{eq}$) that is an invariant manifold with respect to the *MICRO* time evolution. There is no time evolution that takes place on $\mathcal{M}_{eq}$. The upper index $(eMI)$ means that the quantity belongs to *equilibrium level* and is obtained from an analysis taking place on *MESO level*. This notation was already introduced in the text following Eq.(\[classftr\]). We note that the MaxEnt principle on *MICRO level* (i.e. $MICRO \rightarrow equilibrium$ Postulate II), as well as the equilibrium Postulate II in the classical equilibrium thermodynamics (see Section \[RDET\]), does not really address the time evolution leading to equilibrium sates. The *MICRO* time evolution $f_0({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\mapsto f_t({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})=f_0(({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})_{-t})$ introduced in the $MICRO\rightarrow equilibrium$ postulate above leaves the Gibbs entropy introduced in (\[microftr\]) unchaged (see more in Section \[EX2\]). As on *equilibrium level*, the $MICRO \rightarrow equilibrium$ Postulate II addresses only the final result of such evolution. In Section \[EX2\] we shall address the $MICRO\rightarrow equilibrium$ reducing time evolution. We shall write down explicitly the equations governing it. The Gibbs $MICRO\rightarrow equilibrium$ theory enriches the classical equilibrium thermodynamics in particular in the following two points: (i) it brings a microscopic insight into the meaning of the internal energy, and (ii) it offers a way to calculate the fundamental thermodynamic relation from the knowledge of microscopic interactions. Regarding the first point, we note that in the context of *MICRO level* the internal energy is the energy of the particles modulo the overall mechanical energy. The mechanical origin of the internal energy implies then the mechanical nature of the heat and consequently the energy conservation law involved in $equilibrium$ Postulate II. As for the fundamental thermodynamic relation, the Gibbs equilibrium statistical mechanics (specifically the MaxEnt Principle in Postulate II of the Gibbs theory) provides a mapping between the fundamental thermodynamic relation (\[microftr\]) on *MICRO level* (note that it is the particle energy $E^{(MICRO)}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ through which the individual nature of macroscopic systems is expressed on *MICRO level*) and the equilibrium fundamental thermodynamic relation (\[classftr\]) (note that the quantity through which the individual nature of macroscopic systems is expressed on *equilibrium* level is the entropy $S^{(ee)}(N,V,E))$. Finally, we note that the Gibbs equilibrium theory is not supported by a rigorous analysis of the *MICRO* mechanics. Both the ergodic-like behavior of particle trajectories and the tendency of $S^{(MIe)}(f)=-k_B\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\ln f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ to reach its maximum (allowed by constraints) during the microscopic time evolution remain, for most macroscopic systems, an unproven assumption. The support for the Gibbs theory comes from plausible assumptions, illustrations, and the success of its applications. The same will be then true for the general formulation of reducing dynamics presented below. MESO $\rightarrow$ equilibrium; statics {#RDMMee} ---------------------------------------- So far, we have considered only the ultimate microscopic level (called *MICRO level*) and the ultimate macroscopic level (called *equilibrium level*). Now we take into consideration also mesoscopic levels and formulate a general thermodynamics associated with the passage *MESO* $\rightarrow$ *meso* (that we call hereafter simply thermodynamics). We begin with Postulate 0. We modify it by noting that *equilibrium level* is not the only well established level that is less microscopic than the *MICRO level*. There is in fact a whole family of such levels (for example fluid mechanics and kinetic theory levels). These well established mesoscopic levels differ from the equilibrium level by the fact that, in general, a time evolution takes place on them (we recall that no time evolution takes place on the equilibrium level) and also by the fact that they are applicable also to macroscopic systems subjected to external influences (e.g. the level of fluid mechanics is applicable to the Rayleigh-Bénard system). We thus replace the postulate of the existence of equilibrium states with a more general ***MESO Postulate 0*** *There exist well established mesoscopic levels* . The remaining two postulates are the same as in the Gibbs theory except that the state variable $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ used on the *MICRO level* is replaced by another state variables used on mesoscopic levels. As we have done already in Eq.(\[Fdyn\]), we shall denote it on $MESO$ level by the symbol $x$. For example, $x$ can be one particle distribution function (used in kinetic theory) of hydrodynamic fields (used in fluid mechanics): ***MESO $\rightarrow$ equilibrium Postulate I***. *State variables on MESO level are quantities denoted by the symbol $x$* We recall the $\textit{MESO level}$ is a well established level (i.e. theoretical predictions on *MESO level* agree with results of *MESO* experimental observations). This then means that $x$ is known. ***MESO $\rightarrow$ equilibrium Postulate II (statics)***. \(i) *The fundamental thermodynamic relation consists of a specification of three potential $$\label{MESOftr} N^{(Me)}(x),E^{(Me)}(x),S^{(Me)}(x)$$ denoting the number of moles, energy, and entropy respectively.* The map leading from *MESO* state space $M$ to the state space of the classical equilibrium thermodynamics will be denoted by the symbol $\mathfrak{P}^{(Me)}$, i.e. $$\label{PMe} x\mapsto\mathfrak{P}^{(Me)}(x)=(N^{(Me)}(x),E^{(Me)}(x))$$ (compare with (\[PMIe\])). \(ii) *Equilibrium states are defined as states at which the entropy $S^{(Me)}(x)$ reaches its maximum allowed by constraints (i.e. MaxEnt principle for the MESO $\rightarrow$ equilibrium passage)*. As in previous sections, we introduce thermodynamic potential $$\label{Phi1} \Phi^{(Me)}(x;T,\mu)=-S^{(Me)}(x)+\frac{1}{T}E^{(Me)}(x)-\frac{\mu}{T}N^{(Me)}(x)$$ Equilibrium state $x_{eq}$ are states at which $\Phi^{(Me)}(x;T,\mu)$ reaches its minimum. Consequently, $x_{eq}$ are solutions to $$\label{Phieq} \Phi^{(Me)}_x(x,T,\mu)=0,$$ Such states, called equilibrium states, form equilibrium a manifold denoted by the symbol $\mathcal{M}_{eq}\subset M$. The fundamental thermodynamic relation on *equilibrium level* implied by the fundamental thermodynamic relation (\[MESOftr\]) on *MESO level* is given by $$\begin{aligned} \label{Meimpl} N^{(eM)}(\omega)&=&[N^{(Me)}(x)]_{x=x_{eq}}\nonumber \\ E^{(eM)}(\omega)&=&[E^{(Me)}(x)]_{x=x_{eq}}\nonumber \\ S^{(eM)*}(\mu,T)&=&[\Phi^{(Me)}(x,T.\mu)]_{x=x_{eq}}=-\frac{P}{VT}\end{aligned}$$ where $x_{eq}({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}};T,\mu)$, equilibrium states, are solutions of (\[Phieq\]). The upper index $(eM)$ means that the quantity belongs to *equilibrium level* and is obtained from an analysis taking place on *MESO level*. This notation was already introduced in the text following Eq.(\[classftr\]). Summing up, the difference between the Gibbs equilibrium statistical mechanics and the mesoscopic equilibrium theory formulated above is in the Postulate 0, in the fundamental thermodynamic relation, and in the arguments supporting the theory. Postulate 0 includes now also existence of mesoscopic levels. Regarding the fundamental thermodynamic relation, all three potentials $N^{(Me)}(x),E^{(Me)}(x)$, and $S^{(Me)}(x)$ have to be specified. The same three potentials have to be also specified in the Gibbs theory (see (\[microftr\])) but two of them, namely $N^{(MIe)}$ and $S^{(MIe)}$, are universal. On *MESO level*, neither of them is universally applicable. For example, let $x$ be one particle distribution function. The fundamental thermodynamic relation (\[microftr\]), but now transposed to the level of kinetic theory (i.e. the N-particle distribution function is replaced by one particle distribution function), leads to the fundamental thermodynamic relation representing ideal gas on *equilibrium* level (recall that if $f$ in (\[microftr\]) is replaced by one particle distribution function then the only energy is the kinetic energy); in order to include more complex macroscopic systems, e.g. van der Waals gas, one has to modify both the energy - by introducing a mean field energy - and entropy - see more in [@Gr71] where also the corresponding reducing dynamics is specified). As for the supporting arguments, they now mainly come from relating the *MESO* equilibrium theory to the Gibbs theory. *MESO* equilibrium theories are indeed an organic part of the Gibbs equilibrium statistical mechanics. They arise as its simplified versions applicable to particular families of macroscopic systems. MESO $\rightarrow$ equilibrium; reducing dynamics {#RDMMde} -------------------------------------------------- An important advantage of investigating the passage $MESO \rightarrow equilibrium $ instead of the passage $MICRO \rightarrow equilibrium $ is that we can more easily investigate reducing dynamics. We have seen in Section \[RDMee\] that in order to pass from $MICRO$ dynamics to *equilibrium level*, we need assumptions (that, at least in general, remain unproven) about ergodic-type behavior of microscopic trajectories. On the other hand, mesoscopic-type experimental observations include also direct observations of the approach to equilibrium. Based on results of such observations, mathematical formulations of particular examples of the reducing dynamics $MESO \rightarrow equilibrium $ have been developed (for example the Navier-Stokes-Fourier equations of fluid mechanics and the Boltzmann kinetic equation of gas dynamics). The Boltzmann kinetic equation was then the first time evolution equation for which the passage $kinetic\,\,theory \rightarrow equilibrium$ was explicitly investigated (by Ludwig Boltzmann). In investigations of the reducing dynamics representing the passage $MESO \rightarrow equilibrium $ we can therefore use result obtained independently in several particular examples of well-established mesoscopic dynamical theories. The abstract formulation of reducing dynamics $MESO \rightarrow equilibrium$ presented below has emerged as a common mathematical structure of such well established theories. The first step was made by Clebsch [@Clebsch], who realized that the particle dynamics and the Euler fluid mechanics share the structure of Hamiltonian dynamics. The investigation initiated by Clebsch then continued in particular in the works of Arnold [@Arnold], Marsden and Weinstein [@MW]. Independently, Landau and Ginzburg [@LG] and Cahn and Hilliard [@CH] have recognized a common structure of gradient dynamics in the part of the time evolution that is represented in (\[GENERIC\]) by the second term on its right-hand side. Time evolution equations involving both the Hamiltonian and the gradient part had appeared first in [@DV], in [@Grmboulder] (that was presented at the AMS-IMS-SIAM Joint Summer Research Conference in the Mathematical Sciences on Fluids and Plasmas:Geometry and Dynamics, held at the University of Colorado, Boulder, CO, USA, 17–23 July 1983) and in [@Morr], [@Kauf], [@GrPhysD], [@BEd]. In [@GO], [@OG] the abstract equation (\[GENERIC\]) has been called GENERIC. Its formulation in the context of the contact geometry is presented in [@Grmcontact] Specific realizations of (\[GENERIC\]) on many examples of *MESO* levels can be found in [@Obook], [@Grmadv]. Now we proceed to the formulation. Postulate 0 and Postulate I remain the same as in Section \[RDMMee\]. In Postulate II we replace the static MaxEnt principle with dynamics MaxEnt principle. We explicitly specify the dynamics making the maximization of entropy. The third postulate *MESO $\rightarrow$ equilibrium Postulate II (statics)* in Section \[RDMMee\] is now replaced with dynamic postulate ***MESO $\rightarrow$ equilibrium Postulate II (dynamics)*** \(i) *remains the same as in MESO $\rightarrow$ equilibrium Postulate II (statics)* \(ii) *The time evolution making the passage MESO $\rightarrow$ equilibrium is governed by the GENERIC equation* $$\label{GENERIC} \dot{x}=[TL(x)x^*-[\Xi_{x^*}(x,X^{(CR)}(x^*))]_{x^*=\Phi^{(Me)}_x}$$ In the rest of this section we explain the meaning of the symbols appearing in (\[GENERIC\]) and prove that the time evolution governed by (\[GENERIC\]) brings $x$ indeed to the equilibrium states $x_{eq}\in\mathcal{M}_{eq}$ that are solutions of $\Phi^{(Me)}_x=0$. The first term on the right hand side of (\[GENERIC\]) represents the part of the time evolution of $x$ that is on *MESO level* directly inherited from *MICRO level*. It generates the Hamiltonian time evolution. The operator $L$, transforming the covector $x^*$ into a vector, is a Poisson bivector. This means that $\{A,B\}=<A_x,LB_x>$ is a Poisson bracket, i.e. $\{A,B\}=-\{B,A\}$ and the Jacobi identity $\{A,\{B,C\}\}+\{B,\{C,A\}\}+\{C,\{A,B\}\}=0$ holds. By the symbols $A,B,C$ we denote sufficiently regular real valued functions of $x$, $<,>$ denotes pairing in $M$. We recall that on *MICRO level* the Hamiltonian time evolution is generated by energy $E(x)$. We recall that in the particular case if $x=(r,v)^T$, where $r$ is the particle position vector, $v$ particle momentum, and $()^T$ means transpose of $()$, then $L=\left(\begin{array}{cc}0&1\\-1&0\end{array}\right)$, and the equation governing the time evolution of $(r,v)$ is $\left(\begin{array}{cc}\dot{r}\\ \dot{v}\end{array}\right)=\left(\begin{array}{cc}0&1\\-1&0\end{array}\right) \left(\begin{array}{cc}E_r\\ E_v\end{array}\right)$. In order to keep the energy as a sole generator of dynamics also on *MESO level*, we require that the operator $L^{(Me)}$ is degenerate in the sense that $\{A,S^{(Me)}\}=\{A,N^{(Me)}\}=0$ for all $A$. By using the terminology established in investigations of Hamiltonian systems, the potentials $S^{(Me)}$ and $N^{(Me)}$ are required to be Casimirs of the Poisson bracket $\{A,B\}$. If this is the case then indeed, the first term on the right hand side of (\[GENERIC\]) is $LE^{(Me)}_x$. A direct consequence of the antisymmetry and the degeneracy of $L$ is that this Hamiltonian part of the time evolution leaves the energy and the generating potential $\Phi^{(Me)}$ (see (\[Phi1\])) unchanged (i.e. $(\dot{\Phi}^{(Me)})_{Hamilton}=<\Phi^{(Me)}_x,L\Phi^{(Me)}_x>=<E^{(Me)}_x,LE^{(Me)}_x>=0$). Examples of the operator $L$ in many *MESO levels* can be found in [@Grmadv]. Before leaving the Hamiltonian part of the time evolution we make a comment about the role that the Jacobi identity plays in it. If the second term on the right hand side of (\[GENERIC\]) is absent then the equation $\dot{x}=LE^{(Me)}_x$ governing the time evolution can also be written in the form $\dot{A}=\{A,E^{(Me)}\}$ holds for all $A$. If we now replace $A$ with $\{A,B\}$ we obtain $\dot{\{A,B\}}=\{\{A,B\},E^{(Me)}\}$. But $\dot{\{A,B\}}= \{\dot{A},B\}+\{A,\dot{B}\}=\{\{A,E^{(Me)}\},B\}+\{A,\{B,E^{(Me)}\}\}$. The Jacobi identity guarantees that these two time derivatives of $\{A,B\}$ are equal and thus the Poisson bracket remains unchanged during the time evolution. If we now consider the time evolution governing by (\[GENERIC\]) involving also the second term on the right hand side, the Poisson bracket is not preserved during the time evolution even if the Jacobi identity holds. The role of the Jacobi identity in GENERIC time evolution is thus much less important that in the Hamiltonian time evolution. Hereafter, we shall call a time evolution GENERIC time evolution even if the Jacobi identity remains unproven. The second part on the right hand side of (\[GENERIC\]) is the part that arises due to the fact that *MESO level* is not *MICRO level*. This means that some microscopic details that are seen on *MICRO level* are ignored on *MESO level*. This ignorance then influences the time evolution in such a way that the potential $\Phi^{(Me)}$ approaches its minimum. By $\Xi(x,X)$, called a dissipation potential, we denote a sufficiently regular and real valued function of $x\in M$ and of $X$ that is called a thermodynamic force. Its specification $X=X^{(CR)}(x^*)$ as a function of $x^*$ is called a *constitutive relation*. The superscript “CR” means Constitutive Relation. We assume that the dissipation potential satisfies the following properties: $$\begin{aligned} \label{Xi} &&\Xi(x,0)=0\nonumber \\ && \Xi\,\,reaches\,\,its\,\,minimum\,\,at\,\,X=0\nonumber\\ &&\Xi\,\,is\,\,a\,\,convex\,\,function\,\,in\,\,a\,\,neighborhood\,\,of\,\,X=0\end{aligned}$$ Regarding the constitutive relations, we assume that $$\label{crelprop} <x^*,\Xi_{x^*}(x,X^{(CR)}(x^*))>=\alpha <X^{(CR)},\Xi_{X^{(CR)}}(x,X^{(CR)})>$$ where $\alpha >0$ is a parameter. In addition, we require that the dissipation potential $\Xi$ is degenerate in the following sense: $$\begin{aligned} \label{degXi} <[x^*]_{x^*=E^{(Me)}_x},\Xi_{x^*}>=<[x^*]_{x^*=N^{(Me)}_x},\Xi_{x^*}>&=&0\nonumber \\ <x^*,[\Xi_{x^*}]_{x^*=E^{(Me)}_x}>=<x^*,[\Xi_{x^*}]_{x^*=N^{(Me)}_x}>&=&0\end{aligned}$$ The simplest example of the dissipation potential $\Xi$ satisfying (\[Xi\]) is the quadratic potential $\Xi=<X\Lambda X>$, where $\Lambda $ is a matrix with required degeneracy and positive definite if applied on vectors outside its nullspace. More general potentials arise in particular in chemical kinetics (see [@Grmchem]). It has been suggested in [@Beretta] to regard $\Lambda$ as a metric tensor. This interpretation brings then Riemannian geometry to dissipative dynamics. We emphasize that this geometrical viewpoint is limited to the quadratic dissipation potential. In the case of nonlinear dissipation potentials (for example those arising in chemical kinetics - see also Section \[EX2\]), the geometrical interpretation is still possible but the classical Riemannian geometry has to be replaced by a more general geometry. As an example of the constitutive relation satisfying (\[crelprop\]) we mention the Fourier constitutive relation in the investigation of heat transfer. In this example $x^*=\frac{1}{T({{\boldmath \mbox{$r$}}})}$, where $T({{\boldmath \mbox{$r$}}})$ is the local temperature and the constitutive relation is $X^{(CR)}(x^*)=\nabla \left(\frac{1}{T({{\boldmath \mbox{$r$}}})}\right)$. Direct calculations lead to $<x^*,\Xi_{x^*}>\\=<\frac{1}{T({{\boldmath \mbox{$r$}}})},\Xi_{\frac{1}{T({{\boldmath \mbox{$r$}}})}}>=\int d{{\boldmath \mbox{$r$}}}\frac{1}{T({{\boldmath \mbox{$r$}}})},\Xi_{\frac{1}{T({{\boldmath \mbox{$r$}}})}} =-\int d{{\boldmath \mbox{$r$}}}\frac{1}{T({{\boldmath \mbox{$r$}}})},\nabla\Xi_{\nabla\left(\frac{1}{T({{\boldmath \mbox{$r$}}})}\right)}\\= \int d{{\boldmath \mbox{$r$}}}\nabla\left(\frac{1}{T({{\boldmath \mbox{$r$}}})}\right),\Xi_{\nabla\left(\frac{1}{T({{\boldmath \mbox{$r$}}})}\right)}=<X^{(CR)},\Xi_{X^{(CR)}}>$ provided the boundary conditions guarantee that the integrals over the boundary that arise in by parts integrations (leading to the last equality) equal zero. We see that in this example $\alpha=1$. In the context of chemical kinetics, where the thermodynamic forces $X$ are chemical affinities, the parameter $\alpha\neq 1$ (see Section \[EX1\] and [@Grmchem]; for example, for the dissipation potential (\[XiN\]) the coefficient $\alpha=\frac{1}{2}$ and for the dissipation potential $\Xi$ appearing in the Boltzmann equation (\[intlin\]) the coefficient $\alpha=\frac{1}{4}$). Dissipation potentials and constitutive relations will play an important role also in the investigation of the passage *MESO* $\rightarrow$ *meso* in Section \[RDMmde\] below. It follows directly from (\[GENERIC\]) and from the properties of $L$, $\Xi$, and $X^{CR}$ listed above that $$\label{asGEN} \dot{\Phi}^{(Me)}=-<x^*,\Xi_{x^*}(x,X^{CR})>=-\alpha <X^{CR},\Xi_{X^{CR}}(x,X^{CR})>\leq 0$$ The first equality is the required property (\[crelprop\]) of constitutive relations and the last inequality is a direct consequence of the properties (\[Xi\]). The inequality (\[asGEN\]) allows us to see the thermodynamic potential $\Phi^{(Me)}$ as a Lyapunov function associated with the approach of solutions of (\[GENERIC\]) to $x_{eq}$ given by (\[Phieq\]). We have thus proven that Eq.(\[GENERIC\]) indeed makes the passage *MESO* $\rightarrow$ *equilibrium*. If, in addition, we assume that $L$ and $\Xi$ are degenerate in the sense that $\{S,A\}=0$ for all $A$ (i.e., if we use the terminology of Hamiltonian dynamics, the entropy $S^{(Me)}$ is Casimir of the Poisson bracket $\{A,B\}$), and $<E_x,\Xi_{S^{(Me)}_x}>=0$, $<N^{(Me)}_x,\Xi_{S^{(Me)}_x}>=0$ and $<x^*,\Xi_{E^{(Me)}_x}>=0$, $<x^*,\Xi_{N^{(Me)}_x}>=0 \,\,\forall x^*$ then, in addition to the inequality (\[asGEN\]), also the following equalities (conservation laws) $$\begin{aligned} \label{consEGEN} \dot{E}^{(Me)}(x)&=&0\\\label{consNGEN} \dot{N}^{(Me)}(x)&=&0\end{aligned}$$ hold. MESO $\rightarrow$ meso; reducing dynamics {#RDMmde} ------------------------------------------- In this section we come to the main subject of this paper. We consider externally driven macroscopic systems whose time evolution is governed on *MESO level* by (\[Fdyn\]). External forces prevent approach to equilibrium which means that *equilibrium level* is inaccessible and the approach $MESO \rightarrow equilibrium$ does not exist. Let however the behavior of the externally driven macroscopic systems under consideration be found to be described well also on $meso$ level that is more macroscopic (i.e. it takes into account less details) than *MESO level*. This then means that by investigating solutions of the governing equations on *MESO level* we have to be able to recover the governing equations on *meso level*. In addition, such investigation will also reveal reducing dynamics making the passage $MESO \rightarrow meso$. In Section \[RDMMde\] we have shown how thermodynamics on *equilibrium level* arises from the reducing dynamics $MESO \rightarrow equilibrium$ or $meso \rightarrow equilibrium$. In this section we show how thermodynamics on *meso level* (we shall call it Constitutive Relation *meso* thermodynamics or in short form ***CR meso-thermodynamics*** to distinguish it from $equilibrium\,\, thermodynamics$ discussed above in Sections \[RDET\] - \[RDMMde\]) arises from the reducing dynamics $MESO \rightarrow meso$. We recall that the formulation of thermodynamics presented in Section \[RDMMde\] (i.e. the formulation of thermodynamics implied by the reducing dynamics $MESO \rightarrow equilibrium$) has been supported mainly by the unification that it brings to various versions of mesoscopic thermodynamics that have emerged in the last one hundred fifty years in well studied and essentially independently developed (each on the basis if its own experimental evidence) mesoscopic dynamical theories. We do not find such examples in the context of the passage $MESO \rightarrow meso$. We do find however important results in nonequilibrium thermodynamics, like for instance dissipation thermodynamics (see references in [@GPK]) and extended thermodynamics (see e.g. [@Joubook], [@MullRugg]). We expect them to become, in some form, a part of the general formulation of *CR meso-thermodynamics*. Our goal is thus to formulate *CR meso-thermodynamic* in such a way that equilibrium thermodynamics, dissipation thermodynamics, and extended thermodynamics make appearance as its different aspects. ### Motivating example {#motex} Before formulating the three postulates of $MESO\rightarrow meso$ thermodynamics, we work out a particular example. First, we present the physical idea and then we formulate it mathematically . We begin with a given *meso level* represented by (\[Gdyn\]). For example, we can think of Eq.(\[Gdyn\]) as standing for the Navier-Stokes-Fourier set of equations. The corresponding to it more detailed *MESO level* represented by Eq.(\[Fdyn\]) will be constructed as an extension of (\[Gdyn\]). Following [@MullRugg], [@Joubook], the extension from *meso* to *MESO* is made, roughly speaking, by replacing the second term on the right hand side of (\[GENERIC\]) (i.e. the dissipative term) with a new state variable (denoted by the symbol $J$ and interpreted physically as a flux corresponding to the state variable $y$). The time evolution of $J$ is then governed by a newly introduced equation that involves dissipative term and is coupled to the time evolution of $y$. We require that $J$ dissipates rapidly to a quasi-stationary state $z_{qeq}(y)$ at which it becomes completely enslaved to $y$. At such quasi-stationary state, the newly constructed *MESO* dynamics reduces to the original *meso* dynamics represented by (\[Gdyn\]). We shall make now an additional requirement. Having realized that all equations governing the reducing time evolution $MESO \rightarrow equilibrium$ possess the structure (\[GENERIC\]), we require that in the absence of the dissipative term the time evolution of $(y,J)$ is Hamiltonian. In this example we restrict ourselves to *meso* dynamics (\[Gdyn\]) that is GENERIC (\[GENERIC\]) and without the Hamiltonian part. Moreover, we consider only isothermal systems (see also Section \[EX3\]) and, for the sake of simplicity, we omit the potential $N$ representing the number of moles. The time evolution equation (\[Gdyn\]) takes thus the form $$\label{I0} \dot{y}=-[\Xi^{(me)}_{y^*}(y,y^*)]_{y^*=\Phi^{(me)}_y}$$ where $\Phi^{(me)}(y,T)$ respectively $\Xi^{(me)}(y,y^*)$ is the thermodynamic potential respectively the dissipation potential associated with *meso* $\rightarrow$ *equilibrium* passage. The temperature $T$ is a constant. We investigate first the passage *meso* $\rightarrow$ *equilibrium*. We see immediately that (\[I0\]) implies $\dot{\Phi}^{(me)}=-\left[y^*\Xi^{(me)}_{y^*}\right]_{y^*=\Phi^{(me)}_y}\leq 0$ provided $\Xi^{(me)}$ satisfies the properties (\[Xi\]). This thermodynamic potential then implies the fundamental thermodynamic relation on *equilibrium level* $$\label{I20} \Phi^{(em)*}(T)= [\Phi^{(me)}(y,T)]_{y=y_{eq}(T)}$$ where $y_{eq}(T)$ is a solution of $\Phi^{(me)}_y=0$. By the upper index $(em)$ in $\Phi^{(em)*}(T)$ appearing in (\[I20\]) we denote (see the paragraph following Eq.(\[classftr\])) that this quantity belongs to *equilibrium level* and is obtained by MaxEnt reduction from *meso level*. So far, we have looked from *meso level* to *equilibrium level*. Now we look in the opposite direction towards *MESO level* involving more details. We extend the *meso* dynamics (\[I0\]) to *MESO* dynamics by following the physical considerations sketched in the beginning of this section. The state variables $x$ on *MESO level* become $x=(y,J)$, where $J$ is a newly adopted state variable having the physical interpretation of a “flux” of $y$. Equation (\[Fdyn\]) is proposed to have the form $$\begin{aligned} \label{I3} \dot{y}&=& \Gamma [J^*]_{J^*=\Phi^{(Me)}_J}\nonumber \\ \dot{J}&=& -\Gamma^T [y^*]_{y^*=\Phi^{(Me)}_y} - [\Theta^{(Me)}_{J^*}(y,J^*)]_{y^*=\Phi^{(Me)}_y;J^*=\Phi^{(Me)}_J}\end{aligned}$$ where $\Gamma$ is an operator, $\Gamma^T$ is its transpose, $\Phi^{(Me)}(y,J)$ is the thermodynamic potential associated with the *MESO* $\rightarrow$ *equilibrium* passage. The dissipation potential $\Theta^{(Me)}(y,J^*)$ is the Legendre transformation of the dissipation potential $\Xi^{(Me)}(y,X^*)$ where $X^*=\Theta^{(Me)}_{J^*}$ (i.e. $\Theta^{(Me)}(y,J^*) =[-\Xi^{(Me)}(y,X^*) +X^*J^*]_{X^*=X_0^*(y,J^*)}$, where $X_0^*(y,J^*)$ is a solution of $[-\Xi^{(Me)}(y,X^*) +X^*J^*]_{X^*}=0$). If $\Xi^{(Me)}(y,X^*)$ satisfies the properties (\[Xi\]) then also $\Theta^{(Me)}(y,J^*)$ satisfies them and vice versa. The time evolution equation (\[I3\]) is again GENERIC (\[GENERIC\]) but contrary to (\[I0\]) it has now also the Hamiltonian part\ $\left(\begin{array}{cc}0&\Gamma\\-\Gamma^T&0\end{array}\right)\left(\begin{array}{cc}y^*\\J^*\end{array}\right)$. The operator $\left(\begin{array}{cc}0&\Gamma\\-\Gamma^T&0\end{array}\right)$ is skew symmetric for any operator $\Gamma$ but the corresponding to it bracket does not necessarily satisfy the Jacobi identity for any $\Gamma$. In view of the remark that we made in Section \[RDMMde\] about the role of the Jacobi identity in GENERIC, we still consider (\[I3\]) as being GENERIC. At this point we note that the extension that we made above differs from extensions made in [@MullRugg], [@Joubook] by requiring that the nondissipative part of the extended equation is Hamiltonian. As a consequence, the flux appearing on the right hand side of the first equation in (\[I3\]) is conjugate to the flux appearing on the left hand side of the second equation of (\[I3\]). The fact that this feature of the extension is not seen in Refs.[@MullRugg] and [@Joubook] is that the master structure for extensions is in Refs.[@MullRugg] and [@Joubook] the classical Grad-hierarchy reformulation of the Boltzmann equation that addresses only a very special physical system (namely the ideal gas) and thus, in terms of our formulation, only a very special class of functions $\Phi^{(Me)}(y,J)$ and $\Phi^{(me)}(y)$. First, we again establish the passage *MESO* $\rightarrow$ *equilibrium*. It directly follows from (\[I3\]) that $$\label{eprodM} \dot{\Phi}^{(Me)}=-\left[J^*\Theta^{(Me)}_{J^*}\right]_{J^*=\Phi^{(Me)}_J}\leq 0$$ provided $\Theta^{(Me)}$ satisfies the properties (\[Xi\]). In the same way as on *meso* level we arrive at the fundamental thermodynamic relation on *equilibrium* level $$\label{I21} \Phi^{(eM)*}(T)=[\Phi^{(Me)}(y,J,T)]_{y=y_{eq}(T); J=J_{eq}(T)}$$ where $y_{eq}(T)$ and $J_{eq}(T)$ are solutions to $\Phi^{(Me)}_y=0$ and $\Phi^{(Me)}_J=0$. The thermodynamic potential $\Phi^{(Me)}(y,J)$ represents a more detailed picture of the physics taking place in the macroscopic system under consideration than the picture represented by $\Phi^{(me)}(y)$. Depending on the particular forms of $\Phi^{(Me)}(y,J)$ and $\Phi^{(me)}(y)$, some of the details taken into consideration on *MESO level* may or may not show up in the equilibrium fundamental thermodynamic relation on *equilibrium level*. In general, the *equilibrium level* fundamental thermodynamic relations (\[I20\]) and (\[I21\]) are not identical. Next, we reduce (\[I3\]) to (\[I0\]). Let the operator $\Gamma$, the dissipation potential $\Theta^{(Me)}$, and the thermodynamic potential $\Phi^{(Me)}$ be such that $J$ evolves in time more rapidly than $y$. If this is the case then we regard the time evolution governed by (\[I3\]) as proceeding in two stages. In the first stage (the reducing evolution), the time evolution of $J$ is governed by the second equation in (\[I3\]) in which $y$ (and thus also $y^*$) are fixed. This reducing (fast) time evolution is thus governed by $$\label{I4} \dot{J}=-\Phi^{(Mm)}_{J^*}$$ where $$\label{I5} \Phi^{(Mm)}(X^{(CR)*}(y^*),J^*)=\Theta^{(Me)}(J^*)-X^{(CR)*}(y^*)J^*$$ with the constitutive relation $$\label{CR1} X^{(CR)*}(y^*)=-\Gamma^T y^*$$ In order to distinguish the conjugates with respect to the entropy $\Phi^{(Me)}$ (i.e. $y^*=\Phi^{(Me)}_y; J^*=\Phi^{(Me)}_J$) from conjugates with respect to the dissipation potential $\Theta$, we do not use the upper index star to denote $\Theta_{J^*}$ but we use, following the traditional notation established in nonequilibrium thermodynamics, $X^*=\Theta_{J^*}$. Still following the traditional terminology of nonequilibrium thermodynamics, we call $X^*$ the thermodynamic force corresponding to the thermodynamic flux $J^*$. Now we turn our attention to solutions of (\[I4\]). We see immediately that $$\label{I100} \dot{\Phi}^{(Mm)}=-\Phi^{(Me)}_{JJ}(\Phi^{(Mm)}_{J^*})^2\leq 0$$ which means that $J$ tends, as $t\rightarrow \infty$, to $J_{qeq}^*(y^*)$ that is a solution of $$\label{I101} \Theta^{(Me)}_{J^*}(J^*)=X^{(CR)*}(y^*)$$ We see that the potential $\Theta^{(Me)}$ plays different roles in the analysis of $MESO \rightarrow meso$ and in the analysis of $MESO\rightarrow equilibrium$. In the former analysis it plays the same role as the thermodynamic potential $\Phi^{(Me)}$ plays in the investigation of the approach $MESO \rightarrow equilibrium$ governed by (\[I3\]). In the latter analysis it plays the role that is closely related to the entropy production (see (\[eprodM\])). The relation $$\label{Mmth} \Phi^{(mM)*}(y^*)=\Phi^{(Mm)}(X^{(CR)*}(y^*),J_{qeq}^*(y^*))=-\Xi^{(Me)}(y,X^{(CR)*}(y^*))$$ is the fundamental thermodynamic relation on *meso level* implied by the fast time evolution governed by (\[I4\]). If we insert $J_{qeq}^*$ (i.e. solution of (\[I101\])) into the first equation in (\[I3\]) we arrive at $$\label{Xxrelation} \Xi^{(me)}(y,y^*)=[\Xi^{(Me)}(y,X)]_{X=X^{(CR)*}(y,y^*)}$$ The analysis presented above can be summed up in two results. *Result 1* Equation (\[I4\]) governs the reducing time evolution (i.e. the time evolution making the reduction $MESO\rightarrow meso$) and (\[Mmth\]) is the fundamental thermodynamic relation on *meso* level implied by it. *Result 2* The reducing time evolution equation (\[I4\]) is explicitly related to the *MESO* time evolution equation (\[I3\]) and to the *meso* time evolution equation (\[I0\]). The *MESO* dynamics (\[I3\]) is split into (fast) reducing dynamics (\[I4\]) followed by (slow) reduced dynamics (\[I0\]). ### General formulation {#GF} In this section we formulate Result 1 in a more general context. Result 2 requires a detail specification of *MESO* dynamics and a detail analysis of the phase portrait $\mathcal{P}^{MESO}$ that it generates. Except for a few simple illustrations presented in Section \[EX\], we shall not attempt in this paper to formulate Result 2 in general terms. Our objective is to adapt the three Postulates of $MESO\rightarrow equilibrium$ thermodynamics (formulated in Sections \[RDET\], \[RDMee\], \[RDMMee\], \[RDMMde\] above) to $MESO\rightarrow meso$ thermodynamics. First we note an important difference between the reductions $MESO\rightarrow equilibrium$ and $MESO\rightarrow meso$. In the former reduction the target level is *equilibrium level*, i.e. a level of description on which no time evolution takes place. In such reducing dynamics, the fundamental thermodynamic relations consist of equilibrium state variables $\omega$ expressed in terms of the state variables used on the initial level and the entropy driving the reduction (see (\[microftr\]) and (\[MESOftr\])). In the latter reduction the target level is *meso level* on which the time evolution does take place. The fundamental thermodynamic relation corresponding to $MESO\rightarrow meso$ reduction must again include the state variables $y$ expressed in terms of $x$ but it must also include the vector field $g$ on *meso level* (see (\[Gdyn\]) ) expressed in terms of $x$. We present now a setting in which we subsequently formulate the fundamental thermodynamic relation of $MESO \rightarrow meso$ thermodynamics. We begin with *MESO* dynamics (\[Fdyn\]) and with the map $$\label{PMm} \mathfrak{P}^{(Mm)}:M\rightarrow N; x\mapsto y=y(x)$$ allowing to express the state variables on *meso level* in terms of the state variables on *MESO level* (compare with (\[PMIe\]) and (\[PMe\]). We apply the map $\mathfrak{P}^{(Mm)}$ on (\[Fdyn\]) and obtain $$\label{Gdynn} \dot{y}=\mathfrak{P}(G(x))$$ Hereafter, we shall write the right hand side of (\[Gdynn\]) in the form $$\label{J} \mathfrak{P}(G(x))=\Gamma({{\boldmath \mbox{$J$}}}(x))$$ where $\Gamma$ is a fixed operator and ${{\boldmath \mbox{$J$}}}(x)=(J_1(x),...,J_n(x))$ are quantities called ***thermodynamic fluxes***. For example, if (\[Fdyn\]) is the Boltzmann kinetic equation (with the one particle distribution function $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$, where ${{\boldmath \mbox{$r$}}}$ is the position vector and ${{\boldmath \mbox{$v$}}}$ momentum of one particle) and the the map $\mathfrak{P}^{(Mm)}$ is a projection on the first five moments in ${{\boldmath \mbox{$v$}}}$ then $\Gamma =-\frac{\partial}{\partial {{\boldmath \mbox{$r$}}}}$ and ${{\boldmath \mbox{$J$}}}$ are higher order moments. In chemical kinetics (see [@Grmchem]) $\Gamma$ is the stoichiometric matrix. With (\[J\]), the time evolution equation (\[Gdynn\]) takes the form $$\label{JJJ} \dot{y}=\Gamma({{\boldmath \mbox{$J$}}}(x))$$ Its right hand side does not represent, at least in general, a vector field on $N$. In order to be that, it has to be closed, i.e. evaluated at $x=c_{cl}(y)$ $$\label{Gdynnc} \dot{y}=[\Gamma({{\boldmath \mbox{$J$}}}(x))]_{x=\hat{x}(y)}$$ The map $y\mapsto \hat{x}(y)$ is a map $N\rightarrow M$ called a ***closure map***. In the formulation of the $MESO \rightarrow meso$ thermodynamics we shall limit ourselves to the *meso* dynamics (\[Gdyn\]) that has the form (\[Gdynnc\]) with no restriction on the closure map. It will be the reducing dynamics that will determine it. We are now in position to formulate three Postulates of $MESO\rightarrow meso$ thermodynamics. We shall follow closely the formulation of the reducing dynamics *MESO* $\rightarrow$ *equilibrium* presented in Section \[RDMMde\]. ***MESO $\rightarrow$ meso Postulate 0*** is the same as the *MESO* $\rightarrow$ *meso* Postulate 0 in Section \[RDMMde\]. ***MESO $\rightarrow$ meso Postulate I*** *$x\in M$ are state variables on MESO level, $y\in N$ are state variables on meso level and (\[Gdynnc\]) with unrestricted closure map is a family of the time evolution equations on meso level.* ***MESO $\rightarrow$ meso Postulate II*** \(i) *The fundamental thermodynamic relation consists of the specification of the following quantities* $$\label{Mmcr} y(x),{{\boldmath \mbox{$J$}}}(x),\Phi^{(0Mm)}(x)$$ The first quantity in (\[Mmcr\]) is the map $\mathfrak{P}^{(Mm)}:M\rightarrow N$ which expresses the state variables $y$ used on *meso level* in terms of state variables $x$ used on the more microscopic *MESO level*. The reducing time evolution introduced below leaves the space $N$ unchanged. The quantities ${{\boldmath \mbox{$J$}}}=(J_1,...,J_n)$ are the thermodynamic fluxes appearing in the target *meso* dynamics (\[Gdynnc\]). The final quantity $\Phi^{(0Mm)}$ is the thermodynamic potential $\Phi^{(0Mm)}: M\rightarrow \mathbb{R}$. Following (\[Phi\]) and (\[Phi1\]), we write it in the form $$\label{Phi0} \Phi^{(0Mm)}(x,\theta)=-S^{(Mm)}(x)+\frac{1}{\theta}W^{(Mm)}(x)$$ The motivating example discussed above in Section \[motex\] and the reducing dynamics discussed below indicate that $[S^{(Mm)}(x)]_{x_{qeq}}$ (where $x_{qeq}$ is $t\rightarrow \infty$ solution of the reducing dynamics) has the physical interpretation of the entropy production on *meso level* and $[W^{(Mm)}(x)]_{x_{qeq}}$ has the physical interpretation of the work per unit time performed by external forces. The quantity $\theta$ is a temperature or a quantity having the physical dimension of the temperature. \(ii) *The $MESO\rightarrow meso$ reducing time evolution is governed by* $$\label{CRGENERIC} \dot{x}=[\theta L^{(Mm)}(x)x^*-[\Xi^{(Mm)}_{x^*}(x,x^*)]_{x^*=\Phi^{(Mm)}_x}$$ where $$\label{PhiMm} \Phi^{(Mm)}(x,{{\boldmath \mbox{$X$}}})=\Phi^{(0Mm)}(x,\theta)+\sum_{i=1}^{n}X_iJ_i(x)$$ is the thermodynamic potential. The quantities ${{\boldmath \mbox{$X$}}}=(X_1,...,X_n)$ are called thermodynamic forces corresponding to the thermodynamic fluxes ${{\boldmath \mbox{$J$}}}=(J_1,...,J_n)$. As in (\[GENERIC\]), the operator $L^{(Mm)}$ is a Poisson bivector and the $\Xi^{(Mm)}$ is a dissipation potential. Both $L^{(Mm)}$ and $\Xi^{(Mm)}$ are required to be degenerate so that the space $N$ remains invariant under the time evolution governed by (\[CRGENERIC\]). The same considerations as the ones that led us in Section \[RDMMde\]) to the conclusion that solutions to (\[GENERIC\]) have the property $x\rightarrow x_{eq}$ as $t\rightarrow \infty$, lead us the the conclusion that solutions of (\[CRGENERIC\]) have the property $x\rightarrow x_{qeq}$ as $t\rightarrow \infty$, where $x_{qeq}$ is a solution to $$\label{qeq} \Phi^{Mm)}_x=0$$ These states are time independent (i.e. steady) states with respect to the reducing dynamics but they are, in general, not steady states with respect to both the original *MESO* dynamics and the reduced dynamics. Provided the thermodynamic potential $\Phi^{(Mm)}$ is specified, $x_{qeq}$ depends on $({{\boldmath \mbox{$X$}}},\theta)$. In order $x_{qeq}$ could play the role of the closure $\hat{x}(y)$, $({{\boldmath \mbox{$X$}}},\theta)$ have to be specified as functions of $y$. We shall call such specification a ***constitutive relation*** $({{\boldmath \mbox{$X$}}}^{(CR)}(y),\theta^{(CR)}(y))$. The asymptotic solution $x_{qeq}$ of (\[CRGENERIC\]) with ${{\boldmath \mbox{$X$}}}={{\boldmath \mbox{$X$}}}^{(CR)}(y)$ and $\theta =\theta^{(CR)}(y)$ will be denoted $x_{cl}(y)$. The lower index “cl” denotes “closure”. We shall comment about constitutive relations at the end of this section. Now, we assume that the constitutive relations are known The fundamental thermodynamic relation on *meso level* implied by the fundamental thermodynamic relation (\[Mmcr\]) on *MESO level* is the following: $$\begin{aligned} \label{Mmimpl} {{\boldmath \mbox{$J$}}}^{(CR)}({{\boldmath \mbox{$X$}}}^{(CR)},\theta^{(CR)})&=& [{{\boldmath \mbox{$J$}}}(x)]_{x=x_{cl}} \\\label{thr} y&=&[y(x)]_{x=x_{cl}}\nonumber \\ S^{(mM)*}({{\boldmath \mbox{$X$}}}^{(CR)},\theta^{(CR)})&=&[\Phi^{(Mm)}]_{x=x_{cl}}\end{aligned}$$ The relation (\[Mmimpl\]) is the specification of the reduced dynamics. The unspecified closure $\hat{x}(y)$ appearing in (\[Gdynnc\]) is specified: $\hat{x}(y)=x_{cl}({{\boldmath \mbox{$X$}}}^{(CR)},\theta^{(CR)})$. The first line in (\[thr\]) is the same as the first two lines in the fundamental thermodynamic relations (\[MIimpl\]) and (\[Meimpl\]) implied by $MICRO\rightarrow equilibrium$ thermodynamics. The second line in (\[thr\]) is again the same as the third lines in (\[MIimpl\]) and (\[Meimpl\]). It is a thermodynamic relations on *meso* level implied by the reducing dynamics (\[CRGENERIC\]). We emphasize that this relation is not implied by the reduced dynamics. As for the notation, we use the upper index $(mM)$ to denote that the quantity is formulated on *meso level* and is implied by dynamics on *MESO level* (see the explanation of the notation in the text after (\[classftr\])). If we compare the fundamental thermodynamic relation (\[Mmimpl\]), (\[thr\]) implied by $MESO\rightarrow meso$ with the fundamental thermodynamic relations (\[MIimpl\]) and (\[Meimpl\]) implied by $MICRO\rightarrow equilibrium$ and $MESO\rightarrow equilibrium$, we see that the new feature in (\[Mmimpl\]), (\[thr\]) is the reduced dynamics (\[Mmimpl\]). Indeed, the reduced dynamics in the approach to $MESO\rightarrow equilibrium$ is no dynamics and thus there is no need to specify it. If, on the other hand, we compare (\[Mmimpl\]), (\[thr\]) with standard investigations of reductions that put into focus only the reduced dynamics, the second line in (\[thr\]) is new. It represents new thermodynamics implied by reducing dynamics. We also emphasize that the fundamental thermodynamic relation (\[Mmimpl\]), (\[thr\]) exists independently of whether the states in the reduced dynamics are steady or time dependent. Finally, we return to the constitutive relations $({{\boldmath \mbox{$X$}}}^{(CR)}(y),\theta^{(CR)}(y))$ introduced in the text after Eq.(\[qeq\]). First, we note that in the context of $MESO\rightarrow equilibrium $ investigations in Sections \[RDMee\], \[RDMMee\] and \[RDMMee\], constitutive relations are specifications of $\omega^*$. This means that in constitutive relations arising in $MESO\rightarrow equilibrium $ investigations we are expressing the conditions under which the macroscopic systems under consideration are investigated. This is also true in the context of $MESO\rightarrow meso$ investigations but with two new features. First, the conditions involve now also external forces. The imposed external forces are expressed in some of the forces $X$. We shall denote them by ${{\boldmath \mbox{$X$}}}^{(ext)}$. Second, the remaining forces, denoted ${{\boldmath \mbox{$X$}}}^{(int)}$ must be specified, as well as the free energy $\Phi^{(0Mm)}$ by solving *MESO* time evolution equation (\[Fdyn\]) (i.e. constructing the phase portrait $\mathcal{P}^{MESO}$, and extracting from it slower changing pattern representing the reduced *meso* time evolution (see also discussion in Section \[RD\]). This, of course, can be done only if (\[Fdyn\]) is more specified. We have done it in the example discussed in Section \[motex\] and we shall make other illustrations in the next Section \[EX\]. At this point we only mention that it is in the constitutive relations where the entropy $S^{(me)}$ enters the analysis. The entropy $S^{(Mm)}$ then typically becomes closely related to the production of the entropy $S^{(me)}$. Recall for example the Fourier and Navier-Stokes constitutive relations in fluid mechanics (see more in Section \[EX4\]). They are expressed in terms of the conjugate state variables with respect to the local entropy that, in the classical fluid mechanics, plays the role of $S^{(me)}$. We have also seen the similar constitutive relations in Section \[motex\]. We end this section with a few remarks. More comments and illustrations are then in Section \[EX\]. The CR fundamental thermodynamic relation (\[Mmimpl\]) is a relation involving only the state variables and the material parameters used on *meso level* (\[Gdyn\]). From the physical point of view, we expect that even if it is not directly related to Eq.(\[Gdyn\]), it reflects important properties of solutions of (\[Gdyn\]). This is because both Eq.(\[Gdyn\]) and the relation (\[Mmimpl\]) address the same physics even if expressed on different levels of description. In particular, we anticipate, on the physical ground, that the presence of bifurcations in solutions to (\[Gdyn\]) expressing mathematically the presence of sudden changes in behavior (e.g. the onset of convection in the Rayleigh-Bénard system) is manifested in the CR fundamental thermodynamic relation (\[Mmimpl\]) as phase transitions. This anticipation is based on the experimentally observed growth of fluctuations in meso-measurements of macroscopic systems in situations in which their behavior changes dramatically (we shall call them critical situations). From this observation we then conclude that in critical situations the “distance” between *meso* and *MESO levels* diminishes and the critical behavior manifests itself on both *meso* and *MESO levels*. Since the CR fundamental thermodynamic relation is inherited from the *MESO level*, we expect to see the critical behavior also in it. Even without specifying the CR thermodynamic potential $\Phi^{(Mm)}$, the fact that the constitutive relations arise from minimizing it implies Maxwell-type reciprocity relations (that, from the mathematical point of view, express symmetry of the second derivatives of $\Phi^{(Mm)}$) among the thermodynamic fluxes and forces. In the case when the potential $\Phi^{(Mm)}$ is quadratic then these reciprocity relations become Onsager’s relations. Examples of reciprocity relations that arise in chemical kinetics are worked out in Section III B in [@GrmPavKlika]. We ask now the following question. Given an externally driven macroscopic system, how do we find the CR thermodynamical potential $\Phi^{(Mm)}$ (see (\[PhiMm\])) corresponding to it? If we ask the same question, but with externally unforced macroscopic systems and with the thermodynamic potential $\Phi^{(Me)}$ (see (\[Phi1\]) ) replacing externally driven macroscopic systems and the CR thermodynamic potential $\Phi^{(Mm)}$, then the answer is the following. On the most macroscopic level (that is for externally unforced systems the level of classical equilibrium thermodynamics - see Section \[RDET\]), the only way we can identify the thermodynamic potential $\Phi^{(ee)}$ is by making experimental observations (e.g. observation of the relation among $P,V,T$ and of the specific heat - see Section \[RDET\]). The knowledge of $\Phi^{(ee)}$ on the level of the classical equilibrium thermodynamics can be then transferred, via the local equilibrium assumption, also to the level of fluid mechanics. On the level of kinetic theory, we can take as the point of departure for the search of $\Phi^{(Me)}$ the Boltzmann kinetic equation (playing in this example the role of *MESO* dynamics) and arrive (following Boltzmann) to the Boltzmann entropy by investigating properties of its solutions. On the *MICRO level*, it suffices to know all the mechanical interactions expressed in the energy $E^{MICRO}$ since the entropy on the *MICRO level* is the universal Gibbs entropy (\[microftr\]). On *meso levels*, we may find $\Phi^{(Me)}$ by MaxEnt reduction from the *MICRO level* (i.e. by maximizing the Gibbs entropy subjected to constraints expressing the mapping from *MICRO* to *meso* state spaces) and/or by relating entropy to concepts arising in the information theory and the theory of probability. In Section \[EX2\] we shall suggest a possible universal *MICRO level* CR entropy. Reducing Dynamics: Examples {#EX} =========================== Our objective in this section is to make a few comments and illustrations that will bring a more concrete content to the investigation discussed in previous sections. As for the $MESO\rightarrow equilibrium$ passage, many very specific illustrations can be found in [@Grmadv] and references cited therein and in [@Obook]. In Section \[EX2\], we work out a new illustration in which *MESO level* is replaced by *MICRO level*. In Sections \[EX3\] and \[EX4\] we develop two simple examples illustrating the $MESO\rightarrow meso$ passage. In Section \[EX5\] we use the *CR meso-thermodynamics* to estimate volume fractions at which phase inversion occurs in a blend of two immiscible fluids. In Section \[EX1\] we comment about reductions seen as a pattern recognition in phase portraits. Before proceeding to specific illustrations, we shall comment about the physics and the experimental basis of the general thermodynamics presented above. We begin with the classical equilibrium thermodynamics. This theory has emerged from an attempt to combine mechanics involved in large scale mechanical engines with heat. As it became clear later in the Gibbs equilibrium statistical mechanics, the heat is a manifestation, on the macroscopic scale, of the mechanics on the microscopic (atomic) scale. To combine the large scale mechanics with heat is to combine large scale mechanics with microscopic mechanics. The objective of the classical equilibrium thermodynamics is to incorporate the microscopic mechanics (or heat which, at the time when thermodynamics was emerging, was a rather mysterious concept) into the large scale mechanics by ignoring all that is irrelevant to our direct macroscopic interest. In the classical equilibrium thermodynamics this has been achieved by enlarging the concept of mechanical energy (by introducing a new type of energy, namely the internal energy) and by introducing the concept of entropy together with the MaxEnt principle. The setting of the classical equilibrium thermodynamics is thus a two-level setting: one level (macroscopic) is of our direct interest and the other (microscopic) is not of our direct interest. We cannot however completely ignore it since it influences what happens on the macroscopic level. It is the concept of entropy that on the macroscopic level represents all from the microscopic level that is important for describing the behavior that directly interests us. All the other details involved on the microscopic level are ignored. The essence of the classical equilibrium thermodynamics is thus to provide a relation $MICRO\rightarrow macro$ between two levels of description. Its experimental basis consist of observations showing that indeed the “minimalist” inclusion of the microscopic level offered by the classical equilibrium thermodynamics leads to predictions that agree with the macroscopic experimental observations. In the formulation of general thermodynamics we have extended the classical equilibrium thermodynamics by keeping its two-level $MICRO\rightarrow macro$ setting but we have replaced the *MICRO* and *macro* levels with two general *MESO* and *meso* levels. Thermodynamics (including the classical equilibrium thermodynamics) is a theory of theories or, in other words, a metaphysics. The experimental basis of thermodynamics are meta-observations showing that behavior observed and well described on one level can also be observed and well described on another level. Direct experimental observations, contrary to meat-observations, are observations made on a single level. They provide experimental basis of individual levels. The concept of entropy can only be understood in the two-level $MESO\rightarrow meso$ viewpoint of thermodynamics. The often asked questions like for instance: does the entropy exist for driven systems, should be replaced with the question: can the behavior of the driven system under investigation be described on two separate levels. If the answer to this latter question is yes then the answer to the former question is also yes. Since both well established levels are applicable, solutions of the time evolution on the level involving more details must approach solutions on the second level involving less details. The entropy is then the potential driving the approach. In conclusion of the above comment about the general thermodynamics we note that the very wide scope of thermodynamics (on the one hand it is a metaphysics and on the other hand it is a very practically oriented engineering tool) is certainly one of the reasons for its attractiveness but it is also a reason (at least one of the reasons) for unusually strong disagreements among its practitioners. Pattern recognition in the phase portrait, Chapman-Enskog method {#EX1} ---------------------------------------------------------------- An archetype example of *MESO* time evolution equation (\[Fdyn\]) is the Boltzmann kinetic equation. An archetype example of $MESO\rightarrow meso$ investigation is the Chapman-Enskog analysis of the passage from the Boltzmann equation to fluid mechanics. In this section we illustrate the pattern recognition viewpoint of reductions on the $MICRO\rightarrow MESO$ derivation of the Boltzmann equation and on the Chapman-Enskog method. ### MICRO $\rightarrow$ MESO introduction of the Boltzmann equation {#derBE} We emphasize that our objective is not to derive rigorously the Boltzmann equation from *MICRO* mechanics but only to illustrate how it can arise in the pattern recognition process in $\mathcal{P}^{MICRO}$. In order to be able to recognize patterns in phase portraits $\mathcal{P}^{MICRO}$ we have to generate it (or at least to obtain some pertinent information about it). Since $\mathcal{P}^{MICRO}$ is a collection of particle trajectories we have to find the trajectories, i.e. we have solve the *MICRO* time evolution equations. It is important to realize that it is not the *MICRO* vector field (i.e. the *MICRO* time evolution equations) that is our starting point in in pattern recognition process but it the collection of trajectories that it generates (i.e. solutions to the *MICRO* time evolution equations). In the case of a dilute ideal gas (i.e. a macroscopic system composed of particles that do not interact except for occasional binary collisions) the particle trajectories can be seen as a composition of straight lines (representing free particle motion) and two intersecting lines (representing binary collisions). Intersections of three (or more ) lines at one point (representing ternary (or higher order) collisions) are, due to the dilution, very rare and we therefore ignore them. We choose one particle distribution function $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ as the *meso* state variable and try to recognize its time evolution (in particular the vector field generating it) that can be seen as a pattern in $\mathcal{P}^{MICRO}$. We begin with the straight line ${{\boldmath \mbox{$r$}}}\rightarrow {{\boldmath \mbox{$r$}}}+\frac{{{\boldmath \mbox{$v$}}}}{m}t$. We can see this line as a trajectory generated by $\dot{{{\boldmath \mbox{$r$}}}}=\frac{{{\boldmath \mbox{$v$}}}}{m}$ and $\dot{{{\boldmath \mbox{$v$}}}}=0$. By $m$ we denote the mass of one particle and $t$ denotes the time. This particle time evolution induces the time evolution $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})\rightarrow f({{\boldmath \mbox{$r$}}}-\frac{{{\boldmath \mbox{$v$}}}}{m}t,{{\boldmath \mbox{$v$}}})$ in one particle distribution functions. This time evolution is then generated by the vector field $$\label{ffl1} \frac{\partial f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})}{\partial t}=-\frac{\partial}{\partial {{\boldmath \mbox{$r$}}}}({{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}}))$$ We have thus arrived at the vector field representing the first feature of particle trajectories. We turn now to the second feature, i.e. to two intersecting lines representing binary collisions. Formally, we regarded this feature as two straight lines, corresponding to momenta $({{\boldmath \mbox{$v$}}}_1,{{\boldmath \mbox{$v$}}}_2)$, meeting at the position with coordinate ${{\boldmath \mbox{$r$}}}_1$ and continuing as two straight lines corresponding to momenta $({{\boldmath \mbox{$v$}}}'_1,{{\boldmath \mbox{$v$}}}'_2)$. The ingoing momenta $({{\boldmath \mbox{$v$}}}_1,{{\boldmath \mbox{$v$}}}_2)$ and the outgoing momenta $({{\boldmath \mbox{$v$}}}'_1,{{\boldmath \mbox{$v$}}}'_2)$ are related by the relations $$\begin{aligned} \label{conBE1} v_1^2+v_2^2&=&(v_1')^2+(v_2')^2\nonumber \\ {{\boldmath \mbox{$v$}}}_1+{{\boldmath \mbox{$v$}}}_2&=&{{\boldmath \mbox{$v$}}}_1'+{{\boldmath \mbox{$v$}}}_2'\end{aligned}$$ expressing the mechanics of the collision. More details about the collision mechanics (that would make the relation between the ingoing and outgoing momenta one-to-one) are ignored and are not a part of the second feature. In terms of one particle distribution functions we express it therefore as the time evolution generated by the gain-loss balance, or in other words, by considering $({{\boldmath \mbox{$v$}}}'_1,{{\boldmath \mbox{$v$}}}'_2)\leftrightarrow({{\boldmath \mbox{$v$}}}_1,{{\boldmath \mbox{$v$}}}_2)$ as a chemical reaction obeying the constraint (\[conBE1\]). We shall see in Section \[EX2\] that the vector field such gain-loss balance is given by $$\begin{aligned} \label{intlin1} \frac{\partial f({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)}{\partial t}&=&-\Xi^{(BE)}_{f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)}\nonumber \\ &&=\int d2\int d1'\int d2'\widetilde{W}^{(BE)}(f(1')f(2')-f(1)f(2))\end{aligned}$$ where $\Xi^{(BE)}$ is the dissipation potential, $\widetilde{W}^{(BE)}$ is a quantity appearing in it (see details in Section \[EX2\] below), and $f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)$ is a conjugate of $f({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)$ with respect to a entropy $S^{(BE)}(f)$ (i.e. $f^*=S^{(BE)}_f$). With a particular choice of these quantities, the right hand side of (\[intlin1\]) becomes the classical Boltzmann collision operator (see Section \[EX2\]). Both features of the particle phase portrait $\mathcal{P}^{MICRO}$ are thus expressed in the time evolution of one particle distribution functions as the sum of the vector fields (\[ffl1\]) and (\[intlin1\]). The kinetic equation that we are obtaining in this way is the Boltzmann kinetic equation. The nonclassical formulation in which the Boltzmann equation is emerging from our derivation has several advantages. One of them is that the H-theorem (i.e. $\dot{S}^{(BE)}\geq 0$) is in it manifestly visible. Another advantage is that we have in fact derived a generalization of the Boltzmann equation since we do not have to choose $S^{(BE)}(f)=-k_B\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}}) \ln f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$. The entropy $S^{(BE)}(f)$ can be a more general potential $\int d{{\boldmath \mbox{$r$}}}\int d{{\boldmath \mbox{$v$}}}c(f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}}))$, where $c$ is an unspecified but sufficiently regular concave function $\mathbb{R}\rightarrow\mathbb{R}$. With such more general entropy $S^{(BE)}(f)$ we still have the H-theorem $\dot{S}^{(BE)}\geq 0$ since, as we convince ourselves by a direct verification, the time evolution generated by (\[ffl1\]) does not change $S^{(BE)}(f)$. In addition, we also see that the step in the above introduction of the Boltzmann equation where the time reversibility brakes and the dissipation emerges is our ignorance of details of trajectories during binary collisions. ### Chapman-Enskog method {#CEmeth} Let $\mathcal{P}^{MESO}$ and $\mathcal{P}^{meso}$ be the phase portraits corresponding to the *MESO* dynamics (\[Fdyn\]) and the *meso* dynamics (\[Gdyn\]) respectively. Our problem is to recognize $\mathcal{P}^{meso}$ as a pattern inside of $\mathcal{P}^{MESO}$. While this viewpoint of the $MESO \rightarrow meso$ reduction does provide a good intuitive understanding of the process, it does not provide a practical way to proceed. The archetype method offering such procedure is the Chapman-Enskog method (see e.g. [@GKHilb]). This method was originally developed for reducing the Boltzmann kinetic equation to the Navier-Stokes-Fourier hydrodynamic equations but it can be applied to any $MESO \rightarrow meso$ passage. The pattern recognition process becomes in the context of the Chapman-Enskog method in the process of identifying a manifold $\mathcal{M}\subset M$ that satisfies the following two requirements: (i) $\mathcal{M}$ is in one-to-one relation to $N$, (ii) $\mathcal{M}$ is quasi-invariant (i.e. $\mathcal{M}$ is “as much as possible” invariant with respect to the *MESO* time evolution taking place on $M$). We shall sketch below the geometrical essence of the method in three steps.\ *Chapman-Enskog, Step 1* By using an insight into the physics involved in the *MESO* dynamics, we write the *MESO* vector field $G$ as a sum of $G_0$, playing the dominant role, and $G_1$ that is seen as a perturbation (i.e. we write $G=G_0+G_1$). The splitting of the vector field $G$ induces then splitting of the search of the quasi invariant manifold $\mathcal{M}\subset M$ into two stages. A first approximation $\mathcal{M}^{(0)}$ of $\mathcal{M}$ (called zero Chapman-Enskog approximation) is identified in the first stage (the second step in the Chapman-Enskog method) with neglecting $G_1$ (i.e. we consider $G=G_0$). In the second stage (the third step in the Chapman-Enskog method) the manifold $\mathcal{M}^{(0)}$ is deformed into $\mathcal{M}^{(1)}$ that is called a first Chapman-Enskog approximation of $\mathcal{M}$. In the case of (\[Fdyn\]) being the Boltzmann kinetic equation, $G_0$ is the Boltzmann collision term (since the pieces of the trajectories involving binary collisions are seen as being dominant in the phase portrait $\mathcal{P}^{MESO}$).\ *Chapman-Enskog, Step 2* In this step we identify $\mathcal{M}^{(0)}$. We define it as a manifold on which the dominant vector field $G_0$ disappears (i.e. we solve the equation $[G_0]_{\mathcal{M}^{(0)}}=0$). The quantities that parametrize $\mathcal{M}^{(0)}$ are then chosen to be the *meso* state variables $y$ expressed in terms of the *MESO* state variables $x$. We thus obtain a mapping $\Pi:M\rightarrow N; x\mapsto y$. This mapping subsequently induces a one-to-one mapping $\Pi^{(\mathcal{M})}:\mathcal{M}^{(0)}\rightarrow N$. Next, the vector field $[G]_{\mathcal{M}^{(0)}}$ is projected (by the projection induced by $\Pi^{(\mathcal{M})}$ ) on the tangent space of $\mathcal{M}^{(0)}$. We denote the projected vector field by the symbol $G^{(0)}$. Finally, the vector field $G^{(0)}$ is projected (again by the projection induced by $\Pi^{(\mathcal{M})}$) on the tangent space of $N$. This is then the vector field on $N$, denoted by $g^{(0)}$ and called a zero Chapman-Enskog approximation of $G$ on $N$. In the case of the *MESO* dynamics (\[Fdyn\]) being the Boltzmann kinetic theory, the mapping $\Pi$ is the standard mapping from one particle distribution functions to hydrodynamic fields, $\mathcal{M}^{(0)}$ is the manifold whose elements are local Maxwell distribution functions, and $g^{(0)}$ is the right hand side of the Euler (reversible and nondissipative) hydrodynamic equations.\ *Chapman-Enskog, Step 3* The first Chapman-Enskog approximation $\mathcal{M}^{(1)}$ of $\mathcal{M}$ is found in this step. We note that the manifold $\mathcal{M}^{(0)}$ is not an invariant manifold since the vectors $[G]_{\mathcal{M}^{(0)}}$ do not lie in the tangent spaces attached to $x_0\in \mathcal{M}^{(0)}$. We want to make it more invariant. We therefore deform $\mathcal{M}^{(0)}$ into $\mathcal{M}^{(1)}$ $( x_0\mapsto x_1)$ in such a way that $G^{(1)}\equiv [G]_{\mathcal{M}^{(0)}}$, where $G^{(1)}$ is the vector field $G$ attached to the points $x_1$ and projected on $\mathcal{M}^{(1)}$. We note that the manifold $\mathcal{M}^{(1)}$ is still not invariant (since, in general, $[G]_{\mathcal{M}^{(1)}}\neq [G]_{\mathcal{M}^{(0)}}$) but it is expected to be “more” invariant than $\mathcal{M}^{(0)}$ since the vector field $G_1$ is just a perturbation of $G_0$ (and consequently the deformation $\mathcal{M}^{(0)}\rightarrow \mathcal{M}^{(1)}$ is small). The vector field $g^{(1)}$ projected on $N$ is the first Chapman-Enskog approximation of $G$. In the case of (\[Fdyn\]) being the Boltzmann equation, the vector field $g^{(1)}$ is the right hand side of the Navier-Stokes-Fourier (irreversible and dissipative) hydrodynamic equations. If both *MESO* dynamics (\[Fdyn\]) and *meso* dynamics (\[Gdyn\]) are known and well established (i.e. they both have emerged from direct derivations involving *MESO* measurements and *meso* measurements respectively) then the Chapman-Enskog type derivation of (\[Gdyn\]) from (\[Fdyn\]) brings an additional information. First, the domain of applicability of (\[Gdyn\]) inside of the domain of applicability of (\[Fdyn\]) is identified, and second, mapping $\xi \mapsto \zeta$ emerges (i.e. the material parameters $\zeta$ with which individual features of the systems under consideration are expressed on the *meso level* become functions of the material parameters $\xi$ used for the same purpose on the *MESO level*). Examples of applications of the Chapman-Enskog method in many types of mesoscopic dynamics (including for instance the dynamics describing chemical reactions) can be found in [@Gorbbook], [@GKHilb]. We make now three additional comments about the Chapman-Enskog method. *Comment 1* We note that there is no thermodynamics in the Chapman-Ensog method. How can we bring it to it? Following the viewpoint of thermodynamics presented in previous sections, we have to turn attention not only to the pattern (i.e. in this case to the submanifold $\mathcal{M}_1$) but also to the reducing time evolution bringing $x\in M$ to it. As we have seen, the reducing tome evolution is generated by a potential so, at leat, we should try to identify the potential. As for the manifold $\mathcal{M}_0$, the reducing time evolution is the Boltzmann equation without $G_1$ (i.e. Eq.(\[intlin1\]) and the potential is obviously the Boltzmann entropy. Indeed, $\mathcal{M}_0$ can be obtained by MaxEnt reduction of the Boltzmann entropy. The manifold on which $S^{(BE)}(f)$ reaches its maximum subjected to constraints representing the fluid mechanics fields expressed in terms of the one particle distribution function is exactly the submanifold $\mathcal{M}_0$. For example in [@Gorbbook], this is the way the submanifold $\mathcal{M}_0$ is introduced. The second step in the Chapman-Enskog method can be thus seen as a part of the investigation of reducing dynamics. Can we follow this path and interpret thermodynamically also the third step in the Chapman-Enskog method (i.e. the deformation of $\mathcal{M}_0$ to $\mathcal{M}_1$)? In order to make such interpretation, we look for a potential $S^{(BE)}_1(f)$, that satisfies the following properties: (i) $S^{(BE)}_1(f)$ is a deformation of $S^{(BE)}(f)$, (ii) its maximum is reached at $\mathcal{M}_1$, and (iii) it generates the time evolution in which $\mathcal{M}_1$ is approached (similarly as $S^{(BE)}(f)$ generates the time evolution $\mathcal{M}_0$ is approached. The partial results related to this problem that are reported in Section 4.2 of [@Grmadv] indicate that the potential obtained in this way is indeed the CR-entropy generating the reducing time evolution that is involved in the passage from kinetic theory to fluid mechanics. *Comment 2* For the kinetic equation (\[intlin1\]) the submanifold $\mathcal{M}_0$ is an invariant manifold. For the full Boltzmann kinetic equation (i.e. kinetic equation combining (\[ffl1\]) and (\[intlin1\])) neither $\mathcal{M}_0$ nor $\mathcal{M}_1$ are invariant manifolds. In fact, as it has been shown by Grad in [@Grad], and Desvillettes and Villani in [@Vill], the only invariant manifold is the manifold $\mathcal{M}_{eq}$ of equilibrium sates (i.e. time independent solutions of the full Boltzmann equation). Both manifolds $\mathcal{M}_0$ and $\mathcal{M}_1$ are quasi-invariant manifolds. Grad, Desvillettes and Villani have proven that solutions to the full Boltzmann equation may come very close to $\mathcal{M}_0$ and $\mathcal{M}_1$ (that is why we can call these manifolds quasi-invariant manifolds) but they never fall on neither of them. They only fall eventually on the submanifold $\mathcal{M}_{eq}$ that is a submanifold of both $\mathcal{M}_0$ and $\mathcal{M}_1$. *Comment 3* Reduction to the submanifold $\mathcal{M}_0$ results in the Euler fluid mechanics equations and reduction to its deformation $\mathcal{M}_1$ results in the Navier-Stokes-Fourier fluid mechanics equations. Both these fluid mechanics equations are particular realizations of (\[GENERIC\]) and thus both are physically meaningful. In principle, it is possible to continue the deformations of $\mathcal{M}_0$. Similarly as we have made the deformation $\mathcal{M}_0\rightarrow\mathcal{M}_1$ we can make the next deformation $\mathcal{M}_1\rightarrow \mathcal{M}_2$. In other words, we can proceed to the second Chapman-Enskog approximation. Will be the resulting reduced time evolution again physically meaningful (in the sense that it will be a particular realization of (\[GENERIC\]))? Experience collected in the investigations of higher order Chapman-Enskog approximations (e.g. the investigation of the linearized Boltzmann equation in [@GRHELV]) seems to indicate that the answer to this question is negative. MICRO $\rightarrow$ equilibrium reducing dynamics {#EX2} ------------------------------------------------- Many illustrations of GENERIC equation (\[GENERIC\]) in kinetic theory, fluid mechanics, and solid mechanics of simple and complex fluids can be found in [@Obook], [@Grmadv] and references cited therein. In this section we develop an additional new illustration. We return to the Gibbs equilibrium statistical mechanics presented in Section \[RDMee\] and ask the following question. What is the GENERIC time evolution of the N-particle distribution function $f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})$ (i.e. the time evolution governed by (\[GENERIC\]) with $x=f_N(1,...,N)$) that makes the maximization of the Gibbs entropy (see (\[microftr\])) postulated MaxEnt in the point (iv) of the *MICRO $\rightarrow$ equilibrium Postulate II*? We first introduce one such time equation and then discuss its possible non uniqueness. We use hereafter a shorthand notation $1\equiv ({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1), 2\equiv ({{\boldmath \mbox{$r$}}}_2,{{\boldmath \mbox{$v$}}}_2),...)$. We look for a dissipation potential $\Xi^{(N)}$ which brings N-particle distribution functions $f_N(1,...,N)$ to the Gibbs distribution $(f_N)_{eq}$ (i.e. to $f_N$ for which the thermodynamic potential $\Phi$ given in (\[Phi\]) reaches its minimum). In other words, we look for $\Xi^{(N)}$ for which solutions to $$\label{Nkin} \frac{\partial f_N(1,2,...,N)}{\partial t}=-\Xi^{(N)}_{f^*_N(1,2,...,N)}$$ approach, as $t\rightarrow\infty$, $(f_N)_{eq}$. Inspired by the dissipation potential arising in the Guldberg-Waage chemical kinetics (see [@Grmchem]) and the dissipation potential generating the Boltzmann collision integral [@GrB], we propose $$\begin{aligned} \label{XiN} \Xi^{(N)}&=&\int d1...\int dN\int d1'...\int dN'W^{(N)}(f_N,1,...,N,1',...,N')\nonumber \\ &&\times\left(e^{\frac{1}{2}X^{(N)}}+e^{-\frac{1}{2}X^{(N)}}-2\right)\end{aligned}$$ where the thermodynamic forces are given by $$\label{Nforce} X^{(N)}=\frac{1}{k_B}(f^*_N(1,2,...,N)-f^*_N(1',2',...,N')),$$ $f^*_N=S_{f_N}$, $S(f_N)$ is the Gibbs entropy (\[microftr\]), $$\label{12} (1,2,...,N)\rightleftarrows (1',2',...,N')$$ are one-to-one transformations in which the microscopic energy $E^{MICRO}(1,...N)$ remains constant, i.e. $$\label{12con} E^{MICRO}(1,2,...,N)=E^{MICRO}(1',2',...,N'),$$ and $W^{(N)}\geq 0$ are nonegative material parameters that are different from zero ($W^{(N)}\neq 0$) only if the constraint (\[12con\]) holds and $W$ is symmetric with respect to $(1,2,...,N)\rightarrow (1',2',...,N')$. In the Guldberg-Waage chemical kinetics (see [@Grmchem]), the transformation (\[12\]) is interpreted as a chemical reaction. We shall demonstrate below that the Boltzmann collision operator is the right hand side of (\[Nkin\]) with $N=$, dissipative forces $X^{(1)}$ given in (\[X1\]) and the transformation (\[12\]) given in (\[binreacx\]) and (\[conBE\]). Before proving that solutions to (\[Nkin\]) approach $(f_N)_{eq}$, we write the time evolution equation (\[Nkin\]) explicitly. With the Gibbs entropy (\[microftr\]), Eq.(\[Nkin\]) takes the form $$\begin{aligned} \label{Nkinexp} \frac{\partial f_N(1,2,...,N)}{\partial t}&=&-\Xi^{(N)}_{f^*_N(1,2,...,N)}\nonumber \\ &&=\int d1'...\int dN'\widetilde{W}^{(N)} (f_N(1',...,N')-f_N(1,...,N))\nonumber \\\end{aligned}$$ where $\widetilde{W}^{(N)}=\frac{W^{(N)}}{2k_B(f_N(1,...,N)f_N(1',...,N'))^{\frac{1}{2}}}$. The Legendre transformation $\Theta^{(N)}(J)$ of $\Xi^{(N)}(X)$ is $$\begin{aligned} \label{Theta} \Theta^{(N)}(J)&=&2\int d1...\int dN\int d1'...\int dN' W\nonumber \\ &&\times\left[\hat{J}\ln\left(\hat{J}+\sqrt{1+(\hat{J})^2}\right)-\left(\sqrt{1+(\hat{J})^2} -1\right)\right]\end{aligned}$$ where $\hat{J}=\frac{J}{W}$. Now we prove that solutions to (\[Nkin\]) (or (\[Nkinexp\])) approach, as $t\rightarrow \infty$, the Gibbs distribution $(f_N)_{eq}$. First, we see that the right hand side of (\[Nkin\]) equals zero if $X=0$. In view of (\[12con\])), equation $X=0$ is solved by $f_N=(f_N)_{eq}$. Since the thermodynamic potential $\Phi$ plays the role of the Lyapunov function for the approach to $(f_N)_{eq}$ (see (\[asGEN\])), we see that solutions to (\[Nkin\]) (which takes the form (\[Nkinexp\]) provided the entropy is the Gibbs entropy,) approach, as $t\rightarrow \infty$, the Gibbs distribution $(f_N)_{eq}$. The time evolution governed by (\[Nkin\]) indeed brings macroscopic systems to states investigated in the Gibbs equilibrium statistical mechanics (see Section \[RDMee\]. The dissipation potential $\Xi^{(N)}$ given in (\[XiN\]) (or equivalently its Legendre transformation $\Theta^{(N)}$ given in (\[Theta\])) can be therefore regarded as the universal CR-entropy on *MICRO level* similarly as the Gibbs entropy (\[microftr\]) is the universal entropy on *MICRO level*. Consequently, we can find CR-entropies on *meso levels* similarly as we can find entropies on *meso levels*. We can either try to extract them from the time evolution (generating $meso \rightarrow equilibrium$ passage - in the case of entropy, or $MESO \rightarrow meso$ passage - in the case of CR-entropy) or we can attempt to reduce them (by MaxEnt) from the universally valid expressions (for the Gibbs entropy (\[microftr\]) in the case of entropy and for the CR-entropy (\[XiN\]) in the case of CR-entropy). We now show that the dissipation potential (\[XiN\]) can be seen as a natural extension of the dissipation potential generating the Boltzmann collision operator arising in one particle kinetic theory. At the end of this section we then investigate other possible vector fields that can describe approach to equilibrium. The time evolution equation (\[Nkin\]) has a well defined meaning for any $N\geq 2$. For $N=1$, i.e. for the level of one particle kinetic theory, we cannot make the transformation (\[12\]) and we cannot therefore directly use (\[Nkin\]). In order to be able to introduce transformations of the type (\[12\]) in one particle kinetic theory, we need a partner. We shall denote the coordinates of the particle by $({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)$ and of its partner by $({{\boldmath \mbox{$r$}}}_2,{{\boldmath \mbox{$v$}}}_2)$. From the physical point of view, we regard the transformation (\[12\]) in the context of one particle kinetic theory as a binary collision between the particle and its partner. We therefore write (\[12\]) in the form $$\label{binreacx} ({{\boldmath \mbox{$v$}}}_1,{{\boldmath \mbox{$v$}}}_2)\rightleftarrows ({{\boldmath \mbox{$v$}}}'_1,{{\boldmath \mbox{$v$}}}'_2),$$ with the constraint $$\begin{aligned} \label{conBE} v_1^2+v_2^2&=&(v_1')^2+(v_2')^2\nonumber \\ {{\boldmath \mbox{$v$}}}_1+{{\boldmath \mbox{$v$}}}_2&=&{{\boldmath \mbox{$v$}}}_1'+{{\boldmath \mbox{$v$}}}_2'\end{aligned}$$ replacing the constraint (\[12con\]). The binary collision are assumed to take place at a fixed point with the spatial coordinate ${{\boldmath \mbox{$r$}}}_1$. The constraints (\[conBE\]) express the conservation of energy and momentum in the collisions. With the new transformation (\[binreacx\]) we then replace the thermodynamic force (\[Nforce\]) by $$\label{X1} X^{(1)}=\frac{1}{k_B}(f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)+f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_2)-f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}'_1)-f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}'_2)),$$ The time evolution equation (\[Nkin\]) takes now the form $$\begin{aligned} \label{intlin} \frac{\partial f({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)}{\partial t}&=&-\Xi^{(1)}_{f^*({{\boldmath \mbox{$r$}}}_1,{{\boldmath \mbox{$v$}}}_1)}\nonumber \\ &&=\int d2\int d1'\int d2'\widetilde{W}^{(1)}(f(1')f(2')-f(1)f(2))\end{aligned}$$ where $\widetilde{W}^{(1)}=\frac{W^{(1)}}{2k_B(f(1)f(2)f(1')f(2'))^{\frac{1}{2}}}$, $W^{(1)}$ is symmetric with respect to the transformation ${{\boldmath \mbox{$v$}}}_1 \leftrightarrows {{\boldmath \mbox{$v$}}}_2, \, {{\boldmath \mbox{$v$}}}'_1 \leftrightarrows {{\boldmath \mbox{$v$}}}'_2$ and $({{\boldmath \mbox{$v$}}}_1,{{\boldmath \mbox{$v$}}}_2)\rightleftarrows ({{\boldmath \mbox{$v$}}}'_1,{{\boldmath \mbox{$v$}}}'_2)$. Equation (\[intlin\]) is the Boltzmann kinetic equation without the free flow term (i.e. the right hand side of (\[intlin\]) is the Boltzmann collision operator - see more in [@GrPhysD], [@Grmadv]). Since the Boltzmann collision dissipation appears to be essentially a special case of the dissipation introduced in (\[Nkin\]), we can indeed regard the dissipation potential $\Xi^{(N)}$ in (\[XiN\]) as a natural extension of the dissipation potential generating the classical Boltzmann binary collision dissipation. There is however an interesting difference between the Boltzmann dissipation in (\[intlin\]) and the dissipation appearing in (\[Nkin\]). The former is weaker than the latter since the Boltzmann dissipation drives solutions to local equilibrium while the dissipation appearing in (\[Nkin\]) drives solutions to the total equilibrium. In general, we say that the dissipation generated by the vector field $\left(vector\,field\right)_1$ is stronger than the dissipation generated by the vector field $\left(vector\,field\right)_2$ if the inequality $\dot{\Phi}\leq 0$ holds for both vector fields but $\mathcal{M}_1\subset \mathcal{M}_2$. By $\mathcal{M}_i$ we denote the manifold whose elements are states approached as $t\rightarrow\infty$ in the time evolution generated by the vector field $\left(vector\,field\right)_i$; $i=1,2$. Let us now consider two vector fields: one is given by the right hand side of (\[intlin\]) and the other by the right hand side of $$\label{ffl} \frac{\partial f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}})}{\partial t}=-\frac{\partial}{\partial {{\boldmath \mbox{$r$}}}}({{\boldmath \mbox{$v$}}}f({{\boldmath \mbox{$r$}}},{{\boldmath \mbox{$v$}}}))$$ From the physical point of view, (\[ffl\]) is one particle kinetic equation representing a gas of completely noninteracting particles with no collisions. It is a (continuity) Liouville equation corresponding to the particle dynamics $\dot{{{\boldmath \mbox{$r$}}}}={{\boldmath \mbox{$v$}}}; \,\dot{{{\boldmath \mbox{$v$}}}}=0$. We note that the vector field (\[ffl\]) is nondissipative (i.e. Eq.(\[ffl\]) implies $\dot{\Phi}=0$) while, as we have shown above, the vector field (\[intlin\]) is dissipative (i.e. Eq.(\[intlin\]) implies $\dot{\Phi}\leq 0$) and the manifold $\mathcal{M}_{leq}$ corresponding to it is the manifold composed of local Maxwell distribution functions. Grad in [@Grad], and Desvillettes and Villani (in full generality) in [@Vill], have proven the following result. The manifold $\mathcal{M}_{teq}$ corresponding to the sum of the vector fields (\[ffl\]) and (\[intlin\]) (i.e. to the vector field appearing in the Boltzmann kinetic equation) is the manifold composed of total Maxwell distribution functions. Since $\mathcal{M}_{teq}\subset \mathcal{M}_{leq}$, we see that we can make dissipation generated by a vector field stronger just by adding to it an appropriate nondissipative vector field. This result, if transposed to the setting of N-particle dynamics for $N\geq 2$, indicates that vector fields with weaker dissipation than the vector field (\[Nkinexp\]) can possibly still drive solutions to the Gibbs equilibrium distribution function $(f_N)_{eq}$ provided the nondissipative vector field arising in the Liouville N-particle equation is added to them. What could be the vector fields that have a weaker dissipation than the vector field (\[Nkinexp\])? One way to construct them is to keep the dissipation potential (\[XiN\]), to keep the thermodynamic force (\[Nforce\]), to keep the interaction (\[12\]) but to introduce stronger constraints so that solutions to $X^{(N)}=0$ form a smaller manifold. For example, we can replace the constraint (\[12con\]) with the constraint: $(1',...,N')$ is just a reordering of $(1,...,N)$. In the case of $N=2$, this constraint becomes $(1',2')=(2,1)$. For such dissipation vector field the manifold $\mathcal{M}$ of states approached as $t\rightarrow\infty$ is the manifold of symmetric distribution functions. The time evolution generated by this vector field drives distribution functions to symmetric distribution functions. The time evolution is making the symmetrization. The following question then arises. Is this symmetrization dissipation, if combined with the nondissipative Liouville vector field, strong enough to drive solutions to the Gibbs equilibrium distribution $(f_N)_{eq}$? If the answer is negative (as it is probably the case), the next question is then to identify the dissipation potential with the weakest possible dissipation that, if combined with the Liouville vector field, does drive solutions to $(f_N)_{eq}$. In this paper we leave these questions unanswered. Before leaving this section we note that another interesting variation of the constraint (\[conBE\]) arises in the investigation of granular gases (i.e. gases composed of particles of macroscopic size). In this case the collisions are inelastic so that the first line in (\[conBE\]) is missing. Granular gas is an interesting example of an externally driven macroscopic system that can naturally be investigated on the level of kinetic theory (see e.g. [@granular]). Its thermodynamic investigation could be then based on the CR-thermodynamic potential with the dissipation potential $\Xi^{(1)}$ given in (\[XiN\]), and with two imposed forces: thermodynamic force $X^{(1)}$ (see (\[X1\])) representing the inelastic collisions, and a mechanical force $X^{(mech)}$ representing for example shaking. equilibrium $\rightarrow$ equilibrium (imposed temperature) {#EX3} ----------------------------------------------------------- The external force in this example is the energy exchange with thermal bath that is kept at a constant temperature $\mathfrak{T}$. In this case the externally driven macroscopic system evolves to an equilibrium state. This means that in this investigation of the $MESO \rightarrow meso$ passage the *meso level* is the equilibrium level. The difference between the $MESO \rightarrow equilibrium$ passage investigated in Section \[RDMMde\] and the $MESO \rightarrow equilibrium$ passage investigated in this example is that the state variables $y_{eq}$ at the equilibrium level are not $(E,V,N)$ (as in Section \[ET\]) but $(E^*,V,N)$, where $E^*=S_E=\frac{1}{T}$ is the conjugate of $E$. The CR thermodynamic potential $\Psi$ driving the evolution is, in this example, the thermodynamic potential (\[Phi1\]) with $T=\mathfrak{T}$, i.e. $\Phi(x,\mathfrak{T},\mu)=-S(E,V,N)+\frac{1}{\mathfrak{T}}E-\frac{\mu}{\mathfrak{T}}N$. The CR-GENERIC equation (\[CRGENERIC\]) is, in this example, the GENERIC time evolution (\[GENERIC\]) with $T=\mathfrak{T}$ and with the degeneracies of $L$ and $\Xi$ that guarantee the mass conservation (\[consNGEN\]) but not the energy conservation (\[consEGEN\]). The resulting fundamental thermodynamic relation is the Legendre transformation $(E,N,V)\rightarrow(\frac{1}{T},N,V)$ of the fundamental thermodynamic relation $S=S(E,V,N)$ implied by the GENERIC reducing time evolution discussed in Section \[RDMMde\]. The above analysis becomes particularly interesting if we choose the *MESO level* to be the equilibrium level with state variables $(E,V,N)$. In this case of $MESO \rightarrow meso$ passage both *MESO* and *meso levels* are *equilibrium levels*. They differ only in state variables: on *MESO level* the state variables are $(E,N,V)$ and on *meso level* $(E^*,V,N)$ The reducing time evolution in this $equilibrium \rightarrow equilibrium$ passage is the time evolution making the Legendre transformation $(E,N,V)\rightarrow(\frac{1}{T},N,V)$. The thermodynamic potential generating it is $$\label{phie} \Phi(E,N,V,\mathfrak{T})=-S(E,N,V)+\frac{1}{\mathfrak{T}}E$$ and the time evolution equation (\[CRGENERIC\]) becomes $$\label{ee} \dot{E}=-[\Xi_X(E,X)]_{X=-S_E+\frac{1}{\mathfrak{T}}}$$ that, if we choose the quadratic dissipation potential $\Xi(E,X)=\frac{1}{2}\Lambda X^2$, where $\Lambda>0$ is a material parameter, becomes $\dot{E}= -\Lambda (-S_E+\frac{1}{\mathfrak{T}})$. Cattaneo $\rightarrow$ Fourier (imposed temperature gradient) {#EX4} ------------------------------------------------------------- In this example the external force is an imposed temperature gradient. We denote it by the symbol $\nabla\frac{1}{\mathfrak{T}}$. This force prevents approach to *equilibrium level*. The most macroscopic level (i.e. the level with least details) on which macroscopic systems subjected to temperature gradient can be described is the level of fluid mechanics (we shall call it hereafter FM-level) on which the state variables are: $x=(\rho({{\boldmath \mbox{$r$}}}),{{\boldmath \mbox{$u$}}}({{\boldmath \mbox{$r$}}}),e({{\boldmath \mbox{$r$}}}))$, where ${{\boldmath \mbox{$r$}}}$ is the position vector, $\rho({{\boldmath \mbox{$r$}}})$ is the mass field (mass per unit volume at ${{\boldmath \mbox{$r$}}}$), ${{\boldmath \mbox{$u$}}}({{\boldmath \mbox{$r$}}})$ is the momentum field, and $e({{\boldmath \mbox{$r$}}})$ the energy field. In this example we shall limit ourselves only to the state variable $e({{\boldmath \mbox{$r$}}})$. All other state variables are assumed to be already at equilibrium. In this setting we now investigate the passage $MESO \rightarrow FM$. First, we recall that in the absence of the imposed temperature gradient (i.e. if $\nabla\frac{1}{\mathfrak{T}}=0$, the macroscopic systems under consideration will approach to *equilibrium level* and we can therefore consider the level of fluid mechanics (called FM-level) as *MESO level* and investigate the passage $FM \rightarrow equilibrium$. The GENERIC equation (\[GENERIC\]) representing this passage is well known and can be found for example in [@Grmadv]. Now we switch on the external force (i.e. $\nabla\frac{1}{\mathfrak{T}}\neq 0$) and investigate the $MESO \rightarrow FM$ passage. We proceed to find the CR-GENERIC equation representing it. The state variables that evolve in the reducing time evolution is the heat flux $J^{(h)}$. The CR relation is the Fourier constitutive relations: $X^{(h)}_i=\nabla_i\frac{1}{S^{(leq)}_E}$, where $S^{(leq)}(\rho({{\boldmath \mbox{$r$}}}),{{\boldmath \mbox{$u$}}}({{\boldmath \mbox{$r$}}}),e({{\boldmath \mbox{$r$}}}))$ is the local equilibrium entropy on the *FM level*. We use hereafter the indices $i=1,2,3;\, j=1,2,3$ and the summation convention. The CR thermodynamic potential in this example is $$\label{CRFM} \Phi^{(MFM)}(J^{(h)}; (\nabla\frac{1}{\mathfrak{T}}))=-S^{(0MFM)}(J^{(h)})+(\nabla\frac{1}{\mathfrak{T}})_iJ^{(h)}_i$$ In the CR-GENERIC equation (\[CRGENERIC\]) we neglect the Hamiltonian time evolution (i.e. we put $\mathcal{L}\equiv 0$) and, for the sake of simplicity, choose the dissipation potential $\Xi^{(FMe)}(J^{(h)}),X^{(h)})=\frac{1}{2}\int d{{\boldmath \mbox{$r$}}}(\Lambda^{(h)}X_i^{(h)}X_i^{(h)}$, where $\Lambda^{(h)}>0$ is a material parameter. With these specifications the CR-GENERIC equation (\[CRGENERIC\]) becomes $$\label{delt} \dot{J}^{(h)}_i=-\Lambda^{(h)}(-S^{(0FMe)}_{J^{(h)}_i}+(\nabla\frac{1}{\mathfrak{T}})_i)$$ The fundamental thermodynamic relation on the *FM level* implied by the *MESO level* CR-GENERIC equation (\[delt\]) is $$\label{frFM} S^{(FMM)*}(e({{\boldmath \mbox{$r$}}});(\nabla\frac{1}{\mathfrak{T}}))=\left[\Phi^{(MFM)}(J^{(h)}; (\nabla\frac{1}{\mathfrak{T}}))\right]_{S^{(0MFM)}_{J^{(h)}_i}=(\nabla\frac{1}{\mathfrak{T}})_i}$$ where $\Phi^{(MFM)}$ is the CR thermodynamic potential (\[CRFM\]). We note that Eq.(\[delt\]) is the well known Cattaneo equation [@Cattaneo] provided the imposed external force $\nabla\frac{1}{\mathfrak{T}}$ is replaced by $\nabla\frac{1}{T}$, where $T({{\boldmath \mbox{$r$}}})$ is the local temperature. There is however an important difference between the role it plays in extended thermodynamic theories in [@MullRugg], [@Joubook] and in this paper. In the context of our investigation it is the equation describing approach of the Cattaneo extended fluid dynamics (playing the role of *MESO level*) to the classical fluid mechanics with the Fourier constitutive relation (playing the role of *meso level*). The Cattaneo time evolution driven by the CR thermodynamic potential (\[CRFM\]), implying the CR fundamental thermodynamic relation (\[frFM\]) on the level of classical fluid mechanics, describes $MESO\rightarrow meso$ passage. In the extended theories investigated in [@MullRugg], [@Joubook] the Cattaneo equation is the equation arising in the $MESO\rightarrow equilibrum$ passage. It is just an extra equation (governing the time evolution of the extra state variable and coupled to the other time evolution equations) in the set of extended fluid mechanics equations whose solutions are required to approach equilibrium states. This means that the physical systems under investigation in [@MullRugg], [@Joubook] are externally unforced. Thermodynamics of immiscible blends; phase inversion {#EX5} ---------------------------------------------------- In this section we apply CR-thermodynamics to immiscible blends. We recall that the extension of the classical equilibrium thermodynamics of single component macroscopic systems to multicomponent miscible blends led Gibbs to the completion of the mathematical formulation of equilibrium thermodynamics. Further extensions to immiscible blends require to leave the realm of equilibrium thermodynamics and to enter CR-thermodynamics. Imposed external forces (e.g. imposed flows in the mixing process) prevent approach to equilibrium states. Moreover, extra variables addressing morphology of the interfaces among the components are needed to characterize their states. We shall not attempt in this paper to make a systematic investigation of CR-thermodynamics of immiscible blends. We shall concentrate only on one particular problem and use thermodynamics to investigate it. The immiscible blend that we consider is composed of two immiscible fluids (component “1”, and component “2”). The problem that we investigate is phase inversion. Let initially the component “1” form a continuous phase in which the component “2” is dispersed. This means that the component “2” resides inside drops encircled completely by the component “1”. Every two points in the component “1” can be joined by a line that lies completely inside the component “1”. We shall now increase the amount of the component “2”. We anticipate that at some volume fraction $\phi_2$ of the second component the roles of the two components change, the second component becomes the continuous phase and the first component becomes the dispersed phase. At the critical state at which the change occurs both components form a continuous phase. The morphology at the critical state is called a co-continuous morphology. The problem that we want to investigate is to estimate the critical value of $\phi_2$ as a function of the properties of the components (as for instance the viscosity, elasticity etc.) and of the blending conditions (i.e. the externally imposed forces). The co-continuous morphology is in particular very important in applications involving blends of polymer melts (see e.g. [@Favis]). In order to have a specific example in mind, we can think of immiscible blends of oil and water. If water is the continuous phase then the blend is milk, if oil is the continuous phase then the blend is butter. One way to approach the problem of phase inversion is by attempting to formulate a dynamical model of immiscible blends. In this paper we shall not take this route. We turn directly to thermodynamics. The concept with which we begin is the CR thermodynamic potential (\[PhiMm\]). Next, we regard phase inversion as phase transition. In view of the comment that we made at the end of Section \[GF\], this means that at the point of phase inversion $$\label{Phinver} \Phi^{(mM)}_{1/2}=\Phi^{(mM)}_{2/1}$$ where $\Phi^{(mM)}_{i/k}$ is the CR thermodynamic potential when the component “i” forms the continuous phase and the component “k” the disperse phase. This is the equation that answers our question. It remains now only to specify the CR thermodynamic potential $\Phi^{(mM)}$. We shall limit ourselves to most simple specifications that nevertheless illustrate the power of CR-thermodynamics. Following (\[Psi\]), we write $$\label{Pimbl} \Phi^{(mM)}= -S^{(0mM)}+\frac{1}{T_0}W$$ where we consider $S^{(0mM)}$ to be simply the local entropy production due to the presence of the dispersed phase, $W$ the work involved in elastic deformations of the dispersed droplets, and $T_0$ the temperature of the blend. Our problem now is to estimate $S^{(0Mm)}$ and $W$ in the mixture in which $\phi_1$ and $\phi_2$ are not too different. Let the component “1”, forming the continuous phase, be a fluid of viscosity $\eta_1$. The main contribution to the entropy production $S^{(0Mm)}_{1/2}$ comes from the flow in the continuous phase “1”. $S^{(0mM)}_{1/2}$ is thus $\alpha_2\eta_1\phi_2 \dot{\gamma}^2$, where $\alpha_2$ is a parameter depending on the shape of the inclusion and $\dot{\gamma}$ is the absolute value of the shear rate. The main contribution to the work $W_{1/2}$ is assumed to come also from the continuous phase, i.e. $W_{1/2} \sim H_1D_1^2\dot{\gamma}$, where $H_1$ is the elastic constant and $D_1$ is the deformation displacement of the matrix). We therefore obtain $$\label{P1} S^{(0mM)}_{1/2}=\alpha_2\phi_2\eta_1\dot{\gamma}^2;\,\,\, S^{(0mM)}_{2/1}=\alpha_1\phi_1\eta_2\dot{\gamma}^2$$ and $$\label{W} W_{1/2}=\phi_1H_1D_1^2\dot{\gamma};\,\,\, W_{2/1}=\phi_2H_2D_2^2\dot{\gamma}$$ By inserting these relations to (\[Pimbl\]) and (\[Phinver\]) we arrive finally at the estimate of the critical volume fraction $$\label{result} \frac{\phi_1}{\phi_2}=\frac{\alpha_2\eta_1\dot{\gamma}+\frac{1}{T_0}H_2D_2^2}{\alpha_1\eta_2\dot{\gamma}+\frac{1}{T_0}H_1D_1^2}$$ The above estimate appears to be an extension of several empirical formulas that can be found in the literature. For example, if we neglect the elasticity of the two fluids (i.e. we put $W_{1/2}=W_{2/1}=0$), or if the mixing is very vigorous (i.e. if $\dot{\gamma}$ is large so that the terms in (\[result\]) involving the viscosity are much larger than the terms involving the elastic energy), or also if $T_0$ is large then (\[result\]) becomes $\frac{\phi_1}{\phi_2}=\frac{\alpha_2\eta_1}{\alpha_1\eta_2}$. If, in addition, we neglect the shape factor (i.e. $\alpha_1=\alpha_2=1$) then we arrive at the estimate $\frac{\phi_1}{\phi_2}=\frac{\eta_1}{\eta_2}$ which is indeed the empirical formula introduced in [@phinv]. Concluding remarks {#CR} ================== Reducing dynamics $MESO\rightarrow meso$ is a dynamics bringing a mesoscopic level of description (called *MESO level*) to another mesoscopic level of description (called *meso level*) that involves less details. By identifying the reducing dynamics with thermodynamics we have been able to formulate a general thermodynamics that encompasses the classical equilibrium thermodynamics (corresponding to $equilibrium\rightarrow equilibrium$), the equilibrium statistical mechanics (corresponding to $MICRO\rightarrow equilibrium$), mesoscopic equilibrium thermodynamics (corresponding to $MESO\rightarrow equilibrium$), and thermodynamics of externally driven systems (corresponding to $MESO\rightarrow meso$). The general thermodynamics is presented in three postulates. The first postulate (called Postulate 0 in order to keep as much as possible the traditional terminology) states that there exist well established mesoscopic levels of description. By well established we mean well tested with experimental observations. This postulate generalizes the postulate of the existence of the equilibrium states that serves as a basis of the classical equilibrium and the Gibbs statistical equilibrium thermodynamics. The second postulate (called Postulate I) is about state variables used on the *MESO* and *meso levels* and about potentials needed to formulate mechanics. Again, this postulate generalizes the classical Postulate I in the classical equilibrium thermodynamics. The third postulate (Postulate II) addresses the process (called a preparation process) in which the macroscopic systems are prepared to states at which the *meso* description is found to agree with a certain family of experimental observations forming the experimental basis of the *meso* description. The time evolution making the preparation process is called reducing time evolution. In the classical equilibrium thermodynamics this postulate is the static Maximum Entropy principle (static MaxEnt principle) specifying only the final result of the preparation process. In the general *MESO* and *meso* descriptions, it is dynamic MaxEnt principle postulating equation governing the time evolution making the preparation processes. Two important results arise on the *meso level* from the static or the dynamic MaxEnt principles. First, it is the *meso level* time evolution (the time evolution reduced from the time evolution on the *MESO level*). Second, it is the fundamental thermodynamic relation that is constructed from the potential generating the reducing time evolution. The generating potential has the physical interpretation of entropy if the *meso level* in the approach $MESO \rightarrow meso$ is the equilibrium level and entropy production (or related to it quantity) if the *meso level* in the approach $MESO \rightarrow meso$ is a general *meso level*. In the classical, or the Gibbs statistical, equilibrium thermodynamics (i.e. if the meso-level in the approach $MESO \rightarrow meso$ is the equilibrium level) the reduced *meso* dynamics is in this case no dynamics. The fundamental thermodynamic relation is, in the context of the classical or the Gibbs statistical equilibrium thermodynamics, the classical equilibrium fundamental thermodynamic relation. In the context of *MESO* and *meso* descriptions of externally driven macroscopic systems it is a new relation on the *meso level* representing its thermodynamics. This thermodynamics is not directly related to the *meso* dynamics (as, indeed, the classical equilibrium fundamental thermodynamic relation is in no relation to no dynamics at equilibrium). It represents an extra information about macroscopic systems. The *meso* dynamics is the *MESO* dynamics seen on the *meso level* and the thermodynamics is an information extracted from the way how the details (that are seen on the *MESO level* but are invisible on the *meso level*) are being forgotten. The fourth postulate (Postulate III of the classical equilibrium thermodynamics) addresses the value of entropy at zero absolute temperature. Investigations of macroscopic systems at such extreme conditions are outside the scope of this paper. We are not extending this postulate to mesoscopic dynamical theories. In conclusion, we have demonstrated that if we limit ourselves to one fixed *meso level* (e.g. the level of fluid mechanics) then the dynamic and the thermodynamic modeling represent two essentially independent ways to investigate externally driven macroscopic systems. In particular, the validity and the pertinence of the thermodynamic *meso* models does not depend on establishing their relation to the dynamic *meso* models. If however we make our investigation simultaneously on two well established levels, one *MESO level* (that involves more details than the *meso level*) and the other the chosen *meso level* then we can derive both the *meso* dynamics and the *meso* thermodynamics from the *MESO* dynamics. The derivation consists of splitting the *MESO* time evolution into reducing time evolution (providing the *meso* thermodynamics) and reduced time evolution that becomes the *meso* time evolution. In most investigations of reductions the attention is payed only the the reduced dynamics. We hope that this paper will stimulate investigations in both reduced and reducing dynamics. What are the arguments supporting the three postulates of the general thermodynamics? In the case of $equilibrium\rightarrow equilibrium$ passage they become the standard postulates of the classical equilibrium thermodynamics. In the case of $MICRO\rightarrow equilibrium$ passage they become a formulation (equivalent to many other existing formulations) of the Gibbs equilibrium statistical mechanics. In the case of $MESO\rightarrow equilibrium$ passage, a large body of supporting evidence has been collected (see in particular [@Grmadv] and references cited therein and in [@Obook]). 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--- abstract: 'Small non-autonomous perturbations around an equilibrium of a nonlinear delayed system are studied. Under appropriate assumptions, it is shown that the number of $T$-periodic solutions lying inside a bounded domain $\Omega\subset \R^N$ is, generically, at least $|\chi \pm 1|+1$, where $\chi$ denotes the Euler characteristic of $\Omega$. Moreover, some connections between the associated fixed point operator and the Poincaré operator are explored.' author: - 'P. Amster' - 'M. P. Kuna' - 'G. Robledo' title: Multiple solutions for periodic perturbations of a delayed autonomous system near an equilibrium --- Introduction ============ Let $\Omega\subset \R^N$ be a bounded domain with smooth boundary. An elementary result from the theory of ODEs establishes that if a smooth function $G:\overline\Omega\to \R^N$ is inwardly pointing over $\partial\Omega$, that is $$\label{hart-weak} \langle G(x),\nu(x)\rangle <0 \qquad x\in \partial\Omega,$$ where $\nu(x)$ denotes the outer normal at $x$, then the solutions of the autonomous system of ordinary differential equations $$u'(t)=G(u(t))$$ with initial data $u(0)=u_0\in \overline \Omega$ are defined and remain inside $\Omega$ for all $t>0$. [Now, let us denote the space of $T$–periodic continuous functions as $$C_T:=\{u\in C(\R,\R^N):u(t+T)=u(t)\}$$ and, for given $p\in C_{T}$, consider the non-autonomous system $$u'(t)=G(u(t)) + p(t).$$]{} [If $\overline\Omega$ has the fixed point property, then the above system has at least one $T$-periodic orbit, provided that $\|p\|_\infty$ is small.]{} This is a straightforward consequence of the fact that the time-dependent vector field $G(x)+ p(t)$ is still inwardly pointing for all $t$; hence, the set $\overline \Omega$ is invariant for the associated flow and thus the Poincaré operator given by $Pu_0:=u(T)$ is well defined for $u_0\in \overline\Omega$ and satisfies $P(\overline\Omega)\subset \overline\Omega$. More generally, observe that, when (\[hart-weak\]) is assumed, the homotopy defined by $h(x,s):= sG(x) - (1-s)\nu(x)$ with $s\in [0,1]$ does not vanish on $\partial\Omega$; whence $$deg_B(G,\Omega,0) = deg_B(-\nu,\Omega,0),$$ where $deg_B$ stands for the Brouwer degree. Thus, it follows from [@hopf] that $deg_B(G,\Omega,0)=(-1)^N\chi(\Omega)$, where $\chi(\Omega)$ denotes the Euler characteristic of $\Omega$. It is worthy to recall (see *e.g.*,[@wecken]) that if $\overline \Omega$ has the fixed point property, then $\chi(\Omega)$ is different from $0$. This follows easily in the present setting from the fact that if $\chi(\Omega)=0$ then one can construct a field $G$ satisfying (\[hart-weak\]) that does not vanish in $\Omega$. If $\overline\Omega$ has the fixed point property, then there exist (non-constant) $T$-periodic solutions of all periods which, in turn, implies that $G$ vanishes, a contradiction. Interestingly, the converse of the result in [@wecken] is not true; that is, one can easily find $\Omega$ with nonzero Euler characteristic such that $\overline \Omega$ has not the fixed point property. For such a domain, the Poincaré map has obviously a fixed point (because $G$ vanishes in $\Omega$). This yields the conclusion that a fixed point-free map in $C(\overline \Omega,\overline\Omega)$ cannot belong to the closure of the set of all the Poincaré maps associated to the homotopy class of $-\nu$. Now suppose, independently of the value of $\chi(\Omega)$, that $G$ vanishes at some point $e\in \Omega$, namely, that $e$ is an equilibrium point of the autonomous system. It is well known that if $M:=DG(e)$ is nonsingular, then the degree of $G$ over any small neighbourhood $V$ of $e$ is well defined and coincides with $s(M)$, where $$\label{sM} s(M):= sgn ({\rm det}(M)).$$ Thus, if $s(M)$ is different from $(-1)^N\chi(\Omega)$, then the excision property of the degree implies that the system has at least another equilibrium point in $\Omega\setminus \overline V$. Furthermore, it follows from Sard’s lemma that, for almost all values $\overline p$ in a neighbourhood of $0\in \R^N$, the mapping $G + \overline p$ has at least $\Gamma$ different zeros in $\Omega$, with $$\label{Gamma} \Gamma=\Gamma(M):=|\chi(\Omega)- (-1)^{N} s(M)| + 1.$$ Thus, one might expect that if $p\in C(\R,\R^N)$ is $T$-periodic and $\|p\|_\infty$ is small, then the number of $T$-periodic solutions of the non-autonomous system is generically greater or equal to $\Gamma$. Here, ‘generically’ should be understood in the sense of Baire category, that is, the property is valid for all $p$ (close to the origin) in the space of continuous $T$-periodic except for a meager set. It can be shown, indeed, that the fixed point index of the Poincaré map $P$ at $e$ is equal to $(-1)^Ns(M)$ and, moreover, a homotopy argument shows that the degree of $P$ over $\Omega$ is equal to $\chi(\Omega)$. Details are omitted because the result follows from the main theorem of the present paper. For several reasons, the situation is different for the delayed system $$\label{ec} u'(t) = g(u(t),u(t-\tau))$$ where, for simplicity, we shall assume that $g:\overline\Omega\times \overline\Omega\to \R^N$ is continuously differentiable. In the first place observe that, due to the delay, the condition that the field $G(x):=g(x,x)$ is inwardly pointing does not necessarily avoid that solutions with initial data $x_0:=\phi\in C([-\tau,0],\overline\Omega)$ may eventually abandon $\overline\Omega$. However, taking into account that $$|u(t_0-\tau)- u(t_0)| \le \tau \max_{t\in [t_0-\tau,t_0]} |u'(t)|,$$ it follows that the flow-invariance property, now over the set $C([-\tau,0],\overline\Omega)$, is retrieved under the stronger assumption $$\label{hart} \langle g(x,y),\nu(x)\rangle < 0 \qquad (x,y)\in \mathcal A_\tau (\Omega)$$ where $$\mathcal A_\tau (\Omega):= \{ (x,y)\in \partial\Omega\times \overline\Omega: |y-x|\le \tau\|g\|_{\infty}\}.$$ In the second place, the previous considerations regarding the Poincaré map become less obvious, since the latter is now defined not over $\overline\Omega$ but over the metric space $C([-\tau,0],\overline\Omega)$. In connection with this fact, we recall that the characteristic equation for the autonomous linear delayed systems is transcendental (also called quasipolynomial equation), so there exist typically infinitely many complex characteristic values. Throughout the paper, we shall assume as before that system (\[ec\]) has an equilibrium point $e\in \Omega$, that is, such that $g(e,e)=0$. This necessarily occurs when $\chi(\Omega)\neq 0$, although this latter condition shall not be imposed. Denote by $A,B\in \R^{N\times N}$ the respective matrices $D_xg(e,e)$ and $D_yg(e,e)$. Again, if $A+B$ is nonsingular and $s(A+B)$ is different from $(-1)^N\chi(\Omega)$, then the system has at least one extra equilibrium point in $\Omega$; furthermore, the number of equilibria in $\Omega$ is generically greater or equal to $\Gamma$. This is readily verified by writing the set of all the functions $g\in C^1(\overline\Omega\times\overline\Omega,\R^N)$ satisfying (\[hart\]) as the union of the closed sets $$X_n:=\left\{g\in C^1(\overline\Omega\times\overline\Omega,\R^N): \langle g(x,y),\nu(x)\rangle \le -\frac 1n \quad\hbox{for $(x, y)\in \mathcal A_\tau (\Omega)$} \right\}$$ and noticing that $X_n\cap \mathcal C$ is nowhere dense, where $ \mathcal C$ denotes the set of those functions $g$ such that $0$ is a critical value of the corresponding $G$. Our goal in this work is to extend the preceding ideas for non-autonomous periodic perturbations of (\[ec\]), namely the problem $$\label{nonaut} u'(t) = g(u(t),u(t-\tau)) + p(t)$$ with [$p\in C_{T}$]{}. As a basic hypothesis, we shall assume that the linearisation at the equilibrium, that is, the system $$\label{linear} u'(t) = Au(t)+ Bu(t-\tau)$$ has no nontrivial $T$-periodic solutions. This clearly implies, in particular, the above-mentioned condition that $A+B$ is invertible. From the Floquet theory for DDEs, it is known that the latter condition is also sufficient for nearly all positive values of $T$ (*i.e.*, except for at most a countable set). For the sake of completeness, this specific consequence of the Floquet theory shall be shown below (see Remark \[remark1\]). Our main result reads as follows. \[main\] Let the equilibrium $e$ and the matrices $A$ and $B$ be as before and assume that the linear system (\[linear\]) has no nontrivial $T$-periodic solutions. Then: - There exists $r>0$ such that [for any $p\in C_{T}$]{} with $\|p\|_\infty<r$ the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution. - If moreover (\[hart\]) holds and $ s(A+B) \neq (-1)^N\chi(\Omega) $ with $s$ defined as in (\[sM\]), then (\[nonaut\]) has at least two $T$-periodic solutions. - Furthermore, there exists a residual set $\Sigma_r\subset C_T$ such that if $p\in \Sigma_r\cap B_r(0)$, then the number of $T$-periodic solutions is at least $\Gamma(A+B)$, where $\Gamma$ is given by (\[Gamma\]). The next result is an immediate consequence of [Theorem \[main\] combined with the preceding comments]{}. \[corol\] Let $e, A$ and $B$ be as before and assume that $A+B$ is invertible. Then for nearly all $T>0$ there exists $r=r(T)>0$ such that if [$p\in C_{T}$]{} with $\|p\|_\infty<r$ then the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution. If moreover (\[hart\]) holds and $ s(A+B) \neq (-1)^N\chi(\Omega), $ then the number of $T$-periodic solutions is at least $2$ and generically $\Gamma(A+B)$. [For small delays, the condition that (\[linear\])]{} has no nontrivial $T$-periodic solutions can be formulated explicitly in terms of the matrix $A+B$ : \[smalldelay\] Let $e, A$ and $B$ be as before and assume that $\frac{2k\pi}Ti$ is not an eigenvalue of the matrix $A+B$ for all $k\in\N_0$. Then for each $\tau$ small enough there exists $r=r(\tau)$ such that the non-autonomous problem (\[nonaut\]) has at least one $T$-periodic solution for any [$p\in C_{T}$]{} with $\|p\|_\infty<r$. If moreover (\[hart-weak\]) holds for $G(x):=g(x,x)$ and $s(A+B)\ne (-1)^N\chi(\Omega)$, then (\[nonaut\]) has at least two $T$-periodic solutions and generically $\Gamma(A+B)$. It is worthy mentioning that if $\Omega$ is for example a ball, then the condition $s(A+B)\neq (-1)^N\chi(\Omega)$ implies that the equilibrium is unstable. As we shall see, this can be regarded as a consequence of the fact that the Leray-Schauder index of the fixed point operator defined in the proof of our main theorem is $(-1)^{N+1}$. This connection can be deduced from a version of the Krasnoselskii relatedness principle, which implies that the mentioned index coincides except for a $(-1)^N$ factor with that of the Poincaré operator. As shown in Proposition \[poinc-stab\], this implies, in turn, that the equilibrium cannot be stable. The paper is organised as follows. In the next section, we prove some basic facts concerning the linearised problem (\[linear\]); in particular, we give a necessary and sufficient condition in order to ensure that it has no nontrivial $T$-periodic solutions. In section \[dem\] we present a proof of Theorem \[main\] by means of an appropriate fixed point operator. The next two sections are devoted to a proof In section \[sec-delay\], we give a proof Corollary \[smalldelay\]. In section \[poincare\], we make some considerations on the stability of the equilibrium and the indices, on the one hand, of the fixed point operator defined in section \[dem\] and of the Poincaré map, on the other hand. Finally, a simple application of the main results for a singular system is introduced in section \[exam\]. Linearised system ================= In this section, we shall prove some basic facts concerning the linear system (\[linear\]). To this end, let us introduce some notation. For $k\in \mathbb N_0$, define $$\lambda_k:= \frac{2k\pi}T$$ and $$\varphi_k(t):= \cos(\lambda_k t) \qquad \psi_k(t):= \sin (\lambda_k t).$$ It is readily verified that $$\varphi_k(t-\tau)= \varphi_k(t)\varphi_k(\tau) + \psi_k(t)\psi_k(\tau)$$ $$\psi_k(t-\tau)= \psi_k(t)\varphi_k(\tau) - \varphi_k(t)\psi_k(\tau)$$ and $$\varphi_k'= -\lambda_k \psi_k,\qquad \psi_k'=\lambda_k\varphi_k.$$ For an element $u\in C_T$, we may consider its Fourier series, namely $$u = a_0 + \sum_{k=1}^\infty (\varphi_k a_k +\psi_k b_k)$$ in the $L^2$ sense, with $a_k, b_k\in \R^N$. Furthermore, recall that if $u$ is smooth (*e.g.*, of class $C^2$) then the series and its term-by-term derivative converge uniformly to $u$ and $u'$ respectively. \[lema\] Let $u\in C_T$ and define $$\label{matrices} X_k:=A+\varphi_k(\tau)B, \qquad Y_k:=\lambda_kI + \psi_k(\tau) B.$$ Then $u$ is a solution of (\[linear\]) if and only if $$\label{matr-ident} \left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right) \left(\begin{array}{c} a_k\\ b_k \end{array} \right) = \left(\begin{array}{c} 0\\ 0 \end{array} \right)$$ for all $k\in \mathbb N_0$. Since $\varphi_k'(t), \varphi_k(t-\tau), \psi_k'(t)$ and $\psi_k(t-\tau)$ belong to ${\rm\bf span}\{ \varphi_k(t),\psi_k(t)\}$, it follows that $u$ is a solution of of (\[linear\]) if and only if $$(A+B)a_0=0$$ and $$\varphi_k'(t) a_k +\psi_k'(t) b_k = A(\varphi_k(t) a_k +\psi_k(t) b_k) + B(\varphi_k(t-\tau) a_k + \psi_k(t-\tau) b_k)$$ for all $k>0$. The latter identity, in turn, is equivalent to $$\begin{array}{ccc} \lambda_k b_k & = & [A+ \varphi_k(\tau) B] a_k - \psi_{k}(\tau) Bb_k \\ {} \\ -\lambda_k a_k & = & \psi_{k}(\tau)Ba_k + [A+ \varphi_k(\tau) B] b_k, \end{array}.$$ that is, $$X_ka_{k} -Y_kb_{k}= Y_ka_{k} + X_kb_{k}= 0.$$ Because $X_0=A+B$ and $Y_0=0$, we deduce that $u$ is a solution of (\[linear\]) if and only if (\[matr-ident\]) holds for all $k\in \mathbb N_0$. \[no-nontrivial\] (\[linear\]) has no nontrivial $T$-periodic solutions if and only if $$\label{nec-suf} h_k:={\rm det}\left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right)\neq 0$$ for all $k\in \mathbb N_0$. \[remark1\] 1. Because $A+B$ is invertible, it is clear that for nearly all $T>0$ condition (\[nec-suf\]) is satisfied for all $k$. Indeed, it suffices to observe that $h_k$, regarded as a function of $T\in (0,+\infty)$, is an analytic function and, consequently, it has at most a countable number of zeros. 2. It can be shown that $h_k\ge 0$; in particular, its roots have even multiplicity. The proof is straightforward when $A$ and $B$ commute, since in this case $${\rm det}\left(\begin{array}{cc} X_k & -Y_k\\ Y_k & X_k \end{array} \right) = {\rm det}(X_k ^2+Y_k ^2).$$ The conclusion then follows, because for any pair of square real matrices $X, Y$ such that $XY=YX$ it is verified that $${\rm det}(X ^2+Y^2)= {\rm det}[(X+iY)(X-iY)] = {\rm det}(X+iY)\overline{{\rm det}(X+iY)}\ge 0.$$ [A proof for the non-commutative case is given below in section \[dem\], step \[directo-fourier\]. ]{} It is noticed that (\[nec-suf\]) may hold for non-invertible matrices $X_k$ and $Y_k$: for instance, observe that $$\left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)^2 + \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)^2 = I.$$ 3. \[k\_0\] Since $\lambda_k\to +\infty$ it follows, for $k$ large, that $$h_k={\rm det}(Y_k){\rm det}(Y_k + X_kY_k^{-1}X_k)\simeq \lambda_k^{2N} > 0.$$ In particular, there exists $k_0$ such that if $u$ is a $T$-periodic solution of (\[linear\]) then $a_k=b_k=0$ for $k>k_0$. This means that $u$ is a (vector) trigonometric polynomial. Incidentally, observe that, because the family $\{\varphi_k, \psi_k\}$ is uniformly bounded, the constant $k_0$ may be chosen independent of $\tau$. [ In other words, if we consider the linear operator $L:C_T\to C_T$, given by $Lu(t):=u'(t) - Au(t) - Bu(t-\tau)$, then $ {\rm ker}(L)\subset {\rm \bf span}\{\varphi_k,\psi_k\}_{0\le k\le k_0}$. Observe furthermore that ${\rm Im}(L)$ consists of all the Fourier series $a_0+ \sum_{k>0}(\varphi_ka_k + \psi_kb_k)$ such that $a_0\in {\rm Im}(A+B)$ and $(a_k,b_k)\in {\rm Im}(M_k)$, where $M_k$ is the matrix defined in (\[matr-ident\]). This yields a direct proof of the well-known fact that $L$ is a zero-index Fredholm operator. Moreover, it is verified that $(a_k,b_k)\in {\rm ker}(M_k)\iff (-b_k,a_k) \in {\rm ker}(M_k)$, a fact that will be of relevance in the proofs of our results.]{} Proof of the main theorem {#dem} ========================= For convenience, a little of extra notation shall be introduced. For a function $u\in C_T$, let us write $$\mathcal Iu(t):= \int_0^t u(s)\, ds, \qquad \overline u:= \frac 1T {\mathcal Iu (T)}.$$ Moreover, denote by $\mathcal N$ the Nemitskii operator associated to the problem, namely $$\mathcal Nu(t):= g(u(t),u(t-\tau)).$$ Without loss of generality we may assume $e=0$ and fix $T>0$ such that (\[linear\]) has no nontrivial $T$-periodic solutions. For simplicity, we shall assume from the beginning that all the assumptions are satisfied; it shall be easy for the reader to deduce the existence of one solution near the equilibrium when (\[hart\]) is not satisfied. Define the open bounded set $U=\{u\in C_T:u(t)\in \Omega\,\hbox{ for all $t$} \}$ and the compact operator $K:\overline U\to C_T$ defined by $$Ku(t):= \overline u -t\, \overline {\mathcal Nu} + \mathcal I\mathcal Nu(t) - \overline{\mathcal I\mathcal Nu}.$$ We shall prove that the Leray-Schauder degree of $I-K$ is equal to $(-1)^N\chi(\Omega)$ over $U$ and to $s(A+B)$ over $B_\rho(0)$ for small values of $\rho>0$. To this end, let us proceed in several steps: 1. Let $K_0u:= \overline u -\frac T2 \overline {\mathcal Nu}$ and define, for $s\in [0,1]$, the operator given by $K_s:=s K +(1-s)K_0$. We claim that $K_s$ has no fixed points on $\partial U$. Indeed, for $s>0$ it is clear that $u\in\overline U$ is a fixed point of $K_s$ if and only if $u'(t)=s\mathcal Nu(t)$, that is: $$u'(t)= sg(u(t),u(t-\tau)).$$ Suppose there exists $t_0$ such that $u(t_0)\in\partial\Omega$, then we deduce, as before, $$|u(t_0-\tau)- u(t_0)| \le \tau \max_{t\in [t_0-\tau,t_0]} |u'(t)| \le \tau \|g\|_ \infty$$ and by (\[hart\]) we obtain $$0= \langle u'(t_0),\nu(u(t_0))\rangle = s\langle g(u(t_0),u(t_0-\tau)),\nu(u(t_0))\rangle <0,$$ a contradiction. On the other hand, we observe that the range of $K_0$ is contained in the set of constant functions, which can be identified with $\R^N$; thus, the Leray-Schauder degree of $I-K_0$ can be computed as the Brouwer degree of its restriction to $\overline U\cap \R^N = \overline \Omega$. Furthermore, for $u(t)\equiv u\in \overline \Omega$ it is clear that $K_0u= u - \frac T2 G(u)$, which does not vanish on $\partial \Omega=\partial U\cap \R^N$. By the homotopy invariance of the degree, we conclude that $$deg(I-K,U,0)=deg \left(\frac T2G,\Omega,0\right)=(-1)^N\chi(\Omega).$$ 2. Let $K_L$ be the operator associated to the linearised problem, defined by $$K_Lu(t):= \overline u -t\,\overline {\mathcal N_Lu} + \mathcal I\mathcal N_Lu(t) - \overline{\mathcal I\mathcal N_Lu},$$ with $\mathcal N_Lu(t):= Au(t) + Bu(t-\tau).$ As before, it is seen that $K_Lu=u$ if and only if $u$ is a solution of (\[linear\]); hence, it follows from the assumptions that $K_L$ has no nontrivial fixed points. Furthermore, the degree of $I-K_L$ coincides with the degree of $I-K$ on $B_\rho(0)$ when $\rho$ is small. This is a well-known fact but, for the reader’s convenience, a simple proof is sketched as follows. Since the degree is locally constant, we may assume that $g$ is of class $C^2$ near $(0,0)$, then [for some $C>0$,]{} $$\|Kv-K_Lv\|_{\infty} \le C\|\mathcal Nv-\mathcal N_Lv\|_\infty = o(\rho).$$ Because $K_L$ is compact, it is verified that, for some $\theta>0$, $$\|v-K_Lv\|_{\infty}\ge \theta \rho$$ for all $v\in \partial B_\rho(0)$. Indeed, due to linearity, it suffices to prove the claim for $\rho=1$. By contradiction, suppose there exists a sequence $\{v_n\}\subset \partial B_1(0)$ such that $\|v_n-K_Lv_n\|_{\infty}\to 0$, then passing to a subsequence we may assume that $\{K_Lv_n\}$ converges to some $v$. Then $v_n\to v$ which, in turn, implies that $\|v\|_{\infty}=1$ and $v=K_Lv$, a contradiction. It follows that if $\rho>0$ is small then $sK + (1-s)K_L$ has no fixed points on $\partial B_\rho(0)$ for $s\in [0,1]$ because $$\|v - sKv - (1-s)K_Lv\|_{\infty} \ge \|v - K_Lv\|_{\infty} - \|K_Lv-Kv\|_{\infty} \ge \theta \rho - o(\rho)>0$$ for $v\in \partial B_\rho(0)$. Thus, the degree of $I-K$ is well defined and coincides with the degree of $I-K_L$ over $B_\rho(0)$. 3. Claim: $deg(I-K_L,B_\rho(0),0) = s(A+B)$. \[directo-fourier\] Indeed, for $u$ as before it is seen by direct computation that $$u-K_Lu=\tilde a_0 + \sum_{k\ge 1} (\varphi_k\tilde a_k + \psi_k\tilde b_k)$$ where $$\tilde a_0= \mathcal M_0 a_0$$ and $$\left( \begin{array}{c} \tilde a_k \\ \tilde b_k \end{array}\right) = \mathcal M_k \left( \begin{array}{c} a_k \\ b_k \end{array}\right)$$ with $$\mathcal M_0:= \frac T2(A+B)\qquad \hbox{and }\,\, \mathcal M_k:= \frac 1{\lambda_k} \left( \begin{array}{cc} Y_k & X_k \\ -X_k & Y_k \end{array}\right)\quad \hbox{for}\, k>0.$$ Hence, the degree coincides with the sign of the determinant of the block matrix $$\left( \begin{array}{ccccc} \mathcal M_0 & 0 & 0 & \ldots & 0 \\ 0 & \mathcal M_1 & 0 & \ldots & 0 \\ 0 & 0 & \mathcal M_2 & \ldots & 0\\ \ldots & \ldots & \ldots & \ldots & \ldots\\ 0 & 0 & 0 & \ldots & \mathcal M_J \end{array}\right)$$ for $J$ sufficiently large. Thus, the proof follows in a straightforward manner from the fact that ${\rm det}(\mathcal M_k) >0$ for all $k>0$. We remark that the latter property holds even when $A$ and $B$ do not commute (see Remark \[remark1\]). Indeed, identifying the pairs $(a,b)\in \R^N\times \R^N$ with vectors $a+ib\in \mathbb C^N$, a matrix of the form $\left( \begin{array}{cc} X & -Y \\ Y & X \end{array}\right)$ may be called a [$\mathbb C$-linear matrix]{}. Thus, we need to prove that if $\mathcal M$ is an arbitrary invertible $\mathbb C$-linear matrix, then the algebraic multiplicity of each eigenvalue $\sigma<0$ of $\mathcal M$ is even. It is known that this value can be computed as the dimension of the kernel of the matrix $(\mathcal M-\sigma I)^m$, where $m$ is the minimum integer such that ${\rm ker}(\mathcal M-\sigma I)^m = {\rm ker}(\mathcal M-\sigma I)^{m+1}$. Now observe that the set of $\mathbb C$-linear matrices is a subring of $\R^{2N\times 2N}$; thus, $(\mathcal M-\sigma I)^m$ is again a $\mathbb C$-linear matrix. In particular, if $(a,b)\in {\rm ker}(\mathcal M-\sigma I)^m$ then $(-b,a)\in {\rm ker}(\mathcal M-\sigma I)^m$ and the result follows. 4. *Existence of two solutions for small $p$*. From the previous steps and the fact that the degree is locally constant we deduce that $$deg(I-K,U,\hat p)=(-1)^N\chi(\Omega),\qquad deg(I-K,B_\rho(0),\hat p)=s(A+B)$$ when $\|\hat p\|_\infty$ is small. Now the excision property of the Leray-Schauder degree implies $$deg(I-K,B_\rho(0),\hat p)=s(A+B)\ne 0,$$ and $$deg(I-K,U\backslash B_\rho(0),\hat p)=(-1)^N\chi(\Omega)- s(A+B) \ne 0.$$ Thus, there exists $\hat r>0$ such that the equation $(I-K)u=\hat p$ has at least two solutions for $\|\hat p \|_\infty <\hat r$. Finally, for each $p\in C_T$ define $$\hat p (t):= \mathcal I p(t) - \overline{\mathcal Ip} - t\overline p,$$ then clearly $\|\hat p\|_\infty\le c\|p\|_\infty$ for [some $c>0$]{}. The result is then deduced from the fact that if $u-Ku=\hat p$, then $u$ is a $T$-periodic solution of [(\[nonaut\])]{}. $$u'(t)=g(u(t),u(t-\tau))+p(t).$$ 5. *Genericity.* The last part of the proof follows as a consequence of the following particular case of the Sard-Smale Theorem [@smale]: Let $\mathcal F:X \to Y$ be a $C^1$ Fredholm map of index $0$ between Banach manifolds, i.e. such that $D\mathcal F(x):T_x X \to T_{\mathcal F(x)} Y$ is a Fredholm operator of index $0$ for every $x\in X$. Then the set of regular values of $\mathcal F$ is residual in $Y$. At this point, we notice that the argument is a bit subtle: when applied to $\mathcal F:=I-K$, the Sard-Smale Theorem implies the existence of a residual set $\Sigma\subset C_T$ such that the mapping $\mathcal F-\hat p$ has at least $\Gamma - 1$ zeros in $U\setminus B_\rho(0)$ for $\hat p\in \Sigma\cap B_{ \hat r}(0)$. [Indeed, it is readily seen that $K$ is of class $C^1$ and $DK(u)$ is compact for all $u$. Thus, $\mathcal F=I-K$ is a zero-index Fredholm operator. If $\hat p$ is a regular value, that is, $D\mathcal F(u)$ is surjective for every preimage $u \in \mathcal F^{-1}(\hat p)$ then, since the index is $0$, it is also injective and from the open mapping theorem we conclude that $D\mathcal F(u)$ is an isomorphism. Hence, the number of such preimages in $U\setminus B_{\rho}(0)$ is greater or equal than $|deg(I-K,U\setminus B_{\rho}(0),0)|$. This follows by taking small neighbourhoods $N_u$ around each of these values $u$ such that $\mathcal F:N_u\to \mathcal F(N_u)$ is a diffeomorphism. Because there are no other zeros of $\mathcal F -\hat p$ in $U\setminus B_{\rho}(0)$, the degree is the sum of the degrees $d_u$ over each of these neighbourhoods. The claim then follows from the fact that $d_u=\pm 1$ for each $u$.]{} However, although the mapping $p\mapsto \hat p$ defined before establishes an isomorphism $J:C_T\to C_T^1$, it might happen that $J^{-1}(\Sigma\cap C^1_T)$ is not a residual set. The difficulty is overcome for example by considering the same operator $K$ as before, now defined over the set $$\hat U:= \{u\in C^1_T: u(t)\in \Omega,\, \|u'\|_\infty < \|g\|_{\infty} + 1\} \subset C^1_T.$$ Details are left to the reader. [Notice that]{} 1. The existence of a solution near the equilibrium can be also proved in a direct way by the Implicit Function Theorem. 2. Condition (\[hart\]) alone implies the existence of generically $|\chi(\Omega)|$ solutions. 3. Analogous conclusions are obtained if the sign of (\[hart\]) is reversed. In this case, $G$ is homotopic to $\nu$ and hence $deg(I-K,U,0)= \chi(\Omega)$. However, in this latter case the considerations about the Poincaré operator become less clear, because it is not guaranteed that solutions with initial values $\phi$ with $\phi(t)\in \overline \Omega$ remain inside $\Omega$. Small delays {#sec-delay} ============ As mentioned in the introduction, condition (\[hart\]) implies that the vector field $G(x)=g(x,x)$ is inwardly pointing over $\partial \Omega$, although the converse is not true; the need of a condition stronger than (\[hart-weak\]) is due to the presence of the delay. However, if only (\[hart-weak\]) is assumed, then Theorem \[main\] is still valid for all $\tau<\tau ^*$, where $\tau ^*$ depends only on $\|g\|_\infty$. More precisely, by continuity we may fix $\ee>0$ such that (\[hart\]) holds for all $x\in \partial \Omega$ and all $y\in\overline\Omega$ with $|y-x|<\ee$ and take $\tau* := \frac \ee{\|g\|_\infty}$. In this section, we show that the problem for small $\tau$ can be seen as a perturbation of the non-delayed case, thus giving the explicit sufficient condition for the non-existence of nontrivial $T$-periodic solutions of (\[linear\]) expressed in Corollary \[smalldelay\]. We shall make use of the following lemmas: \[lambdas\] $1$ is a Floquet multiplier of the system $u'(t)=Mu(t)$ if and only if $-\lambda_k^2$ is an eigenvalue of $M^2$ for some $k\in\N_0$, that is, if and only if $\pm i\lambda_k$ are eigenvalues of $M$ for some $k$. The result follows by direct computation, or from Lemma \[lema\] with $\tau=0$. For example, when $M$ is triangularizable (or, equivalently, when all its eigenvalues are real), $1$ is not an eigenvalue of the system $u'(t)=Mu(t)$ if and only if $M$ is nonsingular; in this particular case, the conclusion follows directly, because the system uncouples and the result is obviously true for a scalar equation. \[Floq\] Assume that $1$ is not a Floquet multiplier of the linear ODE system $u'(t)=(A+B)u(t)$. Then the DDE system (\[linear\]) has no nontrivial $T$-periodic solutions, provided that $\tau$ is small. Suppose that $u_n\in C_T$ is a nontrivial solution for $\tau_n\to 0$. Without loss of generality, it may be assumed that $\|u_n\|_\infty=1$ and hence $\|u_n'\|_\infty\le C$ for some constant $C$. Thus, we may assume that $u_n$ converges uniformly to some $u\in C_T$ with $\|u\|_\infty=1$. Because $\|u_n(t-\tau_n)-u_n(t)\|\le C\tau_n\to 0$, it becomes clear that $u_n'$ converges uniformly to $(A+B)u$ which, in turn, implies $u'=(A+B)u$, a contradiction. A more direct proof of Lemma \[Floq\] follows just by considering Remark \[remark1\].\[k\_0\] and Lemma \[lambdas\]. Indeed, in the context of Lemma \[lema\] it suffices to check that $h_k\ne 0$ only for a finite number of values of $k$. By continuity, this is true for small $\tau$, because ${\rm det} [(A+B)^2 + \lambda_k^2 I]\neq 0$ for all $k$. However, the previous proof has an interest in its own because it can be extended in a straightforward manner to the non-autonomous case. : As a consequence of the preceding lemma, the conclusions of Theorem \[main\] hold for small $\tau$, provided that the linearisation has no nontrivial $T$-periodic solutions for the non-delayed case. Thus, in view of Lemma \[lambdas\], the proof is complete. $\square$ Poincaré operator {#poincare} ================= In this section, we shall make some considerations regarding the Poincaré operator associated to the system. Let us firstly observe that if $\chi(\Omega)=1$ (for example, if $\Omega$ is homeomorphic to a ball), then the condition $s(A+B) \neq (-1)^N\chi(\Omega)$ in Theorem \[main\] simply reads $(-1)^N {\rm det}(A+B)<0$. This, in turn, implies that the equilibrium is unstable. [Indeed, consider the characteristic function $h(\lambda)= {\rm det}\left(\lambda I - A - Be^{-\lambda\tau} \right)$, then $h(0)= (-1)^N {\rm det}(A+B)<0$ and $h(\lambda) =\lambda^N$ for $|\lambda|\gg 0$. In particular, this implies the existence of a characteristic value $\lambda>0$. ]{} We shall show that, in the present context, the instability of the equilibrium when $(-1)^N {\rm det}(A+B)<0$ is due to the fact, proved in section \[dem\], that the index of the fixed point operator $K$ at $e$ (i. e. the degree of $I-K$ over small balls around $e$) is equal to $(-1)^{N+1}$. When $\tau=0$, this can be regarded as a direct consequence of the following properties: 1. $deg(I-K, B_\rho(e),0)$ with $B_\rho(e)\subset C_T$ is equal to $(-1)^Ndeg_B(I-P, B_\rho(e),0)$ with $B_\rho(e)\subset\R^N$, where $P$ is the Poincaré map. 2. If the equilibrium is stable, then the index of $P$ is $1$. The first property is a particular case of a *relatedness principle* due to Krasnoselskii (see [@krasno]). The second property is well-known and can be found for example in [@K]. For more details see [@rafa], where sufficient conditions for the validity of the converse statement are also obtained. Our goal in this section consists in understanding the connections between the instability of the equilibrium and the index of the fixed point operator defined in the proof of the main theorem. With this aim, let us define the Poincaré operator for the delayed case as follows. Let $\tau\le T$ and consider a general autonomous system $$\label{general} u'(t)=F(u_t)$$ with $F: C([-\tau,0])\to \R^N$ locally Lipschitz, *i.e.*: for all $R>0$ there exists a constant $L$ such that $$\|F(\phi)-F(\psi)\|\le L\|\phi-\psi\|_\infty$$ for all $\phi,\psi\in \overline {B_R(0)}\subset C([-\tau,0],\mathbb R^n)$. The notation $u_t$ expresses, as usual, the mapping defined by $u_t(\theta):=u(t+\theta)$ for $\theta\in [-\tau,0]$. Denote by ${\rm dom}(P)\subset C([-\tau,0])$ the set of those functions $\phi$ such that the unique solution $u=u(\phi)$ of the problem with initial condition $\phi$ is defined up to $t=T$, then $P:{\rm dom}(P)\to C([-\tau,0])$ is defined by $$P\phi(s):=u(T+s).$$ Clearly, the $T$-periodic solutions of the problem can be identified with the fixed points of $P$. We shall see that, as in the non-delayed case, if the linearisation has no nontrivial $T$-periodic solutions then the index $i(P)$ of the operator $P$ at a stable equilibrium is equal to $1$. To this end, assume without loss of generality that $e=0$ and observe that stability implies that ${\rm dom}(P)$ is a neighbourhood of $0$. It is worth noticing that, in the general setting, extra conditions are required in order to prove the compactness of $P$ (see *e.g*. [@liu]), so the Leray-Schauder degree may be not well defined; however, it is verified that the stability assumption implies that $P$ is compact over small neighbourhoods of $0$. More precisely: \[compact\] Let $F$ be as before and assume that for some open $U\subset C([-\tau,0])$ there exists $R>0$ such that if $\phi\in U$ then the solution $u$ with initial condition $\phi$ is defined and satisfies $|u(t)| <R$ for all $t\in [0,T]$. Then $P$ is well defined and compact over $U$. Let $B\subset U$ be bounded and observe, in the first place, that $P(B)$ is bounded. Moreover, if $u$ is a solution with initial condition $\phi\in B$, then $$u(t)= \phi(0) + \int_0^t F(u_s)\, ds.$$ Enlarging $R$ if necessary, we may assume $B\subset B_R(0)$, then $\|u_s\|_\infty <R$ for all $s\in [0,T]$. Given $t_1<t_2$ in $[-\tau,0]$, since $\tau\le T$ it is verified that $$|P\phi(t_2)-P\phi(t_1)| \le \int_{T+t_1}^{T+t_2} |F(u_s)|\, ds.$$ Let $L$ be the Lipschitz constant corresponding to $R$, then $$|F(\phi)|\le |F(0)| + L\|\phi\|_\infty\le C + LR,$$ where $C:=|F(0)|$. Hence $|P(t_2)-P(t_1)|\le (C+LR)(t_2-t_1)$ and the result follows from the Arzelà-Ascoli Theorem. For example, the assumptions of the previous lemma are satisfied if $F$ has linear growth, that is $$|F(\phi)|\le \gamma\|\phi\|_\infty + \delta.$$ Furthermore, extra assumptions are required to ensure the non-existence of nontrivial periodic solutions near $0$; this is why we shall impose this fact as an extra condition (see Proposition \[poinc-stab\] below), which is clearly satisfied for example when the stability is asymptotic. For simplicity, we shall also assume that $F$ is Fréchet differentiable at $0$, that is, $$F(\phi)= D_\phi(0)\phi + \mathcal R(\phi)$$ with $\|\mathcal R(\phi)\|_\infty \le o(\|\phi\|_\infty)\|\phi\|_\infty$. Thus, it is readily verified that the linearisation of $P$ at the origin coincides with the Poincaré operator associated to the linearised system $u'(t)=D_\phi(0)u_t$. \[poinc-stab\] In the previous setting, assume that $0$ is a stable equilibrium of (\[general\]) such that its linearisation has no nontrivial $T$-periodic solutions. Then $i(P)=1$. Without loss of generality, we may assume that $P$ is compact on $\overline V$ for some neighbourhood $V$ of $0$. It follows from the assumptions that the index of $P$ is well defined and coincides with the index of its linearisation $P_L$. According to Theorem 13.8 in [@brown], $deg(I-P_L,B_\rho(0),0)$ is equal to $(-1)^\alpha$, where $\alpha$ is the sum of the (finite) algebraic multiplicities of the (finitely many) eigenvalues $\sigma$ of $P_L$ satisfying $\sigma>1$. If $deg(I-P_L,B_\rho(0),0) = -1$, then $P_L$ has an eigenfunction $\phi$ with eigenvalue $\sigma>1$. If $u$ is the corresponding solution of the linearised problem with initial condition $u=\phi$ on $[-\tau,0]$ then $u$ can be extended to $\R$ in a $(T,\sigma)$-periodic fashion, that is, with $u(t+T)=\sigma u(T)$ for all $t$ (see [@pinto]). In particular, $u(t)$ is unbounded for $t>0$. In other words, $0$ is unstable for the linearised problem which, in turn, implies that it cannot be stable for the original problem (see *e.g.* [@hale]). In order to complete the picture for system (\[ec\]), it would be interesting to prove that, indeed, the index of the Poincaré operator at the equilibrium when the linearisation has no nontrivial solutions is $(-1)^Ns(A+B)= (-1)^Ni(K)$. Here, we shall simply verify that the claim holds when the delay is small; the analysis of the general case and [a version of the Krasnoselskii relatedness principle for delayed systems shall be the subject of a forthcoming paper.]{} To this end, let us start with a direct computation for the non-delayed case: \[degP\] Let $M\in \mathbb R^{N\times N}$ and let $P_M$ be the Poincaré operator associated to the linear ODE system $u'(t)=Mu(t)$ for some fixed $T$. If $1$ is not a Floquet multiplier, then $$deg_B(I-P_M,V,0) = (-1)^Ns(M)$$ for any neighbourhood $V\subset \R^N$ of the origin. By definition, $$(I-P_M)(u)= \left(I-e^{TM}\right)u.$$ Write $M$ in its (possibly complex) Jordan form $M=C ^{-1}JC$, where $J$ is upper triangular. Then $${\rm det}\left(I- e^{TM}\right) = {\rm det}\left(I- e^{TJ}\right) = \prod_{j=1}^N \left(1-e ^{\lambda_jT}\right),$$ where $\lambda_j$ are the eigenvalues of $M$. Now observe that if $\lambda=a+ib\notin \R$, then $$\left(1-e ^{\lambda T}\right)\left(1-e ^{\overline{\lambda}T}\right) = 1 + e ^{aT}\left(e^{aT} -2\cos (bT)\right) >0.$$ Thus, complex eigenvalues do not affect the sign of ${\rm det}\left(I- e^{TM}\right)$, as well as it happens with the sign of ${\rm det}(M)$ because $\lambda\overline\lambda =|\lambda|^2$. The result follows now from the fact that, for $\lambda\in\R$, $$sgn\left(1 - e^{\lambda T}\right) = -sgn (\lambda).$$ An alternative (somewhat exotic) proof follows from the relatedness principle. Indeed, we may consider the operator $K_L$ in the proof of Theorem \[main\] with $A=M$ and $B=0$, then $deg_B(I-P,V,0)=(-1)^Ndeg(I-K_L, V,0) = (-1)^Ns(M)$. The conclusion for small $\tau$ is obtained now by a continuity argument. Indeed, fix $r>0$ and $P_L$ as before. The solutions of (\[linear\]) with initial value $\phi\in B_r(0)$ are uniformly bounded; thus, by Gronwall’s lemma we deduce that $\|P-P_0\| = O(\tau)$, where the operator $P_0$ is defined by $P_0(\phi)(t)\equiv v(T)$, with $v$ the unique solution of the system $v'(t)=(A+B)v(t)$ satisfying $v(0)=\phi(0)$. Moreover, recall that if $\tau$ is small then $P_L$ is homotopic to $P_0$; thus, the result follows from Lemma \[degP\]. Example: a system of DDEs with singularities {#exam} ============================================ A simple example is presented here in order to illustrate our main results. Let $0\le J_0\le J \ne 0$ and $$g(x,y):= -dx + |y|^2\left( \sum_{j=1}^{J_0} a_j\frac{x-v_j}{|x-v_j|^{\alpha_j}} + \sum_{j=J_0+1}^{J} a_j\frac{y-v_j}{|y-v_j|^{\alpha_j}} \right)$$ where $d,a_j>0$, $\alpha_j>2$ and $v_j\in \R^N\backslash\{0\}$ are pairwise different vectors. A simple computation shows that $$\langle g(x,x),x\rangle < 0 \qquad |x|\gg 0$$ and $$\langle g(x,x),v_j-x\rangle < 0 \qquad |x-v_j|\ll 1$$ for $j=1,\ldots, J$. Moreover, $g(0,0)=0$ and $$A=D_xg(0,0)=-dI, \quad B=D_yg(0,0)=0.$$ Thus, taking $\Omega:=B_R(0)\backslash \cup_{j=1}^J B_\eta(v_j)$ where $R\gg 0$ and $\eta\ll 1$, Corollary \[smalldelay\] applies. Since $\chi(\Omega)= 1-J < 1 = (-1)^Ns(A+B)$, we conclude that the number of $T$-periodic solutions of (\[nonaut\]) for small $\tau$ and $\|p\|_\infty$ is generically $J+1$. Acknowledgements {#acknowledgements .unnumbered} ================ The first two authors were partially supported by projects CONICET PIP 11220130100006CO and UBACyT 20020160100002BA. The first author wants to thank Prof. J. Barmak for his thoughtful comments regarding the fixed point property and the Euler characteristic. 1 R. F. Brown, *A Topological Introduction to Nonlinear Analysis*. First edition, Birkhäuser (2004). J. K. Hale and S. M. Verduyn Lunel, *Introduction to Functional Differential Equations*, Springer, New York (1993). H. Hopf, [*Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten*]{}. Math. Ann. 96 (1926/1927), pp. 225–250. J. Liu, G. N’Guérékata and Nguyen Van Minh, [*Topics on Stability and Periodicity in Abstract Differential Equations*]{}. World Scientific, Singapore (2008). M. A. Krasnoselskii, [*The operator of translation along the trajectories of differential equations*]{}. Amer. Math. Soc., Providence, RI, (1968). M. A. Krasnoselskii, P.P Zabreiko, [*Geometrical methods of nonlinear analysis*]{}. Springer-Verlag, Berlin (1984). R. Ortega, [*Topological degree and stability of periodic solutions for certain differential equations*]{}, J. London Math. Soc. (2) 42 (1990), pp. 505–516. M. Pinto, [*Pseudo-almost periodic solutions of neutral integral and differential equations with applications*]{}. Nonlinear Anal. 72(12) (2010), pp. 4377–-4383. S. Smale, [*An infinite dimensional version of Sard’s theorem*]{}. American Journal of Mathematics 87 (1965), pp. 861–866. H. Smith, [*An Introduction to Delay Differential Equations with Applications to the Life Sciences*]{}, Springer-Verlag, New York (2011). F. Wecken, [*Fixpunktklassen*]{}. Ill, Math. Ann. 118-119 (1941-1943), pp. 544–577. Pablo Amster and Mariel Paula Kuna *E-mails*: [email protected] – [email protected]. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina and IMAS-CONICET. Gonzalo Robledo *E-mail*: [email protected]. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653 Santiago, Chile.

ArXiv

--- abstract: 'Consider a continuous word embedding model. Usually, the cosines between word vectors are used as a measure of similarity of words. These cosines do not change under orthogonal transformations of the embedding space. We demonstrate that, using some canonical orthogonal transformations from SVD, it is possible both to increase the meaning of some components and to make the components more stable under re-learning. We study the interpretability of components for publicly available models for the Russian language (RusVectōrēs, fastText, RDT).' author: - | Alexey Zobnin\ National Research University Higher School of Economics,\ Faculty of Computer Science,\ [email protected] bibliography: - 'paper.bib' title: 'Rotations and Interpretability of Word Embeddings: the Case of the Russian Language' --- Introduction ============ Word embeddings are frequently used in NLP tasks. In vector space models every word from the source corpus is represented by a dense vector in $\mathbb{R}^d$, where the typical dimension $d$ varies from tens to hundreds. Such embedding maps similar (in some sense) words to close vectors. These models are based on the so called distributional hypothesis: similar words tend to occur in similar contexts [@harris1954distributional]. Some models also use letter trigrams or additional word properties such as morphological tags. There are two basic approaches to the construction of word embeddings. The first is count-based, or explicit [@levy2014linguistic; @dhillon2015eigenwords]. For every word-context pair some measure of their proximity (such as frequency or PMI) is calculated. Thus, every word obtains a sparse vector of high dimension. Further, the dimension is reduced using singular value decomposition (SVD) or non-negative sparse embedding (NNSE). It was shown that truncated SVD or NNSE captures latent meaning in such models [@landauer1997solution; @murphy2012learning]. That is why the components of embeddings in such models are already in some sense canonical. The second approach is predict-based, or implicit. Here the embeddings are constructed by a neural network. Popular models of this kind include word2vec [@mikolov2013efficient; @mikolov2013distributed] and fastText [@bojanowski2016enriching]. Consider a predict-based word embedding model. Usually in such models two kinds of vectors, both for words and contexts, are constructed. Let $N$ be the vocabulary size and $d$ be the dimension of embeddings. Let $W$ and $C$ be $N \times d$-matrices whose rows are word and context vectors. As a rule, the objectives of such models depend on the dot products of word and context vectors, i. e., on the elements of $WC^T$. In some models the optimization can be directly rewritten as a matrix factorization problem [@levy2014neural; @cotterell2017explaining]. This matrix remains unchanged under substitutions $W \mapsto W S, \quad C \mapsto C {S^{-1}}^T$ for any invertible $S$. Thus, when no other constraints are specified, there are infinitely many equivalent solutions [@fonarev2017riemannian]. Choosing a good, not necessarily orthogonal, post-processing transformation $S$ that improves quality in applied problems is itself interesting enough [@mu2017all]. However, only word vectors are typically used in practice, and context vectors are ignored. The cosine distance between word vectors is used as a similarity measure between words. These cosines will not change if and only if the transformation $S$ is orthogonal. Such transformations do not affect the quality of the model, but may elucidate the meaning of vectors’ components. Thus, the following problem arises: *what orthogonal transformation is the best one for describing the meaning of some (or all) components?* It is believed that the meaning of the components of word vectors is hidden [@gladkova2016intrinsic]. But even if we determine the “meaning” of some component, we may loose it after re-training because of random initialization, thread synchronization issues, etc. Many researchers [@luo2015online; @ruseti2016using; @andrews2016compressing; @jang2017elucidating] ignore this fact and, say, work with vector components directly, and only some of them take basis rotations into account [@tsvetkov2016correlation]. We show that, generally, re-trained model differ from the source model by almost orthogonal transformation. This leads us to the following problem: *how one can choose the canonical coordinates for embeddings that are (almost) invariant with respect to re-training?* We suggest using well-known plain old technique, namely, the singular value decomposition of the word matrix $W$. We study the principal components of different models for Russian language (RusVectōrēs, RDT, fastText, etc.), although the results are applicable for any language as well. Related Work ============ Interpretability of the components have been extensively studied for topic models. In [@chang2009reading; @lau2014machine] two methods for estimating the coherence of topic models with manual tagging have been proposed: namely, word intrusion and topic intrusion. Automatic measures of coherence based on different similarities of words were proposed in [@aletras2013evaluating; @nikolenko2016topic]. But unlike topic models, these methods cannot be applied directly to word vectors. There are lots of new models where interpretability is either taken into account by design [@luo2015online] (modified skip-gram that produces non-negative entries), or is obtained automagically [@andrews2016compressing] (sparse autoencoding). Lots of authors try to extract some predefined significant properties from vectors: [@jang2017elucidating] (for non-negative sparse embeddings), [@tsvetkov2016correlation] (using a CCA-based alignment between word vectors and manually-annotated linguistic resource), [@rothe2016word] (ultradense projections). Singular vector decomposition is the core of count-based models. To our knowledge, the only paper where SVD was applied to predict-based word embedding matrices is [@mu2017all]. In [@arora2017simple] the first principal component is constructed for sentence embedding matrix (this component is excluded as the common one). Word embeddings for Russian language were studied in [@kutuzov2015texts; @Kutuzov2015; @panchenko2015russe; @arefyev2015evaluating]. Theoretical Considerations ========================== Singular value decomposition ---------------------------- Let $m \ge n$. Recall [@jolliffe2002principal] that a singular value decomposition (SVD) of an $m\times n$-matrix $M$ is a decomposition $M = U \Sigma V^T$, where $U$ is an an $m \times n$ matrix, $U^T U = I_{n}$, $\Sigma$ is a diagonal $n \times n$-matrix, and $V$ is an $n \times n$ orthogonal matrix. Diagonal elements of $\Sigma$ are non-negative and are called singular values. Columns of $U$ are eigenvectors of $M M^T$, and columns of $V$ are eigenvectors of $M^T M$. Squares of singular values are eigenvalues of these matrices. If all singular values are different and positive, then SVD is unique up to permutation of singular values and choosing the direction of singular vectors. Buf if some singular values coincide or equal zero, new degrees of freedom arise. Invariance under re-training ---------------------------- Learning methods are usually not deterministic. The model re-trained with similar hyperparameters may have completely different components. Let ${M_1}$ and ${M_2}$ be the word matrices obtained after two separate trainings of the model. Let these embeddings be similar in the sense that cosine distances between words are almost the same, i. e., ${M_1}{M_1}^T \approx {M_2}{M_2}^T$. Suppose also that singular values of each ${M_i}$ are different and non-zero. Then one can show that ${M_1}$ and ${M_2}$ differ only by the (almost) orthogonal factor. Indeed, left singular vectors in SVD of ${M_i}$ are eigenvectors of ${M_i}{M_i}^T$. Hence, matrices $U$ and $\Sigma$ in SVD of ${M_1}$ and ${M_2}$ can be chosen the same. Thus, ${M_2}\approx {M_1}Q$, where $Q Q^T = I_d$. Here $Q$ can be chosen as $V_1 V_2^T$ where $V_i$ are matrices of right singular vectors in SVD of ${M_i}$. Interpretability measures ------------------------- One of traditional measures of interpretability in topic modeling looks as follows [@newman2010automatic; @lau2014machine]. For each component, $n$ most probable words are selected. Then for each pair of selected words some co-occurrence measure such as PMI is calculated. These values are averaged over all pairs of selected words and all components. The other approaches use human markup. Such measures need additional data, and it is difficult to study them algebraically. Also, unlike topic modeling, word embeddings are not probabilistic: both positive and negative values of coordinates should be considered. Let all word vectors be normalized and $W$ be the word matrix. Inspired by [@nikolenko2016topic], where vector space models are used for evaluating topic coherence, we suggest to estimate the interpretability of $k$th component as $$ \operatorname{interp}_k W = \sum_{i,j=1}^N W_{i,k} W_{j,k} \left(W_i \cdot W_j \right).$$ The factors $W_{i,k}$ and $W_{j,k}$ are the values of $k$th components of $i$th and $j$th words. The dot product $\left(W_i \cdot W_j\right)$ reflects the similarity of words. Thus, this measure will be high if similar words have similar values of $k$th coordinates. What orthogonal transformation $Q$ maximizes this interpretability (for some, or all components) of $WQ$? In matrix terms, $$\operatorname{interp}_k W =(W^T W W^T W)_{k, k},$$ and $$\operatorname{interp}_k WQ = \left(Q^T W^T W W^T W Q\right)_{k,k}$$ because $Q$ is orthogonal. The total interpretability over all components is $$\begin{gathered} \sum_{k=1}^d \operatorname{interp}_k WQ = \sum_{k=1}^d \left(Q^T W^T W W^T W Q \right)_{k,k} = \\ = \operatorname{tr}Q^T W^T W W^T W Q = \operatorname{tr}\left(W^T W W^T W\right) = \sum_{k=1}^d \operatorname{interp}_k W,\end{gathered}$$ because $\operatorname{tr}Q^T X Q = \operatorname{tr}Q^{-1} X Q = \operatorname{tr}X$. It turns out that *in average* the interpretability is constant under any orthogonal transformation. But it is possible to make the first components more interpretable due to the other components. For example, $$(Q^T W^T W W^T W Q)_{1, 1} = \left(q^T W^T W q\right)^2$$ is maximized when $q$ is the eigenvector of $W^T W$ with the largest singular value, i. e., the first right singular vector of $W$ [@jolliffe2002principal]. Let’s fix this vector and choose other vectors to be orthogonal to the selected ones and to maximize the interpretability. We arrive at $Q = V$, where $V$ is the right orthogonal factor in SVD $W = U \Sigma V^T$. Experiments =========== \[singular\_values\] {width="12cm"} \[overlapping\] {width="12cm"} \[alignment\_shifts\] {width="12cm"} {width="12cm"} Canonical basis for embeddings ------------------------------ We train two fastText skipgram models on the Russian Wikipedia with default parameters. First, we normalize all word vectors. Then we build SVD decompositions[^1] of obtained word matrices and use $V$ as an orthogonal transformation. Thus, new “rotated” word vectors are described by the matrix $WV = U \Sigma$. The corresponding singular values are shown in Figure 1, they almost coincide for both models (and thus are shown only for the one model). For each component both in the source and the rotated models we take top 50 words with maximal (positive) and bottom 50 words with minimal (negative) values of the component. Taking into account that principal components are determined up to the direction, we join these positive and negative sets together for each component. We measure the overlapping of these sets of words. Additionally, we use the following alignment of components: first, we look for the free indices $i$ and $j$ such that $i$th set of words from the first model and $j$th set of words from the second model have the maximal intersection, and so on. We call the difference $i - j$ the alignment shift for the $i$th component. Results are presented in Figures 2 and 3. We see that at least for the first part of principal components (in the rotated models) the overlapping is big enough and is much larger that that for the source models. Moreover, these first components have almost zero alignment shifts. Other principal components have very similar singular values, and thus they cannot be determined uniquely with high confidence. Normalized interpretibility measures for different components (calculated for 50 top/bottom words) for the source and the rotated models are shown in Fig. 4. Principal components of different models ---------------------------------------- We took the following already published models: - RusVectōrēs[^2] lemmatized models (actually, word2vec) trained on different Russian corpora [@KutuzovKuzmenko2017]; - Russian Distributional Thesaurus[^3] (actually, word2vec skipgram) models trained on Russian books corpus [@Panchenko:17:RDT]; - fastText[^4] model trained on Russian Wikipedia [@bojanowski2016enriching]. For each model we took $n = 10000$ or $n = 100000$ most frequent words. Each word vector was normalized in order to replace cosines with dot products. Then we perform SVD $W = U \Sigma V^T$ and take the matrix $W V = U \Sigma$. For each of $d$ components we sort the words by its value and choose top $t$ “positive” and bottom $t$ “negative” words ($t=15$ or 30). For clarity, every selection was clustered into buckets with the simplest greedy algorithm: list the selected words in decreasing order of frequency and either add the current word to some cluster if it is close enough to the word (say, the cosine is greater than $0.6$), or make a new cluster. The cluster’s vector is the average vector of its words. Intuitively, the smaller the number of clusters, the more interpretable the component is. Similar approach was used in [@ramrakhiyani2017measuring]. Tables in the Appendix show the top “negative” and “positive” words of the first principal components for different models. We underline that principal components are determined up to the direction, and thus the separation into “negative” and “positive” parts is random. The full results are available at <https://alzobnin.github.io/>. We cluster these words as described above; different clusters are separated by semicolons. We see the following interesting features in the components: - stop words: prepositions, conjunctions, etc. (RDT 1, fastText 1; in RusVectōrēs models they are absent just because they were filtered out before training); - foreign words with separation into languages (fastText 2, web 2), words with special orthography or tokens in broken encoding (not presented here); - names and surnames (RDT 8, fastText 3, web 3), including foreing names (fastText 9, web 6); - toponyms (not presented here) and toponym descriptors (web 7); - fairy tale characters (fastText 6); - parts of speech and morphological forms (cases and numbers of nouns and adjectives, tenses of verbs); - capitalization (in fact, first positions in the sentences) and punctuation issues (e. g., non-breaking spaces); - Wikipedia authors and words from Wikipedia discussion pages (fastText 5); - other different semantic categories. We also made an attempt to describe obtained components automatically in terms of common contexts of common morphological and semantic tags using MyStem tagger and semantic markup from Russian National Corpus. Unfortunately, these descriptions are not as good as desired and thus they are not presented here. Conclusion ========== We study principal components of publicly available word embedding models for the Russian language. We see that the first principal components indeed are good interpretable. Also, we show that these components are almost invariant under re-learning. It will be interesting to explore the regularities in canonical components between different models (such as CBOW versus Skip-Gram, different train corpora and different languages [@smith2017offline]. It is also worth to compare our intrinsic interpretability measure with human judgements. Acknowledgements ================ The author is grateful to Mikhail Dektyarev, Mikhail Nokel, Anna Potapenko and Daniil Tararukhin for valuable and fruitful discussions. Appendix {#appendix .unnumbered} ======== Top/bottom words for the first few principal components for different Russian models {#topbottom-words-for-the-first-few-principal-components-for-different-russian-models .unnumbered} ------------------------------------------------------------------------------------ Note misspellings in 4a. [^1]: With numpy.linalg.svd it took up to several minutes for 100K vocabulary. [^2]: <http://rusvectores.org/ru/models/> [^3]: <https://nlpub.ru/Russian_Distributional_Thesaurus> [^4]: <https://github.com/facebookresearch/fastText/blob/master/pretrained-vectors.md>

ArXiv

--- abstract: 'A global picture of a random particle movement is given by the convex hull of the visited points. We obtained numerically the probability distributions of the volume and surface of the convex hulls of a selection of three types of self-avoiding random walks, namely the classical Self-Avoiding Walk, the Smart-Kinetic Self-Avoiding Walk, and the Loop-Erased Random Walk. To obtain a comprehensive description of the measured random quantities, we applied sophisticated large-deviation techniques, which allowed us to obtain the distributions over a large range of the support down to probabilities far smaller than $P = 10^{-100}$. We give an approximate closed form of the so-called large-deviation rate function $\Phi$ which generalizes above the upper critical dimension to the previously studied case of the standard random walk. Further we show correlations between the two observables also in the limits of atypical large or small values.' author: - Hendrik Schawe - 'Alexander K. Hartmann' - 'Satya N. Majumdar' bibliography: - 'lit.bib' title: 'Large Deviations of Convex Hulls of Self-Avoiding Random Walks' --- Introduction ============ The standard random walk is a simple Markovian process, which has a history as a model for diffusion. There are many exact results known [@hughes1996random]. If memory is added to the model, e.g., to interact with the past trajectory of the walk, analytic treatment becomes much harder. A class of self-interacting random walks that we will focus on in this study, are *self-avoiding* random walks, which live on a lattice and do not visit any site twice. This can be used to model systems with excluded volume, e.g., polymers whose single monomers can not occupy the same site at once [@Madras2013]. There are more applications which are not as obvious, e.g., a slight modification of the *Smart-Kinetic Self-Avoiding Walk* traces the perimeter of critical percolation clusters [@weinrib1985kinetic], while the *Loop-Erased Random Walk* can be used to study spanning trees [@manna1992spanning] (and vice versa [@Majumdar1992Exact]). One of the central properties of random walk models is the exponent $\nu$, which characterizes the growth of the end-to-end distance $r$ with the number of steps $T$, i.e., $r \propto T^\nu$. While this has the value $\nu = 1/2$ for the standard random walk, its value is larger for the self-avoiding variations, which are effectively pushed away from their past trajectory. In two dimensions, this value (and other properties) can often be obtained by the correspondence to Schramm-Loewner evolution [@cardy2005sle; @lawler2002scaling; @Lawler2011; @Kennedy2015]. But between two dimensions and the upper critical dimension, above which the behavior is the same as the standard random walk, Monte Carlo simulations are used to obtain estimates for the exponent $\nu$. Here we want to study the convex hulls of a selection of self-avoiding walk models featuring larger values of $\nu$. The convex hull allows one to obtain a global picture of the space occupied by a walk, without exposing all details of the walk. As an example, convex hulls are used to describe the home ranges of animals [@mohr1947; @worton1987; @boyle2009]. Namely, we will look at the *Smart-Kinetic Self-Avoiding Walk* (SKSAW), the classical *Self-Avoiding Walk* (SAW) and the *Loop-Erased Random Walk* (LERW), since they span a large range of $\nu$ values and are well established in the literature. About the convex hulls of standard random walks we already know plenty properties. The mean perimeter and area are known exactly since over 20 years [@Letac1980Expected; @Letac1993explicit] for large walk lengths $T$, i.e., the Brownian Motion limit. Since then simpler and more general methods were devised, which are based on using Cauchy’s formula with relates the support function of a curve to the perimeter and the area enclosed by the curve [@Majumdar2009Convex; @Majumdar2010Random]. More recently also the mean hypervolume and surface for arbitrary dimensions was calculated [@Eldan2014Volumetric]. For discrete-time random walks with jumps from an arbitrary distribution, the perimeter of the convex hull for finite (but large) walk lengths $T$ were computed explicitly [@grebenkov2017mean]. For the case of Gaussian jump lengths even an exact combinatorial formula for the volume in arbitrary dimensions is known [@kabluchko2016intrinsic]. For the variance there is an exact result for Brownian bridges [@Goldman1996]. Concerning the full distributions, no exact analytical results are available. Here sophisticated large-deviation simulations were used to numerically explore a large part of the full distribution, i.e., down to probabilities far smaller than $10^{-100}$ [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim]. Despite this increasing interest in the convex hulls of standard random walks, there seem to be no studies treating the convex hulls of self-avoiding walks. To fill this void, we use Markov chain Monte Carlo sampling to obtain the distributions of some quantities of interest over their whole support. To connect to previous studies [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim] we also compare the aforementioned variants to the standard random walk on a square lattice (LRW). We are mainly interested in the full distribution of the area $A$ and the perimeter $L$ of $d=2$ dimensional hulls for walks in the plane, since the effects of the self-interactions are stronger in lower dimensions. Though, we will also look into the volume $V$ in the $d=3$ dimensional case. In the past study on standard random walks [@schawe2017highdim] we found that the full distribution can be scaled to a universal distribution using only the exponent $\nu$ and the dimension for large walk lengths $T$. For the present case, where a walk might depend on its full history, one could expect a more complex behavior. Nevertheless, our results presented below show convincingly that also for self-interacting walks the distributions are universal and governed mainly by the exponent $\nu$, except for some finite-size effects, which are probably caused by the lattice structure. Further we use the distributions to obtain empirical large-deviation rate functions [@Touchette2009large], which suggests that a limiting rate function is mathematically well defined. We also give an estimate for the rate function, which is compatible with the known case of standard random walks and with all cases under scrutiny in this study. Models and Methods {#sec:mm} ================== This sections gives a short overview over the used models and methods, with references to literature more specialized on the corresponding subject. Where we deem adequate, also technical details applicable for this study are mentioned. Sampling Scheme --------------- To generate the whole distribution of the area or perimeter of the convex hull of a random walk over its full support, a sophisticated Markov chain Monte Carlo (MCMC) sampling scheme is applied [@Hartmann2002Sampling; @Hartmann2011]. The Markov chain is here a sequence of different walk configurations. The fundamental idea is to treat the observable $S$, i.e., the perimeter, area or volume, as the energy of a physical system which is coupled to a heat bath with adjustable “temperature” $\Theta$ and to sample its equilibrium distribution using the Markov chain. This can be easily done using the classical Metropolis algorithm [@metropolis1953equation]. Therefore the current walk configuration is changed a bit (the precise type of change is dependent on the type of walk, we are looking at and is explained in the following sections). The change is accepted with the acceptance probability $$\begin{aligned} \label{eq:pacc} p_\mathrm{acc} = \min\{1,\operatorname{e}^{-\Delta S / \Theta}\} \end{aligned}$$ and rejected otherwise. The $\Theta$ will then bias the configuration towards specific ranges of $S$. Configurations at small and negative $\Theta$ will show larger than typical $S$, small and positive $\Theta$ show smaller than typical $S$ and large values independent of the sign sample configurations from the peak of the distribution. Fig. \[fig:saw\_temp\_cmp\] shows typical walk configurations of the self-avoiding walk at different values of $\Theta$. In a second step, histograms of the equilibrium distribution $P_\Theta(S)$ are corrected for the bias introduced via $\Theta$ as follows. $$\begin{aligned} P(S) = \operatorname{e}^{S / \Theta} Z(\Theta) P_\Theta(S) \end{aligned}$$ The free parameter $Z(\Theta)$ can be obtained by enforcing continuity and normalization of the distribution. We do not present further details here, because the algorithm [@Hartmann2002Sampling] has been applied and explained in detail several times, also in a very general form [@Hartmann2014high]. In particular, the algorithm was already used successfully in other studies looking at the large deviation properties of convex hulls of random walks [@Claussen2015Convex; @Dewenter2016Convex]. ![\[fig:saw\_temp\_cmp\] (color online) Typical SAW configurations with $T=200$ steps and their convex hulls at different temperatures $\Theta$. $\Theta = \pm \infty$ corresponds to a typical configuration without bias. ](SAW_temp_cmp){width="0.7\linewidth"} Lattice Random Walk (LRW) ------------------------- All of the self-interacting random walks, which are the focus of this study, are typically treated on a lattice. Hence, we will start by introducing the simple, i.e., non-interacting, isotropic random walk on a lattice. For simplicity we will use a square lattice with a lattice constant of $1$. A realization consists of $T$ randomly chosen discrete steps ${\bm{\delta}}_i$. Here we use steps between adjacent lattice sites, i.e., $d$-dimensional Cartesian base vectors ${\bm{e}}_i$, which are drawn uniformly from $\{\pm{\bm{e}}_i\}$. The realization can be defined as the tuple of the steps $({\bm{\delta}}_1, ..., {\bm{\delta}}_T)$ and the position at time $\tau$ as $$\begin{aligned} {\bm{x}}(\tau) = {\bm{x}}_0 + \sum_{i=1}^\tau {\bm{\delta}}_i. \end{aligned}$$ Here we set the start point ${\bm{x}}_0$ at the coordinate origin. The set of visited sites is therefore $\mathcal P = \{{\bm{x}}(0), ..., {\bm{x}}(T)\}.$ The central quantity of the LRW is the average end-to-end distance $$r = \sqrt{\langle ({\bm{x}} (T)- {\bm{x}}_0)^2 \rangle} \,,$$ where $\langle \ldots \rangle$ denotes the average over the disorder. It grows polynomially and is characterized by the exponent $\nu$ via $r \propto T^\nu$. For the LRW it is $\nu = 1/2$, which is typical for all diffusive processes. As the change move for the Metropolis algorithm, we replace a randomly chosen ${\bm{\delta}}_i$ by a new randomly drawn displacement. Since our quantity of interest is the convex hull, i.e., a global property of the walk, we do not profit much from local moves, e.g., crankshaft moves. Thus we use this simple, global move. Smart-Kinetic Self-Avoding Walk (SKSAW)\[sec:sksaw\] ---------------------------------------------------- The Smart-Kinetic Self-Avoiding Walk (SKSAW) [@weinrib1985kinetic; @kremer1985indefinitely] is probably the most naive approach to a self-avoiding walk. It grows on a lattice and never enters sites it already visited. Since it is possible to get trapped on an island inside already visited sites, this walk needs to be *smart* enough to never enter such traps. In $d=2$ it is possible to avoid traps using just local information in constant time using the *winding angle* method [@kremer1985indefinitely]. In conjunction with hash table backed detection of occupied sites, a realization with $T$ steps can be constructed in time $\mathcal O(T)$. This method will typically yield longer stretched walks than the LRW, due to the constraint that it needs to be self-avoiding. This can be characterized by the exponent $\nu$, which is larger than $1/2$ in $d=2$. The sketch Fig. \[fig:saw\_prob\] shows that this ensemble does not contain every configuration with the same probability but prefers closely winded configurations. This is also visible in Fig. \[fig:SKSAW\]. This is characterized by the exponent $\nu=4/7$ [@Kennedy2015] which is larger than the $\nu$ for LRW, but smaller than for the SAW. Also note that it is conjectured that the upper critical dimension is $d=3$ [@kremer1985indefinitely], i.e., $\nu = 1/2$ for all $d \ge 3$ – possibly with logarithmic corrections in $d=3$. Therefore only $d=2$ is simulated in this study. ![\[fig:saw\_prob\] Decision tree visualizing the probability to arrive at certain configurations following the construction rules of the SKSAW. Not all possible configuration have the same probability, hence this rules define a different ensemble than SAW. ](saw_prob) While it is easy to draw realizations from this ensemble uniformly, i.e., simple sampling, it is not so straight forward to apply the MCMC changes. If one just changes single steps like for the LRW, and accepts if it is self-avoiding or rejects if it is not, one will generate all self-avoiding walk configurations with equal probability. Our approach to generate realizations according to this ensemble handles the construction of the walk as a *black box*. It acts on the random numbers used to generate a realization from scratch. During the MCMC at each iteration one random number is replaced by a new random number and a SKSAW realization is regenerated from scratch using the modified random numbers [@Hartmann2014high]. This change is then accepted according to Eq.  and undone otherwise. Self-Avoiding Random Walk (SAW) ------------------------------- While the above mentioned SKSAW does produce self-avoiding walks, SAW denotes another ensemble. The ensemble where realizations are drawn uniformly from the set of all self-avoiding configurations. It is not trivial to sample from this distribution efficiently. The black box method used for SKSAW is not feasible, since the construction of a SAW takes time exponential in the length with simple methods like dimerization [@dimerization; @Madras2013]. It is possible to perform changes directly on the walk configuration and accept them according to Eq. , but their rejection rate is typically quite high and the resulting configurations are very similar [@Madras2013], which makes this inefficient. The state of the art method to sample SAW is the *pivot algorithm* [@Madras2013]. It chooses a random point and uses it as the pivot for a random symmetry operation, i.e., rotation or mirroring. If the resulting configuration is not self avoiding, it is rejected. Otherwise we accept it with the temperature dependent acceptance probability Eq. . As mentioned previously, the exponent $\nu=3/4$ [@lawler2002scaling] is larger than for the SKSAW. Since the upper critical dimension for SAW is $d=4$, this study will also look at $d=3$, where an exact value of $\nu$ is not known and the best estimate is $\nu = 0.587597(7)$ [@Clisby2010Accurate], though our focus is on $d=2$ for this type. While there are highly efficient implementations of the pivot algorithm [@Clisby2010Accurate; @clisby2010efficient] the time complexity of the problem at hand is dominated by the time needed to construct the convex hull, thus we go with the simple hash table based $\mathcal O(T)$ approach [@Madras2013]. Loop-Erased Random Walk (LERW) ------------------------------ The LERW [@Lawler1980Self] uses a different approach to achieve the self-avoiding property. It is built as a simple LRW but each time a site is entered for the second time, the loop that is formed, i.e., all steps since the first entering of this site, is erased. While this ensures no crossings in the walk, the resulting ensemble is different from the SAW ensemble and the walks are longer stretched out, as characterized by the larger exponent $\nu = 4/5$ [@Lawler2011; @Guttmann1990Critical; @Majumdar1992Exact]. Similar to the SAW the upper critical dimension is $d=4$ and an estimate for $d=3$ is $\nu = 0.61576(2)$ [@Wilson2010]. For construction – similar to SKSAW – we need to keep all used random numbers and change them in the MCMC algorithm. This leads to a dramatically higher memory consumption than simple sampling, where each loop can be discarded as soon as it is closed. Convex Hulls ------------ We will study the *convex hulls* $\mathcal C$ of the sites visited by the random walk $\mathcal P$. The convex hull of a point set $\mathcal P$ is the smallest polytope containing all Points $P_i \in \mathcal P$ and all line segments $(P_i, P_j)$. Some example hulls are shown in Fig. \[fig:rw\]. Convex hulls are one of the most basic concepts in computational geometry [^1] with noteworthy application in the construction of Voronoi diagrams and Delaunay triangulations [@brown1979Voronoi]. For point sets in the $d=2$ plane, we use Andrew’s *Monotone Chain* [@Andrew1979Another] algorithm for its simplicity and *Quickhull* [@Bykat1978Convex] as implemented by *qhull* [@Barber1996thequickhull] for $d=3$. Both algorithms have a time complexity of $\mathcal O(T \ln T)$. In $d=2$ Andrew’s Monotone Chain algorithm results in ordered points of the convex hull. Adjacent points $(i, j)$ in this ordering are the line segments of the convex hull. Quickhull results in the simplical facets of the convex hull. To obtain the perimeter of a $d=2$ convex hull, we sum the lengths of its line segments $L_{ij}$. To calculate the area and the volume, we use the same fundamental idea. In both cases we subdivide the area/volume into simplexes, i.e., triangles for the area and tetrahedra for the volume. Therefore we choose an arbitrary fixed point $p_0$ inside of the convex hull and construct a simplex for each facet $f_m$, i.e., for $d=2$ each line segment of the hull $f_m = (i, j)$ forms a triangle $(i, j, p_0)$ and each triangular face $f_m = (i, j, k)$ of a $d=3$ dimensional polyhedron, forms a tetrahedron with $p_0$. The volume of a triangle is trivially $$\begin{aligned} A_{ijp_0} = \frac{1}{2} \operatorname{dist}(f_m, p_0) L_{ij}, \end{aligned}$$ where $\operatorname{dist}(f_m, p_0)$ is the perpendicular distance from a facet $f_m$ to a point $p_0$. Since the union of all triangles built this way, is the whole polygon, the sum of their areas is the area of the polygon. Similar the volume of a polyhedron is the sum of the volumes of all tetrahedra constructed from its faces. The volume of the individual tetrahedra is given by $$\begin{aligned} V_{ijkp_0} = \frac{1}{3} \operatorname{dist}(f_m, p_0) A_{ijk}. \end{aligned}$$ For random walks on a lattice with $T$ steps of length $1$ in $d$ dimensions the maximum volume is $$\begin{aligned} \label{eq:max} S_\mathrm{max} = \frac{(T/{{\ensuremath{d_{\mathrm{e}}}}})^{{\ensuremath{d_{\mathrm{e}}}}}}{{{\ensuremath{d_{\mathrm{e}}}}}!} \end{aligned}$$ for $T$ divisible by the effective dimension ${{\ensuremath{d_{\mathrm{e}}}}}$ of the observable, e.g., 2 for the area of a planar hull or 3 for the volume in three dimensions. For example, the configuration of maximum area corresponds to an L-shape, i.e., $A_\mathrm{max} = \frac{T^2}{8}$. This form can be derived by the general volume of an $d$-dimensional simplex defined by its $d+1$ vertices ${\bm{v}}_i$ [@Stein1966Volume] $$\begin{aligned} V = \frac{1}{d!} \det{({\bm{v}}_1 - {\bm{v}}_0, \ldots, {\bm{v}}_d - {\bm{v}}_0)}. \end{aligned}$$ Without loss of generality, we set ${\bm{v}}_0$ to be the coordinate origin. To achieve maximum volume all ${\bm{v}}_i, i>0$ need to be orthogonal and of equal length. Thus a random walk going $T/d$ steps along some base vector ${\bm{e}}_i$ and continuing with $T/d$ steps in direction ${\bm{e}}_{i+1}$ has a convex hull defined by the tetrahedron specified by ${\bm{v}}_i = \sum_{j=1}^{i} \frac{T}{d} {\bm{e}}_j$. The matrix $M = ({\bm{v}}_1, \ldots, {\bm{v}}_d)$ is thus triangular and its determinant is the product of its diagonal entries $M_{ii} = \frac{T}{d}$ which leads directly to Eq. . An exception occurs in $d=2$ where the perimeter is $L_\mathrm{max} = 2T$. Results ======= The focus of this work lies on $d=2$ dimensional SAW and LERW. The results for higher dimensions and for SKSAW are generated with less numerical accuracy. The LRW results also have a lower accuracy as their purpose is mainly to scrutinize the effect of the lattice structure underlying all considered walk types in comparison to the non-lattice results from [@schawe2017highdim]. Also not all combinations are simulated, but only those listed with a value in Table \[tab:kappa\]. The same raw data is evaluated for equidistant bins and logarithmic bins. And the respective variants are shown according to the scaling of the $x$-axis. Correlations ------------ To get an intuition for how the configurations with atypical large areas $A$ or perimeters $L$ look like, we visualize the correlation between these two observables as scatter plots in Fig. \[fig:scatter\]. ![\[fig:scatter\] (color online) The top row shows data from simulations biasing towards larger (and smaller) than typical perimeters $L$. The bottom row biases the area $A$. The left column shows data from LRW and the right from SAW both with $T=512$ steps. The results of simple sampling are shown in black. Note that only very narrow parts are covered by simple sampling for the LRW. ](scatter_SAWvsLRW) Since the smallest possible SAW is an (almost) fully filled square, there can not be instances below some threshold, which explains the gaps on the left side of the scatter plots and of the distributions shown in the following section. In the center of the scatter plots, which is already in probability regions far beyond the capabilities of simple sampling methods, the behavior becomes strongly dependent on the bias. If biasing for large perimeters (top) the area shows a non-monotonous behavior. First, somehow larger perimeters come along typically with larger areas for entropic reasons, i.e., there are less configurations which are long and thin, and more bulky, which have a larger area. Though, for the far right tail, the only configurations with extreme large perimeters are almost line like and have thus a very small area. Also note that the excluded volume effect of the SAW leads to overall larger areas at the same perimeters. On the other hand, when biasing for large areas (bottom) the configurations with largest area, which are L-shaped (cf. Fig. \[fig:saw\_temp\_cmp\]), unavoidably have quite large perimeters, hence the scatter plots show an almost linear correlation between area and perimeter. Since the configurations of large areas naturally avoid self intersections, since steps on already visited points do not enlarge the convex hull, the differences between LRW and SAW diminish in the right tail. Note that with the large-area bias, no walks with the very extreme perimeters exist, for the reason already mentioned. Note however that these scatter plots are very dependent on which observable we are biasing for. In principle we observe that small perimeters are strongly correlated with small areas while for large but not too large perimeters, there is a broad range of area sizes possible. For extremely large perimeters, the area must be small. For a comprehensive analysis, one would need a full two dimensional histogram, wich could be obtained using Wang Landau sampling, but which is beyond the scope of this study and would require a much larger numerical effort. Nevertheless, from looking at Fig. \[fig:scatter\] one can anticipate that the two dimensional histogram would exhibit a strong correlation for small values of $L$ and a broad scatter of the accessible values of $A$ for larger but not too large values of $L$. Moments and Distributions ------------------------- The distributions of the different walk types differ considerably. This can be observed in Fig. \[fig:compare\], where distributions of the area $A$ for all types with $T=1024$ steps are drawn. The main part of the distribution shifts to larger values for larger value of $\nu$ as expected and the probability of atypically large areas is boosted even more in the tails. ![\[fig:compare\] (color online) Distribution of all scrutinized walk types with $T=1024$ steps. The vertical line at $A_\mathrm{max}=131072$ denotes the maximum area (Eq. ), i.e., SAW and LERW are sampled across their full support and SKSAW and LRW are not. The inset shows the peak region. The gap on the left is due to excluded volume effects, i.e., there are no configurations with area below some threshold, since this would require self-intersection. ](compareTypes2D) In the right tail, the distributions seem to bend down. Below, where we show results for different walk sizes $T$, we see that this is a finite-size effect of the lattice structure and the fixed step length. This can be seen also as follows: Since the lattice together with the fixed step length sets an upper bound on the area, the probability plummets near this bound for entropic reasons, i.e., there are for any walk length $T$ only 8 configurations with maximum area (due to symmetries) such that all self-avoiding types will meet at this point. (not visible because the bins are not fine enough) This is supported from Ref. [@Claussen2015Convex] which shows that the distribution $P(A)$ for standard random walks with Gaussian jumps, i.e., without lattice or fixed step length, do not bend down and have an exponential right tail. We conclude that the deviation from this are thus caused by this difference. First we will look at the rescaled means $\mu_S = {\ensuremath{\left<S\right>}} / T^{{{\ensuremath{d_{\mathrm{e}}}}}\nu}$, where $S$ is an observable and ${{\ensuremath{d_{\mathrm{e}}}}}$ its effective dimension, as introduced above in Eq. . The scaling is a combination of the scaling of the end-to-end distance $r \propto T^\nu$ and the typical scaling that a $d$-dimensional observable scales as $r^d$ with a characteristic length $r$. ![\[fig:means\] (color online) Scaled means $\mu_A = {\ensuremath{\left<A\right>}} / T^{2\nu}$ and $\mu_L = {\ensuremath{\left<L\right>}} / T^{\nu}$ for different walk types. The lines are fits to extrapolate the asymptotic values shown in Table \[tab:measuredMu\] according to Eq. . Errorbars of the values are smaller than the line of the fit and not shown for clarity. ](means) Nevertheless, due to finite-size corrections, the ratios $\mu_S={\ensuremath{\left<S\right>}} / T^{{{\ensuremath{d_{\mathrm{e}}}}}\nu}$ will still depend on the walk length. Thus, the measured estimates $\mu_S=\mu_S(T)$ at specific walk lengths $T$ need to be extrapolated to get an estimate of the asymptotic value $\mu_S^\infty = \lim_{T\to \infty}\mu_S(T)$. For the extrapolation we use [@schawe2017highdim] $$\begin{aligned} \label{eq:extrapolation} \mu_S(T) = \mu_S^\infty + C_1 T^{-1/2} + C_2 T^{-1} + o(T^{-1}). \end{aligned}$$ This choice is motivated by a large $T$ expansion for the area $A$ (${{\ensuremath{d_{\mathrm{e}}}}}=2$) of the convex hulls of standard random walks ($\nu=1/2$) with Gaussian jumps [@grebenkov2017mean] $$\begin{aligned} \frac{{\ensuremath{\left<A\right>}}}{T} = \frac{\pi}{2} + \gamma \sqrt{8\pi}\, T^{-1/2} + \pi(1/4+\gamma^2)\, T^{-1}+ o(T^{-1}), \end{aligned}$$ where the constant $\gamma= \zeta(1/2)/\sqrt{2\pi}=-0.58259\dots$. A natural guess for a generalization to oberservables of a different effective dimension ${{\ensuremath{d_{\mathrm{e}}}}}$ [@schawe2017highdim] and different walk types would be a similar behavior with different coefficients like Eq. . Indeed, using this form to estimate the asymptotic means $\mu_S^\infty$ of the observable $S$ yields good fits, as visible in Fig. \[fig:means\]. In fact, for the fit quality we obtain $\chi_\mathrm{red}^2$ values between $0.4$ and $1.7$ (the fit ranges for SKSAW begin at $T=512$, for LRW, SAW and LERW at $T=128$, hinting at more severe corrections to scaling for the former). We assume that the scaling is thus valid for arbitrary random walk types. The resulting fit parameters are shown in Table \[tab:measuredMu\]. For standard random walks with Gaussian jumps the asymptotic means $\mu^\infty_{S,\mathrm{Gaussian}}$ are known [@Eldan2014Volumetric]. These results can be used to predict the corresponding values for LRW. First consider the following heuristic argument for a $d=2$ square lattice. On average a random walk takes the same amount of steps in $x$ and $y$ direction, such that on average two steps displace the walker by $\sqrt{2}$, i.e., the diagonal of a square. In contrast a Gaussian walker with variance $1$ will be displaced on average by $1$ every step. To make both types comparable, we can increase the lattice constant to $\sqrt{2}$, which leads to an average displacement of $1$ per step for the LRW. Using the same argumentation for higher dimensions, we can use the trivial scaling with the lattice constant $S^{{\ensuremath{d_{\mathrm{e}}}}}$ and the length of the diagonal of a unit hypercube $d^{1/2}$, to derive a general conversion: $$\begin{aligned} \label{eq:lattice} \mu_{S,\mathrm{LRW}}^\infty = \mu_{S,\mathrm{Gaussian}}^\infty / d^{{{\ensuremath{d_{\mathrm{e}}}}}/ 2}. \end{aligned}$$ These known results are listed next to our measurements in Table \[tab:measuredMu\] and are within errorbars compatible with our measurements. ------------------------ -------------- ------------- ------------- -------------- -- \[0.05cm\] LRW (exact) $3.5449...$ $0.7854...$ $2.0944...$ $0.21440...$ LRW $3.5441(7)$ $0.7852(2)$ $2.0945(4)$ $0.21445(4)$ SKSAW $4.5355(12)$ $1.2642(5)$ - - SAW $0.8233(7)$ $0.7714(1)$ $2.069(2)$ $0.1996(2)$ LERW $2.1060(3)$ $0.2300(1)$ $1.6436(2)$ $0.13908(3)$ ------------------------ -------------- ------------- ------------- -------------- -- : \[tab:measuredMu\] Asymptotic mean values extrapolated from simulational data and the exactly known values for the standard random walk (LRW). The columns labeled with $\mu_L^\infty$ and $\mu_A^\infty$ are for $d=2$, those labeled with $\mu_{\partial V}^\infty$ and $\mu_V^\infty$ are for $d=3$. For $d=3$ we did not simulate the SKSAW, see Section \[sec:sksaw\]. Also SAW has lower accuracy because of fewer samples in $d=3$. Since we have data for the whole distributions, a natural question is, whether this scaling does apply over the whole support of the distribution. There is evidence that this is true for the convex hulls of standard random walks [@Claussen2015Convex] in arbitrary dimensions [@schawe2017highdim]. That means the distributions of an observable $S$ for different walk lengths $T$ should collapse onto one universal function $$\begin{aligned} \label{eq:scaling} P(S) = T^{-{{\ensuremath{d_{\mathrm{e}}}}}\nu} \widetilde{P}(ST^{-{{\ensuremath{d_{\mathrm{e}}}}}\nu}). \end{aligned}$$ Fig. \[fig:scaling\] shows the distributions of the $d=2$ area of all considered random walk types scaled according to Eq. . The curves collapse well in the peak region and in the intermediate right tail. In the far right tail clear deviations from a universal curve are obvious, which are the mentioned finite-size effects caused by the lattice. The distributions look qualitatively similar, though with weaker finite size effects, i.e., a better collapse, for the perimeter $L$ (not shown). In $d=3$, where we have studied the volume, the results also look similar but exhibit stronger finite-size effects (not shown). Using the full distributions at different values of the walk length $P_T$, we can test if it obeys the large deviation principle, i.e., if $\Phi$ exists, such that the distribution scales as $$\begin{aligned} \label{eq:largeDev} P_T \approx \operatorname{e}^{-T\Phi} \end{aligned}$$ for large values of $T$ [@Touchette2009large]. To simplify comparison, the support of the rate function is usually normalized to $[0, 1]$. Here we achieve this by using the maximum Eq. . Solving Eq.  for $\Phi$ results in $$\begin{aligned} \Phi(S/S_\mathrm{max}) = -\frac{1}{T} \ln P(S/S_\mathrm{max}). \end{aligned}$$ We plot this for a selection of our results in Fig. \[fig:rate\]. From these plots, $\Phi$ seems to approximately follow a power law in the intermediate right tail, while the finite-size effects caused by the lattice play a major role in the far right tail, which “bends up” consequently. Assuming that the rate function behaves approximately as a power law, which seems consistent with our data shown in Fig. \[fig:rate\], i.e., $$\begin{aligned} \Phi(s) \propto s^{\kappa}, \label{eq:power-law} \end{aligned}$$ the exponent $\kappa$ can be estimated by combining the definition of $\Phi$ Eq.  with the scaling assumption Eq.  as follows, note that for clarity we use here $S_\mathrm{max} \propto T^{{{\ensuremath{d_{\mathrm{e}}}}}}$. $$\begin{aligned} \exp{\ensuremath{\left(-T \Phi(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}})\right)}} \sim \frac{1}{T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}} \widetilde{P}(S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}) \end{aligned}$$ The $1/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$ term on the right-hand side can be ignored next to the exponential. Since the right-hand side is a function of $S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$, the left-hand side must also be only dependent on $S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}$. This can be achieved by assuming $-\nu {{\ensuremath{d_{\mathrm{e}}}}}\kappa + {{\ensuremath{d_{\mathrm{e}}}}}\kappa = 1$, as one can easily see by using Eq. (\[eq:power-law\]): $$\begin{aligned} \intertext{Starting from the left-hand side} & \exp{\ensuremath{\left(-T^{1} \Phi(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}})\right)}} \\ \propto& \exp{\ensuremath{\left(-T^{1} {\ensuremath{\left(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}}\\ =& \exp{\ensuremath{\left(-T^{-\nu {{\ensuremath{d_{\mathrm{e}}}}}\kappa + {{\ensuremath{d_{\mathrm{e}}}}}\kappa} {\ensuremath{\left(S/T^{{{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}}\\ =& \exp{\ensuremath{\left(-{\ensuremath{\left(S/T^{\nu {{\ensuremath{d_{\mathrm{e}}}}}}\right)}}^{\kappa}\right)}} \end{aligned}$$ From this we can conclude $$\begin{aligned} \label{eq:kappa} \kappa = \frac{1}{d_{\mathrm{e}}(1-\nu)}, \end{aligned}$$ which simplifies to the case of the standard random walk above the critical dimension of the given walk type [@schawe2017highdim] $$\begin{aligned} \kappa_g = \frac{2}{d_{\mathrm{e}}}. \end{aligned}$$ To compare this crude estimate with the results of our simulations, we do a point-wise extrapolation of the empirical rate functions for fixed walk lengths $T$ as done before in [@Claussen2015Convex; @Dewenter2016Convex; @schawe2017highdim]. For the point-wise extrapolation, we use measurements $\Phi_T$ for multiple values of the walk length $T$ at fixed values of $S/S_\mathrm{max}$. Since our data are discrete due to binning, the values of $\Phi_T$ are obtained by cubic spline interpolation. With these data points, which can be thought of as vertical slices through the plots of Fig. \[fig:rate\], we extrapolate the $T\to\infty$ case with a fit to a power law with offset $$\begin{aligned} \Phi = a T^b + \Phi_\infty. \end{aligned}$$ The extrapolated values are marked with black dots in Fig. \[fig:rate\]. Since finite-size effects have major impact on the tails due to the lattice structure, we expect that our estimate is only valid for the intermediate right tail of our simulational data. To estimate sensible uncertainties, we fit different ranges of our data and give the center of the range of the obtained $\kappa$ as our estimate with an error including the extremes of the obtained $\kappa$. The black lines in Fig. \[fig:rate\] are our expected values, which are in all examples compatible with some range of our extrapolated data. All exponents $\kappa$ we calculated, together with our expectations, are listed in Table \[tab:kappa\]. A more detailed discussion of the examples shown in Fig. \[fig:rate\] follows. In Fig. \[fig:rate:LRW\] the LRW is shown, which is equivalent to Brownian motion in the large $T$ limit for which Ref. [@Claussen2015Convex; @schawe2017highdim] showed the rate function to behave like a power law with exponent $\kappa=1$ for the area in $d=2$. Using the above mentioned procedure we obtain $\kappa = 0.99(2)$ which is in perfect agreement with the expectation $\kappa = 1$. Fig. \[fig:rate:SKSAW\] shows the same for the SKSAW. The obtained asymptotic rate function’s exponent $\kappa = 1.28(12)$ is compatible with our expectation, though the stronger finite-size effects, lead to larger uncertainties of our estimate. Fig. \[fig:rate:SAW\] shows the same but for the volume of the SAW in $d=3$ dimensions. The finite-size effects are apparently stronger for the volume in $d=3$, as the slope of the right-tail rate function gets less steep with increasing system size. Fig. \[fig:rate:LERW\] for the perimeter of a $d=2$ dimensional LERW. In contrast to the area and volume the far right tail of the perimeter seems to bend down instead of up, albeit slightly. Though in the intermediate right tail, the rate function seems to behave as expected. --------------- --------------- ------------ --------------- ------------ \[0.1cm\] LRW $1 $ $0.99(2)$ $2 $ - SKSAW $\frac{7}{6}$ $1.28(12)$ $\frac{7}{3}$ - SAW $2 $ $2.2(4)$ $4 $ $4.11(14)$ SAW $d=3$ $0.809... $ $0.92(11)$ $1.214... $ - LERW $\frac{5}{2}$ $2.57(24)$ $5 $ $4.82(19)$ LERW $d=3$ $0.867... $ $0.89(9)$ $1.299... $ - --------------- --------------- ------------ --------------- ------------ : Comparison of expected and measured rate function exponent $\kappa$. The value is the center of multiple fit ranges and the error is chosen such that the largest and the smallest result is enclosed. []{data-label="tab:kappa"} In general, our data supports the convergence to a limiting rate function, which, mathematically speaking, means that the *large-deviation principle* holds. This means that the distributions are somehow well behaved and might be accessible to analytical calculations. Though the estimate for what the rate function $\Phi$ actually is, can possibly be improved. However, since our estimate for $\kappa$ is always compatible with our measurements it appears plausible that also for interacting walks the distribution of the convex hulls is governed by the scaling behavior of the end-to-end distance, as given by the exponents $\nu$. Conclusions {#sec:conclusion} =========== We numerically studied the area and perimeter of the convex hulls of different types of self-avoiding random walks in the plane and to a lesser degree the volume of their convex hulls in $d=3$ dimensional space. By applying sophisticated large-deviation algorithms, we calculated the full distributions, down to extremely small probabilities like $10^{-400}$. We also obtained corresponding rate functions of these observables. Our data support a convergence of the rate functions, which means the large-deviation principle seems to hold. We observed a generalized scaling behavior, which was before established for standard random walks. Thus, although the self-avoiding types of walk exhibit a more complicated behavior as compared to standard random lattice walks, and although the limiting scaled distributions of their convex hull’s volume and surface look quite different for the various walk cases, in the end the convex hull behavior seems to be still governed by the single end-to-end distance scaling exponent $\nu$. We also observed, rather expectedly, that the two observables area and perimter are highly correlated for small values. For large but not too large values of the perimeter, many different values of the area are possible, but statistically dominated by rather small values of the area. Extremly large values of the perimeter are only feasible with shrinking area. Finally, we gave estimates for the large $T$ asymptotic mean values of the mentioned observables. These might be of interest for attempts to calculate these values analytically. For future studies it could be interesting to look closer into the correlations between different observables that we briefly noticed. For a more throughout study, it would be useful to obtain full two-dimensional histograms. This work was supported by the German Science Foundation (DFG) through the grant HA 3169/8-1. HS and AKH thank the LPTMS for hospitality and financial support during one and two-month visits, respectively, where considerable part of the projects were performed. The simulations were performed at the HPC clusters HERO and CARL, both located at the University of Oldenburg (Germany) and funded by the DFG through its Major Research Instrumentation Programme (INST 184/108-1 FUGG and INST 184/157-1 FUGG) and the Ministry of Science and Culture (MWK) of the Lower Saxony State. We also thank the GWDG Göttingen for providing computational resources. [^1]: 3 of the first 4 examples for static problems of computational geometry in the Wikipedia can utilize convex hulls for their solution (<https://en.wikipedia.org/wiki/Computational_geometry>, 12.01.2018).

ArXiv

--- author: - Xiangyu Cao - Alexandre Nicolas - Denny Trimcev - Alberto Rosso bibliography: - 'yield.bib' title: ' Soft modes and strain redistribution in continuous models of amorphous plasticity: the Eshelby paradigm, and beyond?' --- ![(a) Sketch of the macroscopic shear stress response of a disordered solid subject to a quasistatic deformation, with depictions of common deformation protocols. Stress fluctuations are not represented in the sketch. (b) Representations of the new strain variables $e_1$, $e_2$, and $e_3$. In this work we consider pure shear along the $e_2$ direction and $\gamma$ identifies to the average of $e_2$.[]{data-label="fig:Macroscopic_response"}](figure1.png){width="0.8\columnwidth"} Apply a dab of toothpaste onto a toothbrush and slightly tilt the brush. The paste will respond to the small shear stresses $\Sigma$ thus created in its bulk by deforming elastically. In contrast, when you squeeze a toothpaste tube, the stresses in the material exceed a critical yield value $\Sigma_y$, and the paste starts to flow. This “liquid”-like phase under shear is observed not only in pastes and (concentrated) suspensions, but also in other soft solids such as emulsions and foams [@coussot2014yield]. Other disordered materials such as metallic glasses also depart from an elastic behavior under large enough stresses, but then break instead of flowing. In the athermal limit, the change observed at $\Sigma_y$ is a dynamical phase transition known as yielding transition. To study it, a standard experimental protocol consists in slowly[^1] deforming the material and monitoring its macroscopic stress. For small deformations, the response is linear and elastic. For larger ones, the deformation becomes macroscopically irreversible, due to the onset of plasticity. Three distinct plastic responses can be observed, as shown in Fig. \[fig:Macroscopic\_response\]: (i) the stress grows monotonically and saturates at a steady-state value $\Sigma_y$; (ii) the stress overshoots $\Sigma_y$ and, upon reaching $\Sigma_\text{max}>\Sigma_y$, drops rapidly to the stationary value $\Sigma_y$ [@divoux2011stress] (note that it is still unclear if the material fails globally at $\Sigma_\text{max}$ – as in a spinodal transition [@zapperi1997first; @procaccia2017mechanical] – or through a large sequence of finite-size avalanches) (iii) at $\Sigma_\text{max}$, the material breaks and the stress drops to zero. Microscopically, the mechanism underlying the irreversible plastic response is the localised rearrangement of a few particles (droplets in emulsions, bubbles in foams), a process called shear transformation (ST) [@argon1979plastic]. Recently, more detailed investigation has revealed that these ST do not arise randomly in the material, but display spatial correlations [@chikkadi2012shear; @nicolas2014spatiotemporal]. It is now widely believed that these correlations stem from the elastic deformation induced by the ST, which has a peculiar quadrupolar shape, as predicted by Eshelby half a century ago [@eshelby1957determination; @schall2007structural]. This quadrupolar kernel has been observed in atomistic simulations of several model glasses [@maloney2006amorphous; @puosi2014time]. Moreover, the plastic ST instability is preceded by the emergence of localisation in the low-frequency modes of the vibrational spectrum [@tanguy10mode; @manning11soft; @charbonneau16universal]. The localised soft spots tend to coincide with the subsequent ST. Interestingly, the displacement field around a soft spot displays a long-range tail, decaying as $r^{1-d}$, with $d$ the spatial dimension, which is consistent with the quadrupolar shape observed after the plastic event. On the basis of this picture of localised plastic rearrangements, elasto–plastic models (EPM) have proposed to coarse-grain disordered solids into a collection of blocks alternating between an elastic regime and plastic events interacting via a quadrupolar kernel. Following similar endeavours for the study of earthquakes [@chen1991self], these models have succeeded in capturing the presence of strongly correlated dynamics in these systems (avalanches, possible shear bands, etc.) [@baret2002extremal; @budrikis2013avalanche; @nicolas2017deformation; @lin2014scaling; @lin15prl; @gueudre17scaling]. However, a clear connection between the microscopic description and these coarse-grained models is missing. In particular, the universality of the quadrupolar propagator used in EPM may still be questioned and the discreteness of EPM precludes the study of vibrational modes. An alternative approach is provided by continuum models that extend the free energy description of solids beyond the perfect elastic limit. In these models plasticity is introduced by means of a disordered potential which displays many local minima, as explained in Section \[sec:model\]. Such models, possibly pioneered by Kartha and co-workers [@kartha95tweed], were intensively studied by Onuki [@onuki2003plastic] and Jagla [@jagla07shear]. This paper intends to use the continuum approach to bridge the gap between atomistic simulations and discrete EPM [^2], with an emphasis on the initial soft modes and the actual response to ST. Considering two-dimensional (2D) materials subjected to pure shear, we find that the low-frequency modes are always peaked in point-like “soft spots”, where the next ST will take place. This extreme localisation is at variance with short-range depinning models, where soft modes have a finite localisation length which can be tuned by playing with the disorder strength [@cao2017localisation; @Tanguy2004localisation]. A closer analysis shows a halo of finite displacements around the soft spots pointing in the radial direction, with a $1/r$ radial decay and a two-fold azimuthal symmetry (this corresponds to a $1/r^2$ decay with four-fold azimuthal symmetry in the strain field), due to the elastic embedding of the impurities, see Section \[sec:denny\]. Surprisingly, these halos do not always match Eshelby’s solution. Instead, we find a one-parameter continuous family of kernels depending on the distribution of plastic disorder in the system (see Fig. \[fig:displacement\_th\]). Their shapes are rationalised analytically in Section \[sec:single\_impurity\] by calculating the soft mode associated with a point-like plastic impurity at $r=0$ embedded in an incompressible elastic medium. In polar coordinates, the (non-affine) displacement field $u$ reads: $$u_r(r,\theta) \propto \frac{ \cos(2\theta)} {1 + \delta \cos(4\theta)} \,,\, u_\theta = 0\,, \label{eq:u}$$ where $\theta = 0$ is the principal axis of positive stretch, and $\delta=(\mu_3-\mu_2)/(\mu_3+\mu_2)$ quantifies the plasticity-induced anisotropy in the shear moduli $\mu_2$ and $\mu_3$ (associated with the strains $e_2$ and $e_3$, respectively; see Fig \[fig:Macroscopic\_response\]). For $\delta=0$ we recover the standard quadrupolar (Eshelby-like) propagator. When $\delta \to 1$ we find a fracture-like kernel concentrating the deformation along the diagonal directions. This limit is obtained when the plastic potential softens the material to such an extent that the modulus along these directions vanishes (namely $\mu_2 \to 0$ while $\mu_3$ remains finite). To the best of our knowledge, the fracture-like propagator has not been observed yet, but we speculate that it might be seen in carefully aged glasses, in the marginal state that precedes global failure, when extended regions are on the brink of plastic failure. In the case of a single impurity, the soft mode has exactly the same shape as the final strain field induced by the ST, but also closely (or exactly if $\mu_2=\mu_3$) matches the transient strain field during the plastic event, up to renormalisation (Section \[sec:SPE3\]). {width=".9\textwidth"} Field-based models \[sec:model\] ================================ To begin with, we recall how plasticity is introduced in Continuum Mechanics descriptions of disordered solids [@kartha95tweed; @puglisi2000mechanics; @onuki2003plastic; @lookman03ferro; @jagla07shear; @jagla2017non]. In the spirit of the works of Jagla [@jagla07shear], this is achieved by first writing the free energy of an elastic material and then incorporating the plastic disorder in it. Strain variables and linear elasticity -------------------------------------- Even though glassy materials are discrete at the atomic scale, they can be handled as continua as long as one is interested in length scales larger than a few particle diameters [@tsamados2009local]. To linear order, deformations in a continuous medium are quantified by the strain tensor $$\begin{aligned} &\epsilon_{ij} = \frac12(\partial_j u_{i} + \partial_i u_{j}) \text{ with } i,j= 1,2\text{ in 2D.} \label{eq:tensors}\end{aligned}$$ It is convenient to trade off the strain tensor $\epsilon_{ij}$ for the following three strain variables: the volume distortion $e_1 = (\epsilon_{11} + \epsilon_{22})/2$ and the two independent shear strains $e_2 = (\epsilon_{11} - \epsilon_{22})/2$ and $e_3 = \epsilon_{12}$, represented in Fig. \[fig:Macroscopic\_response\](b). Since the $e_i$ derive from the same displacement field, the commutation rule for partial derivatives, *i.e.*, $\partial_{jk} u_{i} = \partial_{kj} u_{i}$, imposes a constraint on these variables, known as St. Venant condition. In terms of the Fourier transforms $ \hat{e}_i (q) = \int e_i(r) e^{{\mathbf{i}}q.r} {\mathrm{d}}^2 r $, this constraint reads $$\begin{aligned} &g:=\sum_{i=1}^3 Q_i \hat{e}_i = 0 \,,\, \label{eq:StV} \\ &(Q_i)_{i=1}^3 = (-{\left\vertq\right\vert}^2, q_1^2 - q_2^2, 2 q_1 q_2) \,, \label{eq:Ds} \end{aligned}$$ where $q_j = -{\mathbf{i}}\partial_j$ are the differential operators in Fourier space. These conditions turn out to be sufficient to prove the existence of the displacement field. With these new variables, the free energy of a uniform linear elastic solid reads $$F = \int_r B e_1^2 + \mu_2 e_2^2 + \mu_3 e_3^2 \label{eq:elastic_free} \,,$$ where $B$ is the bulk modulus and $\mu_2$ and $\mu_3$ are the shear moduli. The incompressible limit corresponds to $B\rightarrow \infty$ while, for an isotropic material, $\mu_2=\mu_3$. Plasticity ---------- If a disordered solid is strongly sheared, it will start to respond plastically by accumulating irretrievable deformation. This irreversible response is accounted for by introducing an anharmonic contribution to the free energy $F$ of eq. \[eq:elastic\_free\] so that $F$ can have multiple local minima, viz., $$F = \int_r \left[B e_1(r)^2 + V_{2,r}[e_2(r)] + \mu_3 e_3(r)^2 \right] dr \label{eq:free} \,,$$ where $V_{2,r}(e_2)= \mu_2 e_2^2 + W_{r}(e_2)$, with typically $ W_{r}(e_2)\leqslant 0$. Here, and in all the following, we will consider that plasticity only develops along the macroscopic shear direction, chosen to be $e_2$. Schematically, when the local strain $e_2$ at, say, $r=0$ is driven to a local maximum of $V_{2,r=0}$, a plastic event begins in which $e_2(0)$ slides into the next potential valley. This local change is elastically coupled to other regions of the material and may trigger other plastic events at $r^\prime \neq 0$, if $V_{j,r^\prime}(e_j)$ is also anharmonic. It is interesting to remark that, because of the shear-softening contribution $W_r$ to $V_{2,r}$, the local tangential shear modulus $\tilde{\mu}_2(0)= \frac{1}{2} V_{2,r=0}''$ vanishes at the onset of the instability, as observed in atomistic simulations. The softening of $\tilde{\mu}_2$ has the same effect on the local stress-strain relation as a shift of the minimum of the potential $V_{2,r=0}$ by an amount $e^{pl}$ in a material where the shear modulus would remain constant ($\tilde{\mu}_2=\mu_2$), *viz.*, $$\begin{aligned} &\sigma(r) = V_{2,r}^\prime[e_2(r)] = 2 \mu_2 [ e_2(r) - e^{pl} ]\,,\, \\ \text{where } & e^{pl} := - \frac{1}{2\mu_2}W_r^\prime[e_2(r)] \,. \label{eq:epldef}\end{aligned}$$ ![ *Illustrations of possible disordered potentials $V_{2,r}(e_2)$ –* (a,b) Interpretation of a plastic rearrangement from the viewpoint of the potential: While the system sits in a linear elastic region (a), the applied macroscopic stress $\Sigma$ tilts the potential, so much so that the system starts exploring the shear-softening region ($W_r<0$) and is finally able to briskly slide into a deeper minimum (b). (c) Illustration of the disordered potential used to derive rules of elasto-plastic models. The local stress variations during a plastic event are highlighted by green arrows and explained in the main text. []{data-label="fig:V2"}](potentials){width=".7\columnwidth"} Dynamics -------- Turning to the dynamics, the evolution of $e_i$ in the overdamped limit (relevant for foams and concentrated emulsions) is obtained by differentiating the total free energy $F_\mathrm{tot}$, viz. $$2 \eta_i \dot{e}_i = - \frac{\delta F_{\text{tot}}}{\delta e_i} \,,\label{eq:EoM}$$ where $\eta_i$ refers to the effective microscopic viscosity associated with the strain $e_i$. A difference with respect to the equation describing the depinning of an elastic line deserves to be underscored here. In the depinning case, the damping acts on the velocities $\dot{u}_i$, due to friction against a fixed substrate (or medium). Here, there is no such substrate and dissipation is due to the non-uniform velocities within the system; thus, only the velocity gradients are damped, as in the Navier-Stokes equation for fluids. In addition to the terms in eq. \[eq:free\], the total free energy $F_{\text{tot}}$, includes two contributions: the driving along $e_2$, $- \Sigma \, \hat{e}_2(0)$, where $ \Sigma$ corresponds to the macroscopic stress, as well as the St Venant constraint (eq. \[eq:StV\]) for all $q\neq0$ enforced by the Lagrange multipliers $\lambda(q)$, viz., $$F_{\text{tot}} = F + \int_{q \neq 0} \hat{\lambda}(q) \hat{g}(q) - \Sigma\, \hat{e}_2(0) \,. \label{Ftot}$$ The St. Venant constraint is naturally obtained by extremising $F_\text{tot}$ with respect to $\lambda(q)$, i.e., solving $\hat{g}(q) = 0$. The system can be driven in two distinct ways: *strain-controlled* protocols consist in controlling the time evolution of the macroscopic strain $\hat{e}_2(0)$ by properly adjusting $\Sigma$. Conversely, *stress-controlled* protocols result from imposing a constant macroscopic stress $\Sigma$ in eq. \[Ftot\] and leaving $\hat{e}_2(0)$ free. Via eqs. -, the macroscopic stress can be expressed as $\Sigma= \frac{1}{V} \int_r \sigma(r) + \eta_2 \dot \gamma$, where $\dot{\gamma}= \frac{2}{V} \dot{\hat{e}}_2(0) $ is the shear rate. If $\Sigma$ exceeds a value $\Sigma_y$ (set by the potentials $V_{2,r}$), the system will flow forever, with an ever–increasing strain $\hat{e}_2(0) \sim \dot{\gamma} t$ in the stationary state. Otherwise, it will reach a new equilibrium. Incompressible limit -------------------- From now on, we will focus on the incompressible limit $B\rightarrow \infty$, but equations for compressible systems are provided in Appendix \[sec:compressible\]. In this limit, $e_1=0$ so the St. Venant constraint yields $$2 q_1 q_2 \hat{e}_3(q) = (q_2^2 - q_1^2) \hat{e}_2(q) \,. \label{eq:StV-incomp}$$ As a consequence, the Fourier modes $\hat{e}_2(q)$ vanish whenever $q_1=0$ or $q_2=0$, with ${\left\vertq\right\vert}>0$. In real space, this means that the integral of $e_2$ along each horizontal or vertical line is constant and equal to the average strain, regardless of the mechanical law. Similarly, the integral of $e_3$ along each $\pm \pi/4$-direction line is also constant. These constraints on the strains are more fundamental than their counterparts for the stresses (or elastic strains), which are largely used in elasto-plastic models. The latter constraints impose that $\sigma_2$ ($\sigma_3$) have constant average along $\pm \pi/4$ lines (horizontal/vertical lines, respectively), where the stress components ${\sigma}_j$’s are in direct correspondence to the strain variables $e_j$, but these constraints are derived under the assumption of mechanical equilibrium. In addition, the displacement field $u_i$, which is easier to interpret than $e_2$, reduces to: $$\hat{u}_1 = {\mathbf{i}}q_1^{-1} \hat{e}_2 \,,\, \hat{u}_2 = -{\mathbf{i}}q_2^{-1} \hat{e}_2 \, \label{eq:integration}$$ if $q_1 q_2\neq 0$ and $\hat{u}_1,\hat{u}_2=0$ otherwise. Note that the zero mode $q=(0,0)$ of $\hat{u}_j$ corresponds to a global translation of the system. Hereafter, we assume isotropic viscosities, which are set to unity, viz., $\eta_2= \eta_3= 1$. Thus, the evolution of $e_2$, eq. \[eq:EoM\], turns into $$\begin{aligned} \label{eq:motion} 2 \dot{e}_2 = & - \frac{\delta F}{\delta e_2} + Q_2 \frac{ Q_2 \frac{\delta F}{\delta e_2} + Q_3 \frac{\delta F}{\delta e_3}}{ Q_2^2 + Q_3^2} + \Sigma \end{aligned}$$ and simplifies to $$2 \dot{e}_2 = -2\mu_2 e_2 -W_r^\prime(e_2) + \Sigma - \mathcal{G}_q [ W_r^\prime(e_2) - 2 \delta \mu e_2 ] , \label{eq:EoM_inc_real}$$ where we have used the shorthands $\delta \mu =\mu_3 - \mu_2$ and $$\mathcal{G}_q=\frac{-(q_1^2 - q_2^2)^2}{q^4} \label{eq:Gq}$$ for $q\neq 0$, and $\mathcal{G}_{q=0} = 0$. For systems with finite inertia, the foregoing overdamped equation can be generalised: $$\begin{aligned} 2 \rho \ddot{e}_2 = -q^2 \left[ \frac{1+\mathcal{G}_q}2 W_r^\prime(e_2) + \mu_2 e_2 - \delta \mu \, \mathcal{G}_q e_2 + \dot{e}_2 \right] \,, \end{aligned}$$ where $\rho$ is the density (see Appendix \[app:underdamped\]). Mathematically, one may notice that, for any given $q \neq 0$, the incompressible St Venant constraint  is satisfied on the line $e_3= -Q_2 Q_3^{-1} e_2$ of the plane $(\hat{e}_2(q),\hat{e}_3(q))$ and that the action of the Lagrange multiplier $\hat{\lambda}(q)$ is equivalent to taking only the tangential component of $\nabla F:= \left[ \delta F / \delta \hat{e}_2, \delta F / \delta \hat{e}_3 \right] $ along this line, via a projection of $\nabla F$ onto the *unit* tangential vector $\boldsymbol{t} := (Q_3/q^2,-Q_2/q^2)$ \[note that $Q_2^2 + Q_3^2 = q^4$ through eq. \]. Therefore, it is incorrect to just express $F$ as a function of $e_2$ on this line, *viz.*, $\tilde{F}(e_2) := F(e_2, -Q_2 Q_3^{-1} e_2)$ and then omit the Lagrange multiplier. Indeed, differentiating $\tilde{F}$ with respect to $e_2$ is equivalent to projecting $\nabla F$ onto the *non-normalised* tangential vector $\boldsymbol{\tilde{t}} = \left[ 1,-Q_2 Q_3^{-1} e_2\right]$. To obtain the correct dynamics without using multipliers, one should parametrise $F$ with the properly rescaled tangential coordinate $$\hat{e} (q):= \left[ \hat{e}_2(q), \hat{e}_3(q) \right] \cdot \boldsymbol{t} = q^2 Q_3^{-1} \hat{e}_2 (q) \,, \label{eq:rescaled_strain}$$ where $\cdot$ denotes the inner product.The rescaled coordinate $\hat{e}$ will prove useful in Sec. \[sec:denny\]. Response to a single plastic event \[sec:single\_impurity\] =========================================================== Although the equations established in the previous section fully define the model once the disorder potential $W_r$ is chosen, solving the full problem is complex. It is thus enlightening to start by considering the simple case where the plastic disorder is confined to an ‘impurity’ of size $a \to 0$ around $r=0$, while the rest of the material is elastic, viz., $$\begin{aligned} V_{2,r}(e_2) =& \mu_2 e_2^2(r) + a^2 \delta_{2D}(r) W(e_2(0)) - \Sigma\, \hat{e}_2(0) \,, \label{eq:V2oneimp}\end{aligned}$$ where $W$ is a disordered potential and $\delta_{2D}(r)$ is the Dirac distribution in 2D, evaluated at position $r$. Note that, for convenience, we have incorporated the driving contribution $- \Sigma\, \hat{e}_2(0)$ to the total free energy $F_\mathrm{tot}$ of eq.  into $V_{2,r}$. Beginning with the stable configuration sketched in Fig. \[fig:V2\](a), the driving gradually tilts the potential until the configuration becomes unstable (Fig. \[fig:V2\](b)). This triggers an instability whereby the local strain $e_2(0,t=0)$ evolves rapidly with time $t$ and modifies the local plastic disorder $W^\prime[e_2(0,t)]$ and plastic strain $e^{pl}(t)=- W^\prime[e_2(0,t)] / 2\mu_2$. Finally, the next stable configuration is attained, after a plastic strain $e^{pl}= e^{pl}_\star$ has been cumulated. In elasto–plastic models, the mechanical equilibration time is neglected everywhere in the material, except in the plastic impurity [@nicolas2017deformation]; thus, the strain field $e_2$ is always an equilibrium configuration for a given plastic strain $e^{pl}$, but this plastic strain may need a finite time to reach its final value $e^{pl}_\star$. In this section, we first derive the equilibrium configurations for the continuous models under study here and then show that, up to a normalisation coefficient, they coincide with the soft modes of the system just before the onset of the instability. Finally, we explore the transient dynamics during the plastic event. During the whole evolution, the dynamics are governed by the equation of motion . For Fourier modes $q \neq 0$ and for a single impurity, this equation reduces to: $$\dot{\hat{e}}_2(q) = \hat{e}_2(q) \left( - \mu_2 + \mathcal{G}_q \delta \mu \right) + ( 1 + \mathcal{G}_q ) \mu_2 a^2 e^{pl} \, \label{eq:EoM2}$$ with $\delta \mu = \mu_3 - \mu_2$. Equilibrium configuration ------------------------- To find the equilibrium configuration, $\dot e_2=0$ is set to zero in eq. . This immediately leads to $$\begin{aligned} \hat{e}_2(q) &= a^2 \, e^{pl}_* \frac {1+\mathcal{G}_q} {1+\mathcal{G}_q\,(1-\mu_3/\mu_2)}\,\mathrm{for}\,q\neq 0\, , \label{eq:e2q} \\ \hat{e}_2(0) &= a^2\, e^{pl}_* + \frac{V\,\Sigma}{2 \mu_2},\end{aligned}$$ where $V$ is the volume of the system and $e^{pl}_*$ must be determined self-consistently by integrating both sides of eq.  and noticing that $\int_q \hat{e}_2(q) dq \propto e_2(0)$ (for explicit results, see Appendix \[sec:C\]). Regarding the $q=0$-mode (derived from eq.  ), i.e., the extensive total strain $\hat{e}_2(0)$, one may remark that it will remain constant in a strain-controlled protocol, in which the macroscopic stress $\Sigma$ will thus decrease by $2\mu_2 a^2\, e^{pl}_* /V$, whereas in a stress-controlled protocol $\hat{e}_2(0)$ will increase by $a^2\, e^{pl}_*$ because of plasticity. For simplicity, the formulae below are given in the case of a strain-controlled protocol with $\hat{e}_2(0)=0$. In real space, the Fourier expression of eq.  translates to $$e_2(r, \theta) = \frac{ C }{r^2} \frac{\delta + \cos(4\theta)}{(1 + \delta \cos(4\theta))^2} + C_0 \delta(r) \,, \label{eq:e2r}$$ where the anisotropy parameter $\delta = \delta \mu/(\mu_3 + \mu_2)$ was introduced below eq. , and the constants $C_0$ and $C$ are given in Appendix \[app:Fourier\] (eq. ), along with details of the derivation. Finally, we can integrate $e_2(r, \theta)$ to obtain the noteworthy formula of eq. \[eq:u\] for the displacement field $u(r,\theta)$, as detailed in Appendix \[sec:ur\]. Let us mention that these calculations (for an incompressible material) can be generalised to the compressible regime ($B<\infty$, see Appendix \[sec:compressible\]) by substituting $\mathcal{G}_q$ in eq.  with $$\label{eq:kernelB} \mathcal{G}_{q}^{B<\infty} = \frac{B}{B + \mu_3} \mathcal{G}_q \,.$$ More general expressions giving the elastic field generated by an *extended* impurity can be found in the literature on the mechanics of anisotropic solids[@yang1976generalized; @kinoshita1971elastic; @dunn1997inclusions], but, being more general, they are also (much) more complex. Connection with elasto–plastic models {#sec:EPM} ------------------------------------- How do the foregoing results compare with the propagator and rules implemented in elasto-plastic models? To address this question, we set $a=1$ and focus on the isotropic ($\mu_2 = \mu_3$) and incompressible ($B=+\infty$) case. The elastic strain $\hat{e}_2^{el}(q) := \hat{e}_2 - e^{pl}_* $ then boils down to $e_*^{pl} \mathcal{G}_q$ by virtue of eq. , where the quadrupolar elastic propagator $\mathcal{G}_q=\frac{-(q_1^2 - q_2^2)^2}{q^4}$ used in EPM is made apparent. This quadrupolar propagator was derived by Picard and co-workers [@picard04elastic; @nicolas2014universal] using a quite different approach, by writing mechanical equilibrium ($\nabla \cdot \boldsymbol\sigma=0$, where $\boldsymbol\sigma$ is the stress tensor) in an incompressible medium with a plastic eigenstrain $e^{pl}_*$ at the origin. It can also be regarded as the point-wise limit of an Eshelby inclusion [@eshelby1957determination], *i.e.*, a circular inclusion that would spontaneously deform into an ellipse in free space. In real space, the propagator $\mathcal{G}_q$ becomes $$\mathcal{G}(r,\theta) = -\mathrm{cos}(4\theta)/\pi r^2 - \delta(r)/ 2 \label{eq:prop_EPM}$$ in polar coordinates \[see eq. \]. The elastic strain $e_2^{el}(r=0)$ at the origin is thus depressed by $e^{pl}_*/2$ to mitigate the local surge of the plastic strain $e^{pl}_*$, and a quadrupolar halo surrounds the plastic event. Bearing the foregoing considerations in mind, it is easy to understand that the rules of elasto-plastic cellular automata [@picard2005slow; @lin14epl; @nicolas2017deformation] ensue from the choice of the piecewise quadratic potential $V_{2,r}$ sketched in Fig. \[fig:V2\] (c) and the neglect of the transient dynamics before mechanical equilibration. Indeed, with this choice, while $e_2(r)$ evolves in a continuous region of $V_{2,r}$, the material is locally Hookean, with $\dot{e}_2(r)=\dot{e}_2^{el}(r)$. Upon reaching a discontinuity of $V_{2,r}$, $e_2(r)$ falls into another quadratic branch of the potential, which corresponds to a finite jump of the local plastic strain by an amount $e_*^{pl}(r)$. Using eq. \[eq:prop\_EPM\], this local distortion generates a quadrupolar halo of elastic stress and a depression of the local elastic stress by $e_*^{pl}(r)/2$, viz., $$\begin{aligned} \sigma(r) & \to \sigma(r) - \mu_2 e^{pl}_*(r) \pm \mu_2 e^{pl}_*(r)/V \, \\ \sigma(r') & \to \sigma(r') + 2 \mu_2 e^{pl}_*(r) \mathcal{G}(r^\prime - r) \pm \mu_2 e^{pl}_*(r)/V \label{eq:CA2}\end{aligned}$$ where the sign $\pm$ is positive (negative) for the stress-controlled (strain-controlled, respectively) protocol. Soft modes {#sec:SPE2} ---------- One of the advantages of the continuous approach under study over discrete elasto–plastic automata is that, instead of describing a sequence of mechanical equilibria, it contains the whole dynamics. This, in particular, gives access to the soft modes, whose study was so far restricted to atomistic simulations [@tanguy10mode; @manning11soft; @charbonneau16universal; @yang2017correlations] and, to a lesser extent, experiments in which the fluctuations in the particle positions can be imaged [@henkes2012extracting]. In order to perform a linear stability analysis, we assume that $$W(e_2) = - \mu_0 e_2^2 + O(e_2^3) \,,\, \mu_0 > 0$$ near the equilibrium position $e_2 = 0$, see Fig. \[fig:V2\] (a). Decomposing the dynamics of $\hat{e}_2(q,t)$ into a linear sum of Laplace modes, viz., $\hat{e}_2(q,t)= \int_0^{\infty} e^{\omega t} \tilde{e}_2(q,\omega) d\omega$ (so that $\Re(\omega) > 0$ refers to *unstable* modes, contrary to the convention of Ref. [@charbonneau16universal; @cao2017localisation]), and inserting into eq. \[eq:EoM\_inc\_real\], we arrive at the following equation: $$\left[ \mu_2 - \delta \mu \mathcal{G}_q + \omega \right] \tilde{e}_2(q, \omega) = \mu_0 (1 + \mathcal{G}_q ) a^2 e_2^o(\omega)\, ,\label{eq:softmodegen}$$ where $e_2^o(\omega)$ is a shorthand for $\tilde{e}_2(r=0, \omega)$. (For inertial systems, an additional term $2 \rho \omega^2 q^{-2}$ is present between the brackets on the left-hand side, as detailed in Appendix \[app:underdamped\]). This equation determines the shape of the dynamical modes. Indeed, if $e_2^o(\omega)=0$, then $\tilde{e}_2(q, \omega)$ must vanish at all wavenumbers, except those which cancel the prefactor on the left-hand side of eq. \[eq:softmodegen\]. Bearing in mind that $\mathcal{G}_q \in [-1,0]$, this cancellation is possible if and only if $\omega \in [-\mu_3,-\mu_2]$; there is thus a continuous range of admissible growth rates $\omega$. On the other hand, if $e_2^o(\omega) \neq 0$, integrating $\tilde{e}_2(q, \omega)$ over $q$ from eq. \[eq:softmodegen\] and noticing that $\int_q \tilde{e}_2(q,\omega) \propto e_2^o(\omega)$ gives the following closure relation for $\omega$: $$e_2^o(\omega) = e_2^o(\omega) \frac{\mu _0}{\mu _2+\omega } \mathcal{I}\left( \frac{\mu _2-\mu _3}{\mu _2+\omega } \right) \,$$ where the integral function $\mathcal{I}(x)$ is made explicit in eq.  of the Appendix. The closure relation is only satisfied for $\omega = \omega_\star$, with $$\omega_\star = -\mu_2+\frac{\mu _0^2}{2 \mu _0 + \mu _3 - \mu _2} > - \mu_2 \,. \label{eq:omega}$$ Since $\omega_\star$ is the maximal growth rate, it is associated with the most unstable mode. Following the same steps as before to compute this eigenmode from eq. , we find that it derives from the following displacement field: $$\begin{aligned} \label{eq:delta'} u_{r,\star} \propto \frac{1}{r} \frac{\cos (2\theta)}{1 + \delta_\star \cos (4\theta) } \,, u_{\theta,\star} = 0 \,. \end{aligned}$$ in polar coordinates. Here $ \delta_\star = \frac{\mu _3-\mu _2}{\mu _2+\mu _3+2 \omega_\star }$ quantifies the anisotropy of the elastic medium, as the parameter $\delta$ defined below eq. . Another consequence of eq.  concerns the situations of marginal stability, when $\omega_\star \to 0$, or $\mu_0 \to \mu_2 + \sqrt{\mu_2 \mu_3}$ by eq. . These situations are on no account mathematical curiosities, but occur whenever a plastic instability is about to take place during the *quasi-static* deformation of disordered solids; this instability is triggered by the marginally stable soft mode [@cao2017localisation]. In this case ($\omega_\star \to 0$), the two anisotropy parameters, $\delta$ and $\delta_\star$, converge and the soft mode (given by eq. \[eq:delta’\]) coincides with the equilibrium configuration that will be reached after the development of this plastic instability, up to a proportionality coefficient. Besides, this mode is not affected by the presence of inertia, which becomes negligible for $\omega \to 0$. Therefore, in quasi-static protocols, we can identify the marginal soft mode and the equilibrium configuration. ![Time evolution of the displacement field $u(r,\theta)$ in response to a single plastic event for a highly anisotropic incompressible material at fixed macroscopic strain $\overline{e_2}=0$, with $\mu_2 = 1/19$ and $\mu_3 = 1$ (so that $\delta = 0.9$, as in Fig. \[fig:displacement\_th\], (c,f)). We chose an anharmonic potential $W(e_2) = -\mu_0 e_2^2 - \frac32 e_2^3 + e_2^4$, where $\mu_0 = \mu_2 + \sqrt{\mu_2 \mu_3}$ so that $e_2 \equiv 0$ is a marginally stable equilibrium. The initial condition is a small number times the soft mode $\hat{e}_2(q) = 0.04 a^2 (1+\mathcal{G}_q)/( \mu_2 + (\mu_2 - \mu_3) \mathcal{G}_q )$ and the system is evolved according to eq.  for a duration $T \approx 150$; the evolutions of $e_2(0,t)$ with time and of $W$ as a function of $e_2(0)$ are plotted in insets (ii) and (i), respectively. The main plot shows snapshots of the normalised displacement field at $t = T/10, 2T/10, \dots, T$, computed from $\hat{e}_2$ using the methods of Appendices \[sec:ur\] and \[app:Fourier\]. The colors represent time in the way indicated by the insets. The final configuration is plotted with dashed line since it overlaps with the initial one.[]{data-label="fig:evolution"}](evolution.pdf){width=".9\columnwidth"} Transient dynamics {#sec:SPE3} ------------------ We close this section by discussing the transient dynamics during the plastic event, i.e., we wonder how the marginal soft mode gradually unfurls and fades into the new equilibrium state, which is of similar shape. In the special case of isotropic elasticity, i.e., $\mu_2 = \mu_3$, one can check, using eq. \[eq:EoM2\], that the shape is conserved during the *whole* dynamics, viz., $\hat{e}_2(q,t) = e^{pl}(t) a^2 (1 + \mathcal{G}_q)$, even if the instability originated in an excited mode ($\omega_\star<0$) and not in the marginal soft mode. Only the normalisation constant evolves (in the strain-controlled protocol with $\gamma = 0$): $$\begin{aligned} \frac{{\mathrm{d}}e^{pl}}{{\mathrm{d}}t} = \mu_2 e^{pl} - \frac12 W'\left(\frac{e^{pl}}{2}\right) \,.\end{aligned}$$ The anisotropic case, $\mu_2 \neq \mu_3$, is more involved and requires to solve the equation of motion eq.  numerically, which can be done efficiently by noticing that $\hat{e}_2(q)$ depends only on $q_2/q_1$. Perturbing a marginally stable equilibrium configuration $e^0_2$ along a soft mode and letting the system relax at fixed macroscopic strain $\gamma = 0$, we observe that the transient between the initial and final stages is short-lived and only weakly deviates from the equilibrium shape given by eq. (\[eq:u\]) (up to normalisation), as shown in Fig. \[fig:evolution\]. In particular, in the strongly anisotropic case $\mu_2 \ll \mu_3$, the equilibrium ‘fracture’-like profile is apparent during the whole evolution and clearly distinct from the quadrupolar shape found in an isotropic medium. In summary, the propagators of eq.  are robust signatures of the elastic medium anisotropy, even during the transient dynamics. Response of a disordered medium with many impurities {#sec:denny} ==================================================== The previous section has shed light on the response of a (possibly anisotropic) elastic medium to a single plastic impurity, with $W_r \propto \delta_{2D}(r)$, and unveiled a continuous family of elastic propagators. Now, we extend the study to fully disordered media, with many plastic impurities (generic $W_r$). In this case, the problem becomes analytically intractable, but deserves to be investigated on account of its relevance for collective effects in disordered solids under shear, including avalanches of rearrangements. A recent analysis of the yielding behaviour of these materials close to the critical point was notably proposed on the basis of a very similar model by Jagla [@jagla2017non]. Incidentally, even the numerical study of these models presents difficulties. In particular, the fluctuating sign of the propagator $\mathcal{G}_q$ imposes to carefully follow the order of the rearrangements. This is in stark contrast with the related, but distinct [@lin2014scaling; @lin14epl], problem of elastic line depinning, where the propagator is non-negative (hence the existence of Middleton theorems [@MiddletonPRL]) and efficient algorithms can be devised [@werner02rough]. To proceed, we will inspect the low-frequency excitations of an equilibrium configuration in this generic case by numerically computing the eigenvectors of the so called dynamical matrix, under the assumption of spatially uncorrelated plastic disorder. Dynamical matrix {#sec:matrix} ---------------- To start with, we write the general equation of motion (\[eq:EoM\_inc\_real\]) in terms of the properly rescaled [^3] strain variable $e= \mathcal{F}^{-1} e_2$ introduced in eq. (\[eq:rescaled\_strain\]), with $ \mathcal{F}= Q_3 / q^2$, $$\dot{e} = -\mu_2 e - \frac{1}{2} \mathcal{F} W_r^\prime\left( \mathcal{F} e \right) + \frac{\Sigma}{2\mathcal{F}} + (1 - \mathcal{F}^2) (\mu_2-\mu_3) e \, , \label{eq:edot_overdamped}$$ where we have used $\mathcal{F}^2 = 1+ \mathcal{G}_q$. A small deviation $\delta e$ away from an equilibrium configuration $e^{(0)}$ thus decays as $\delta \dot{e}(r) = -\sum_{r^\prime} \mathcal{M}_{rr^\prime} \delta e(r^\prime)$ in discrete space, where the dynamical matrix $\mathcal{M}_{rr^\prime}$ reads $$\begin{aligned} \mathcal{M}_{rr^\prime} = \mu_2 \delta_{rr^\prime} - \mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F} + (\delta_{rr'} - \mathcal{F}^2) (\mu_2-\mu_3)& \, \nonumber \\ \text{with } \mathcal{D}_{rr'} = -\frac{1}{2} \delta_{rr'} W''_r\left(\mathcal{F} e^{(0)}(r)\right) \,.& \label{eq:D}\end{aligned}$$ From now on, we focus on the case $\mu_2 = \mu_3$. Then, $$\mathcal{M}_{rr^\prime} = \mu_2 \delta_{rr^\prime} - \mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F} \label{eq:Mdef}$$ is the sum of the scalar matrix $\mu_2 \delta_{rr^\prime}$ and a matrix product between $\mathcal{F}$ (diagonal in Fourier space) and $\mathcal{D}_{rr^\prime}$ (diagonal in real space). Accordingly, rescaling the plastic disorder strength as $\mathcal{D}_{rr^\prime} \leadsto k \mathcal{D}_{rr^\prime} $ has no effect on the excitation modes, bar an affine transformation of the eigenvalues $\omega \leadsto k(\omega + \mu_2) - \mu_2$. Once again, the contrast with respect to the equation describing the depinning of an elastic interface should be noted. In that case, the dynamical matrix is a *sum* of a propagator $\mathcal{G}^{\mathrm{(d)}}$ accounting for the elastic couplings within the interface (usually a fractional Laplacian that is diagonal in Fourier space) and a disorder matrix $\mathcal{D}^{\mathrm{(d)}}_{rr^\prime} := \delta_{rr'} W_r^{\mathrm{(d)}\prime\prime}\left(u^{(0)}(r)\right)$ obtained by deriving the disorder potential $W_r^{\mathrm{(d)}}$ with respect to the local [displacement]{} $u(r)$. The disorder strength can, and does, affect the shape of the eigenmodes, in particular, their localisation length [@cao2017localisation]. Random approximation {#sec:anderson} -------------------- ![The Probability density function of the Weibull law $\text{Pdf}(D_r) = \kappa D_r^{\kappa-1} e^{-D_r^\kappa}$ for parameters $\kappa = 1, 2,5$. []{data-label="fig:weibull"}](weibull.pdf){width=".9\columnwidth"} {width="85.00000%"} In light of the foregoing observation that the dynamical matrix has the same eigenvectors as $\mathcal{F} \mathcal{D}_{rr^\prime} \mathcal{F}$, where $\mathcal{D}_{rr'}$ is given by eq. (\[eq:D\]), we are interested in gaining insight into $D_r := - W''_r\left(\mathcal{F} e^{(0)}(r)\right) / 2$. A priori, it will depend on the specific equilibrium configuration under study and the spatial correlations of $W_r$. However, the spatial correlations of the disorder will be discarded here, which will spare us the meticulous search of equilibria. Albeit uncontrolled, this approximation was recently found to preserve key properties of the eigenmodes at the depinning transition [@cao2017localisation]. Indeed, over an ensemble of equilibria, the values taken by $D_r$ at different $r$ are only weakly correlated and they will be considered random. More precisely, we will handle $D_r$ as independent random variables drawn from a Weibull distribution of parameter $\kappa$, which means that $\left(D_r \right)^\kappa$ is exponentially distributed (see Fig. \[fig:weibull\]). This enforces $D_r\geqslant 0$ (because plasticity tends to soften the material). The choice of a Weibull distribution is arbitrary, but it will prove convenient, in that it allows us to tune the dispersion of plastic disorder via the parameter $\kappa$. Considering the limit-cases, when $\kappa \leq 1$, the distribution of $D_r$ is peaked at $D_r = 0$ and, apart from large outliers, $D_r \sim 0$. To the contrary, when $\kappa \gg 1$, a peak around $1$ emerges and the whole distribution concentrates in this peak. These distributions of $D_r$, combined with the formula of eq. \[eq:Mdef\], are used to populate random matrices $\mathcal{M}_{rr^\prime}$ (of size $256^2 \times 256^2$, *i.e.*, $r = (x,y), x,y = -128,\dots,127$ and similarly for $r^\prime$). The lowest-frequency eigenvector (softest mode) of each matrix is found numerically by the power/Lancsoz iteration method and is integrated via eq. \[eq:integration\] to get the associated displacement field. Some representative displacement fields for different values of $\kappa$ are plotted in Fig. \[fig:fits\], with the peak value shifted to the origin, for easier comparison with Fig. \[fig:displacement\_th\]. Two major conclusions can be drawn from the observation of these plots (among many similar plots). First, irrespective of $\kappa$, we always find ‘soft’ modes that are localised, with displacements $u$ and strains $e_2$ clearly peaked at an individual site. In other words, no collective pinning (with a peak region spread over many sites) is seen. At a distance $r$ away from the peak, the radial displacement $u_r$ and the strain $e_2$ decay as a power laws $\propto 1/r$ and $1/r^2$, respectively, while the azimuthal component $u_\theta$ remains very small compared to $u_r$. Secondly, the overall shapes of the displacement fields are strongly reminiscent of the elastic propagators derived in Section \[sec:single\_impurity\] for the single-impurity problem and range from the standard quadrupolar propagator (for small $\kappa$) to the fracture-like propagator (for large $\kappa$). In fact, the theoretical expressions of eq. (\[eq:delta’\]) turn out to fit the numerical displacements quite well, if one is given the freedom to adjust the anisotropy parameter $\delta_\star$ of the elastic medium, although here $\mu_2=\mu_3$. How can we rationalise these findings? If all sites in the system were decoupled, the softest mode would just be a local excitation of the softest site (where $D_r$ is maximal). But, because of the elastic embedding of the site (and, more formally, the St Venant constraint), this excitation is coupled with an elastic deformation of the surroundings, as though it were an impurity. For small $\kappa$, the disorder $D_r$ is close to zero on most sites (but not on the softest one, of course), so the medium is virtually a perfect isotropic solid, hence the standard quadrupolar shape of the mode, that is well captured by an anisotropy parameter $\delta_\star \to 0$. On the other hand, for larger $\kappa$, most sites display considerable plasticity-induced softening, with $D_r \approx 1$ (while $D_r>1$ on the softest site). This global softening along $e_2$ is tantamount to a lowering of the shear modulus $\mu_2$. Indeed, the dynamical matrix of eq. (\[eq:D\]) is not altered by simultaneously reducing $D_r$ by its spatial average $\overline{D_r}$ and lowering $\mu_2$ to $\mu_2-\overline{D_r}$. Therefore, the impurity is effectively coupled with an anisotropic elastic medium characterised by a finite $\delta_\star$ and, in the limit of large anisotropy ($\kappa \gg 1$), a fracture-like propagator can emerge, with $\delta_\star \to 1$. Still, it is noteworthy that, even in this regime where the $D_r$ are narrowly distributed around 1, the observed fields remain dominated by single impurities. Discussion ========== In this work, we have studied an intermediate class of models describing the plastic deformation of disordered solids, that operate at a coarse-grained scale similar to that of elasto–plastic models (EPM) while still describing the non-linear elastic properties of atomistic systems. The models considered focus on the (continuous) strain fields in the material and incorporate them into a Ginzburg-Landau free energy which combines a purely elastic part and a plastic disorder potential. We investigated the equilibrium configurations, the soft modes and the equilibration dynamics associated with this free energy. The last two aspects are lost in EPM, which hop between mechanically equilibrated configurations, following rules that we could clarify. In contrast, the transient dynamics during mechanical equilibration are present in the continuous models under study, whose global relaxation thus depends on the deformation rate.[^4] When plastic disorder is spatially confined in a single impurity, we were able to derive analytically the soft mode and the equilibrium configuration of the system, which coincide (up to a rescaling factor). Our first important result is that the quadrupolar propagator routinely used in EPM is not ubiquitous. Indeed, in the presence of strong elastic anisotropy, we found a new, fracture-like propagator, in which the deformation concentrates along the easy directions. A continuous family of propagators, obeying a simple formula  depending on one anisotropy parameter, interpolates between the quadrupolar propagator and the fracture-like one. Remarkably, these single-impurity calculations keep being relevant in fully disordered systems, whose lowest-frequency excitations were numerically found to localise around the softest (point-like) site, even for weakly dispersed disorder. Moreover, the deformation halo around this soft site replicates the foregoing family of propagators, as the distribution of plastic disorder is varied. The fracture-like propagator is recovered for finite, narrowly distributed plastic disorder. Indeed, the latter softens the material along one shearing direction and thus renders the surroundings of the soft site effectively anisotropic. Accordingly, one may expect to find the fracture-like propagator whenever extended regions of the material collectively soften on the brink of failure, without immediately failing. In the presently studied models, this situation is precluded when the disorder potential is piecewise parabolic with cusps, but should be possible with any smooth potential. Recently, Jagla showed that these two classes of potentials led to different critical exponents at the yielding transition for the flow curve [@jagla2017non], but the shape of the propagator in each case and its possible relevance were not studied. In atomistic simulations and experiments on glasses, non-standard elastic propagators (in the soft mode or the actual stress redistribution) have not been reported either, to the best of our knowledge. Certainly, specific conditions are required to observe collective softening of the material along *one* direction, such as high mechanical homogeneity in the initial configuration, and they may not be met often. In addition, the noise and fluctuations in the numerical and experimental data, notably due to the granularity of the material at the microscale, may complicate the distinction of a new propagator, all the more so as the paradigm resting on Eshelby’s solution is overwhelming. Nevertheless, we may tentatively expect to see it in carefully aged (ultra-stable [@berthier2017origin]) glasses just before their dramatic macroscopic failure; the incipience of a shear band[@nguyen16shearband; @tanguy10mode] may also reflect the presence of collective softening. So we advocate to test fits to the different propagators in future simulations and experiments. Acknowledgments {#acknowledgments .unnumbered} --------------- We thank S. Bouzat and A. B. Kolton for collaboration on related projects, E. Jagla, T. de Geus, P. Le Doussal, A. Tanguy, S. Cazayus-Claverie, and M. Wyart, for insightful discussions. The authors acknowledge support from a Simons Investigatorship, Capital Fund Management Paris and LPTMS (X.C.), a LabEx-ICFP scholarship and CNRS (D.T.), and an ANR grant ANR-16-CE30-0023-01 THERMOLOC (A.R.). *Conflict of interest.* There are no conflicts of interest to declare. Compressible material {#sec:compressible} ===================== In this Appendix, the results derived in the main text for the incompressible case are extended to compressible systems ($B<\infty$). To this end, the equation of motion is generalised to compressible systems, while one still assumes that the viscosities acting on the different strain variables are equal, $\eta=1$ . Extremising the free energy of eq.  with respect to the Lagrange multipliers leads to $\sum_j Q_j \dot{\hat{e}}_j(q) = 0$, for $q \neq 0$. The equation of motion then straightforwardly generalises to $$\label{eq:motiongen} \dot{\hat{e}}_j(q) = -\frac{\delta F}{\delta \hat{e}_j(q)} - Q_j \lambda \,,\, \lambda = -\frac{\sum_{k=1}^3 Q_k \frac{\delta F}{\delta \hat{e}_j(q)} }{\sum_{k=1}^3 Q_k^2 } \,.$$ for $j = 1, 2, 3$ and $q\neq0$. For a single impurity, the above equation can be recast into a matrix form, in terms of the 3-vector $\mathbf{e}(q) =\begin{bmatrix} {\hat{e}}_1(q) & {\hat{e}}_2(q) & {\hat{e}}_3(q) \end{bmatrix}$: $$\begin{aligned} & \dot{\mathbf{e}} = -\mathbf{P} \left( \mathbf{M} \mathbf{e} + \mathbf{v} \right) \label{eq:vectorform}\end{aligned}$$ where $ \mathbf{v}$ is a $3$-vector, and $\mathbf{P}$ and $\mathbf{M}$ are $3\times3$ matrices defined as follows: $$\begin{aligned} & \mathbf{P}_{jk} = \delta_{j,k} - \frac{ Q_j Q_k}{\sum_{\ell=1}^3 Q_\ell^2 }\,,\, \mathbf{M}_{jk} = \delta_{j,k} \mu_k \,,\, \mu_1 := B\,, \nonumber \\ & \mathbf{v} = \begin{bmatrix} 0 & \mu_2 a^2 e^{pl} & 0 \end{bmatrix} \,. \nonumber\end{aligned}$$ At equilibrium, $\dot{\mathbf{e}}$ vanishes in eq. . The strain modes $\mathbf{e}$ that cancel the right-hand side of eq.  and satisfy the St Venant constraint are $$\begin{aligned} \hat{e}_1(q) &= -a^2 \, e^{pl}_* \frac{Q_2}{Q_1} \frac{ \mu_3/(B + \mu_3) }{ {1+\mathcal{G}_q^{B<\infty}\,(1-\mu_3/\mu_2)} } \nonumber \\ \hat{e}_2(q) &= a^2 \, e^{pl}_* \frac {1+\mathcal{G}_q^{B<\infty}} {1+ \mathcal{G}_q^{B<\infty} \,(1-\mu_3/\mu_2)} \label{eq:e2qgen} \\ \hat{e}_3(q) &=- a^2 \, e^{pl}_* \frac{Q_2}{Q_3} \frac{ 1 + \mathcal{G}_q^{B<\infty} - \mu_3/(B + \mu_3) }{ {1+\mathcal{G}_q^{B<\infty}\,(1-\mu_3/\mu_2)} } \nonumber\end{aligned}$$ where $e^{pl}_*$ must be determined self-consistently, as detailed in Appendix \[sec:C\], and $$\mathcal{G}_q^{B<\infty} = \frac{B}{B + \mu_3} \mathcal{G}_q \,.$$ Plastic strain in the equilibrium situation {#sec:C} =========================================== Section \[sec:single\_impurity\] exposed how the shear softening induced by a single plastic impurity at $r=0$, quantified by $e^{pl}(t=0) = - W'(e_2(0,t=0)) / 2 \mu_2$, generates an elastic deformation field $e_2(r,t)$, which deforms the impurity and possibly further softens the material at $r=0$. Equilibrium is reached when $$e^{pl}(t) \to e^{pl}_*= -\frac{1}{2 \mu_2} W'(e_2(0)),$$ where $e_2(0)$ forms part of a mechanically equilibrated strain field $e_2(r)$ (no time dependence), or $\hat{e}_2(q)$ in Fourier space. The Fourier components $\hat{e}_2(q)$ depend on $e^{pl}$ via eq.  (in the incompressible case). Integrating these components over $q$ to get $ e_2(0) \propto \int \hat{e}_2(q) dq$ leads to $$e^{pl}_*= -\frac{1}{2 \mu_2} W'\left(\beta e^{pl}_* + \overline{e_2} \right) \,. \label{eq:plastic1}$$ where $\beta= (1+ \sqrt{\mu_3 / \mu_2})^{-1}$. Note that special attention was paid to the zero-mode $\hat{e}_2(0)$, which is proportional to the average strain $\overline{e_2}$, because its value depends on the deformation protocol, as explained in the main text. These results can be generalised to the compressible case $B<\infty$, where the Fourier components $\hat{e}_2(q)$ obey eq.  instead of eq. \[eq:e2q\]. It turns out that eq.  still holds, provided that one replaces $\beta$ with $$\frac{B}{B+\mu _3} \mathcal{I}(x) + \frac{\mu_3}{(B+\mu _3)\sqrt{1-x}}.$$ Here, we have introduced the shorthand $x=\frac{B \left(\mu _2-\mu _3\right)}{\mu _2 \left(B+\mu _3\right)}$ and the following integral (that can be calculated by Cauchy’s residue theorem for $x<1$), $$\label{eq:integral} \mathcal{I}(x) = \int_{0}^{2\pi} \frac{{\mathrm{d}}\phi}{2 \pi} \frac{\sin (2\phi)^2 }{1 - x \cos(2\phi)^2} = \frac{1}{\sqrt{1-x}+1}.$$ Fourier transforms in polar coordinates \[app:Fourier\] ======================================================= The calculations in the main text heavily rely on the (convenient) use of Fourier transforms (F.T.), notably in polar coordinates, with $$(x,y)= (r \cos \theta, r \sin \theta) \,,\, q = (\rho \cos \phi, \rho \sin \phi).$$ This Appendix collects some results that are useful to derive the real-space expressions from their F.T. First of all, for functions with an $1/r^2$ dependence (such as the elastic strain generated by a plastic impurity), the following property applies: $$\begin{aligned} \cos(n \theta) r^{-2} \stackrel{\text{F.T.}} \longrightarrow 2 \pi {\mathbf{i}}^n\cos(n \phi) n^{-1} \,,\, n = 1,2,\dots\,. \label{eq:Fourier}\end{aligned}$$ *Proof.* To derive this result, we recall the Jacobi–Anger identity: $$e^{{\mathbf{i}}q x} = \sum_{m\in {\mathbb{Z}}} {\mathbf{i}}^m e^{{\mathbf{i}}m (\theta -\phi)} J_m(\rho r)$$ $J_m(y)$ is the Bessel $J$ function. Now let $G(x) = e^{{\mathbf{i}}n \theta} r^{-2}$, with some $n > 0$, then its Fourier transform is $$\begin{aligned} & \hat{G}(q) = \int_{0}^{\infty} r^{-1} {\mathrm{d}}r \int_0^{2\pi} {\mathrm{d}}\theta G(x) e^{{\mathbf{i}}k x}\nonumber \\ = & \int_{0}^{\infty} r^{-1} {\mathrm{d}}r \int {\mathrm{d}}\theta G(\theta) \sum_{m\in {\mathbb{Z}}} {\mathbf{i}}^m e^{{\mathbf{i}}m (\phi - \theta)} J_m(\rho r) \nonumber \\ =& 2 \pi {\mathbf{i}}^{n} e^{{\mathbf{i}}n \phi} \int_{0}^{\infty} r^{-1} {\mathrm{d}}r J_n( \rho r ) = 2 \pi G_n {\mathbf{i}}^{n} e^{{\mathbf{i}}n \phi} n^{-1} $$ where in the last line we used the identity $$\int_{0}^{\infty} r^{-1} {\mathrm{d}}r J_n(r) = n^{-1} \,,\, n = 1, 2, \dots$$ This means that, for $n> 0$, $$e^{{\mathbf{i}}n \theta} r^{-2} \stackrel{\text{F.T.}} \longrightarrow 2 \pi {\mathbf{i}}^{n} n^{-1} e^{{\mathbf{i}}n \phi} \,,\,$$ which proves eq.  by parity considerations $\blacksquare$ Now, we specifically turn to the derivation of the real-space field $e_2(r)$ from its F.T. eq.  $$\hat{e}_2(q) \propto \frac {1+\mathcal{G}_q} {1+x\mathcal{G}_q},$$ where $x=1-\mu_3/\mu_2$ and $\mathcal{G}_q = -\cos(2\phi)^2$ by virtue of eq. . We start by writing the following identity (obtained *e.g.* via Cauchy residues theorem) $$\frac {1+\mathcal{G}_q} {1+x\mathcal{G}_q} = K + 2 K' \sum_{n=1}^\infty \cos(4n \phi) z^{n} \,,\, \label{eq:Cauchy}$$ where $K = (\sqrt{1-x}+1)^{-1}$, $K' = -\sqrt{1-x}/x$ and $z =x / (1 +\sqrt{1-x})^2$. Then we apply an inverse F.T. to eq.  term by term with the help of eq.  and arrive at $$e_2(r) \propto \frac{2}{\pi} \frac{\sqrt{1-x}}{x-2} \frac{\delta + \cos(4\theta)}{(1 + \delta \cos(4\theta))^2} \frac{1}{r^2} + \frac{\delta_{2D}(r)}{1 + \sqrt{1-x}} \,. \label{eq:inverseF}$$ where $\delta = \frac{x}{x-2}$. This result is consistent with eq. , provided that $$C = -\frac{2 a e_*^{pl}}{ \pi} \frac{\sqrt{\mu_2 \mu_3}}{\mu_2+\mu_3} \,,\, C_0 = \frac{a^2 e_*^{pl}}{\sqrt{\mu_3/\mu_2}+1} \,. \label{eq:CC0}$$ Incidentally, the formula for the isotropic case is recovered for $x = 0$, $$1+\mathcal{G}_q = \frac{1}{2} - \frac{1}{2}\cos(4\phi) \stackrel{\text{F.T.}^{-1}}{\longrightarrow} \frac{\delta(r)}{2} - \frac{ \cos(4\theta)} {\pi r^{2}} \,.\label{eq:EshelbyFourier}$$ Displacement field and strain {#sec:ur} ============================= We explicit the relation between displacement field and strain tensor, for a displacement field which points in the radial direction, and whose amplitude $\Vert u \Vert = F(\theta) / r$. The corresponding strain tensor is as follows \[see eq.  and text below\]: $$\label{eq:e2diff} e_1 = 0 \,,\, e_2 = - \frac1{2r^2}\frac{{\mathrm{d}}\left[\sin (2 \theta ) F(\theta ) \right]} {{\mathrm{d}}\theta}\,.$$ So the displacement is incompressible for any $F$. Comparing to eq.  yields a differential equation for $F(\theta)$, whose general solution is:$$F(\theta) = -C \frac{\cos(2\theta)}{1+\delta \cos(4\theta)} + c_1 \csc(2\theta) \,,$$ where $c_1$ is an integral constant which we set to $0$ (since $\csc(2\theta)$ is divergent at $\theta = 0, \pi/2, \pi, 3\pi/2$), leading to eq. . Eq.  was also used to obtain the displacement field in the numerical simulation of transient dynamics in Section \[sec:SPE3\]. Generalisation to underdamped systems {#app:underdamped} ===================================== The main text focuses on the overdamped limit relevant for foams and concentrated emulsions, notably. Here, the results are generalised to all damping regimes. Following the approach of [@lookman03ferro], this is achieved by complementing the Euler-Lagrange equations of motion with a strain-rate-dependent Rayleigh dissipation term $R = \eta \int_r \left[\dot{e}_2^2 + \dot{e}_3^2 \right] = \eta \int_r \dot{e}^2$, where $\eta=1$ is the viscosity and the rescaled strain $e$ was introduced in eq. . The resulting equations read $$\frac{d}{dt} \frac{ \delta \mathcal{L} }{\delta \dot{e}} - \frac{ \delta \mathcal{L} }{\delta e} = - \frac{\delta R}{\delta \dot{e}} \,,$$ where the Lagrangian $$\mathcal{L}= T- F$$ is the difference between the kinetic energy $T=\frac{1}{2}\int_r \rho \dot{u}^2$ and the free energy $$F = \int_r \left[W_{r}[e_2(r)] + \mu_2 e_2(r)^2 + \mu_3 e_3(r)^2 \right] \,.$$ Since displacement is a (non-local) function of the strain field, the kinetic energy $T$ can be expressed in terms of the strain rate, instead of the velocity. Adapting the results of [@lookman03ferro] (Sec. III.A) to an incompressible system, we obtain $$T = 2 \rho \int_q \frac{ |\dot{\hat{e}}|^2 (q) }{q^2} \,.$$ The Lagrange-Rayleigh equations then simplify to $$\begin{aligned} 2 \rho \ddot{e} = - q^2 \left( \frac12 \frac{Q_3}{q^2} \frac{\partial W_{r}}{\partial e_2(r)} + \mu_2 e + \delta \mu \frac{Q_2^2}{q^4} e + \dot{e} \right) \,, \label{eq:e} \end{aligned}$$ where $\delta \mu = \mu_3 - \mu_2$. Note that the overdamped dynamics of eq.  are recovered for $\rho \to 0$. Linearising the above equation and writing $\hat{e}(q,t)= \int_0^{\infty} e^{\omega t} \tilde{e}(q,\omega) d\omega$ yields $$2 \rho \omega^2 \tilde{e} = q^2 \left( \mathcal{F} \mathcal{D} \mathcal{F} + \delta \mu \, \mathcal{G}_q - \mu_2 - \omega \right) \tilde{e} \,, \label{eq:app_EoM}$$ where $\mathcal{F} = Q_3 / q^2$, $\mathcal{G}_q=\mathcal{F}^2-1$, and $\mathcal{D} = -\frac{1}{2} \delta_{rr'} W''_r\left(\mathcal{F} e^{(0)}(r)\right)$ were already defined in Sec. \[sec:denny\]. Only in the marginal case $\omega \to 0$ do the resulting eigenmodes coincide with those of the overdamped dynamical matrix of eq. . Otherwise, the propagation of sound waves affects the transient dynamics. The foregoing statement can be made more explicit in the case of a single impurity, i.e., $W_r =a^2 \delta(r) W$. While the equilibrium configuration does not depend on the damping, the transient dynamics do. In particular, the vibrational modes from eq. , expressed in terms of the original strain variable $e_2$, read $$\tilde{e}_2(q) \propto \frac{1 +\mathcal{G}_q}{2 \rho \omega^2 q^{-2} + \omega + \mu_2 - \delta \mu \, \mathcal{G}_q},$$ which matches the (overdamped) eigenmodes of eq.  only for the marginally stable mode at $\omega = 0$. [^1]: One may regard the deformation as slow if the stress-strain curve does not substantially vary when the driving rate is reduced. [^2]: See [@salman2011minimal] for a related endeavour to connect continuous models with discrete automata in the context of crystal plasticity. [^3]: Using the properly rescaled variable $e$ allows us to obtain a Hermitian dynamical matrix, whereas a non–Hermitian one is obtained if one considers the variable $e_2$. This is reminiscent of the symmetrization transform from the Fokker–Planck to the Schroedinger equation. [^4]: Nevertheless, the possibility to relax to different energy basins *locally*, depending on the deformation rate, is not taken into account here.

ArXiv

--- abstract: '@Tempelmeier2007 considers the problem of computing replenishment cycle policy parameters under non-stationary stochastic demand and service level constraints. He analyses two possible service level measures: the minimum no stock-out probability per period ($\alpha$-service level) and the so called “fill rate”, that is the fraction of demand satisfied immediately from stock on hand ($\beta$-service level). For each of these possible measures, he presents a mixed integer programming (MIP) model to determine the optimal replenishment cycles and corresponding order-up-to levels minimizing the expected total setup and holding costs. His approach is essentially based on imposing service level dependent lower bounds on cycle order-up-to levels. In this note, we argue that Tempelmeier’s strategy, in the $\beta$-service level case, while being an interesting option for practitioners, does not comply with the standard definition of “fill rate”. By means of a simple numerical example we demonstrate that, as a consequence, his formulation might yield sub-optimal policies.' author: - 'Roberto Rossi, Onur A. Kilic, S. Armagan Tarim' bibliography: - 'note.bib' title: 'A note on Tempelmeier’s $\beta$-service measure under non-stationary stochastic demand' --- Introduction ============ The increasing pace in new product developments has resulted in shorter product life-cycles through which demands do not follow stationary patterns [@Kurawarwala1996; @Graves2008]. Henceforth, inventory problems addressing non-stationary stochastic demands have gained a growing interest from both researchers and practitioners [@citeulike:7928534]. A recent paper, @Tempelmeier2007, addresses the replenishment cycle policies under non-stationary stochastic demand and $\beta$-service level (i.e., “fill rate”) constraints. The *fill rate* is the fraction of demand satisfied immediately from stock on hand. $\beta$-service level constraints therefore specify the minimum prescribed fraction of customer demand that should be met routinely, without backorders or lost sales. Tempelmeier’s work constitutes an interesting development that extends results such as those presented by @citeulike:7766622 to the non-stationary stochastic demand case. More specifically, Tempelmeier extends the model proposed in @Tarim2004, by replacing the $\alpha$-service level constraints — which enforce a minimum no-stockout probability per period — with a new set of constraints based on the inverse first-order loss function. It is stated that the resultant formulation provides the optimal replenishment cycle plan under $\beta$-service level constraints. In this research note, we argue that the $\beta$-service level formulation proposed in @Tempelmeier2007 does not comply with the standard definition of $\beta$-service level found in the literature. In what follows, we provide the formal definition of $\beta$-service level (Section \[sec:definition\]) and we discuss its application in an inventory system controlled with a replenishment cycle policy. Then we discuss the formulation proposed in @Tempelmeier2007 (Section \[sec:tempelmeier\]) for computing optimal replenishment cycle policy parameters under $\beta$-service level constraints and show, by means of a simple numerical example, that this formulation may yield suboptimal policy parameters (Section \[sec:example\]). $\beta$-service measure {#sec:definition} ======================= The $\beta$-service level is a well established service measure used in many practical applications and has been covered by many textbooks on inventory control [see e.g. @Silver1998; @Axsater2006]. [@Axsater2006] defines $\beta$-service level as the fraction of demand satisfied immediately from stock on hand. This definition is formalized within the context of finite horizon inventory models as follows [see e.g. @Chen2003; @Thomas2005]: $$\label{beta} 1 - \operatorname{E}\left\{\frac{\text{Total backorders within the planning horizon}}{\text{Total demand within the planning horizon}}\right\}.$$ The replenishment cycle policy divides the finite planning horizon into a number of, say $m$, consecutive replenishment cycles. We can re-write (\[beta\]) by taking those into account as $$\label{beta_cycle} 1 - \operatorname{E}\left\{\frac{\sum_{i=1}^m\text{Total backorders within the $i$'th replenishment cycle}}{\sum_{i=1}^m\text{Total demand within the $i$'th replenishment cycle}}\right\}.$$ @Tempelmeier2007’s formulation {#sec:tempelmeier} ============================== For the ease of exposition, here we only provide the formulation of the $\beta$-service level constraints. The reader is referred to [@Tempelmeier2007] for the rest of the model. The set of constraints proposed by @Tempelmeier2007 to impose $\beta$-service level are as follows: $$\label{cons_tempelmeier} \operatorname{E}\{I_t\}\geq\sum_{j=1}^t\left[F^{-1}_{Y^{(t-j+1,t)}}(\beta)-\sum_{i=t-j+1}^{t}\operatorname{E}\{D_i\}\right]P_{tj},\quad t=1,\ldots,T$$ where, $I_t$ is the net inventory position at the end of period $t$; $F^{-1}_{Y^{(t-j+1,t)}}$ is the inverse loss function of the total demand in periods $(t-j+1,\ldots,t)$; $D_t$ is the random demand in period $t$; and, $P_{tj}$ is the binary indicator variable that is equal to 1 if the last replenishment before period $t$ takes place in period $t-j+1$ and to 0 otherwise. Following Tempelmeier, the expected net inventory position is assumed to be non-negative; however, it should be noted that relaxing non-negativity constraints on expected net inventory positions may yield better $\beta$-service plans in terms of expected cost. This issue is beyond the scope of this note and therefore not addressed here. Eq.(\[cons\_tempelmeier\]) is binding only if the indicator variable $P_{tj}$ is equal to 1. Let us consider a replenishment cycle covering periods $(t'-j'+1,\ldots,t')$, i.e. $P_{t'j'}=1$. Then the binding part of the constraint reads: $$\label{cons_tempelmeier2} \operatorname{E}\{I_{t'}\} + \sum_{i=t'-j'+1}^{t'}\operatorname{E}\{D_i\} \geq F^{-1}_{Y^{(t'-j'+1,t')}}(\beta).$$ The replenishment at period $t'-j'+1$ covers the interval $(t'-j'+1,\ldots,t')$. The left hand side of the inequality represents the order-up-to level for period $t'-j'+1$. The constraint clearly imposes a lower bound on the order-up-to level for this cycle. Therefore, when these constraints are used, the same $\beta$-service level is imposed on each and every cycle within the planning horizon. This corresponds to the following definition of $\beta$-service level: $$\label{beta_tempelmeier} 1 - \max_{i=1,\ldots,m}\left[\operatorname{E}\left\{\frac{\text{Total backorders in replenishment cycle $i$}}{\text{Total demand in replenishment cycle $i$}}\right\}\right].$$ It is clear that Eq.(\[beta\_cycle\]) is different from Eq.(\[beta\_tempelmeier\]). The original definition imposes a $\beta$-service level throughout the whole planning horizon, whereas @Tempelmeier2007’s definition imposes a $\beta$-service level on each replenishment cycle within the planning horizon independently. The main difference is that, the former allows the decision maker to have $\beta$-service levels smaller than the specified level for individual cycles, while guaranteeing the specified level for the whole of the planning horizon, whereas the latter guarantees the specified $\beta$-service level for each replenishment cycle. It should be noted that Tempelmeier’s strategy may be favorable for practitioners, since it allows a better control of the fill-rate provided to customers in each cycle. In practice, enforcing a given fill rate over the whole planning horizon, rather than on each cycle separately, guarantees a lower cost at the expense of a varying individual replenishment cycle fill rates. Managers may therefore be interested in paying an additional price in order to have a better control over the fill rate provided in each cycle. For a thorough discussion on theoretical vs versus applied models in inventory control [see @citeulike:8061205]. A numerical example {#sec:example} =================== Let us now consider a limit situation in which we aim to compute optimal non-stationary $(R,S)$ policy parameters for a 2-period planning horizon. The fixed ordering cost is 0, implying that the optimal plan has a replenishment in each period. The holding cost is 1. Period demands are normally distributed $N(\mu,\sigma)$ with parameters $N_1(1000,200)$ and $N_2(2000,200)$. We enforce a $\beta$-service level constraint with $\beta=0.98$. According to [@Tempelmeier2007], the minimum expected buffer stock level for period 1 is $181$ units (corresponding to an order-up-to-level, $R_1$, of 1181 units), which guarantees a $\beta$-service level of exactly $0.98$. Furthermore the minimum expected buffer stock level for period 2 is 99 units (corresponding to an order-up-to-level, $R_2$, of 2099 units), which guarantees a $\beta$-service level of exactly $0.98$. From the first-order loss function, it is easy to see that the expected amount of items backordered in periods 1 and 2 are $19.90$ and $39.86$, respectively. It follows that the overall fill rate is $[(1000+2000)-(19.90+39.86)]/(1000+2000)=0.98$. The expected total holding cost of this solution is $280$ (i.e., 181 for period 1 and 99 for period 2). A fill rate of 0.98 can be achieved also with the following plan, which guarantees a lower expected total holding cost. For the first replenishment cycle, we fix an expected buffer stock level of 171, giving an order-up-to-level of $1171$. This expected buffer stock level is lower than the minimum expected buffer stock level allowed in Tempelmeier’s model. In fact, it only guarantees, for the first replenishment cycle, a fill rate of 0.96 and an expected number of backorders of 21.79. For the second replenishment cycle, we target a buffer stock level of 105, giving an order-up-to-level of 2105. This expected buffer stock level is higher than the minimum expected buffer stock level allowed in Tempelmeier’s model. It guarantees, for the second replenishment cycle, a fill rate equal to 0.98 and an expected number of backordered items equal to 38.03. In this alternative plan the overall $\beta$-service level over the 2-period planning horizon is $[(1000+2000)-(21.79+38.03)]/(1000+2000)=0.98$ as targeted. However, the required service level is attained with a lower expected total holding cost of 276 (i.e., 171 for period 1 and 105 for period 2).

ArXiv

--- abstract: 'We study a two-dimensional tight-binding lattice for excitons with on-site disorder, coupled to a thermal environment at infinite temperature. The disorder acts to localise an exciton spatially, while the environment generates dynamics which enable exploration of the lattice. Although the steady state of the system is trivially uniform, we observe a rich dynamics and uncover a dynamical phase transition in the space of temporal trajectories. This transition is identified as a localisation in the dynamics generated by the bath. We explore spatial features in the dynamics and employ a generalisation of the inverse participation ratio to deduce an ergodic timescale for the lattice.' author: - Sam Genway - Igor Lesanovsky - 'Juan P. Garrahan' bibliography: - 'anderson.bib' - 'dicke.bib' title: 'Localisation in space and time in disordered-lattice open quantum dynamics' --- Probing the dynamics of quantum systems out of equilibrium is a big challenge of current research in physics. As well as being of fundamental interest, a particular application is the study of exciton transport, relevant to materials ranging from thin-film dyes [@Tennakone2013] and conjugated polymers [@Bolinger2011; @Bardeen2011] to semiconductor nanostructures [@Wheeler2013; @Scholes2006]. Exciton transport is also of great importance in light harvesting materials [@Cheng2009; @Scholes2000; @Yang2002] such as the Fenna-Matthews-Olson complex [@Fenna1974]. Of particular interest is the interplay between disorder, which leads to exciton-localisation effects, and dissipation which facilitates exciton transport [@Nejad2011; @Nejad2013; @Vlaming2013]. A complete understanding of such systems is still being sought [@Xiong2012] and this motivates the exploration of the rich dynamical features which emerge generically in dissipative disordered systems. In this work, we seek to understand general features in the dynamics of an exciton in a large disordered lattice coupled to an infinite-temperature thermal environment. While disorder acts to localise excitons spatially [@Anderson1958; @Cutler1969; @Lee1985], the environment generates dynamics which allow the entire lattice to be explored. The dynamical phenomena which arise in such systems is studied using a “thermodynamics of trajectories” formalism [@Ruelle2004; @Garrahan2007; @Lecomte2007; @Merolle2005; @*Baule2008; @*Gorissen2009; @*Jack2010; @*Giardina2011; @*Nemoto2011; @*Chetrite2013]. Using this method, we will show that while the steady state of the model at infinite temperature is trivial, with all regions equally likely to be occupied, the dynamics show complex features including a dynamical phase transition in the space of trajectories. The transition takes the form of a localisation transition in time: there is an *inactive* phase, where the exciton remains localised in a particular state, and an *active* state, where the environment induces a rapid change between states and the exciton explores all space. In Fig. \[fig1\] we show this effect becomes more pronounced as the strength of the disorder is increased. Such active-inactive trajectory transitions are characteristic of glasses and other classical and quantum systems with pronounced dynamical metastability [@Garrahan2007; @Hedges2009; @*Speck2012; @*Speck2012b; @Garrahan2010; @Garrahan2011; @*Ates2012]. Our findings suggest that a general feature of the dynamics in disordered systems coupled to environments is the existence of an increasingly super-Poissonian temporal distribution for the jumps between lattice sites as disorder is increased. Remarkably, this dynamical behaviour exists even at infinite temperature. ![(Colour online.) Excitonic occupation of regions of an $N = n\times n$ disordered lattice, with $N=10^4$ sites, coupled to an infinite temperature bath. Shown is the lattice-site occupation $O_m(t)$ for different times $t$ (left to right) and different disorder strengths $d$ (top to bottom). See main text for details.[]{data-label="fig1"}](traj_lowres2.png){width="8.57cm"} We study a two-dimensional tight-binding model with on-site energies chosen randomly from a Gaussian distribution. We are interested in the parameter space in which the disorder is sufficiently strong such that all eigenstates are localised within the size of the lattice. Specifically, we consider a square lattice with $N=n\times n$ sites and periodic boundary conditions with Hamiltonian $$H = \sum_m \varepsilon_m {|m\rangle}{\langle m|} + J\sum_{{\langle mm'\rangle}} {|m\rangle}{\langle m'|} = \sum_i E_i {|i\rangle}{\langle i|}\,.$$ Each state ${|m\rangle}$ has a wavefunction centred on a site with label $m$ and corresponding energy $\varepsilon_m$ drawn randomly from a Gaussian distribution, with variance ${d^2}$ and zero mean. The size of $d$ will set the disorder strength. The site index $m$ is related to the coordinates $(x,y)$ of the lattice site via $m=x+n(y-1)$, with $1\le m \le N$. In the second term, ${\langle mm'\rangle}$ denotes a sum over nearest neighbours and we will choose units for energy such that the hopping integral $J$ equals unity. We will use indices $i$ and $j$ for eigenstates of $H$, where $H{|i\rangle} = E_i {|i\rangle}$. The effect of dissipation is introduced by coupling the system to a bath of harmonic modes with Hamiltonian $$H_b = \sum_k \omega_k b_k^\dag b_k\,.$$ These couple to the system via the coupling Hamiltonian $$H_{sb} = S \otimes B = \sum_m c_m {|m\rangle}{\langle m|} \otimes \sum_k h_k (b_k+b_k^\dag) \,,$$ where the parameters $c_m$ are also selected randomly from a Gaussian distribution with zero mean and a variance we will specify. Under standard manipulations (Born, Markov and secular approximations), we find a master equation diagonal in the basis of eigenstates $\dot{P}_i = (\mathbb{W})_{ij} P_j$, where $P_i$ is the occupation probability of the eigenstate ${|i\rangle}$. The master operator $\mathbb{W}$ has elements $(\mathbb{W})_{ij}$ given by $$(\mathbb{W})_{ij} = W_{j\rightarrow i} - r_i\delta_{i,j} \label{eq:W}$$ where the transition rates $W_{j\rightarrow i}$ are given by $$W_{j\rightarrow i} = J(\omega_{ji})\,\, |{\langle j|}S{|i\rangle}|^2\,. \label{eq:Wij}$$ and $J(\omega_{ji}) = 2\pi \sum_k |h_k|^2 \delta(\omega_k - \omega_{ji})$ is the spectral density of the bath with $\omega_{ji} = E_j - E_i$. We will study the case of a bath with temperature $T=\infty$ such that the rates satisfy $W_{i\rightarrow j} = W_{j\rightarrow i}$. In this work, we consider an Ohmic bath with $J(\omega) = \omega$; this choice fixes the variance of the parameters $c_m$. At long enough times, we anticipate that all knowledge of the initial location of the exciton will be lost and the probability of finding the exciton anywhere in the lattice will be uniform in accordance with the $T=\infty$ distribution. To ascertain how long the exciton has spent in different regions of the lattice we integrate the eigenstate occupation probabilities $P_i(t)$ over time and define $O_i(t) = \int_0^t dt' P_i(t')$. We express these occupation times in the local basis as $O_m(t) = \int_0^t dt' \sum_i |{\langle m | i \rangle}|^2 P_i(t')$ ![(Colour online.) Histograms of the number of jumps $k$ in time intervals $t=300/J$ for simulations (as in Fig. \[fig1\]) with $10^8$ jumps in total. Plotted is the number of time intervals in which $k$ jumps occur, $N_\text{total}(k)$, for different strengths of disorder $d$ (labelled). Shown (dashed line) is a fit to the $d=1$ points assuming a Poisson distribution.[]{data-label="fig2"}](graph_dist.png){width="8.57cm"} Plotted in Fig. \[fig1\] are snapshots at different times of three trajectories at different disorder strengths $d$, all prepared in the same local initial state. At small $d$ it is clear that the exciton moves almost uniformly in space and time, with the lattice having been occupied uniformly after short times. Conversely, as $d$ is increased we find the exploration of the lattice becomes far from uniform in time, with large dwell times in certain regions and quick jumps between other regions. This effect becomes increasingly pronounced as $d$ is increased and it will be studied in greater detail later in the paper. First we look at the statistics for jumps between states, captured by the probability $\pi_t(K)$ that there are $K$ jumps between states in a time $t$. Shown in Fig. \[fig2\] are histograms reflecting this distribution for long time intervals $t=300/J$ at different values of $d$. While at small $d=1$, the distribution is close to Poissonian, as $d$ is increased, it becomes progressively broad, indicating that *rare trajectories*, where $K$ is much smaller or larger than the mean, becoming increasingly likely. We note that at long times the probability distribution takes the large-deviation form [@Touchette2009] $\pi_t(K) \simeq e^{-t\varphi(K/t)}$ where $\varphi(k)$ is a large-deviation function of the average jump rate, or *activity*, $k=K/t$. The associated moment generating function is also of large-deviation form $Z_t(s) = \sum_K \pi_t(K) e^{-sK} \simeq e^{t\theta(s)}$, with $s$ a conjugate field to the number of jumps $K$. The function $\theta(s)$ is analogous to (minus) a free energy for trajectories, with discontinuities in the derivatives of $\theta(s)$ corresponding to dynamical (or trajectory) phase transitions [@Garrahan2007; @Lecomte2007; @Garrahan2010]. The activity $k_s = -\partial_s\theta(s)$ will be used as an order parameter, with $k_{s=0}=k$ the average jump rate of the physical problem (*i.e.* with no $s$ field applied). We can find $\theta(s)$ as largest eigenvalue [@Lecomte2007] of the modified master operator $\mathbb{W}_s$, described by $$(\mathbb{W}_s)_{ij} = e^{-s} W_{j\rightarrow i} - r_i\delta_{i,j}\,. \label{eq:Ws}$$ This operator generates the dynamics of $s$-biased ensembles of trajectories via $\partial_t P_i(s) = \sum_j(\mathbb{W}_s)_{ij} P_j(s)$. The eigenstate of $\mathbb{W}_s$ corresponding to the largest eigenvalue $\theta(s)$ gives the occupation probabilities $P_i(s)$ (of level $i$) associated with the trajectories that dominate at a certain $s$. When $s=0$ this is the stationary state which, since we are studying the case of an infinite temperature environment, has equal probability for all levels, such that $P_i(s=0) = 1/N$ for all $i$. At $s\ne 0$, $P_i(s)$ indicate the occupations for rare trajectories which are more ($s<0$) or less ($s>0$) *active* than those of the average dynamics. ![(Colour online.) (a) Dynamical phase diagram for the $N=100\times 100$ disordered lattice in Fig. \[fig1\] as a function of inverse disorder strength and thermodynamic field $s$ (see main text). (b,c,d) Steady-state lattice-site occupations in the long-time limit, $P_m(s)$. Shown are the occupation probabilities for the parameter values as labelled on (a). (e) A slice through the dynamical phase diagram with parameter values labelled on (a). []{data-label="fig3"}](phase_lowres5.png){width="8.57cm"} We now apply this thermodynamic approach to the dynamics in disordered exciton-lattices. We obtain the dynamical phase diagram for the model  by plotting the activity $k_s$ as a function of the inverse disorder strength $d^{-1}$ and the $s$-field. The data, found from exact diagonalisation, are shown in Fig. \[fig3\](a). We find a first-order phase boundary for $s>0$, which appears to tend to $s=0$ in the limit of infinite disorder strength. We will show that existence of such a transition in a single-particle system is due to states becoming increasingly disconnected at higher disorder. The is existence of a transition is consistent with the increasingly long tails on $\pi_t(K)$ as $d$ is increased, shown in Fig. \[fig2\]. The large-$s$ phase has $k_s=0$ where, for these inactive rare trajectories, the exciton does not jump between eigenstates. Therefore, the exciton must remain localised. This result we confirm in Fig. \[fig3\](d), where we see that the $s$-biased steady-state occupation probability is zero everywhere apart from close to a particular site. In contrast, Fig. \[fig3\](c) shows the uniform ($T=\infty$) distribution across the lattice at $s=0$. This distribution is only slightly perturbed from uniform if $s$ is decreased further to $s=-0.02$, as demonstrated in Fig. \[fig3\](b). The transition in the dynamics can be understood as a localization transition in the master operator . The transition matrix $\mathbb{W}_s$ of this $T=\infty$ model is Hermitian such that we can draw analogy with a quantum Hamiltonian: at large $d$, where the eigenstates are tightly localised, $\mathbb{W}_s$ is equivalent to the sum of a “hopping” part $e^{-s} W_{j\rightarrow i}$ which determines the jumps between localised states, and “on-site energies” $r_i$. $\mathbb{W}_s$ exhibits a localisation transition which is crossed by tuning $s$. We find the existence of the transition is not associated with finite-size effects, but with the presence of long-range hopping terms $W_{j\rightarrow i}$ which favour a delocalised state [@Rodriguez2003; @Metz2013; @Biddle2011]. In Fig. \[fig4\](a) we explore the effect of $d$ on the log-range hopping terms $W_{j\rightarrow i}$. Plotting the mean values of these matrix elements between states centred on sites separated by $d_{ij}$ lattice spacings, we find that hopping terms beyond nearest-neighbour sites are significantly larger at smaller $d$. In Fig. \[fig4\](b) we plot the distribution of effective on-site terms for the dynamics, $r_i$. Interesingly, this effective on-site disorder has a distribution which changes shape with $d$, but the variance of the elements $r_i$ changes little with $d$. Thus we infer that the transition is controlled by the effective hopping terms, with the transition line approaching $s=0$ at large $d$ due to the increased size of long-range hopping matrix elements. When crossing the transition by tuning the $s$ field, the size of the effective hopping terms in comparison to the effective disorder is decreased as $s$ is increased, such that the localised phase is reached. ![(Colour online.) Statistics of matrix elements of the master operator $\mathbb{W}$ for single realisations with $10^4$ sites. (a) The average rate of transitions between states ${|i\rangle}$ and ${|j\rangle}$ centred on lattice sites a distance of $d_{ij}$ lattice periods apart for two disorder strengths $d=5$ and 20. (b) Histograms of the number, $N_\text{elements}$, of diagonal elements $r_i$ in $\mathbb{W}$ with different magnitudes. While the shape of the distribution depends strongly upon $d$, the variance of the elements only changes from 0.274 to 0.229 as $d$ is increased from $d=5$ to $d=20$.[]{data-label="fig4"}](graph_twoplot.png){width="8.57cm"} We now explore the behaviour of the system in both space and time for different values of the disorder strength $d$. We showed in Fig. \[fig1\] that as the exciton jumps through the lattice, it moves around some regions very quickly and dwells for long times in other regions. In the limit of long times, the exciton will have occupied all regions of the lattice for equal fractions of the evolution time, in accordance with the infinite-temperature thermal distribution. We wish study the time scale for the exciton to explore the lattice and consider how the number of persistent sites not visited by the exciton decreases over time at different values of $d$. For this, we introduce a generalised inverse participation ratio (GIPR) $p_t$ at finite times $t$ defined by $$p_t = \frac{1}{t}\sum_m O^2_t(m)$$ If a single lattice site $m$ is occupied initially, we have $p_{0}=1$. More generally, for an initial state ${|\psi\rangle}$, $p_{0}$ is determined by overlaps of ${|m\rangle}$ with the initial state wavefunction and is given by the conventional inverse participation ratio in the basis of position states [@Gogolin2011; @*Genway2012a], $\sum_m |{\langle m | \psi \rangle}|^4$. At times long enough for the entire lattice to be explored, $p_t$ approaches $p_\infty = 1/N$, reflecting the thermal distribution. This sets ergodic timescale as it requires the exciton to explore the entire lattice. To study the statistics associated with trajectories, we now introduce the label $\alpha$ which identifies a trajectory. We estimate the GIPR from simulated trajectories using Eq. . Fig. \[fig5\](a) illustrates, as a function of time, an average of the GIPR over trajectories, ${\langle p_t\rangle}_\alpha$, for different disorder strengths $d$. In Fig. \[fig5\](b), we show the variance of the GIPR over trajectories, $[\Delta p_t^2]_\alpha = {\langle p_t^2\rangle}_\alpha-{\langle p_t\rangle}^2_\alpha$, for the same values of $d$. We see that for $d=20$, the ergodic timescale is around two orders magnitude longer than at $d=3$. This can be contrasted with the corresponding average jump rate (see Fig. \[fig2\]) which only decreases by a factor of $\apprle 4$. The decay of the GIPR takes an approximate power-law form $p_t \sim t^{-\eta}$ for time scales greater than the mean jump rate $k$ but smaller than the ergodic time. From our numerics, we find $\eta \simeq 0.83$ across the range of $d$ we consider in Fig. \[fig5\]. ![(Colour online.) GIPRs for the simulations in Fig. \[fig1\]. (a) The GIPR averaged over trajectories ${\langle p_t\rangle}_\alpha$. The results are an average of $10^3$ simulations of $10^3$ jumps and 5 trajectories of $10^9$ jumps, to capture accurately the GIPR average across different initial states as well as the long-time, initial-state-independent behaviour. Initial states are selected randomly from the uniform distribution. Shown are different disorder strengths $d$, as labelled. (b) The variance of GIPRs for different trajectories $[\Delta p_t^2]_\alpha$ as a function of time for the same disorder strengths as in (a). (c) GIPRs averaged over trajectories for $d=50$ with different initial states. Shown is a comparison of initial states drawn from the uniform distribution and initial states drawn from the inactive-state distribution with occupation probabilities $P_i(s)$ for $s=3$ (as labelled). Shown *inset* is a magnified region with linear fits. These demonstrate that the curves are separated in time by a factor $10^{0.315}\simeq 2.1$.[]{data-label="fig5"}](multi_relax_lowres2.png){width="8.57cm"} As $d$ is increased, the states where the exciton dwells for a long time become increasingly inactive. This situation is analogous to finite-temperature diffusion of a classical particle in a random potential with a set of deep minima where the particle can get stuck for long times. We expect that as $d$ is increased, the initial state will have a larger effect on the decay of $p_t$, with a slower initial decay if the initial state is inactive. This is confirmed by our data for the GIPR by considering the variation between trajectories and the effect of choosing an inactive initial state. The variance between trajectories $[\Delta p_t^2]_\alpha$ is shown in Fig. \[fig5\](b). We find $[\Delta p_t^2]_\alpha$ increases with $d$, as does the length of time to reach the peak in this variance, consistent with the existence of increasingly inactive regions on the lattice. In Fig. \[fig5\](c), for $d=50$ we compare the decay of the GIPR for initial eigenstates selected from $P_i(s=3)$ with those selected from the thermal distribution. We find that the initial decay of the GIPR decays on a time scale approximately twice as long if we prepare the exciton in a state from the inactive dynamical phase. In summary, we have explored the dynamics of excitons in disordered lattices when coupled to infinite-temperature Markovian baths. While the steady-state occupation probability is simply the uniform distribution, we find rich features in the dynamics generated by the coupling to the thermal environment. We have studied the counting statistics of jumps between states and we find, as the strength of the disorder is increased, the distribution of jumps deviates significantly from the Poissonian form found for weak disorder. This deviation can be understood to relate to a dynamical phase transition in the space of temporal trajectories. Upon increasing the disorder, the dynamical phase boundary imposes a greater effect on the physical dynamics of the system, with the exciton spending increasing periods of time in inactive regions where jumps between states are rare. We also find that dynamical metastability also manifests in spatially heterogeneous dynamics, something which is very prevalent in glass forming systems [@Biroli2013]. This effect was characterised with a GIPR, which captures the lifetime of persistent sites not visited by the exciton. We find the GIPR decays in time as a fixed power-law for all disorder strengths $d$. Of course, what we presented here is nothing else than a single particle problem where all non-trivial features are a consequence of the imposed disorder. However, we may speculate that the combination that we find of metastability, a first-order transition in ensembles of dynamical trajectories, and dynamical heterogeneity will also be present in interacting systems which exhibit many-body localisation. We wish to thank Hoda Hossein-Nejad for discussions. We acknowledge support from the The Leverhulme Trust under grant No. F/00114/BG. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 335266 (ESCQUMA).

ArXiv

--- abstract: 'The IceCube Neutrino Observatory with its 1-km$^3$ in-ice detector and the 1-km$^2$ surface detector (IceTop) constitutes a three-dimensional cosmic ray detector well suited for general cosmic ray physics. Various measurements of cosmic ray properties, such as energy spectra, mass composition and anisotropies, have been obtained from analyses of air showers at the surface and/or atmospheric muons in the ice.' address: 'Humboldt-Universität zu Berlin and DESY' author: - 'H. Kolanoski (for the IceCube Collaboration)' bibliography: - 'ecrs\_pcr2\_hlt\_Kolanoski.bib' title: Cosmic Ray Physics with the IceCube Observatory --- Introduction ============ The IceCube Neutrino Observatory [@achterberg06; @Kolanoski_HLT_icrc2011] is a detector situated in the ice of the geographic South Pole at a depth of about 2000 m. The observatory is primarily designed to measure neutrinos from below, using the Earth as a filter to discriminate against muon background induced by cosmic rays (neutrino results are reported elsewhere in these proceedings [@kappes_HLT_ecrs2012]). IceCube also includes an air shower array on the surface called IceTop extendind IceCube’s capabilities for cosmic ray physics. Construction of IceCube Neutrino Observatory was completed in December 2010. IceCube can be regarded as a cubic-kilometer scale three-dimensional cosmic ray detector with the air showers (mainly the electromagnetic component) measured by the surface detector IceTop and the high energy muons and neutrinos measured in the ice. In particular the measurement of the electromagnetic component in IceTop in coincidence with the high energy muon bundle, originating from the first interactions in the atmosphere, has a strong sensitivity to composition. Here IceCube offers the unique possibility to clarify the cosmic ray composition and spectrum in the range between about 300 TeV and 1 EeV, including the ‘knee’ region and a possible transition from galactic to extra-galactic cosmic rays. Detector ======== #### IceCube: The main component of the IceCube Observatory is an array of 86 strings equipped with 5160 light detectors in a volume of 1 km$^3$ at a depth between 1450m and 2450m (Fig.\[fig:I3Array\]). The nominal IceCube string spacing is 125 m on a hexagonal grid. A part of the detector, called DeepCore, is more densely instrumented resulting in a lower energy threshold. ![Left: The IceCube detector with its components DeepCore and IceTop in the final configuration (January 2011). In this paper we present data taken with the still incomplete detector. We will refer to the configuration as IC79/IT73, for example, meaning 79 strings in IceCube and 73 stations in IceTop. The final detector has the configuration IC86/IT81. Right: View of a cosmic ray event which hits IceTop and IceCube. The size of the colored spots is proportional to the signal in the DOMs, the colors encode the signal times, separately for IceCube and IceTop. []{data-label="fig:I3Array"}](I3Array_vector_Jan2011_modHK_red "fig:"){width="62.00000%"}![Left: The IceCube detector with its components DeepCore and IceTop in the final configuration (January 2011). In this paper we present data taken with the still incomplete detector. We will refer to the configuration as IC79/IT73, for example, meaning 79 strings in IceCube and 73 stations in IceTop. The final detector has the configuration IC86/IT81. Right: View of a cosmic ray event which hits IceTop and IceCube. The size of the colored spots is proportional to the signal in the DOMs, the colors encode the signal times, separately for IceCube and IceTop. []{data-label="fig:I3Array"}](BigEvent.pdf "fig:"){width="37.00000%"} Each string, except those of DeepCore, is equipped with 60 light detectors, called ‘Digital Optical Modules’ (DOMs), each containing a $10''$ photo multiplier tube (PMT) to record the Cherenkov light of charged particles traversing the ice. In addition, a DOM houses complex electronic circuitry supplying signal digitisation, readout, triggering, calibration, data transfer and various control functions. The most important feature of the DOM electronics is the recording of the analog waveforms in $3.3{\,\mathrm{ns}}$ wide bins for a duration of $422{\,\mathrm{ns}}$. With a coarser binning a ‘fast ADC’ extends the time range to 6.4$\mu$s. #### IceTop: The 1-km$^2$ IceTop air shower array [@ITDet-IceCube:2012nn] is located above IceCube at a height of 2835m above sea level, corresponding to an atmospheric depth of about 680 g/cm$^2$. It consists of 162 ice Cherenkov tanks, placed at 81 stations mostly near the IceCube strings (Fig.\[fig:I3Array\]). In the center of the array, a denser station distribution forms an in-fill array with a lower energy threshold (about 100TeV). Each station comprises two cylindrical tanks, 10 m apart, with an inner diameter of $1.82{\,\mathrm{m}}$ and filled with ice to a height of $90{\,\mathrm{cm}}$. Each tank is equipped with two DOMs which are operated at different PMT gains to cover linearly a dynamic range of about $10^5$ with a sensitivity to a single photoelectron (the thresholds, however, are around 20 photoelectrons). DOMs, electronics and readout scheme are the same as for the in-ice detector. Cosmic Ray spectrum {#sec:spectrum} =================== ![First evalution of one year of data taken with the 73-station configuration of IceTop in 2010. The events were required to have more than 5 stations and zenith angles in the range $\cos\theta \geq 0.8$. The spectrum is shown for the two assumptions ‘pure proton’ and ‘pure iron’ for the primary composition. []{data-label="fig:IT73-spectrum-p-Fe"}](IT26_spectrum-v2_2.pdf){width="100.00000%"} ![First evalution of one year of data taken with the 73-station configuration of IceTop in 2010. The events were required to have more than 5 stations and zenith angles in the range $\cos\theta \geq 0.8$. The spectrum is shown for the two assumptions ‘pure proton’ and ‘pure iron’ for the primary composition. []{data-label="fig:IT73-spectrum-p-Fe"}](FullYear_cosZenith_above_08_3and_MoreStations.pdf){width="100.00000%"} ![Composition analysis (IC40/IT40 configuration) [@ITIC40-composition_Abbasi:2012]. Left: Simulated correlation between the energy loss of the muon bundles in the ice (K70) and the shower size at the surface (S125) for proton and iron showers. The shading indicates the percentage of protons over the sum of protons and iron in a bin. The lines of constant primary energy are labeled with the logarithms of the energies. Right: IceCube result for the average logarithmic mass of primary cosmic rays compared to other measurements (references in [@ITIC40-composition_Abbasi:2012]). []{data-label="fig:composition_ITIC40"}](pretty_plot_berries_ICRC_zaxis_v2-eps-converted-to.pdf "fig:"){width="43.00000%"} ![Composition analysis (IC40/IT40 configuration) [@ITIC40-composition_Abbasi:2012]. Left: Simulated correlation between the energy loss of the muon bundles in the ice (K70) and the shower size at the surface (S125) for proton and iron showers. The shading indicates the percentage of protons over the sum of protons and iron in a bin. The lines of constant primary energy are labeled with the logarithms of the energies. Right: IceCube result for the average logarithmic mass of primary cosmic rays compared to other measurements (references in [@ITIC40-composition_Abbasi:2012]). []{data-label="fig:composition_ITIC40"}](compositionplot2-eps-converted-to.pdf "fig:"){width="54.00000%"} Figure \[fig:IT26\_spectrum-v2\_2\] shows the energy spectrum from 1 to 100 PeV [@IT26-spectrum_Abbasi:2012wn] determined from 4 month of data taken in the IT26 configuration in 2007. The relation between the measured shower size and the primary energy is mass dependent. Good agreement of the spectra in three zenith angle ranges was found for the assumption of pure proton and a simple two-component model (see [@IT26-spectrum_Abbasi:2012wn]). For zenith angles below 30[$^{\circ}$]{}, where the mass dependence is smallest, the knee in the cosmic ray energy spectrum was observed at about 4.3PeV with the largest uncertainty coming from the composition dependence (+0.38PeV and -1.1PeV). The spectral index changes from 2.76 below the knee to 3.11 above the knee. There is an indication of a flattening of the spectrum above about 20PeV which was also seen by the experiments GAMMA [@Gamma-Garyaka:2008gs], Tunka [@Kuzmichev_HLT_ecrs2012] and Kaskade-Grande [@Haungs_HLT_ecrs2012]. A first preliminary evaluation of IceTop data from the 2010/11 season with 79 IceCube strings and 73 IceTop stations is shown in Fig. \[fig:IT73-spectrum-p-Fe\]. Cosmic ray composition {#sec:composition} ====================== As mentioned in the introduction, the combination of the in-ice detector with the surface detector offers a unique possibility to determine the spectrum and mass composition of cosmic rays from about 300 TeV to 1 EeV. The first such analysis exploiting the IceTop-IceCube correlation was done on a small data set corresponding to only one month of data taken with about a quarter of the final detector for energies from 1 to 30 PeV [@ITIC40-composition_Abbasi:2012]. From the measured input variables, shower size and muon energy loss (Fig.\[fig:composition\_ITIC40\], left), the primary energy and mass was determined using a neural network. The resulting average logarithmic mass is shown in Fig.\[fig:composition\_ITIC40\], right. These results are still dominated by systematic uncertainties, such as the energy scale of the muons in IceCube and the effects of snow accumulation on the IceTop tanks. A similar analysis of IceTop-IceCube coincidences is in progress using the IC79/IT73 data set taken in 2010 (the energy spectrum obtained with these data is displayed in Fig. \[fig:IT73-spectrum-p-Fe\]). The studies indicate that there will be enough statistics for composition analysis up to about 1 EeV. The systematic uncertainties related to the models can be reduced by including different mass sensitive variables, like zenith angle dependence of shower size [@IT26-spectrum_Abbasi:2012wn], muon rates in the surface detector and shower shape variables (see discussion in [@Kolanoski_HLT_icrc2011]). PeV-gamma rays {#sec:pevgamma} ============== IceCube can efficiently distinguish PeV gamma rays from the background of cosmic rays by exploiting coincident in-ice signals as veto. Gamma-ray air showers have a much lower muon content than cosmic ray air showers of the same energy. Candidate events are selected from those showers that lack a signal from a muon bundle in the deep ice. Results of one year of data, taken in the IC40/IT40 configuration are shown in Fig. \[fig:pevgammarays\_limits\] [@Stijn_icrc2011]. The projected gamma-ray sensitivity of the final detector is also given. ![Limits on the diffuse gamma ray flux relative to the cosmic ray flux from a region within 10[$^{\circ}$]{} from the Galactic Plane (IC40/IT409, purple line). The plot includes also the only other available limits from CASA-MIA [@CASA-MIA-Chantell:1997gs] and the expected one-year sensitivity for the complete IceCube detector (blue dashed line for the whole covered energy range, blue dots for smaller energy bins).\ [ ]{}[]{data-label="fig:pevgammarays_limits"}](pevgammarays_limits.pdf){width="48.00000%"} Transient events {#sec:transients} ================ Transient events such as sun flares or gamma ray bursts, if they generate very high fluxes of low energy particles, could be observed as general rate increases above the noise level in the IceTop DOMs even if they could not be detected individually. This was first demonstrated with the observation of the Dec 13, 2006 Sun flare event [@Sun-flare-Abbasi08]. The detector readout has since then been setup such that counting rates could be obtained at different thresholds allowing to unfold cosmic ray spectra during a flare [@Takao_IT_icrc2011]. Atmospheric muons in the ice {#sec:muons_inice} ============================ In this section analyses of atmospheric muons in IceCube (without requiring air shower detection in IceTop) are presented. The related atmospheric neutrinos, an irreducible background for cosmic neutrino search, are discussed elsewhere in these proceedings [@kappes_HLT_ecrs2012]. Muon spectrum and composition ----------------------------- Atmospheric muon and neutrino spectra measured with IceCube probe shower development of cosmic rays with primary energies above about 10 TeV. To penetrate to the IceCube depth and be detectable the muons have to have energies above about 500 GeV. Methods have been developed to distinguish single, high-energetic muons by their stochastic energy loss [@Berghaus_icrc2011] from muon bundles with rather smooth energy deposition. Figure \[fig:Muon-bundle-spectrum\] shows a cosmic ray spectrum derived from an analysis of muon bundles. The flux is plotted against an energy estimator, $E_{mult}$, which is derived from the measured muon multiplicity in the bundles using the empirical formula $N_{\mu} \sim A^{0.23} E^{0.77}$ with iron as reference nucleus ($A=56$). The data are compared to the predictions from different models. None of the models matches particularly well, especially not at low energies (where threshold effects might cause some experimental uncertainty). The data indicate that some additional component at higher energies is required, for example the extra-galactic ‘mixed component’ in the model “Gaiser-Hillas 3a” (see Fig.1 in [@Gaisser:2012zz] and discussion in [@Berghaus_isvhecri_2012]). There is also an interesting flattening observable above about 10 PeV which might be connected to the flattening observed in the same region in the IceTop spectrum (Figs. \[fig:IT26\_spectrum-v2\_2\] and \[fig:IT73-spectrum-p-Fe\]) and by other experiments (see Section \[sec:spectrum\]). This analysis is complementary to the composition analysis and can be exploited to test the consistency of models in a wide energy range from well below the knee to above some EeV. ![Energy spectrum of primary cosmic rays obtained from muon bundles in IceCube. The energy estimator $E_{mult}$ is derived from the measured muon multiplicity in the bundles which is composition dependent. See explanation in the text.\ [ ]{} []{data-label="fig:Muon-bundle-spectrum"}](Muon-bundle-spectrum.pdf){width="48.00000%"} Muons with high transverse momenta ---------------------------------- At high energies the muons reach the in-ice detector in bundles which are, for primaries above about 1PeV, collimated within radii of the order of some 10m. Most of the muons stem from the soft peripheral collisions with little transverse momentum transfer. Perturbative QCD calculations, however, predict the occurrence of muons with higher transverse momenta in some fraction of the events. Large transverse momenta of muons show up in a lateral separation from the muon bundle. In Fig. \[fig:LS-dist\] this lateral distribution obtained from IC59 data [@LSMuons-Abbasi:2012he] is shown along with a fit by an exponential plus a power law function. The power law part indicates the onset of hard scattering in this regime of $p_T \approx 2-15$ GeV/c, as expected from perturbative QCD. However, the zenith angular dependence shown in Fig. \[fig:LS-zenith\] cannot be described by the commonly used models Sibyll and QGSJET while it is reasonably reproduced by DPMJET. The reasons for these differences have to be understood and could have important implications for air shower simulations. ![\[fig:LS-zenith\] The cosine distribution of the directions of bundles with laterally separated muons compared to simulations using commonly used interaction models.](scale_data_and_fit_paper_plot_simple_log.pdf){width="100.00000%"} ![\[fig:LS-zenith\] The cosine distribution of the directions of bundles with laterally separated muons compared to simulations using commonly used interaction models.](paper_plot_cos_zen_maxmuon_levels_models_l7.pdf){width="100.00000%"} Cosmic ray anisotropy {#sec:anisotropy} ===================== IceCube collects large amounts of cosmic ray muon events, about $10^{11}$ events in every year of running with the full detector. These events have been used to study cosmic ray anisotropies on multiple angular scales, for the first time in the Southern sky [@Abbasi_anisotropy:2010mf; @Abbasi_anisotropy:2011ai; @Abbasi_anisotropy:2011zka]. ![Left: Relative intensity maps for the low-energy (top) and high-energy (bottom) data sets. Right: Projections of the maps unto right ascension in the declination band -75[$^{\circ}$]{} to -25[$^{\circ}$]{}. In the projection plot, the error bars are statistical while the colored boxes indicate the systematic uncertainty. The curves are empirical fits.[]{data-label="fig:IT-aniso-RelInt"}](IT-Aniso-RI-proj.pdf){width="100.00000%"} While the previous analyses exploited data from the in-ice muons only, now also first results from data taken with IceTop are available. The advantage of using IceTop is a better energy resolution which allows a finer energy binning if statistics is sufficient. Figure \[fig:IT-aniso-RelInt\] shows skymaps of relative intensities determined from IceTop data for primary energies centered around 400 TeV and 2 PeV (still with a rather coarse binning). The data were taken over 3 years in the configurations IT59, IT73, IT81. The 400 TeV data confirm the in-ice observations [@Abbasi_anisotropy:2011zka], in particular the change in phase compared to the 20-TeV observations. The new result is that at 2PeV the anisotropy as a function of right ascension has a similar shape as at 400TeV but becomes apparently stronger. As yet the anisotropies observed on multiple angular scales and at different energies have not found an explanation. Theoretical explanations like local magnet fields affecting the cosmic ray streams and/or nearby sources of cosmic rays are discussed. The determination of the energy dependence of anisotropies will be crucial for validating such explanations. Conclusion ========== The presented results on cosmic ray properties, such as energy spectra, mass composition and anisotropies, demonstrate the high, partly unique, potential of the IceCube Observatory for studying cosmic rays physics. The IceCube/IceTop system covers an energy range from well below the knee to the expected onset of an extra-galactic component. References {#references .unnumbered} ==========

ArXiv

--- abstract: 'We present (with proof) a new family of decomposable Specht modules for the symmetric group in characteristic $2$. These Specht modules are labelled by partitions of the form $(a,3,1^b)$, and are the first new examples found for thirty years. Our method of proof is to exhibit summands isomorphic to irreducible Specht modules, by constructing explicit homomorphisms between Specht modules.' author: - | Craig J. Dodge\ Department of Mathematics, University at Buffalo, SUNY,\ 244 Mathematics Building, Buffalo, NY 14260, U.S.A.\ \ Matthew Fayers\ Queen Mary, University of London, Mile End Road, London E1 4NS, U.K. title: Some new decomposable Specht modules --- =12.5cm =8.839cm 0 [This is the second author’s version of a work that was accepted for publication in the Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in\ *J. Algebra* ]{} =8.27in =11.69in 1 Introduction ============ Let $n$ be a positive integer, and let ${\mathfrak{S}_}n$ denote the symmetric group on $n$ letters. For any field $\bbf$, the *Specht modules* form an important family of modules for $\bbf{\mathfrak{S}_}n$. If $\bbf$ has characteristic zero, then the Specht modules are precisely the irreducible modules for $\bbf{\mathfrak{S}_}n$. If $\bbf$ has positive characteristic, the simple $\bbf{\mathfrak{S}_}n$-modules arise as quotients of certain Specht modules. In addition, the Specht modules arise as the ‘cell modules’ for Murphy’s cellular basis of $\bbf{\mathfrak{S}_}n$. A great deal of effort is devoted to determining the structure of Specht modules; in particular, finding the composition factors of Specht modules and the dimensions of the spaces of homomorphisms between Specht modules. In this paper, we consider the question of which Specht modules are decomposable. It is known that in odd characteristic the Specht modules are all indecomposable, so we can concentrate on the case where ${\operatorname{char}}(\bbf)=2$. In fact, since any field is a splitting field for ${\mathfrak{S}_}n$, we can assume that $\bbf=\bbf_2$. In this case, there are decomposable Specht modules, but remarkably few examples are known. Murphy [@gm] analysed the Specht modules labelled by ‘hook partitions’, i.e. partitions of the form $(a,1^b)$, computing the endomorphism ring of every such Specht module (and thereby determining which ones are decomposable). However, in the last thirty years no more progress seems to have been made. Our main result is the discovery of a new family of decomposable Specht modules, the first examples of which were discovered by the two authors independently using computations with GAP and MAGMA. These new decomposable Specht modules are labelled by partitions of the form $(a,3,1^b)$, where $a,b$ are even. So in this paper we make a case study of partitions of this form; we are unable to apply Murphy’s method to determine exactly which of these Specht modules are decomposable, but by considering homomorphisms between Specht modules, we are able to show which irreducible Specht modules arise as summands of these Specht modules. We then apply this result to determine which of our Specht modules have a summand isomorphic to an irreducible Specht module. We now briefly indicate the layout of this paper. In the next section, we recall some basic definitions and results in the representation theory of the symmetric group, which enable us to state our main results in Section \[resultsec\]. In Section \[homsec\] we go into more detail on homomorphisms between Specht modules. In Sections \[uvsec\] and \[uv2sec\] we consider the two classes of irreducible Specht modules which can occur as summands of our decomposable Specht modules. We then apply these results in Section \[whichdec\] to complete the proof of our main results. Finally, we make some concluding remarks in Section \[concsec\]. The authors are indebted to David Hemmer, who first made us aware of each other’s work and initiated this collaboration, and also invited the second author to SUNY Buffalo in September 2011, where some of this work was carried out. This work continued during the ‘New York workshop on the symmetric group’; we are grateful to Rishi Nath of CUNY for inviting us to this conference. The research of the first author was supported in part by NSA grant H98230-10-1-0192. Background results {#backsec} ================== In this section, we summarise some basic results on the representation theory of the symmetric group. For brevity, we specialise some results to characteristic $2$, referring the reader to the literature for general results. We begin by fixing a field $\bbf$; all our modules will be modules for the group algebra $\bbf{\mathfrak{S}_}n$. We assume familiarity with James’s book [@j2]; in particular, we refer the reader there for the definitions of partitions, the dominance order, the permutation modules $M^\la$, the Specht modules $S^\la$ and the simple modules $D^\la$. We shall also briefly use the Nakayama Conjecture [@j2 Theorem 21.11] which describes the block structure of the symmetric group. We also need the following two results; recall that if $\la$ is a partition then $\la'$ denotes the conjugate partition. \[isospecht\] Suppose ${\operatorname{char}}(\bbf)=2$ and $\la$ is a partition such that $S^\la$ is irreducible. Then $S^\la\cong S^{\la'}$. By [@j2 Theorem 8.15] we have $S^\la\cong(S^{\la'})^\ast$, since the sign representation is trivial in characteristic $2$. But by [@j2 Theorem 11.5], all simple modules for the symmetric group are self-dual. \[815hom\] If $\la,\mu$ are partitions of $n$, then $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^\mu)=\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^{\mu'},S^{\la'}).$$ This also follows from [@j2 Theorem 8.15]. Regularisation -------------- We recall here a useful lemma which we shall use later; this is due to James, although it does not appear in the book [@j2]. We concentrate on the special case where $\bbf$ has characteristic $2$, referring to [@j1] for the full result. For any $l\gs1$, the $l$th *ladder* in $\bbn^2$ is $$\call_l=\lset{(i,j)}{i+j=l+1}.$$ If $\la$ is a partition, the *$2$-regularisation* of $\la$ is the partition $\la{^{\operatorname{reg}}}$ whose Young diagram is obtained by moving the nodes in $[\la]$ as high as possible within their ladders. For example, $(8,3,1^6){^{\operatorname{reg}}}=(8,7,2)$, as we see from the following Young diagrams, in which nodes are labelled according to the ladders in which they lie. 0 $$\young(12345678,234,3,4,5,6,7,8)\qquad\qquad \young(12345678,2345678,34)$$ 1 It is a simple exercise to show that $\la{^{\operatorname{reg}}}$ is a $2$-regular partition, and we have the following result. [ ]{}\[jreg\] Suppose $\la$ and $\mu$ are partitions of $n$, with $\mu$ $2$-regular. Then $[S^\la:D^{\la{^{\operatorname{reg}}}}]=1$, while $[S^\la:D^\mu]=0$ if $\mu\ndom\la{^{\operatorname{reg}}}$. In this paper we shall be concerned with the Specht modules labelled by partitions of the form $(a,3,1^b)$; so we compute the regularisations of these partitions. \[reg\] Suppose $a\gs4$ and $b\gs2$. Then $$(a,3,1^b){^{\operatorname{reg}}}= \begin{cases} (a,b+1,2)&(a>b)\\ (b+2,a-1,2)&(a\ls b). \end{cases}$$ Irreducible Specht modules -------------------------- It will be very helpful to know the classification of irreducible Specht modules, which (in characteristic $2$) was discovered by James and Mathas [@jmp2]. If $k$ is a non-negative integer we let $l(k)$ denote the smallest positive integer such that $2^{l(k)}>k$. [ ]{}\[irrspecht\] Suppose $\mu$ is a partition of $n$ and ${\operatorname{char}}(\bbf)=2$. Then $S^\mu$ is irreducible if and only if one of the following occurs: - $\mu_i-\mu_{i+1}\equiv-1\ppmod{2^{l(\mu_{i+1}-\mu_{i+2})}}$ for each $i\gs1$; - $\mu'_i-\mu'_{i+1}\equiv-1\ppmod{2^{l(\mu'_{i+1}-\mu'_{i+2})}}$ for each $i\gs1$; - $\mu=(2^2)$. Note that $\mu$ satisfies the first condition in the theorem if and only if $\mu'$ satisfies the second. In view of Lemma \[isospecht\] (and since we shall only be considering values of $n$ greater than $4$) we may assume that any irreducible Specht module is of the form $S^\mu$ where $\mu$ satisfies the first condition in the theorem. The main results {#resultsec} ================ In this section, we describe the new family of decomposable Specht modules discussed in this paper, and the method we use to prove decomposability. *For the rest of this paper, we assume that $\bbf$ has characteristic $2$.* Computer calculations show that the first few decomposable Specht modules which are not labelled by hook partitions have labelling partitions of the form $(a,3,1^b)$ (and their conjugates) with $a,b$ even positive integers. So in this paper we make a case study of this family of partitions. Our technique is different from that of Murphy [@gm], and is weaker in the sense that we cannot always when tell for certain whether one of our Specht modules is decomposable. However, in the cases where we can show decomposability, we have the advantage of being able to describe one summand explicitly. More specifically, our main result is a determination of exactly which irreducible Specht modules occur as summands of the Specht modules $S^{(a,3,1^b)}$. The technique we use is to consider homomorphisms between Specht modules, and the set-up for computing such homomorphisms is described in Section \[homsec\]. We use homomorphisms between Specht modules in the following way. Suppose $\la,\mu$ are partitions of $n$. Then it is a straightforward result that $S^\mu$ occurs as a summand of $S^\la$ if and only if there are homomorphisms $\gamma:S^\mu\to S^\la$ and $\delta:S^\la\to S^\mu$ such that $\delta\circ\gamma$ is the identity on $S^\mu$. If we assume in addition that $S^\mu$ is irreducible, then by Schur’s Lemma we just need to show that $\delta\circ\gamma$ is non-zero. Some effort has been devoted to computing the space of homomorphisms between two Specht modules, beginning with the paper of the second author and Martin [@fm]. In fact, there is now an explicit algorithm which computes the homomorphism space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^\mu)$ except when ${\operatorname{char}}(\bbf)=2$ and $\la$ is $2$-singular. Even in this exceptional case, this technique can be used to construct some homomorphisms between $S^\la$ and $S^\mu$, though only if $\la$ dominates $\mu$. In our situation, the partitions $\mu$ we shall consider are always $2$-regular, because (as long as $n\neq4$) every irreducible Specht module in characteristic $2$ has the form $S^\mu$ for $\mu$ $2$-regular. So we are able to compute the space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$. Computing the space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^\mu)$ is harder, because $\la$ is $2$-singular, so we fall foul of the exception above. To circumvent this, we use Lemma \[isospecht\], which allows us to take $\delta$ to be an element of ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})$. By Lemma \[815hom\] this has the same dimension as ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la'})$, which we can compute because $\mu$ is $2$-regular. Having established the dimension of this space, we can construct all possible homomorphisms $\delta$, and then check the condition $\delta\circ\gamma\neq0$. In this way, we can find all summands of $S^\la$ which are isomorphic to irreducible Specht modules $S^\mu$. In fact, we can restrict attention to a small set of candidate Specht modules $S^\mu$, as follows. Assuming $S^\mu$ is irreducible and $\mu$ is $2$-regular, $S^\mu$ is isomorphic to the simple module $D^\mu$; therefore in order for $S^\mu$ to appear as a summand of $S^\la$, the decomposition number $[S^\la:D^\mu]$ must be non-zero. Therefore by Theorem \[jreg\], $\mu$ must dominate $\la{^{\operatorname{reg}}}$, and by Lemma \[reg\] this has the form $(x,y,2)$ for some $x,y$. So we may assume that $\mu$ has the form $(u,v,w)$ for some $u>v>w\ls2$. Furthermore, $S^\mu$ and $S^\la$ must lie in the same block of $\bbf{\mathfrak{S}_}n$. Using the Nakayama Conjecture, this means that $u,w$ must be even, while $v$ is odd. So we can restrict attention to $\mu$ of the form $(u,v)$ or $(u,v,2)$ where $u$ is even (and hence $v$ is odd). We apply the technique described above to these two types of partition in Sections \[uvsec\] and \[uv2sec\]. Our main result is the following. \[main\] Suppose $\la=(a,3,1^b)$ is a partition of $n$, where $a,b$ are positive even integers with $a\gs4$, and suppose $\mu$ is a partition of $n$ such that $S^\mu$ is irreducible. Then $S^\la$ has a direct summand isomorphic to $S^\mu$ if and only if one of the following occurs. 1. $\mu$ or $\mu'$ equals $(u,v)$, where $v\equiv3\ppmod4$ and $\mbinom{u-v}{a-v}$ is odd. 2. $\mu$ or $\mu'$ equals $(u,v,2)$, where $\mbinom{u-v}{a-v}$ is odd. Using this result, we can show that most of the Specht modules under consideration are decomposable. Specifically, we have the following result. \[maincor\] Suppose $a,b$ are positive even integers with $a\gs4$, and let $\la=(a,3,1^b)$. Then $S^\la$ has a summand isomorphic to an irreducible Specht module if and only if at least one of the following occurs: - $a+b\equiv0$ or $2\pmod 8$, $a\gs6$ and $b\gs4$; - $a+b\equiv2\ppmod4$ and $\mbinom{a+b-3}{a-3}$ is odd; - $a+b\equiv0\ppmod4$ and $\mbinom{a+b-9}{a-5}$ is odd. Computing the space of homomorphisms between two Specht modules {#homsec} =============================================================== In this section, we explain the set-up for computing the space of homomorphisms between two Specht modules. We begin with a revision of some material from [@j2], before citing some results of the second author and Martin. Homomorphisms from Specht modules to permutation modules -------------------------------------------------------- Suppose $\mu$ and $\la$ are partitions of $n$. Since $S^\la\ls M^\la$, any homomorphism from $S^\mu$ to $S^\la$ can be regarded as a homomorphism from $S^\mu$ to $M^\la$. This is very useful, because if $\mu$ is $2$-regular (or if ${\operatorname{char}}(\bbf)\neq2$), then the space ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,M^\la)$ can be described explicitly. Furthermore, using the Kernel Intersection Theorem below, one can check whether the image of a homomorphism $\theta:S^\mu\to M^\la$ lies in $S^\la$. We now make some more precise definitions. We take $\la,\mu$ as above, but we now allow $\la$ to be any composition of $n$, not necessarily a partition. A *$\mu$-tableau of type $\la$* is a function $T$ from the Young diagram $[\mu]$ to $\bbn$ with the property that for each $i\in\bbn$ there are exactly $\la_i$ nodes of $[\mu]$ mapped to $i$. Such a tableau is usually represented by drawing $[\la]$ with a box for each node $\fkn$, filled with the integer $T(\fkn)$. $T$ is *row-standard* if the entries in this diagram are weakly increasing along the rows, and is *semistandard* if the entries are weakly increasing along the rows and strictly increasing down the columns. We write ${\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ for the set of row-standard $\mu$-tableaux of type $\la$, and ${\calt_{\hspace{-2pt}0}}(\mu,\la)$ for the set of semistandard $\mu$-tableaux of type $\la$. For each $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, James defines a homomorphism $\Theta_T:M^\mu\to M^\la$ (over any field), whose precise definition we do not need here. The restriction of $\Theta_T$ to $S^\mu$ is denoted ${\hat\Theta_{T}}$. Now we have the following. [ ]{}\[semi\] The set $$\lset{{\hat\Theta_{T}}}{T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)}$$ is linearly independent, and spans ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,M^\la)$ if $\mu$ is $2$-regular. The Kernel Intersection Theorem ------------------------------- Now return to the assumption that $\la$ is a partition. As a consequence of Theorem \[semi\], in order to compute ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$ when $\mu$ is $2$-regular, we just need to find all linear combinations $\theta$ of the homomorphisms ${\hat\Theta_{T}}$ for $T\in{\calt_{\hspace{-2pt}0}}(\mu,\la)$ for which the image of $\theta$ lies in $S^\la$. Even when $\mu$ is not $2$-regular, homomorphisms from $S^\mu$ to $S^\la$ can very often be expressed in this way. In order to determine whether the image of such a homomorphism $\theta$ lies in $S^\la$, we use another theorem of James which provides an alternative definition of $S^\la$. For any pair $(d,t)$ with $d\gs 1$ and $1\ls t\ls \la_{d+1}$, there is a homomorphism ${\psi_{d,t}}:M^\la\to M^\nu$, where $\nu$ is a composition depending on $\la,d,t$. Again, we refer the reader to [@j2 §17] for the definition; we warn the reader that the homomorphism ${\psi_{d,t}}$ is called $\psi_{d,\la_{d+1}-t}$ in \[*loc. cit.*\]. The importance of the homomorphisms ${\psi_{d,t}}$ is in the following. [ ]{}\[kit\] Suppose $\la$ is a partition of $n$. Then $$S^\la=\bigcap_{\begin{smallmatrix}d\gs1\\1\ls t\ls\la_{d+1}\end{smallmatrix}}\ker({\psi_{d,t}}).$$ This provides a clear strategy for computing ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$: find all linear combinations $\theta$ of the homomorphisms ${\hat\Theta_{T}}$ such that ${\psi_{d,t}}\circ\theta=0$ for every $d,t$. Fortunately, it is known how to compute the composition ${\psi_{d,t}}\circ{\hat\Theta_{T}}$ when $T\in {\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$. For our next few results, we need to introduce some more notation. For any multiset $A$ of positive integers, let $A_i$ denote the number of $i$s in $A$. If $A,B$ are multisets, we write $A\sqcup B$ for the multiset with $(A\sqcup B)_i=A_i+B_i$ for all $i$. Given a row-standard tableau $T$, we write $T^j$ for the multiset of entries in row $j$ of $T$. [ ]{}\[lemma5\] Suppose $\la,\mu$ are partitions of $n$, $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, $d\in\bbn$ and $1\ls t\ls\la_{d+1}$. Let $\cals$ be the set of all row-standard tableaux which can be obtained from $T$ by replacing $t$ of the entries equal to $d+1$ in $T$ with $d$s. Then $${\psi_{d,t}}\circ\Theta_T=\sum_{S\in\cals}\prod_{j\gs1}\binom{S^j_d}{T^j_d}\Theta_S.$$ The slight difficulty with using this lemma to compute homomorphism spaces is that the tableaux $S$ in the lemma are not always semistandard; so it can be difficult to tell whether a particular linear combination is zero when restricted to $S^\mu$. To circumvent this, we recall another lemma from [@fm] which gives certain linear relations between the homomorphisms ${\hat\Theta_{T}}$, and often enables us to write a homomorphism ${\hat\Theta_{T}}$ in terms of semistandard homomorphisms. [ ]{}\[lemma7\] Suppose $\mu$ is a partition of $n$ and $\la$ a composition of $n$, and $i,j,k$ are positive integers with $j\neq k$ and $\mu_j\gs\mu_k$. Suppose $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, and let $\cals$ be the set of all $S\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ such that: - $S^j_i=T^j_i+T^k_i$; - $S^j_l\ls T^j_l$ for every $l\neq i$; - $S^l=T^l$ for all $l\neq j,k$. Then $${\hat\Theta_{T}} = (-1)^{T^k_i}\sum_{S\in\cals}\prod_{l\gs1}\binom{S^k_l}{T^k_l}{\hat\Theta_{S}}.$$ Informally, a tableau in $\cals$ is a tableau obtained from $T$ by moving all the $i$s from row $k$ to row $j$, and moving some multiset of entries different from $i$ from row $j$ to row $k$. One very simple case of Lemma \[lemma7\] which we shall apply frequently is the following: if $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$ and for some $i,j,k$ we have $T^j_i+T^k_i>\max\{\mu_j,\mu_k\}$, then ${\hat\Theta_{T}}=0$. Lemma \[lemma7\] turns out to be very useful for expressing a tableau homomorphism in terms of semistandard homomorphisms. However, we shall occasionally need to use the following alternative. \[newsemilem\] Suppose $\la$ and $\mu$ are partitions of $n$, and $T$ is a row-standard $\la$-tableau of type $\mu$. Suppose $i\gs1$, and $A,B,C$ are multisets of positive integers such that $|B|>\la_i$ and $A\sqcup B\sqcup C=T^i+T^{i+1}$. Let $\calb$ be the set of all pairs $(D,E)$ of multisets such that $|D|=\la_i-|A|$ and $B=D\sqcup E$. For each such pair $(D,E)$, define $T_{D,E}$ to be the row-standard tableau with $$T_{D,E}^j= \begin{cases} A\sqcup D&(j=i)\\ C\sqcup E&(j=i+1)\\ T^j&(\text{otherwise}). \end{cases}$$ Then $$\sum_{(D,E)\in\calb}\prod_{i\gs1}\binom{A_i+D_i}{D_i}\binom{C_i+E_i}{E_i}{\hat\Theta_{T_{D,E}}}=0.$$ This lemma appears in the second author’s forthcoming paper [@garnir] where it is proved in the wider context of Iwahori–Hecke algebras; however, a considerably easier proof exists in the symmetric group case. In [@garnir], Lemma \[newsemilem\] is used to provide an explicit fast algorithm for writing a tableau homomorphism as a linear combination of semistandard homomorphisms. We now give another result which will help us in showing that a linear combination of row-standard homomorphisms is non-zero without having to go through the full process of expressing it as a linear combination of semistandard homomorphisms. This concerns the *dominance order* on tableaux. Suppose $\mu$ is a partition, and $S,T$ are row-standard $\mu$-tableaux of the same type. We say that $S$ dominates $T$, and write $S\dom T$, if it is possible to get from $S$ to $T$ by repeatedly swapping an entry of $S$ with a larger entry in a lower row (and re-ordering with each row). We warn the reader that this is not quite the same as the dominance order described in [@j2 13.8]. For example, the dominance order on ${\calt_{\hspace{-2pt}\operatorname{r}}}\left((3,2),(2^2,1)\right)$ is given by the following Hasse diagram. $$\begin{tikzpicture} \draw(0.5,0.5)--++(0,2.5)--++(2,2)--++(-2,2)--++(-2,-2)--++(2,-2); \tyoung(0cm,0.5cm,223,11) \tyoung(0cm,3cm,123,12) \tyoung(2cm,5cm,122,13) \tyoung(-2cm,5cm,113,22) \tyoung(0cm,7cm,112,23) \end{tikzpicture}$$ Now we have the following lemma. \[semidom\] Suppose $\mu$ is a partition of $n$ and $\la$ a composition of $n$, and $T\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$. If we write $${\hat\Theta_{T}}=\sum_{S\in{\calt_{\hspace{-2pt}0}}(\mu,\la)}a_S{\hat\Theta_{S}},$$ then $a_S\neq0$ only if $S\dom T$. For this proof, we adopt the set-up of [@j2 §13]. If we fix a $\mu$-tableau $t$ (of type $(1^n)$), then $t$ determines a natural bijection between the set of $\la$-tabloids and the set of $\mu$-tableaux of type $\la$. We identify tabloids with tableaux according to this bijection. We also let $e_t$ denote the polytabloid indexed by $t$, which generates $S^\mu$. Given $\la$-tableaux $U,V$, we write $U{\stackrel{\text{row}}\longleftrightarrow}V$ if $V$ can be obtained from $U$ by permuting entries within rows, and we define $U{\stackrel{\text{column}}\longleftrightarrow}V$ similarly. Now given $U\in{\calt_{\hspace{-2pt}\operatorname{r}}}(\mu,\la)$, we have $${\hat\Theta_{U}}(e_t)=\sum_{(V,W)}W,$$ where we sum over all pairs $(V,W)$ of $\la$-tableaux such that $U{\stackrel{\text{row}}\longleftrightarrow}V{\stackrel{\text{column}}\longleftrightarrow}W$ and $V$ has distinct entries in each column. (In general there are also signs determined by the column permutations, but in characteristic $2$ we can neglect these.) Now if $S$ is a semistandard tableau with $U{\stackrel{\text{row}}\longleftrightarrow}V{\stackrel{\text{column}}\longleftrightarrow}S$ for some $V$, then (since the entries in each column of $S$ are increasing) we must have $S\dom U$. On the other hand, if $U$ is semistandard, then the only tableau $V$ such that $U{\stackrel{\text{row}}\longleftrightarrow}V{\stackrel{\text{column}}\longleftrightarrow}U$ is $U$ itself. Hence for any semistandard $U$, the coefficient of $U$ in ${\hat\Theta_{U}}(e_t)$ is $1$. Now suppose $S_0\in{\calt_{\hspace{-2pt}0}}(\mu,\la)$ is such that $a_{S_0}\neq0$ and $S_0$ is minimal (with respect to $\dom$) with this property. Then the coefficient of $S_0$ in $${\hat\Theta_{T}}(e_t)=\sum_{S\in{\calt_{\hspace{-2pt}0}}(\mu,\la)}a_S{\hat\Theta_{S}}(e_t)$$ is $a_{S_0}$. So the coefficient of $S_0$ in ${\hat\Theta_{T}}(e_t)$ is non-zero, and hence $S_0\dom T$. Any $S\in{\calt_{\hspace{-2pt}0}}(\mu,\la)$ for which $a_S\neq0$ dominates some such minimal tableau $S_0$, and so dominates $T$. Using the results in this section, it will be possible to compute ${\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)$ in the cases of interest to us. We remark that it is often easier to express such homomorphisms as linear combinations of non-semistandard homomorphisms; in particular, the conditions ${\psi_{d,t}}\circ\theta=0$ can be easier to check. Of course, when doing this we have to be careful to show that the homomorphisms we construct are non-zero. Composition of tableau homomorphisms ------------------------------------ It will also be important in this paper to compute compositions of homomorphisms between Specht modules. It is well understood how to compose two tableau homomorphisms; indeed, computing this composition is the same as computing the structure constants for the Schur algebra. We give this result, of which Lemma \[lemma5\] is a special case. This result is easy to prove and well known (indeed, ‘quantised’ versions appear in the literature) but it does not seem to appear explicitly. However, translating to the language of the Schur algebra (where $\Theta_T$ corresponds to a basis element $\xi_{i,j}$) it amounts to the Multiplication rule (2.3b) in Green’s monograph [@gr]. Recall that if $S$ is a tableau, then $S^j$ denotes the multiset of entries in row $j$ of $S$, and in particular $S^j_i$ denotes the number of entries equal to $i$ in row $j$ of $S$. If $x_1,x_2,\dots$ are non-negative integers with finite sum $x$, we write $\mbinom{x}{x_1,x_2,\dots}$ for the corresponding multinomial coefficient. \[tabcomp\] Suppose $\la,\mu,\nu$ are compositions of $n$, $S$ is a $\la$-tableau of type $\mu$ and $T$ is a $\mu$-tableau of type $\nu$. Let $\calx$ be the set of all collections $X=(X^{ij})_{i,j\gs1}$ of multisets such that $$|X^{ij}|=S^j_i\quad\text{ for each $i$, $j$,}\qquad\bigsqcup_{j\gs1}X^{ij}=T^i\quad\text{ for each $i$.}$$ For $X\in\calx$, let $U_X$ denote the row-standard $\la$-tableau with $(U_X)^j=\bigsqcup_{i\gs1}X^{ij}$. Then $$\Theta_T\circ\Theta_S = \sum_{X\in\calx}\prod_{i,j\gs1}\binom{X^{1j}_i+X^{2j}_i+X^{3j}_i+\dots}{X^{1j}_i,X^{2j}_i,X^{3j}_i,\dots}\Theta_{U_X}.$$ Irreducible summands of the form $S^{(u,v)}$ {#uvsec} ============================================ In this section, we find all cases where one of our Specht modules $S^{(a,3,1^b)}$ has a summand isomorphic to an irreducible Specht module of the form $S^{(u,v)}$, where $u$ is even and $v$ is odd. Throughout, we continue to assume that $a,b$ are positive even integers with $a\gs4$, and we let $n=a+b+3$. By Theorem \[jreg\] and Lemma \[reg\], $D^{(u,v)}$ cannot appear as a composition factor of $S^{(a,3,1^b)}$ unless $(u,v)\dom(a,3,1^b){^{\operatorname{reg}}}$, which is the partition $(\max\{a,b+2\},\min\{a-1,b+1\},2)$. So we may assume that this is the case, which is the same as saying $v\ls\min\{a+1,b+3\}$. For easy reference, we set out notation and assumptions for this section. **Assumptions and notation in force throughout Section \[uvsec\]:** $\la=(a,3,1^b)$ and $\mu=(u,v)$, where $a,b,u,v$ are positive integers with $a,b,u$ even, $a\gs4$, $u>v$, $n=a+b+3=u+v$ and $v\ls\min\{a+1,b+3\}$. Homomorphisms from $S^\la$ to $S^{\mu'}$ {#hlamu'1} ---------------------------------------- In this subsection we consider $\bbf{\mathfrak{S}_}n$-homomorphisms from $S^\la$ to $S^{\mu'}$. We begin by constructing such a homomorphism in the case where $3\ls v\ls a-1$. Let ${\calu}$ be the set of $\la$-tableaux having the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;\star;\star,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star)$$ in which the $\star$s represent the numbers from $2$ to $u$, and in which - the entries along each row are strictly increasing, - the entries down each column are weakly increasing. Now define $$\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}.$$ \[sigmahom\] With the assumptions and notation above, we have ${\psi_{d,t}}\circ\sigma=0$ for each $d,t$. First take $v<d\ls u$ and $t=1$. If $T\in{\calu}$, then $T$ contains a single $d$ and a single $d+1$. If these occur in the same row or the same column of $T$, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$ by Lemma \[lemma5\] and Lemma \[lemma7\]. Otherwise, there is another tableau $T'\in{\calu}$ obtained by interchanging the $d$ and the $d+1$. By Lemma \[lemma5\] we have ${\psi_{d,1}}\circ({\hat\Theta_{T}}+{\hat\Theta_{T'}})=0$. Hence by summing ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ over all $T\in{\calu}$, we get zero. A similar argument applies in the case $d=v$. If $1\ls d<v$ and $t=2$, then we have ${\psi_{d,t}}\circ{\hat\Theta_{T}}=0$ for each $T\in{\calu}$ just using Lemma \[lemma5\]. Now take $2\ls d<v$ and $t=1$, and consider a tableau $T\in{\calu}$. There are a single $d$ and a single $d+1$ below row $1$. If these lie in the same row or column, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$. Otherwise, let $T'$ be the tableau obtained by interchanging the $d$ and the $d+1$ below row $1$. Then ${\psi_{d,1}}\circ({\hat\Theta_{T}}+{\hat\Theta_{T'}})$, and we are done. We are left with the case $d=t=1$. Applying Lemma \[lemma5\], we find that ${\psi_{1,1}}\circ\theta$ is the sum of homomorphisms labelled by tableaux $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;\star;\star,;1,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star)$$ in which the $\star$s now represent the numbers from $3$ to $u$, and where the entries are strictly increasing along rows and weakly increasing down columns. Now we apply Lemma \[lemma7\] to each of these homomorphisms to move the $1$ from row $3$ to row $2$, and then to reorder rows $3,\dots,b+2$. We obtain a sum of tableaux of the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;\star,;\star,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;\star),$$ but each tableau occurs $b$ times in this way. Since $b$ is even, we have zero. Now we need to check that $\sigma\neq0$, which is not obvious because the tableaux involved are not semistandard. \[sigmanz\] With the notation above, $\sigma\neq0$. The version of this paper published in the Journal of Algebra includes a fallacious proof of Proposition \[sigmanz\]; the proof below replaces it. The authors are grateful to Sinéad Lyle for pointing out the error. We’ll use Lemma \[semidom\]. Consider the semistandard tableau $$S={\text{\footnotesize$\gyoungx(1.2,;1;1;2_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;{b\!\!+\!\!3};{b\!\!+\!\!4},;3,;4,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}.$$ We’ll show that when $\sigma$ is expressed as a linear combination of semistandard homomorphisms, ${\hat\Theta_{S}}$ occurs with non-zero coefficient, and hence $\sigma\neq0$. Given $T\in{\calu}$, consider expressing ${\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms. By Lemma \[semidom\], ${\hat\Theta_{S}}$ can only occur if $S\dom T$; so we can ignore all $T\in{\calu}$ for which $S\ndom T$. In particular, we need only consider those tableaux in ${\calu}$ which have $b+5,\dots,u$ in the first row and $b+3,b+4$ in the top two rows. If we assume for the moment that $v<b+3$, then the tableaux $T\in{\calu}$ that we need to consider are those of the following forms: [$$\begin{aligned} T[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $v<i\ls b+2$;}\\ U[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;i;{b\!\!+\!\!4},;2,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{\hat\imath},|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls i\ls b+2$;}\\ V[i]&={\text{\footnotesize$\gyoungx(1.2,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!4};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;1;i;{b\!\!+\!\!3},;2,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{\hat\imath},|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls i\ls b+2$.}\end{aligned}$$]{} As usual, the ${\hat\imath}$ in the first column indicates that $i$ does not appear in that column. First consider the tableau $T[i]$, and apply Lemma \[lemma7\] to move the $1$ from row $2$ to row $1$. Of the tableaux appearing in the resulting expression, the only ones dominated by $S$ are [$$\begin{aligned} T'[i]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;i;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\\ \intertext{and the tableaux} T'[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;j;{b\!\!+\!\!3};{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls j\ls v$.}\\ \intertext{Applying Lemma \ref{lemma7} to $T'[i]$ to move the $2$ from row $3$ to row $2$, we obtain three tableaux, but two of these are not dominated by $S$. The other one is} T''[i]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;{b\!\!+\!\!3};{b\!\!+\!\!4},;i,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}},\end{aligned}$$]{} and $i-3$ more applications of Lemma \[lemma7\] show that ${\hat\Theta_{T''[i]}}={\hat\Theta_{S}}$. Now consider applying Lemma \[lemma7\] to $T'[i,j]$, to move the $2$ from row $3$ to row $2$. If $j=2$, then neither of the two tableaux obtained is dominated by $S$. If $j>2$, then two of the three tableaux obtained are not dominated by $S$; the third has rows $3$ and $j+1$ both equal to $\young(j)$, so the resulting homomorphism is zero by Lemma \[lemma7\]. So we conclude that ${\hat\Theta_{T[i]}}$ equals ${\hat\Theta_{S}}$ plus a linear combination of homomorphisms indexed by tableaux not dominated by $S$.\ Now we consider applying Lemma \[lemma7\] to $U[i]$, moving the $1$ up from row $2$. The tableaux obtained that are dominated by $S$ are $T'[i]$ and the tableaux [$$\begin{aligned} U'[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;j;i;{b\!\!+\!\!4},;2,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}\tag*{for $2\ls j\ls v$ with $i\neq j$.}\\ \intertext{\parbox{\linewidth}{(Note that if $i<j$ then $i$ and $j$ in the second row should be written the other way round; the case $i=j$ does not occur because the accompanying coefficient would be $\binom21=0$.)\\\hspace*{17pt}If $i=2$, then $U'[i,j]$ is a semistandard tableau different from $S$. If $i>2$ then we apply Lemma \ref{lemma7} to move the $2$ from row $3$ to row $2$; neglecting the tableau not dominated by $S$ and (in the case $j>2$) neglecting the tableau with two rows equal to $\young(j)$, the only tableau we get is}} U''[i,j]&={\text{\footnotesize$\gyoungx(1.2,;1;1;2;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3};{b\!\!+\!\!5};{b\!\!+\!\!6}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};u,;2;j;{b\!\!+\!\!4},;i,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}};\end{aligned}$$]{} $i-3$ more applications of Lemma \[lemma7\] show that ${\hat\Theta_{U''[i,j]}}$ equals a semistandard homomorphism different from ${\hat\Theta_{S}}$. We conclude that ${\hat\Theta_{U[i]}}$ equals ${\hat\Theta_{S}}$ plus a linear combination of homomorphisms indexed by tableaux which are either not dominated by $S$ or semistandard and different from $S$. The homomorphism ${\hat\Theta_{V[i]}}$ is analysed in exactly the same way, interchanging $b+3$ and $b+4$. Putting these cases together, we find that the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is the total number of tableaux of the form $T[i]$, $U[i]$ or $V[i]$, i.e. $(b+2-v)+2(b+1)$, which is odd. It remains to consider the case $v=b+3$. In this case only the tableaux $V[i]$ appear, but the analysis of these tableaux is exactly the same, so the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is the number of tableaux $V[i]$, i.e. $b+1$, which again is odd. It turns out that up to scaling, $\sigma$ is the only homomorphism from $S^\la$ to $S^\mu$. \[cdhomdim1\] With $\la,\mu$ as above, $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})= \begin{cases} 0&(v=1\text{ or }v=a+1)\\ 1&(3\ls v\ls a-1). \end{cases}$$ The construction of the homomorphism $\sigma$ above shows that the dimension of the homomorphism space is at least that claimed. So we just have to show the reverse inequality. By Lemma \[815hom\], we have $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})=\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la'}),$$ and we can use the technique outlined in §\[homsec\] to compute the right-hand side, since $\mu$ is $2$-regular. So suppose $\theta$ is a linear combination of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to M^{\la'}$ such that ${\psi_{d,t}}\circ\theta=0$ for all $d,t$. To begin with, we consider ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ for each $T$. Using Lemma \[lemma5\], this equals zero if $T$ has a $2$ in each row, because the homomorphisms occurring in Lemma \[lemma5\] each appear with a coefficient $\binom21$, which is zero in $\bbf$. Otherwise, ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ is either a single semistandard homomorphism or a sum of two semistandard homomorphisms. Moreover, the semistandard tableaux that occur for the various $T$ are distinct. Hence in order to have ${\psi_{2,1}}\circ\theta=0$, $\theta$ can only involve semistandard homomorphisms ${\hat\Theta_{T}}$ for those $T$ having a $2$ in each row. In particular, $\theta=0$ when $v=a+1$, since in this case there is only one semistandard tableau, whose first row consists entirely of $1$s. Now we consider ${\psi_{2,2}}\circ{\hat\Theta_{T}}$ for each of these $T$. If $T$ has a $2$ and a $3$ in each row, we get ${\psi_{2,2}}\circ{\hat\Theta_{T}}=0$, while if $T$ has a $2$ in each row and two $3$s in the same row, ${\psi_{2,2}}\circ{\hat\Theta_{T}}$ is a semistandard homomorphism. Again, all the semistandards that occur in this way are different, so $\theta$ cannot involve any tableaux of the latter type. In particular, if $v=1$ then $\theta=0$. Next consider ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ where $4\ls d<a$ and $T$ is a semistandard tableau having a $2$ and a $3$ in each row. $T$ contains a single $d$ and a single $d+1$. If these both lie in the same row of $T$, then ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$. Otherwise, ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is a semistandard homomorphism ${\hat\Theta_{T'}}$. If $U$ is another semistandard tableau and ${\psi_{d,1}}\circ{\hat\Theta_{U}}$ is a semistandard homomorphism ${\hat\Theta_{U'}}$, then $T'=U'$ if and only if $U$ is obtained from $T$ by interchanging $d$ and $d+1$; hence any two such tableaux must occur in $\theta$ with equal coefficients. Applying this for all $d\gs4$ and all $T$, we find that $\theta$ must be a scalar multiple of the sum of all semistandard homomorphisms ${\hat\Theta_{T}}$ for $T$ having a $2$ and a $3$ in each row. Hence $\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)\ls1$, and we are done. We provide an example to illustrate the above proof for the benefit of the reader who may not be familiar with this technique. We take $(a,b,u,v)=(4,6,8,5)$. (In fact, the Specht module $S^{(8,5)}$ is reducible, so is ultimately irrelevant to our main theorem, but it serves well for this example.) We suppose we have a linear combination $\theta$ of semistandard homomorphisms such that ${\psi_{d,t}}\circ\theta=0$ for all $d,t$. For this example, we abuse notation by identifying a tableau with the corresponding homomorphism. The first step of the proof is to eliminate most of the possible semistandard homomorphisms by taking $d=2$, $t=1$. For example, Lemma \[lemma5\] gives $${\psi_{2,1}}\circ\,\young(11113346,22578) \,=\, \young(11112346,22578),$$ and no other semistandard tableau can give the semistandard tableau on the right in this way with non-zero coefficient; note that the tableau $$\young(11112346,23578)$$ does give this tableau, but with a coefficient of $\binom21=0$. So since ${\psi_{2,1}}\circ\theta=0$, our initial tableau cannot possibly occur in $\theta$. Arguing in this way, one finds that the only semistandard tableaux which can occur in $\theta$ are those with a $2$ in each row, i.e. those of the form $$\young(1111233\star,2\star\star\star\star),\qquad \young(111123\star\star,23\star\star\star)\quad\text{or}\quad \young(11112\star\star\star,233\star\star).$$ Now the first and last of these three types can be ruled out using the same argument with ${\psi_{2,2}}$. So $\theta$ can only involve tableaux with a $2$ and a $3$ in each row; call these *usable* tableaux. Now we consider ${\psi_{d,1}}\circ\theta$ for $d\gs4$. Now for each usable tableau $T$, ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero (if $d$ and $d+1$ occur in the same row in $T$) or a semistandard homomorphism. Furthermore, these semistandard homomorphisms ‘pair up’; for example, with $d=4$ we have $${\psi_{4,1}}\circ\,\young(11112356,23478)\,={\psi_{4,1}}\circ\, \young(11112346,23578)\,=\, \young(11112346,23478).$$ Since the semistandard tableau on the right can only arise in this way from the two semistandard tableaux on the left, these two semistandard homomorphisms must occur with equal coefficients in $\theta$. Now we observe that we can get from any usable tableau to any other by a sequence of steps in which we interchange the integers $d,d+1$ for various values of $d\gs4$. So if we apply the above argument for all $d\gs4$, we see that all usable tableaux occur with the same coefficient in $\theta$. Homomorphisms from $S^\mu$ to $S^\la$ ------------------------------------- Now we consider homomorphisms from $S^\mu$ to $S^\la$, where $\la,\mu$ are as above. In view of Proposition \[cdhomdim1\], *we assume for the rest of this section that $3\ls v\ls a-1$*. It turns out that all such homomorphisms can be expressed as linear combinations of ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$, where $A,B$ are the following $\mu$-tableaux of type $\la$: $$\begin{aligned} A&= {\text{\footnotesize$\gyoungx(1.2,;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;2;2;2;3;4_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;1;1)$}};\\ B&= {\text{\footnotesize$\gyoungx(1.2,;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;1;1;2;3;4_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};1;2;2)$}}.\end{aligned}$$ Note that our assumptions on the parameters $a,b,u,v$ mean that these tableaux really do exist, i.e. there are enough $1$s to fill the bottom row. \[abnz\] ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ are non-zero, and are linearly independent if $v\ls b+1$. It is straightforward to express ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ as linear combinations of semistandard homomorphisms using a single application of Lemma \[lemma7\]; in each case we get at least one semistandard appearing, so the homomorphisms are non-zero. If in addition $v\ls b+1$, then in the expression for ${\hat\Theta_{A}}$ there is at least one semistandard tableau with two $2$s in the first row; there is no such tableau appearing in the expression for ${\hat\Theta_{B}}$, so ${\hat\Theta_{A}},{\hat\Theta_{B}}$ are linearly independent. \[abhoms\] - If $a-v\equiv3\ppmod4$ or $v=b+3$, then ${\psi_{d,t}}\circ{\hat\Theta_{A}}=0$ for all admissible $d,t$. - If $v\equiv1\ppmod4$, then ${\psi_{d,t}}\circ{\hat\Theta_{B}}=0$ for all admissible $d,t$. - If $a\equiv0\ppmod4$, then ${\psi_{d,t}}\circ({\hat\Theta_{A}}+{\hat\Theta_{B}})=0$ for all admissible $d,t$. Lemma \[lemma5\] immediately gives ${\psi_{d,1}}\circ{\hat\Theta_{A}}={\psi_{d,1}}\circ{\hat\Theta_{B}}=0$ for $d\gs2$. Using the fact that $A,B$ each have an odd number of $1$s in each row, we also get $${\psi_{1,t}}\circ{\hat\Theta_{A}}={\psi_{1,t}}\circ{\hat\Theta_{B}}=0$$ for $t=1,3$. Finally, we have ${\psi_{1,2}}\circ{\hat\Theta_{B}}=0$ if $v\equiv1\ppmod 4$, and ${\psi_{1,2}}\circ{\hat\Theta_{A}}=0$ if $a-v\equiv3\ppmod4$ or $v=b+3$ (where we apply Lemma \[lemma7\] in the latter case), and ${\psi_{1,2}}\circ{\hat\Theta_{A}}={\psi_{1,2}}\circ{\hat\Theta_{B}}$ if $a\equiv0\ppmod4$. \[muladim\] $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la})= \begin{cases} 2&(\text{if }a\equiv0\ppmod4,\ v\equiv1\ppmod4\text{ and }v\ls b+1)\\ 1&(\text{otherwise}). \end{cases}$$ By Lemma \[abnz\] and Proposition \[abhoms\] the dimension of the homomorphism space is at least that claimed. Now we show the reverse inequality, by considering linear combinations of semistandard homomorphisms. Throughout this proof, we’ll write ${\calt[i]}$ for the set of semistandard $\mu$-tableaux of type $\la$ having exactly $i$ $2$s in the first row, for $i=0,1,2,3$, and let $\tau_i=\sum_{T\in{\calt[i]}}{\hat\Theta_{T}}$. Suppose we have a linear combination $\theta$ of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to M^\la$ such that ${\psi_{d,t}}\circ\theta=0$ for all applicable $d,t$. First consider ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ for $T\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$ and $d\gs3$. By Lemma \[lemma5\], ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a semistandard homomorphism (according to whether the $d$ and the $d+1$ in $T$ occur in the same row). If it is non-zero, then there is exactly one other $T'\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$ such that ${\psi_{d,1}}\circ{\hat\Theta_{T}}={\psi_{d,1}}\circ{\hat\Theta_{T'}}$, namely the tableau obtained by interchanging the $d$ and the $d+1$ in $T$. Hence ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ must occur with the same coefficient in $\theta$. Applying this for all $d\gs 3$, we find that for a fixed $i\in\{0,1,2,3\}$, all the homomorphisms ${\hat\Theta_{T}}$ for $T\in{\calt[i]}$ occur with the same coefficient in $\theta$. In other words, $\theta$ is a linear combination of $\tau_0,\tau_1,\tau_2,\tau_3$. We can apply a similar argument in which we consider ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ for $T\in{{\calt_{\hspace{-2pt}0}}(\mu,\la)}$. Again ${\psi_{3,1}}\circ{\hat\Theta_{T}}$ is either zero or a semistandard homomorphism; and if it is non-zero, then the only other $T'$ having ${\psi_{3,1}}\circ{\hat\Theta_{T'}}={\psi_{3,1}}\circ{\hat\Theta_{T}}$ is the tableau obtained by exchanging the $3$ in $T$ with a $2$ in the other row. ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ occur with the same coefficient in $\theta$, and we deduce that $\theta$ must be a linear combination of $\tau_0+\tau_1$ and $\tau_2+\tau_3$. Finally, we consider ${\psi_{1,2}}\circ\theta$. Each $\mu$-tableau $T$ of type $(a+2,1^{b+1})$ contains a single $2$; let $\phi$ denote the sum of ${\hat\Theta_{T}}$ for all those $T$ having the $2$ in row $1$, and $\chi$ the sum of all ${\hat\Theta_{T}}$ for $T$ having the $2$ in row $2$. Using Lemma \[lemma5\] and Lemma \[lemma7\] (and recalling that $a$ is even and $v$ is odd), we have $$\begin{aligned} {\psi_{1,2}}\circ\tau_0&=\mbinom{v-1}2\chi,\\ {\psi_{1,2}}\circ\tau_1&=\mbinom v2\phi,\\ {\psi_{1,2}}\circ\tau_2&=\left(\mbinom{a+2}2+1\right)\phi+\chi,\\ {\psi_{1,2}}\circ\tau_3&=\mbinom{a+2}2\phi.\end{aligned}$$ So if $v\equiv 3\ppmod4$, then ${\psi_{1,2}}\circ(\tau_0+\tau_1)\neq0$, so $\theta$ cannot equal $\tau_0+\tau_1$. If $a\equiv2\ppmod4$, then ${\psi_{1,2}}\circ(\tau_2+\tau_3)\neq0$, so $\theta$ cannot be $\tau_2+\tau_3$. Hence $\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)\ls1$ in these cases. We also have $\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)\ls1$ in the case where $b=d-3$, since in this case ${\calt[2]}$ and ${\calt[3]}$ are empty, so $\tau_2+\tau_3=0$. Composing the homomorphisms --------------------------- Now we complete the analysis of when $S^\mu$ is a summand of $S^\la$, by composing the homomorphisms from the preceding subsections. This will be straightforward, using Proposition \[tabcomp\]. Recall that the space of homomorphisms from $S^\la$ to $S^{\mu'}$ is one-dimensional, spanned by the homomorphism $\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}$. On the other hand, the space of homomorphisms from $S^\mu$ to $S^\la$ has dimension one or two, each homomorphism being a linear combination of the homomorphisms ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$. So it suffices to compute the compositions $\sigma\circ{\hat\Theta_{A}}$ and $\sigma\circ{\hat\Theta_{B}}$. Let $D$ be the $\mu$-tableau $$D=\gyoung(;1;2;3_4{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(4*.25,0);\end{tikzpicture}}};u,;1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};v)$$ of type $\mu'$. Then we have the following. \[uab\] Suppose $T\in{\calu}$, and let $x$ be the entry in the $(2,2)$-position of $T$. Then $${\hat\Theta_{T}}\circ{\hat\Theta_{A}}={\hat\Theta_{D}}, \qquad {\hat\Theta_{T}}\circ{\hat\Theta_{B}}=\begin{cases} {\hat\Theta_{D}}&(x\ls v)\\ 0&(x>v). \end{cases}$$ Furthermore, ${\hat\Theta_{D}}\neq0$. The fact that ${\hat\Theta_{D}}\neq0$ is a simple application of Lemma \[lemma7\]. To show that the compositions of homomorphisms are as claimed, take $T\in{\calu}$ and recall the notation of Proposition \[tabcomp\], with $S$ equal to either $A$ or $B$. Suppose $X\in\calx$. Since each $T^i$ is a proper set, each $X^{ij}$ must be as well. This means that if some integer $i$ appears in two sets $X^{kj},X^{lj}$, then the multinomial coefficient $\mbinom{X^{1j}_j+X^{2j}_j+X^{3j}_j+\dots}{X^{1j}_j,X^{2j}_j,X^{3j}_j,\dots}$ from Proposition \[tabcomp\] will include a factor $\binom21$, which gives $0$. So in order to get a non-zero coefficient in Proposition \[tabcomp\], we must have $X^{1j},X^{2j},X^{3j},\dots$ pairwise disjoint for each $j$, which means that we will have $$X^{11}\sqcup X^{21}\sqcup\dots=\{1,\dots,u\},\qquad X^{12}\sqcup X^{22}=\{1,\dots,v\};\tag*{($\dagger$)}$$ so $U_X$ will equal $D$. If $S=A$, the only way to achieve this is to have $$X^{11}=T^1\setminus\{1,\dots,v\},\quad X^{12}=\{1,\dots,v\},\quad X^{i1}=T^i\text{ for }i\gs2.$$ Thus we have ${\hat\Theta_{T}}\circ{\hat\Theta_{A}}={\hat\Theta_{D}}$. In the case $S=B$, let $y$ be the $(2,3)$-entry of $T$. Then $y>x$. $X^{22}$ must contain either $x$ or $y$, so if $x>v$ then we cannot possibly achieve ($\dagger$). So we get ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}=0$ in this case. If $x\ls v<y$, then the only way to achieve ($\dagger$) is to have $X^{22}=\{1,x\}$ and $X^{12}=\{2,\dots,\hat x,\dots,v\}$, and this yields ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}={\hat\Theta_{D}}$. Finally, if $y\ls v$, then there are three possible ways to achieve ($\dagger$); each of these gives a coefficient of $1$, and again we have ${\hat\Theta_{T}}\circ{\hat\Theta_{B}}={\hat\Theta_{D}}$. This result is very helpful: it tells us that the composition of $\sigma$ with a combination of ${\hat\Theta_{A}}$ and ${\hat\Theta_{B}}$ is a scalar multiple of ${\hat\Theta_{D}}$; hence this composition is non-zero if and only if this scalar is non-zero. In order to use this result, we need to find the number of tableaux in ${\calu}$, and also the number of tableaux in ${\calu}$ in which the $(2,2)$-entry is at most $v$. This is a straightforward count. \[countu\] - The number of tableaux in ${\calu}$ is $\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2$. - The number of tableaux in ${\calu}$ whose $(2,2)$-entry is greater than $v$ is $\mbinom{u-v}{a-v}\mbinom{u-a}2$. Suppose $S^\mu=S^{(u,v)}$ is irreducible, with $u+v=a+b+3$. Throughout this proof, all congruences are modulo $4$. Suppose first that $u,v$ satisfy the given conditions, i.e. $v\equiv3$ and $\mbinom{u-v}{a-v}$ is odd. The second condition implies in particular that $0\ls a-v\ls u-v$, which gives $v\ls\min\{a-1,b+3\}$; so the assumptions in force in this section are satisfied. In addition, Theorem \[irrspecht\] gives $u\equiv2$. We need to show that there are homomorphisms $S^\mu\stackrel\gamma\longrightarrow S^\la\stackrel\delta\longrightarrow S^{\mu'}$ such that $\delta\circ\gamma\neq0$. Since $3\ls v\ls a-1$, we can take $\delta=\sigma$. If $a\equiv0$, take $\gamma={\hat\Theta_{A}}+{\hat\Theta_{B}}$. By Proposition \[abhoms\], $\gamma$ is a homomorphism from $S^\mu$ to $S^\la$. By Lemma \[uab\] and Lemma \[countu\], $$\delta\circ\gamma=\mbinom{u-v}{a-v}\mbinom{u-a}2{\hat\Theta_{D}}.$$ The first term is odd by assumption; the second term is odd because $u-a\equiv2$, and ${\hat\Theta_{D}}\neq0$ by Lemma \[uab\]. If $a\equiv2$, take $\gamma={\hat\Theta_{A}}$. Then $\gamma$ is a homomorphism from $S^\mu$ to $S^\la$, and $$\delta\circ\gamma=\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2{\hat\Theta_{D}}.$$ Again, the first term is odd by assumption, the second term is odd because now $u+v-a-1\equiv2$, and ${\hat\Theta_{D}}\neq0$. Conversely, suppose we have homomorphisms $\gamma,\delta$ such that $\delta\circ\gamma\neq0$. By Proposition \[cdhomdim1\] we can assume that $3\ls v\ls a-1$ and take $\delta=\sigma$. From Proposition \[muladim\] we can take $\gamma$ to be ${\hat\Theta_{A}}$, ${\hat\Theta_{B}}$ or ${\hat\Theta_{A}}+{\hat\Theta_{B}}$, according to the congruences in Proposition \[abhoms\]. Then $\delta\circ\gamma$ will be a scalar multiple of ${\hat\Theta_{D}}$, and the scalar will include $\mbinom{u-v}{u-a}$ as a factor. So this binomial coefficient must be odd, and all that remains is to show that $v\equiv3\ppmod4$. We consider the three cases of Proposition \[abhoms\]. Note that because $v>1$, Theorem \[irrspecht\] gives $u-v\equiv3$. : In this case the coefficient of ${\hat\Theta_{D}}$ in $\delta\circ\gamma$ is $$\mbinom{u-v}{a-v}\mbinom{u+v-a-1}2.$$ The second binomial coefficient must be odd, so $u+v-a\equiv3$. In the case $a-v\equiv3$, this is the same as saying $u\equiv2$, so that $v\equiv3$. In the case $v=b+3$, we have $a=u$, so that again $v\equiv3$. : Now the coefficient of ${\hat\Theta_{D}}$ in $\delta\circ\gamma$ is $$\mbinom{u-v}{a-v}\mbinom{u-a}2.$$ The second binomial coefficient is odd only if $u\equiv a+2$, which is the same as saying $v\equiv 3$. : In this case the coefficient of ${\hat\Theta_{D}}$ in $\delta\circ\gamma$ is $$\mbinom{u-v}{a-v}\left(\mbinom{u+v-a-1}2+\mbinom{u-a}2\right).$$ Since $v\equiv1$, we have $u+v-a-1\equiv u-a$, so that $\mbinom{u+v-a-1}2$ and $\mbinom{u-a}2$ have the same parity. Hence $\delta\circ\gamma=0$, a contradiction. Irreducible summands of the form $S^{(u,v,2)}$ {#uv2sec} ============================================== In this section, we find when one of our Specht modules $S^{(a,3,1^b)}$ has a summand isomorphic to an irreducible Specht module of the form $S^{(u,v,2)}$, where $u$ is even and $v$ is odd. By Theorem \[jreg\] and Lemma \[reg\], $D^{(u,v,2)}$ cannot appear as a composition factor of $S^{(a,3,1^b)}$ unless $(u,v,2)\dom(a,3,1^b){^{\operatorname{reg}}}$. So we may assume that this is the case, which is the same as saying $v\ls\min\{a-1,b+1\}$. We set out notation and assumptions for this section. **Assumptions and notation in force throughout Section \[uv2sec\]:** $\la=(a,3,1^b)$ and $\mu=(u,v,2)$, where $a,b,u,v$ are positive integers with $a,b,u$ even, $a\gs4$, $u>v>2$, $n=a+b+3=u+v+2$ and $v\ls\min\{a-1,b+1\}$. Homomorphisms from $S^\la$ to $S^{\mu'}$ {#homomorphisms-from-sla-to-smu} ---------------------------------------- We begin by constructing a homomorphism from $S^\la$ to $S^{\mu'}$. As in §\[hlamu’1\], we construct this using non-semistandard tableaux. Let ${\calu}$ be the set of $\la$-tableaux having the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;2,;2,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;\star)$$ in which the $\star$s represent the numbers from $v+1$ to $u$, and in which the entries are weakly increasing along the first row and down the first column. Let $\sigma=\sum_{T\in{\calu}}{\hat\Theta_{T}}$. \[sigmahom2\] With the notation and assumptions above, we have ${\psi_{d,t}}\circ\sigma=0$ for all $d,t$. For $d\gs v$ and $t=1$, we use the same argument as that used in several proofs above: for $T\in{\calu}$ either ${\psi_{d,1}}\circ{\hat\Theta_{T}}=0$, or there is a unique other $T'\in{\calu}$ with ${\psi_{d,1}}\circ{\hat\Theta_{T'}}={\psi_{d,1}}\circ{\hat\Theta_{T}}$. The cases where $2\ls d\ls v$ are easier: in this case Lemma \[lemma5\] and Lemma \[lemma7\] imply that we have ${\psi_{d,t}}\circ{\hat\Theta_{T}}=0$ for all $T\in{\calu}$. So we are left with the cases where $d=1$ and $t\in\{1,2,3\}$. For $T\in {\calu}$ we have ${\psi_{1,3}}\circ{\hat\Theta_{T}}$ immediately from Lemma \[lemma5\], while ${\psi_{1,2}}\circ{\hat\Theta_{T}}$ is a homomorphism labelled by a tableau of the form $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;1,;1,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;\star).$$ But this homomorphism is zero by Lemma \[lemma7\]. Finally, ${\psi_{1,1}}\circ{\hat\Theta_{T}}$ is the sum of the homomorphisms labelled by two tableaux $$\gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;1,;2,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;\star),\qquad \gyoung(;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;\star_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};\star,;1;1;2,;1,;3,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;v,;\star,|{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1.5*.125)--++(0,1.5*.25);\end{tikzpicture}}},;\star).$$ But these two homomorphisms are equal by Lemma \[lemma7\], and we are done. Now, as in Section \[hlamu’1\] we have to show that $\sigma\neq0$. Again, we use a dominance argument. \[sigmanz2\] With the notation above, $\sigma\neq0$. We’ll show that when $\sigma$ is expressed as a linear combination of semistandard homomorphisms, the homomorphism ${\hat\Theta_{S}}$ occurs with non-zero coefficient, where $$S= {\text{\footnotesize$\gyoungx(1.2,;1;1;1;2;4;5_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;2;2;3,;3,|2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-2*.125)--++(0,2*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}.$$ Recall that when ${\hat\Theta_{T}}$ is expressed as a linear combination of semistandard homomorphisms, the coefficient of ${\hat\Theta_{S}}$ is zero unless $S\dom T$. The only elements of ${\calu}$ which are dominated by $S$ are the tableaux of the form $$T[i]= {\text{\footnotesize$\gyoungx(1.2,;1;2;3;4_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!3}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;1;1;2,;2,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}$$ for $v+1\ls i\ls b+2$. Consider applying Lemma \[lemma7\] to $T[i]$, to move the two $1$s from row $2$ to row $1$. Of the tableaux appearing in that lemma with non-zero coefficient, the only ones dominated by $S$ are those having no more than four entries less than $4$ in the first row; these are the tableaux $T'[i]$ and $T'[i,j]$ for $4\ls j\ls v$, where $$\begin{aligned} T'[i]&= {\text{\footnotesize$\gyoungx(1.2,;1;1;1;2;4_{3.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3.5*.25,0);\end{tikzpicture}}};v;{b\!\!+\!\!3}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;2;3;i,;2,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}, \\ T'[i,j]&= {\text{\footnotesize$\gyoungx(1.2,;1;1;1;2;4;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{\hat\jmath};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};v;i;{b\!\!+\!\!3};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};u,;2;3;j,;2,;3,;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;v,;{v\!\!+\!\!1},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{\hat\imath},;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13-1*.125)--++(0,1*.25);\end{tikzpicture}}},;{b\!\!+\!\!2})$}}.\end{aligned}$$ So, modulo homomorphisms labelled by tableaux not dominated by $S$, we have $\sigma=\sum_i{\hat\Theta_{T'[i]}}+\sum_{i,j}{\hat\Theta_{T'[i,j]}}$. However, two applications of Lemma \[lemma7\] show that ${\hat\Theta_{T'[i,j]}}=0$ for all $i,j$, and Lemma \[lemma7\] also gives ${\hat\Theta_{T'[i]}}={\hat\Theta_{S}}$. So the coefficient of ${\hat\Theta_{S}}$ in $\sigma$ is $b+2-v$, which is odd; so $\sigma\neq0$. As before, we find that $\sigma$ is the only homomorphism from $S^\la$ to $S^{\mu'}$ up to scaling. \[cd2homdim1\] With $\la,\mu$ as above, $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})=1.$$ The existence of the homomorphism $\sigma$ shows that the space of homomorphisms is non-zero. To show that it has dimension at most $1$, we again use the fact that $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\la,S^{\mu'})=\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^{\la'}).$$ So suppose $\theta$ is a linear combination of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to S^{\la'}$ such that ${\psi_{d,t}}\circ\theta=0$ for all $d,t$. First of all, consider ${\psi_{2,1}}\circ\theta$. Given a semistandard tableau $T$, we can use Lemma \[lemma5\] to compute ${\psi_{2,1}}\circ{\hat\Theta_{T}}$, and then if necessary use Lemma \[lemma7\] (to move a $2$ from row $3$ to row $2$) to express this composition as a linear combination of semistandard homomorphisms. We find that if $T$ has two $2$s in its first row, then ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ involves a semistandard tableau which does not occur in any other ${\psi_{2,1}}\circ{\hat\Theta_{T'}}$; hence the coefficient of ${\hat\Theta_{T}}$ in $\theta$ must be zero. Now we do the same thing with ${\psi_{2,2}}$: in this case we find that if $T$ is a semistandard tableau having two $3$s in its first row, then ${\psi_{2,2}}\circ{\hat\Theta_{T}}$ involves a semistandard homomorphism which does not occur in any other ${\psi_{2,2}}\circ{\hat\Theta_{T'}}$ (except possibly for a tableau $T'$ already ruled out in the paragraph above). So we may restrict attention to those $T$ having at most one $2$ and one $3$ in the first row. Now return to ${\psi_{2,1}}\circ{\hat\Theta_{T}}$, for $T$ of the form $$\begin{aligned} &\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;{x_1}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{x_s},;2;{z_1};{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;3;k), \\ \intertext{where $x_1,\dots,x_s,z_1,\dots,z_t,k$ are the integers $4,\dots,a$ in some order. When we express ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms, we find that the homomorphism labelled by} &\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;{x_1}_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{x_s},;2;2;{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{z_1};k)\end{aligned}$$ occurs with non-zero coefficient; but this homomorphism does not occur in any other ${\psi_{2,1}}\circ{\hat\Theta_{T'}}$ (except for $T'$ having two $3$s in its first row). So for any $T$ of the above form, the coefficient of ${\hat\Theta_{T}}$ in $\theta$ must be zero. Now any semistandard tableau which contributes to $\theta$ must be of one of the following eight forms. $$\begin{aligned} 2 1.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;3;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;3;\star) &\qquad 2.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;2;2;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;3;\star) \\ 3.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;2;2;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;3;3) &\qquad 4.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;2;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;3;\star) \\ 5.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;2;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;\star;\star) &\qquad 6.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;2;2;3;3;\star;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};\star,;\star;\star) \\ 7.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;3;\star;{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};\star,;2;3;\star_3{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3*.25,0);\end{tikzpicture}}};\star,;\star;\star) &\qquad 8.\quad& \gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;3;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;3;\star)\end{aligned}$$ In each case, the $\star$s represent the numbers from $4$ to $a$. Note that in each of these tableaux, the entries $4,\dots,a$ must all occur in different columns (the assumption $v\ls b+1$ means that any column of length at least two has a $1$ at the top). So we can consider the homomorphisms ${\psi_{d,1}}$ for $d\gs3$, and repeat the argument used in the last paragraph of the proof of Proposition \[cdhomdim1\], to show that if $T,T'$ are two tableaux which have their $1$s, $2$s and $3$s in the same positions, then ${\hat\Theta_{T}}$ and ${\hat\Theta_{T'}}$ occur with the same coefficient in $\theta$. Hence $\theta$ is a linear combination of the homomorphisms $\tau_1,\dots,\tau_8$, where $\tau_i$ is the sum of all homomorphisms ${\hat\Theta_{T}}$ for $T$ of type $i$. Once more we can consider ${\psi_{2,1}}\circ\theta$: when we compute ${\psi_{2,1}}\circ\tau_5$, we obtain (in addition to some other semistandard tableaux) the sum of the semistandard tableaux of the form $$\gyoung(;1;1;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;3;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;2;2;2;\star_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};\star,;\star;\star),$$ which do not occur in any other ${\psi_{2,1}}\circ\tau_i$ (note these tableaux do occur when we compute ${\psi_{2,1}}\circ{\hat\Theta_{T}}$ for $T$ of type $4$, but each one occurs twice when we sum over tableaux of type $4$, so the contributions cancel). So $\tau_5$ does not appear in $\theta$. Next we consider ${\psi_{3,1}}\circ\tau_i$ for each $i$. For $i=3,6$ or $8$, we find that ${\psi_{3,1}}\circ\tau_i$ involves semistandard tableaux which do not occur in any other ${\psi_{3,1}}\circ\tau_i$; so $\tau_3,\tau_6,\tau_8$ cannot occur in $\theta$. Moreover, we find that ${\psi_{3,1}}\circ\tau_1={\psi_{3,1}}\circ\tau_7$ and ${\psi_{3,1}}\circ\tau_2={\psi_{3,1}}\circ\tau_4$, and that these two homomorphisms are linearly independent. So $\theta$ must be a linear combination of $\tau_1+\tau_7$ and $\tau_2+\tau_4$. Finally we return once more to ${\psi_{2,1}}\circ\theta$. We find that ${\psi_{2,1}}\circ\tau_1={\psi_{2,1}}\circ\tau_4\neq0$, while ${\psi_{2,1}}\circ\tau_2={\psi_{2,1}}\circ\tau_7=0$. So $\tau_1$ and $\tau_4$ must appear with the same coefficient in $\theta$; so $\theta$ must be a scalar multiple of $\tau_1+\tau_2+\tau_4+\tau_7$, and so the homomorphism space has dimension at most $1$. Homomorphisms from $S^\mu$ to $S^\la$ ------------------------------------- Now we consider homomorphisms from $S^\mu$ to $S^\la$. We begin by constructing a non-zero homomorphism. Let $C$ be the $\mu$-tableau $${\text{\footnotesize$\gyoungx(1.2,;1;1_{3.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(3.5*.25,0);\end{tikzpicture}}};1;2;3_2{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(2*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;1;1_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};1,;2;2)$}}$$ of type $\la$. \[cd2chom\] With $C$ as above, we have ${\psi_{d,t}}\circ{\hat\Theta_{C}}=0$ for all $d,t$, and ${\hat\Theta_{C}}\neq0$. Showing the first statement is very easy, using Lemma \[lemma5\]. The only homomorphisms that occur in that lemma with non-zero coefficient are labelled by tableaux with more than $v$ $1$s in rows $2$ and $3$, and therefore by Lemma \[lemma7\] are zero. Showing that ${\hat\Theta_{C}}\neq0$ is also straightforward using Lemma \[lemma7\]. We apply this lemma to move the $1$s from row $2$ to row $1$, and then again to move the $2$s from row $3$ to row $2$. The tableau $${\text{\footnotesize$\gyoungx(1.2,;1;1;1_{4.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(4.5*.25,0);\end{tikzpicture}}};1;{v\!\!+\!\!2}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{b\!\!+\!\!2},;2;2;2;5_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{v\!\!+\!\!1},;3;4)$}}$$ (for example) labels a homomorphism occurring with non-zero coefficient. \[cd2homdim2\] With $\la,\mu$ as above, $$\dim_\bbf{\operatorname{Hom}}_{\bbf{\mathfrak{S}_}n}(S^\mu,S^\la)=1.$$ The existence of the homomorphism ${\hat\Theta_{C}}$ above shows that the space of homomorphisms is non-zero. So we just need to show the upper bound on the dimension. So suppose $\theta$ is a linear combination of semistandard homomorphisms ${\hat\Theta_{T}}:S^\mu\to S^\la$ such that ${\psi_{d,t}}\circ\theta=0$ for all $d,t$. For $3\ls d\ls b+1$, say that a semistandard tableau $T$ is *$d$-bad* if the entries $d,d+1$ appear in the same column of $T$. Note that this must be column $1$ or $2$, because the assumption $v\ls a-1$ guarantees that any column of length greater than $1$ has a $1$ at the top. We claim that if $T$ is $d$-bad, then ${\hat\Theta_{T}}$ cannot appear in $\theta$. To show this, we consider ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ for every semistandard $T$. If $T$ is not $d$-bad, then by Lemma \[lemma5\] ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a homomorphism labelled by a semistandard tableau with two $d$s in different rows. If $T$ is $d$-bad, then we can express ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ as a linear combination of semistandard homomorphisms using Lemma \[lemma5\] together with Lemma \[newsemilem\]. For example, if $$\begin{aligned} T&=\young(11111122248{10},369{11}{12},57) \\ \intertext{then ${\psi_{6,1}}\circ{\hat\Theta_{T}}={\hat\Theta_{T'}}$, where} T'&=\young(11111122248{10},369{11}{12},56)\end{aligned}$$ and we can semistandardise this using Lemma \[newsemilem\], taking $$A=\{3\},\qquad B=\{5,6,7,9,11,12\},\qquad C=\emptyset$$ to express ${\hat\Theta_{T'}}$ as a sum of fourteen semistandard homomorphisms. Doing this for each $d$-bad tableau $T$, we find that ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is a sum of homomorphisms labelled by semistandard tableaux with the same first row as $T$; furthermore, at least one of these tableaux will have two $d$s in the second row. Moreover, each $d$-bad tableau will yield a tableau of this kind which does not occur for any other $d$-bad tableau $T'$. To see this, suppose first of all that $d,d+1$ occur in the second column of $T$. Then there is no other $d$-bad tableau with the same first row as $T$, so any tableau occurring in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ with two $d$s in the second row can only possibly occur in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$. Alternatively, if $d,d+1$ occur in the first column of $T$, then $T$ has the form $${\text{\footnotesize$\gyoungx(1.2,;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;2;2;{x_1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{x_s},;d;{z_1};{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{d\!\!+\!\!1};k)$}}.$$ There are $v-2$ other $d$-bad tableaux with the same first row as $T$, and they all also have the same $(2,2)$-entry as $T$. Hence when we apply Lemma \[lemma5\] and Lemma \[newsemilem\] (or equivalently Lemma \[lemma7\]), we find that the homomorphism labelled by $$\gyoung(;1_5{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(5*.25,0);\end{tikzpicture}}};1;2;2;2;{x_1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};{x_s},;d;d;{z_2};{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1*.25,0);\end{tikzpicture}}};{z_t},;{z_1};k)$$ occurs in ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ but not in ${\psi_{d,1}}\circ{\hat\Theta_{T'}}$ for any other $T'$. So in $\theta$ the coefficient of ${\hat\Theta_{T}}$ is zero for any $d$-bad tableau. In particular, this means that for any ${\hat\Theta_{T}}$ occurring in $\theta$, ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is either zero or a single semistandard homomorphism. Now we claim that the coefficient of ${\hat\Theta_{T}}$ is zero whenever $T$ has two or three $2$s in its first row. Supposing this is false, take a $T$ with at least two $2$s in its first row such that ${\hat\Theta_{T}}$ appears with non-zero coefficient in $\theta$, and suppose that $T$ is minimal (with respect to the dominance order) subject to this property. Suppose the $(3,1)$-entry of $T$ is greater than $3$; then this entry equals $d+1$ for some $d\gs3$, and the entry equal to $d$ cannot be the $(2,1)$-entry (because $T$ is not $d$-bad). So the $d$ and the $d+1$ in $T$ lie in different rows and different columns, and ${\psi_{d,1}}\circ{\hat\Theta_{T}}$ is the semistandard homomorphism obtained by replacing the $d+1$ with a $d$. The only other semistandard tableau $T'$ such that ${\psi_{d,1}}\circ{\hat\Theta_{T'}}={\psi_{d,1}}\circ{\hat\Theta_{T}}$ is the tableau obtained by interchanging the $d$ and the $d+1$ in $T$, so ${\hat\Theta_{T'}}$ must also occur with non-zero coefficient. But $T\dom T'$, contradicting the choice of $T$. So the $(3,1)$-entry in $T$ must be $3$ (and hence the $(2,1)$-entry is $2$). Now consider the $(3,2)$-entry; call this $d+1$. Then $d\gs4$, and the $d$ in $T$ cannot occur in the $(2,2)$-position (because $T$ is not $d$-bad). So we can repeat the above argument and show that that there is a tableau $T'\domby T$ such that ${\hat\Theta_{T'}}$ occurs in $\theta$; contradiction. We now know that every semistandard homomorphism occurring in $\theta$ has at least two $2$s in the second row. This means in particular that the entries $3,\dots,b+2$ lie in different columns. So we can repeat the argument from Proposition \[cdhomdim1\] and Proposition \[cd2homdim1\] to show that $\theta$ must be a linear combination of $\tau_0$ and $\tau_1$, where $\tau_i$ is the sum of all homomorphisms labelled by semistandard tableau with $i$ $2$s in the first row. If $u=a$, then $\tau_1=0$, and so the space of homomorphisms $S^\mu\to S^\la$ has dimension at most $1$. if $u>a$, then ${\psi_{2,1}}\circ\tau_0={\psi_{2,1}}\circ\tau_1\neq0$, so $\theta$ must be a scalar multiple of $\tau_0+\tau_1$ and again the homomorphism space has dimension at most $1$. Composing the homomorphisms --------------------------- We have constructed homomorphisms $S^\mu\stackrel{{\hat\Theta_{C}}}\longrightarrow S^\la\stackrel\sigma\longrightarrow S^{\mu'}$, and shown that these homomorphisms are unique up to scaling. To complete this section, we just need to compute the composition of these homomorphisms. Let $E$ be the $\mu$-tableau $${\text{\footnotesize$\gyoungx(1.2,;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v;{v\!\!+\!\!1}_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};u,;1;2;3_{1.5}{{\begin{tikzpicture}[baseline=0cm]\draw[thick,densely dotted](0,.13)--++(1.5*.25,0);\end{tikzpicture}}};v,;1;2)$}}$$ of type $\mu'$. Then we have the following. \[comp2\] For $T\in{\calu}$, we have ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}\neq0$, and therefore we have $\sigma\circ{\hat\Theta_{C}}\neq0$ if and only if $\mbinom{u-v}{a-v}$ is odd. It is easy to express ${\hat\Theta_{E}}$ as a linear combination of semistandard homomorphisms using three applications of Lemma \[lemma7\], from which it follows that ${\hat\Theta_{E}}\neq0$. To prove that ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}$, use the notation of Proposition \[tabcomp\], with $S=C$. Suppose $X\in\calx$ is such that the coefficient of ${\hat\Theta_{U_X}}$ in Proposition \[tabcomp\] is non-zero. Since $X^{31}$ must be $\{2\}$, we cannot have $X^{21}=\{2\}$ (because this would give a factor $\binom21$), so $X^{21}=\{1\}$ and hence $X^{23}=\{1,2\}$. Now if $X^{11}$ contains any of the numbers $1,\dots,v$ then again we get a factor $\binom21$. So we have $X^{12}=\{1,\dots,v\}$, which determines $X$, and we find that ${\hat\Theta_{T}}\circ{\hat\Theta_{C}}={\hat\Theta_{E}}$ as required. So we have $\sigma\circ{\hat\Theta_{C}}=|{\calu}|{\hat\Theta_{E}}$, which is non-zero if and only if $|{\calu}|$ is odd. But it is easy to see that $|{\calu}|=\mbinom{u-v}{a-v}$, and the proposition is proved. Suppose $S^\mu=S^{(u,v,2)}$ is irreducible. Suppose first that $\mbinom{u-v}{a-v}$ is odd. This implies in particular that $0\ls a-v\ls u-v$, so $v\ls\min\{a-1,b+1\}$. So the assumptions of this section are valid, and we have homomorphisms ${\hat\Theta_{C}}:S^\mu\to S^\la$ and $\sigma:S^\la\to S^{\mu'}$. By Proposition \[comp2\], $\sigma\circ{\hat\Theta_{C}}\neq0$, so $S^\mu$ is a summand of $S^\la$. Conversely, suppose we have homomorphisms $S^\mu\stackrel\gamma\longrightarrow S^\la\stackrel\delta\longrightarrow S^{\mu'}$ with $\delta\circ\gamma\neq0$. By Propositions \[cd2homdim1\] and \[cd2homdim2\], $\delta$ must be a scalar multiple of $\sigma$, and $\gamma$ must be a scalar multiple of ${\hat\Theta_{C}}$. Hence by Proposition \[comp2\], $\mbinom{u-v}{a-v}$ is odd. Decomposability of Specht modules {#whichdec} ================================= In this section, we prove Corollary \[maincor\], which answers the question of which Specht modules are shown to be decomposable by Theorem \[main\]. First we consider the case where $a+b\equiv0\ppmod8$. \[ab0\] Suppose $n\equiv3\ppmod8$, and $a$ is even, with $6\ls a\ls n-7$. Let $b=n-a-3$. Then $S^{(a,3,1^b)}$ has an irreducible summand of the form $S^{(u,v,2)}$. Using Theorem \[main\], we need to show that there is a pair $u,v$ with $u+v+2=n$ such that $S^{(u,v,2)}$ is irreducible and $\mbinom{u-v}{u-a}$ is odd. By Theorem \[irrspecht\], $(u,v,2)$ is irreducible if and only if $$v\equiv1\pmod4,\qquad u-v\equiv-1\pmod{2^{l(v-2)}},$$ where $l(k)=\lceil\log_2(k+1)\rceil$ for an integer $m$. We use induction on $n$, with our main tool being the following well-known relations modulo $2$ on binomial coefficients: $$\binom{2x}{2y}\equiv\binom{2x+1}{2y}\equiv\binom{2x+1}{2y+1}\equiv\binom xy,\qquad \binom{2x}{2y+1}\equiv0\pmod 2.$$ We consider three cases. : In this case, take $v=5$ (so $u=n-7$). Since $n\equiv3\ppmod4$, we get $u\equiv0\ppmod4$, which means that $u-v\equiv3\ppmod4$, so $S^{(u,v,2)}$ is irreducible. Furthermore, the binomial coefficients $$\binom{u-5}0,\binom{u-5}1,\binom{u-5}2,\binom{u-5}3$$ are all odd, which means that $\mbinom{u-5}{u-a}$ must be odd. : In this case, let $$n'=\frac{n+11}2,\quad a'= \begin{cases} \mfrac{a+6}2&(a\equiv2\ppmod4)\\[5pt] \mfrac{a+4}2&(a\equiv0\ppmod4). \end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$ (note that the conditions on $a$ mean that $n>11$). So by induction there is a pair $u',v'$ such that $$v'\equiv1\pmod4,\qquad u-v\equiv-1\pmod{2^{l(v'-2)}},\qquad\mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Note that because $u'-v'$ is odd and $u'-a'$ is even, this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd. We let $u=2u'-4$ and $v=2v'-5$. Then $u+v+2=n$, and we have $v\equiv1\ppmod4$ and $$u-v=2(u'-v')+1\equiv-1\ppmod{2^{l(v'-2)+1}},$$ with $l(v-2)\ls l(v'-2)+1$. So $S^{(u,v,2)}$ is irreducible. Furthermore $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2,$$ and we are done. : In this case, let $$n'=\frac{n+3}2,\quad a'= \begin{cases} \mfrac{a+2}2&(a\equiv2\ppmod4)\\[5pt] \hbox to \frt{\hfil$\mfrac a2$\hfil}&(a\equiv0\ppmod4). \end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$. So by induction there is a pair $u',v'$ such that $$v'\equiv1\pmod4,\qquad u-v\equiv-1\pmod{2^{l(v'-2)}},\qquad\mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Again, because $u'-v'$ is odd and $u'-a'$ is even, $\mbinom{u'-v'}{u'-a'+1}$ is also odd. We let $u=2u'$ and $v=2v'-1$. Then $u+v+2=n$, $v\equiv1\ppmod4$, and $$u-v=2(u'-v')+1\equiv-1\pmod{2^{l(v'-2)+1}},$$ with $l(v-2)\ls l(v'-2)+1$. So $S^{(u,v,2)}$ is irreducible. Furthermore $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2.\tag*{\raisebox{-10pt}{\qedhere}}$$ The next result addresses most of the cases where $a+b\equiv2\ppmod8$. \[ab1\] Suppose $n\equiv5\ppmod8$, and $a$ is even, with $8\ls a\ls n-7$. Let $b=n-a-3$. Then $S^{(a,3,1^b)}$ has an irreducible summand of the form $S^{(u,v)}$ with $v\gs7$. The proof is very similar to the proof of Proposition \[ab0\]. We need to show that there is a pair $u,v$ such that $S^{(u,v)}$ is irreducible, $v\gs7$, $v\equiv3\ppmod4$ and $\mbinom{u-v}{u-a}$ is odd. The condition for $S^{(u,v)}$ to be irreducible is $u-v\equiv-1\pmod{2^{l(v)}}$. Again, we need three cases. : In this case, take $v=7$ (so $u=n-7$). Since $n\equiv5\ppmod8$, we get $u\equiv6\ppmod8$, which means that $u-v\equiv7\ppmod8$ (so $S^{(u,v)}$ is irreducible), and the binomial coefficients $$\binom{u-7}0,\binom{u-7}1,\binom{u-7}2, \binom{u-7}3$$ are all odd, which means that $\mbinom{u-7}{u-a}$ will be odd. : In this case, let $$n'=\frac{n+5}2,\quad a'= \begin{cases} \mfrac{a+2}2&(a\equiv2\ppmod4)\\[5pt] \mfrac{a+4}2&(a\equiv0\ppmod4). \end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$. So by induction there is a pair $u',v'$ such that $$v'\equiv3\pmod4,\qquad v'\gs7,\qquad u-v\equiv-1\pmod{2^{l(v')}},\qquad \mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Note that because $u'-v'$ is odd and $u'-a'$ is even, this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd. We let $u=2u'-2$ and $v=2v'-3$. Then $u+v=n$, and we have $$u-v=2(u'-v')+1\equiv-1\ppmod{2^{l(v')+1}}$$ and $l(v)\ls l(v')+1$. So $S^{(u,v)}$ is irreducible. Furthermore, $v\gs7$, $v\equiv3\ppmod4$ and $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2,$$ as required. : In this case, let $$n'=\frac{n+13}2,\quad a'= \begin{cases} \mfrac{a+6}2&(a\equiv2\ppmod4)\\[5pt] \mfrac{a+8}2&(a\equiv0\ppmod4). \end{cases}$$ Then $n',a'$ satisfy the conditions of the proposition, and $n'<n$. So by induction there is a pair $u',v'$ such that $$v'\equiv1\pmod4,\qquad v'\gs7,\qquad u-v\equiv-1\pmod{2^{l(v')}},\qquad\mbinom{u'-v'}{u'-a'}\equiv1\pmod2.$$ Because $u'-v'$ is odd and $u'-a'$ is even, this also gives $\mbinom{u'-v'}{u'-a'+1}$ odd. We let $u=2u'-6$ and $v=2v'-7$. Then $u+v=n$, and we have $$u-v=2(u'-v')+1\equiv-1\pmod{2^{l(v')+1}},$$ and $l(v)\ls l(v')+1$. So $S^{(u,v)}$ is irreducible. Furthermore, we have $v\equiv3\ppmod4$, $v\gs7$ and $$\binom{u-v}{u-a}=\binom{2u'-2v'+1}{2u'-2a'(+2)}\equiv\binom{u'-v'}{u'-a'(+1)}\equiv1\pmod2.\tag*{\raisebox{-10pt}{\qedhere}}$$ Now we can prove our main result. Suppose we have a pair $a,b$ of positive even integers with $a\gs4$. If $a\gs6$, $b\gs4$ and $a+b\equiv0\ppmod8$, then the result follows from Proposition \[ab0\]. If $a\gs8$, $b\gs4$ and $a+b\equiv2\ppmod8$, then the result follows from Proposition \[ab1\]. If $a=6$ and $a+b\equiv2\ppmod8$, then by Theorem \[main\] the Specht module $S^{(a+b,3)}$ is an irreducible summand of $S^\la$. The second case of the corollary is precisely the condition for $S^{(a+b,3)}$ to be an irreducible summand of $S^\la$, while the third case is the condition for $S^{(a+b-4,5,2)}$ to be a summand. So in any of the given cases, $S^\la$ certainly has an irreducible Specht module as a summand. To complete the proof, it suffices to show that if $a=4$, $b=2$ or $a+b\equiv4$ or $6\ppmod8$, then the only possible Specht modules which can occur as irreducible summands of $S^\la$ are $S^{(a+b,3)}$ and $S^{(a+b-4,5,2)}$. Suppose $S^\la$ has an irreducible summand $S^{(u,v)}$ with $v>3$. Then $v\equiv3\ppmod4$ and $u-v\equiv7\ppmod8$, which means that $u+v\equiv5\ppmod8$ and hence $a+b\equiv2\ppmod8$. Furthermore, $(u,v)\dom\la{^{\operatorname{reg}}}$, which implies that $a\gs6$ and $b\gs4$. Similarly, if $S^\la$ has an irreducible summand $S^{(u,v,2)}$ with $v>5$, then $v\equiv1\ppmod4$, $u-v\equiv7\ppmod8$ and hence $u+v+2\equiv3\ppmod8$, which gives $a+b\equiv0\ppmod8$. Furthermore, the fact that $(u,v,2)\dom\la{^{\operatorname{reg}}}$ implies that $a\gs6$ and $b\gs4$. Concluding remarks {#concsec} ================== The results in this paper do not give anything like a complete picture; this work is intended as a re-awakening of a long-dormant subject. Given how small the first example of a decomposable Specht module in this paper is, it is surprising that it has taken thirty years for this example to be found. We hope that this paper will be the start of a longer study of decomposable Specht modules. We conclude the paper by making some speculations about decomposable Specht modules; these are based on calculations and observations, but we do not have enough evidence to make formal conjectures. Specht filtrations ------------------ Our main results show that in certain cases summands of Specht modules are isomorphic to irreducible Specht modules. In fact, reducible Specht modules can also occur as summands; for example, the first new decomposable Specht module $S^{(4,3,1^2)}$ found in this paper decomposes as $S^{(6,3)}\oplus S^{(4,3,2)}$, with the latter Specht module being reducible. However, it is certainly not the case that every summand of a decomposable Specht module is isomorphic to a Specht module. But in the cases we have been able to calculate, every summand appears to have a filtration by Specht modules. If this is true in general, it means that our main results are stronger, in that we have found all irreducible summands of Specht modules in our family. In fact, we speculate that every Specht module has a filtration in which the factors are isomorphic to indecomposable Specht modules; this would imply in particular that every indecomposable summand has a Specht filtration. This speculation is certainly true in the case of Specht modules labelled by hook partitions; this follows from [@gm2 §2]. $2$-quotient separated partitions --------------------------------- In [@jm Definition 2.1], James and Mathas make the following definition: a partition $\la$ is *$2$-quotient separated* if it can be written in the form $$(c+2x_c,c-1+2x_{c-1},\dots,d+2x_d,d^{2y_d},(d-1)^{2y_{d-1}+1},\dots,1^{2y_1+1}),$$ where $c+1\gs d\gs0$, $x_c\gs\cdots\gs x_d\gs0$ and $y_1,\dots,y_d\gs0$. (Note that the definition includes the case $c=0$, where we have $\la=(2x_0)$ if $d=0$, or $(1^{2y_1})$ if $d=1$.) Informally, the $2$-quotient separated condition means that the Young diagram of $\la$ can be decomposed as in the following diagram, where horizontal ‘dominoes’ can appear in the first $c-d+1$ rows, and vertical ‘dominoes’ can appear in the first $d$ columns. $$\begin{tikzpicture}[scale=0.4] \draw(0,0)--(0,7)--++(9,0); \foreach \x in {0,1,2,3,4}\draw(\x,\x+2)--++(1,0)--++(0,1); \foreach \x in {0,1}\draw(\x,\x)--++(1,0)--++(0,2); \foreach \x in {0,1,2}\draw(\x+3,\x+4)--++(2,0)--++(0,1); \foreach \x in {0,1}\draw(\x+6,\x+5)--++(2,0)--++(0,1); \end{tikzpicture}$$ The definition of a $2$-quotient separated partition was made as part of the study of decomposition numbers: for the Iwahori–Hecke algebra $\calh_{\bbc,-1}({\mathfrak{S}_}n)$, whose representation theory is very similar to that of ${\mathfrak{S}_}n$ in characteristic $2$, the composition factors of a Specht module labelled by a $2$-quotient separated partition are known explicitly. The reason we recall the definition here is that every known example of a decomposable Specht module is labelled by a $2$-quotient separated partition. (Note that the partition $(a,3,1^b)$ considered in this paper is $2$-quotient separated precisely when $a$ and $b$ are even.) It is interesting to speculate whether the $2$-quotient separated condition is necessary for a Specht module to be decomposable. [99]{} [, ‘[An algorithm for semistandardising homomorphisms]{}’, *J. Algebra* **364** ([2012]{}), [38–51]{}.]{} [, ‘[Homomorphisms between Specht modules]{}’, *Math. Z.* **248** ([2004]{}), [395–421]{}.]{} [, *Polynomial representations of $\operatorname{GL}_n$*, [Lecture notes in mathematics]{} **830**, [Springer-Verlag, New York/Berlin]{}, [1980]{}.]{} [, ‘[On the decomposition matrices of the symmetric groups II]{}’, *J. Algebra* **43** ([1976]{}), [45–54]{}.]{} [, *The representation theory of the symmetric groups*, [Lecture notes in mathematics]{} **682**, [Springer-Verlag, New York/Berlin]{}, [1978]{}.]{} [, ‘[Hecke algebras of type $A$ with $q=-1$]{}’, *J. Algebra* **184** ([1995]{}), [102–58]{}.]{} [, ‘[The irreducible Specht modules in characteristic $2$]{}’, *Bull. London Math. Soc.* **31** ([1999]{}), [457–62]{}.]{} [, ‘[On decomposability of some Specht modules for symmetric groups]{}’, *J. Algebra* **66** ([1980]{}), [156–68]{}.]{} [, ‘[Submodule structure of some Specht modules]{}’, *J. Algebra* **74** ([1982]{}), [524–34]{}.]{}

ArXiv

--- abstract: 'We propose a notion of autoreducibility for infinite time computability and explore it and its connection with a notion of randomness for infinite time machines introduced in [@CaSc] and [@Ca3].' author: - Merlin Carl title: 'A note on Autoreducibility for Infinite Time Register Machines and parameter-free Ordinal Turing Machines' --- Autoreducibility for Infinite Time Register Machines ==================================================== The classical notion of autoreducibility can, for example, be found in [@DoHi]. We consider how this concept behaves in the context of infinitary machine models of computations. For the time being, we focus on Infinite Time Register Machines ($ITRM$s) (see [@ITRM] and [@ITRM2]) and ordinal Turing machines (see [@Ko]) - but the notion of course makes sense for other types like the Infinite Time Turing Machines ($ITTM$s, see [@HaLe]) as well. For $x\in ^{\omega}2$, we define $x_{\setminus n}$ as $x$ with its $n$th bit deleted (i.e. the bits up to $n$ are the same, the further bits are shifted one place to the left). We say that $x$ is $ITRM$-autoreducible iff there is an $ITRM$-program $P$ such that $P^{x_{\setminus n}}(n)\downarrow=x(n)$ for all $n\in\omega$. $x$ is called totally incompressible iff it is not $ITRM$-autoreducible, i.e. there is no $ITRM$-program $P$ such that $P^{x_{\setminus n}}(n)\downarrow=x(n)$ for all $n\in\omega$. If there is such a program, then we say that $P$ autoreduces $x$, $P$ is an autoreduction for $x$ or that $x$ is autoreducible via $P$. $x\in ^{\omega}2$ is $ITRM$-random in the measure sense iff there is no $ITRM$-decidable set $X$ of Lebesgue measure $0$ such that $x\in X$. $x\in ^{\omega}2$ is $ITRM$-random in the meager sense iff there is no $ITRM$-decidable meager set $X$ such that $x\in X$. We refer the reader to [@CaSc] and [@Ca3] for more information on $ITRM$-randomness, including that used in the course of this note. For the notion of $ITRM$-recognizability, we refer the reader to [@ITRM2], [@Ca] or [@Ca2]. No totally incompressible $x$ is $ITRM$-computable or even recognizable. $0^{\prime}_{ITRM}$, the real coding the halting problem for $ITRM$s, is $ITRM$-autoreducible. Clearly, if $P$ computes $x$, then $P$ is also an autoreduction for $x$. If $x$ is recognizable and $P$ recognizes $x$, we can easily retrieve a deleted bit by pluggin in $0$ and $1$ and letting $P$ run on both results to see for which one $P$ stops with output $1$. (The same idea works for finite subsets instead of single bits.) For $0^{\prime}_{ITRM}$, if a program index $j$ is given, it is easy to determine some index $i\neq j$ corresponding to a program that works in exactly the same way (by e.g. adding a meaningless line somewhere), so that the remaining bits allow us to reconstruct the $j$th bit. The autoreducibility of $0^{\prime}_{ITRM}$ also follows from the recognizability of $0^{\prime}_{ITRM}$ (see [@Ca2]). Let $x\in^{\omega}2$, $i\in\omega$. Then $\text{flip}(x,i)$ denotes the real obtained from $x$ by just changing the $i$th bit, i.e. $x\Delta\{i\}$. In the classical setting, no random real is autoreducible. This is still true for $ITRM$s: [\[randomnessimpliestotalincompressibility\]]{} If $x$ is $ITRM$-random, then $x$ is totally incompressible. (For the meager as well as for the measure $0$ interpretation of randomness.) Assume that $x$ is autoreducible via $P$. We show that $x$ is not $ITRM$-random. Let $X$ be the set of all $y$ which are autoreducible via $P$. Obviously, we have $x\in X$. $X$ is certainly decidable: Given $y$, use a halting problem solver for $P$ to see whether $P^{y_{\setminus n}}(n)\downarrow$ for all $n\in\omega$. If not, then $y\notin X$. Otherwise, carry out these $\omega$ many computations and check the results one after the other.\ Since $X$ is $ITRM$-decidable, it is provably $\Delta_{2}^{1}$, which implies that $X$ has the Baire property and thus is measurable. We show that $X$ must be of measure $0$. To see this, assume for a contradiction that $\mu(X)>0$. Note first, that, whenever $y$ is $P$-autoreducible and $z$ is a real that deviates from $y$ in exactly one digit (say, the $i$th bit), then $z$ is not $P$-autoreducible (since $P$ will compute the $i$th bit wrongly). By the Lebesgue density theorem, there is an open basic interval $I$ (i.e. consisting of all reals that start with a certain finite binary string $s$ length $k\in\omega$) such that the relative measure of $X$ in $I$ is $>\frac{1}{2}$. Let $X^{\prime}=X\cap I$, and let $X^{\prime}_0$ and $X^{\prime}_1$ be the subsets of $X^{\prime}$ consisting of those elements that have their $(k+1)$th digit equal to $0$ or $1$, respectively. Clearly, $X^{\prime}_{0}$ and $X^{\prime}_{1}$ are measurable, $X^{\prime}_{0}\cap X^{\prime}_{1}=\emptyset$ and $X^{\prime}=X^{\prime}_{0}\cup X^{\prime}_{1}$. Now define $\bar{X}^{\prime}_{0}$ and $\bar{X}^{\prime}_{1}$ by changing the $(k+1)$th bit of all elements of $X^{\prime}_{0}$ and $X^{\prime}_{1}$, respectively. Then all elements of $\bar{X}^{\prime}_{0}$ and $\bar{X}^{\prime}_{1}$ are elements of $I$ (as we have not changed the first $k$ bits), none of them is $P$-autoreducible (since they all deviate from $P$-autoreducible elements by exactly one bit, namely the $k$th), $\bar{X}^{\prime}_{0}\cap\bar{X}^{\prime}_{1}=\emptyset$ (elements of the former set have $1$ as their $(k+1)$th digit, for elements of $\bar{X}^{\prime}_{1}$ it is $0$) and $\mu(\bar{X}^{\prime}_{0})=\mu(X^{\prime}_{0})$, $\mu(\bar{X}^{\prime}_{1})=\mu(X^{\prime}_{1})$ (as the $\bar{X}^{\prime}_{i}$ are just translations of the $X^{\prime}_{i}$). As no element of the $\bar{X}^{\prime}_{i}$ is $P$-autoreducible, we have $(\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1})\cap X^{\prime}=\emptyset$. Let $\bar{X}^{\prime}:=\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1}$. Then we have\ $\mu_{I}(\bar{X}^{\prime})=\mu_{I}(\bar{X}^{\prime}_{0}\cup\bar{X}^{\prime}_{1})=\mu_{I}(\bar{X}^{\prime}_{0})+\mu_{I}(\bar{X}^{\prime}_{1})=\mu_{I}(X^{\prime}_{0})+\mu_{I}(X^{\prime}_{1})=\mu_{I}(X^{\prime})>\frac{1}{2}$ (where $\mu_{I}$ denotes the relative measure for $I$). So $X^{\prime}$ and $\bar{X}^{\prime}$ are two disjoint subsets of $I$ both with relative measure $>\frac{1}{2}$, a contradiction.\ For the meager version, we proceed similarly, taking $I$ to be an interval in which $X\cap I$ is comeager instead. That such an $I$ exists can be seen as follows: Suppose that $X$ is not meager. As above, $X$ is $ITRM$-decidable, hence provably $\Delta_{2}^{1}$ and therefore has the Baire property. Then, there is an open set $U$ such that $X\setminus U\cup U\setminus X$ is meager. In particular, $U$ is not empty. Hence $X$ is comeager in $U$. As $U$ is open, there is a nonempty open interval $I\subseteq U$. It is now obvious that $X\cap I$ is comeager in $I$, so $I$ is as desired. We then use the same argument as above, noting that two comeager subsets of $I$ cannot be disjoint. [\[Cohengenericsareincompressible\]]{} Let $x$ be Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$. Then $x$ is totally incompressible (in the measure sense). Let $x$ be as in the assumption, and assume for a contradiction that $x$ is $P$-autoreducible. By the forcing theorem for provident sets (see [@Ma]), there must be a condition $p\subseteq x$ such that $p\Vdash\text{`}P\text{ autoreduces }\dot{x}\text{'}$, where $\dot{x}$ is a name for the generic real (i.e. for $\bigcup\dot{G}$, where $\dot{G}$ is the canonical name for the generic filter). Let $i\in\omega\setminus\text{dom}(p)$ and let $x^{\prime}:=\text{flip}(x,i)$. Then $x^{\prime}$ is still Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$ and, as $p\subseteq x^{\prime}$, $p$ forces the $P$-autoreducibility of $x^{\prime}$; however, as $x^{\prime}$ differs from the $P$-autoreducible $x$ by only one bit, $x^{\prime}$ cannot be $P$-autoreducible, a contradiction. Thus $x$ is not $P$-autoreducible. [\[totallycompressiblesarerare\]]{} The set of $ITRM$-autoreducible reals has measure $0$. The proof of Theorem \[randomnessimpliestotalincompressibility\] shows that, for any $ITRM$-program $P$, the set of reals autoreducible via $P$ has measure $0$. As there are only countable many programs, the result follows. Denote by $IC_{ITRM}$ and $RA_{ITRM}$ the set of totally incompressible and $ITRM$-random reals, respectively (in the measure sense, for the time being). In this terminology, we showed above that $RA_{ITRM}\subseteq IC_{ITRM}$. However, the converse of Theorem \[randomnessimpliestotalincompressibility\] fails: [\[totalincompressibilitydoesnotimplyrandomness\]]{} $IC_{ITRM}\neq\subseteq RA_{ITRM}$, i.e. there is a real $x$ such that $x$ is totally incompressible, but not $ITRM$-random (in the measure sense). Let $X$ be an $ITRM$-decidable, comeager set of Lebesgue measure $0$. That such an $X$ exists is rather easy to see: A nice example for a comeager set of measure $0$ is the set of reals for which the zeros and ones in their binary representation are not equally distributed. It is straightforward to implement a decision procedure for this set on an $ITRM$.\ Now, the set of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1}$ is comeager and hence must intersect $X$. Let $x\in X$ be Cohen-generic over $L_{\omega_{\omega}^{CK}+1}$. By Corollary \[Cohengenericsareincompressible\], $x$ is totally incompressible. As $x\in X$ and $X$ is $ITRM$-decidable set of measure $0$, $x$ is not $ITRM$-random (in the measure sense). Thus $x\in RA_{ITRM}\setminus IC_{ITRM}$, as desired. In fact, the set of these reals is comeager, as the set $C$ of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1}$ is comeager, so that $C\cap X$ is also comeager and the proof shows that any element of $C\cap X$ is of this kind. Incompressibility and Randomness -------------------------------- We saw above that the following inclusions hold (where $C^{+}$ denotes the set of Cohen-generic reals over $L_{\omega_{\omega}^{CK}+1})$: $C\subsetneq RA_{ITRM}\subsetneq IC_{ITRM}$ (The first inclusion is proper because genericity for $\Pi_{1}$ and $\Sigma_1$-definable over $L_{\omega_{\omega}^{CK}}$ dense sets is sufficient, but not every such real is generic over $L_{\omega_{\omega}^{CK}+1}$, which requires intersection with every definable dense set, $Pi_{1}/\Sigma_1$ or not. We do not know whether $ITRM$-randomness can be characterized in terms of genericity in a natural way.)\ In this section, we consider the question how similar incompressibility is to randomness, i.e. which of the results obtained for random reals also hold for incompressibles.\ We start with an incompressible variant of the Kucera-Gacs theorem, which, as we recall, fails for $ITRM$-randomness, as no lost melody (an $ITRM$-recognizable real which is not $ITRM$-computable; this was shown in [@Ca3]) is reducible to a random real. [\[incompressiblekuceragacs\]]{} For every real $x$, there is a totally incompressible $y$ such that $x\leq_{ITRM} y$. Given $x$, let $y$ be Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$ and let $z:=x\oplus y$. Then certainly $x\leq_{ITRM}z$. Assume that $z$ is $P$-autoreducible for some program $P$. Hence, by the forcing theorem for provident sets [@Ma], there is a condition $p$ such that $p\Vdash$‘x$\oplus\bigcup\dot{G}$ is $P$-autoreducible’, where $\dot{G}$ is the canonical name for the generic filter. The same hence holds for every $y^{\prime}$ which is Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$ with $p\subseteq y^{\prime}$. Let $i\in\omega\setminus\text{dom}(p)$, $y^{\prime}:=\text{flip}(y,i)$, then $p$ forces the $P$-autoreducibility of $x\oplus y^{\prime}$. By absoluteness of computations, $x\oplus y^{\prime}$ is $P$-autoreducible. However, $x\oplus y^{\prime}$ differs from the $P$-autoreducible $x\oplus y$ in exactly one bit and hence cannot be $P$-autoreducible, a contradiction. Also, in contrast to the theorem that $ITRM$-computability from mutually $ITRM$-random reals implies plain $ITRM$-computability, mutually incompressibles can contain common non-trivial information ($COMP$ denotes the set of $ITRM$-computable reals): $x$ is totally incompressible relative to $y$ ($y$-incompressible, incompressible in $y$) iff there is no program $P$ such that $P^{x_{\setminus n}\oplus y}\downarrow=x(n)$ for all $n\in\omega$. If $x$ is $y$-incompressible and $y$ is $x$-incompressible, then $x$ and $y$ are mutually incompressible. [\[mutualincompressibility\]]{} There are mutually incompressible reals $y,z$ and a real $x\notin COMP$ such that $x\leq_{ITRM}y$ and $x\leq_{ITRM}z$. Let $y^{\prime}$, $z^{\prime}$ be mutually Cohen-generic over $L_{\omega_{\omega}^{CK,x}+1}[x]$, $y:=x\oplus y^{\prime}$, $z:=x\oplus z^{\prime}$ and apply the reasoning of the proof of Theorem \[incompressiblekuceragacs\]. Ordinal Turing Machines ======================= See [@Ko] for an introduction to ordinal Turing machines. For $OTM$s without parameters, define the notions of autoreducibility and total incompressibility as above for $ITRM$s. It turns out that there are no totally incompressible reals in $L$:\ [\[noOTMincompressibles\]]{} Assume $V=L$. Then there are no totally $OTM$-incompressible reals. Let $x\in L$. Our goal is to define a countable sequence $(P_{i}|i\in\omega)$ of programs deciding pairwise disjoint sets $(X_{i}|i\in\omega)$ with $\bigcup_{i\in\omega}X_{i}=\mathfrak{P}(\omega)$ such that if $y,z\in X$ differ only in finitely many bits, $y$ and $z$ are not in the same $X_{i}$. Once that is done, the proof is easy to finish: There is some $i\in\omega$ such that $x\in X_i$, without loss of generality let $i=0$. Then $X_{0}$ is decided by $P_{0}$. Now an autoreduction for $x$ works as follows: Given $n\in\omega$ and $x_{\setminus n}$, plug $0$ and $1$ in for the $i$th bit in $x_{\setminus n}$, getting reals $x_{0}$ and $x_{1}$, respectively, one of which is equal to $x$. Now use $P_{0}$ to decide whether $x_{0}\in X_{0}$ or $x_{1}\in X_{0}$. As $x_{0}$ and $x_{1}$ only differ in one bit and $X_{0}$ does not contain two (distinct) reals differing in only finitely many places, only one of $x_{0}$ and $x_{1}$ can be an element of $X_{0}$, and that is $x$, determining the $n$th digit of $x$.\ Now we construct $(P_{i}|i\in\omega)$ as follows: Let $(S_{i}|i\in\omega)$ be a natural enumeration of the finite sets of integers in order type $\omega$. Write $y\sim z$ iff $y$ and $z$ differ only in finitely many bits. For a real $a$, denote by $[a]_{0}$ the $<_{L}$-smallest real such that $[a]_{0}\sim a$. Then let $X_{i}:=\{[y]_{0}+_{b}S_{i}|y\in\mathfrak{P}^{L}(\omega\}$, where $+_{b}$ denotes the bitwise sum. Clearly, this is a countable partition of the constructible reals. Furthermore, there is a decision procedure for $X_{i}$ on an $OTM$ (which is in fact uniform in $i$) which works as follows: Given a real $a$ in the oracle, we can write $L$ on the tape until we arrive an $L$-level $L_{\alpha}\ni a$. Then, searching $L_{\alpha}$, we can identify $[a]_{0}$. Now compute the set $S$ of bits where $a$ and $[a]_{0}$ differ and compare it to our enumeration of finite subsets of $\omega$ fixed above: If $S=S_{i}$, then $a\in X_{i}$, otherwise $a\notin X_{i}$. Note that there are constructible reals which do not lie in any $OTM$-decidable null set, as the union $Y$ of all $OTM$-decidable null sets is an element in $L$ and, as a countable union of null sets, also a null set in $L$. Hence, at least in $L$, not every (parameter-free) $OTM$-random real is totally incompressible.\ Note that the situation will probably be quite different for Infinite Time Turing Machines ($ITTM$s), as they have neither the power to enumerate $L$ nor the ability to solve their own restricted halting program (like $ITRM$s).\ The $V=L$ hypothesis is probably unnecessarily strong here. However, even in rather mild extensions of $L$, $OTM$-incompressibles do exist: [\[OTMincomprinCohenextension\]]{} Let $x$ be Cohen-generic over $L$. Then $x$ is $OTM$-incompressible in $L[x]$ (and hence, by absoluteness of computations, in the real world). Assume for a contradiction that $x$ is $OTM$-autoreducible, say by the program $P$, where $x=\bigcup{G}$ and $G$ is a Cohen-generic filter over $L$. Then there is a finite $p\in G$ such that $p\Vdash\forall{n\in\omega}P^{x_{\setminus n}}(n)\downarrow=x(n)$. Let $i\in \omega\setminus\text{dom}(p)$, $x^{\prime}:=\text{flip}(x,i)$. Then $x^{\prime}\in L[x]$ is still Cohen-generic over $L$ and $p\subseteq x^{\prime}$ so that $p\Vdash \forall{n\in\omega}P^{x^{\prime}_{\setminus n}}(n)\downarrow=x^{\prime}(n)$. However, flipping a single bit cannot preserve $P$-autoreducibility, a contradiction. Hence $x$ is $OTM$-incompressible. Taking Theorem \[noOTMincompressibles\] and Theorem \[OTMincomprinCohenextension\] together, we get: [\[OTMincomprindependent\]]{} The existence of (parameter-free) $OTM$-incompressible reals is independent from $ZFC$. Consequently, the analogue of Theorem \[randomnessimpliestotalincompressibility\] for $OTM$s fails at least consistently: Every constructible $OTM$-random real provides a counterexample. For an $OTM$-program $P$, the set of $P$-autoreducibles is in general not decidable: [\[OTMPcompressibilityundecidable\]]{} Assume that $V=L$. Then there are $OTM$-programs $P$ such that $X_{P}:=\{x\mid \forall{n\in\omega}P^{x_{\setminus n}}(n)\downarrow=x(n)\}$ (i.e. the set of $P$-autoreducibles) is not $OTM$-decidable. Assume for a contradiction that $X_{P}$ is decidable for every $P$. By the same argument as in the proof of Theorem \[randomnessimpliestotalincompressibility\] then, $\mu(X_{P})=0$ for every $P$. Consequently, no $OTM$-autoreducible real is $OTM$-random, and hence, every $OTM$-random real is $OTM$-incompressible. However, the non-$OTM$-random reals are contained in a countable union of decidable null sets and hence form a null set themselves, so that the $OTM$-random reals have full measure, while, on the other hand, $OTM$-incompressibles do not exist in $L$, a contradiction. Note, however, that $P$-autoreducibility for $OTM$s is semidecidable by simply simultaneously running all $OTM$-programs on a real $x$ and checking whether one of them is an autoreduction. If such a program exists, it will eventually be found; otherwise, the search will not halt.\ We note further that such sets are in general not measurable: Assume $V=L$. Then there is an $OTM$-program $P$ such that the set of $P$-autoreducible reals is not measurable. In fact, there is a recursive set $I\subseteq\omega$ such that $\forall{x}P_{i}^{x}\downarrow=0\vee P_{i}^{x}\downarrow=1$, $S_{i}:=\{x\mid P_{i}^{x}\downarrow=1\}$ is not measurable, $S_{i}\cap S_{j}=\emptyset$ for $i\neq j$ and $\mathfrak{P}(\omega)=\bigcup_{i\in\omega}S_{i}$. Let $(s_{i}|i\in\omega)$ be an enumeration of $^{<\omega}\omega$ in order type $\omega$, denote by $x\sim y$ that $x$ and $y$ differ only at finitely many places, let $[x]_{\sim}$ be the $\sim$-equivalence class of $x$ and let $P_{i}$ be the program described in the proof of Theorem \[noOTMincompressibles\] that works as follows: Given $x$ in the oracle, determine the $<_{L}$-minimal representative $x_{0}$ of $[x]_{\sim}$, then output $x_{0}\Delta s_{i}$ (where $\Delta$ denotes the symmetric difference, that is we flip all the bits at places in $s_{i}$). Denoting, for $i\in\omega$, $E_{i}:=\{x\mid x=x_{0}\Delta s_{i}\}$, we have that $\mathfrak{P}(\omega)=\bigcup_{i\in\omega}E_{i}$, $E_{i}\cap E_{j}=\emptyset$ for $i\neq j$ and $P_{i}$ decides $E_{i}$ for all $i,j\in\omega$. Furthermore, it is well-known that none of the $E_i$ is measurable. Acknowledgements ================ We are indebted to Philipp Schlicht for several very helpful discussions on the subject and in particular for the proof idea for Theorem \[totalincompressibilitydoesnotimplyrandomness\]. M. Carl. The distribution of $ITRM$-recognizable reals. To appear in: Annals of Pure and Applied Logic, special issue from CiE 2012 M. Carl. Optimal Results on $ITRM$-recognizability. Preprint. arXiv:1306.5128v1 M. Carl. Algorithmic Randomness for Infinite Time Register Machines Preprint. arXiv:1401.1734v1 M. Carl, P. Schlicht. Infinite Computations with Random Oracles. Submitted. arXiv:1307.0160v3 M. Carl, P. Schlicht. Infinite Time Algorithmic Randomness. Work in progress. R.G. Downey, D. Hirschfeldt. Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer LLC $2010$ J. Hamkins, A. Lewis. Infinite Time Turing Machines. Journal of Symbolic Logic 65(2), 567-604 (2000) P. Koepke, R. Miller. An enhanced theory of infinite time register machines M. Carl, T. Fischbach, P. Koepke, R. Miller, M. Nasfi, G. Weckbecker. The basic theory of infinite time register machines P. Koepke. Turing computations on ordinals. Bulletin of Symbolic Logic 11 (2005), 377-397 A.R.D. Mathias. Provident sets and rudimentary set forcing. Preprint. Available at https://www.dpmms.cam.ac.uk/ ardm/fifofields3.pdf

ArXiv

--- author: - 'Steven A. Balbus' date: 'Received ; accepted ' title: Nonlinear Scale Invariance in Local Disk Flows --- Introduction ============ Disks with Keplerian rotation profiles are linearly stable by the Rayleigh criterion of outwardly increasing specific angular momentum, but are extremely sensitive to the presence of magnetic fields. A weakly magnetized disk is linearly unstable if its angular velocity decreases outward, a condition met by Keplerian and almost all other astrophysical rotation profiles (Balbus & Hawley 1991). The underlying physics behind this magnetorotational instability (MRI) is well-understood, and the breakdown of the flow into fully developed turbulence has been convincingly demonstrated in a large series of numerical simulations (Balbus 2003 for a review). Not all astrophysical disks need have the requisite minimum ionization level to sustain magnetic coupling, however. Protostellar disks, for example, may have an extended “dead zone” near the midplane on radial scales from $\sim 0.1$ to $\sim 10$ AU (Gammie 1996; Fromang, Terquem, & Balbus 2001). This, along with other similar cases (e.g. CV disks, cf. Gammie & Menou 1998), has led to speculation that there are also hydrodynamical mechanisms by which Keplerian flow is destabilized (Gammie 1996). Before the advent of the MRI, such reasoning was orthodox. The pioneering work of Shakura & Sunyaev (1973), for example, invoked nonlinear, high Reynolds number shear instabilities as a likely destabilizing mechanism that would lead to turbulence (see also Crawford & Kraft 1956). Since, for nonaxisymmetric disturbances, there is still no proof either of linear or nonlinear stability, this mechanism continues to attract adherents (Dubrulle 1993, Richard & Zahn 1999, Richard 2003). Theoretical analysis may have hit an impasse, but the intervening years have in fact seen a stunning rise in the capabilities of numerical simulation. These have shown no indication of local nonlinear rotational instabilities (Hawley, Balbus, & Winters 1999), in Keplerian disks. They do, however, reveal nonlinear shear instabilities when Corilois forces are absent, or when the disk is marginally stable (constant specific angular momentum). Indeed, even linear instability is possible in some Rayleigh-stable disks, provided that global physics is introduced (Goldreich, Goodman, & Narayan 1986; Blaes 1987), a result that has been numerically confirmed (Hawley 1991). The numerical stability findings have been criticized on the grounds that the effective Reynolds number of the codes is too low, and that this damps the nonlinear instabilities: the latter require yet-to-be resolved spatial scales in order to reveal themselves (Richard & Zahn 1999). In this paper, we show that the local disk equations possess a scale invariance that implies any solution to the governing equations must be present on all scales. In other words, for every small scale velocity flow, there is an exact large scale counterpart with the same long term stability behavior. The absence of instability at large scale therefore implies the absence at small scales as well. Conversely, any true small scale instabilities (those present in a shear layer, for example), must also have large scale counterparts, and therefore instability should be found even at crude numerical resolutions. This is indeed the case. Our findings suggest that if nonlinear hydrodynamical instabilities were present in Keplerian disks, such unstable disturbances must involve dynamics beyond the local approximation, and are not an inevitable nonlinear outcome of differential rotation. The Local Approximation ======================= In cylindrical coordinates $(R, \phi, z)$, the fundamental equations of motion for a flow in which viscous effects are negligible are mass conservation \[fun0\] [t]{} + [[ ]{}]{}[[ ]{}]{}([ ]{})= 0, and the dynamical equations, \[fun1\] [v\_Rt]{} + [ ]{}[[ ]{}]{}[[ ]{}]{}v\_R - [v\_\^2R]{} = - [1]{}[PR]{} - [R]{} \[fun2\] [v\_t]{} + [ ]{}[[ ]{}]{}[[ ]{}]{}v\_+ [v\_v\_RR]{} = - [1R]{}[P]{} \[fun3\] [v\_zt]{} + [ ]{}[[ ]{}]{}[[ ]{}]{}v\_z = - [1]{}[Pz]{} -[z]{} Our notation is standard: ${ \mbox{\boldmath{$v$}} }$ is the velocity field, $\rho$ the mass density, $P$ the gas pressure, and $\Phi$ is the Newtonian point mass potential for central mass $M$: = - [GM(R\^2 + z\^2)\^[1/2]{}]{}. $G$ is the gravitational constant. The local limit consists of the following series of approximations. First, we assume that $R$ is large and $z \ll R$, so that \[phis\] - [GMR]{} (1 - [z\^22R\^2]{}) and , Choose a fiducial value of $R$, say $R_0$. Denote the angular velocity as $\Omega(R)$ (we assume a dependence only upon $R$), and let $\Omega_0 = \Omega(R_0)$. We next erect local Cartesian coordinates x = R - R\_0,  y = R\_0(-\_0 t) which corotate with the disk at $R=R_0$. Let \[w\] [ ]{} [ ]{} - R\_0 t [ ]{} be the velocity relative to uniform rotation at $\Omega= \Omega_0$. In the local approximation, the magnitude of ${ \mbox{\boldmath{$w$}} }$ is assumed to be small compared with $R\Omega_0$. The undisturbed angular velocity is Keplerian, \[kep\] \^2 = [GMR\^3]{} Substitution of equations (\[phis\]-\[kep\]) into equations (\[fun1\]–\[fun2\]) and retaining leading order, yields the so-called [*local*]{} or [*Hill*]{} equations (e.g., Balbus & Hawley 1998): \[hill0\] [t]{} + [[ ]{}]{}[[ ]{}]{}([ ]{})= 0, \[hill1\] ( [t]{} + [ ]{}[[ ]{}]{})[w\_R]{} - 2w\_= - x[d\^2d R]{} - [1]{} [Px]{} \[hill2\] ( [t]{} + [ ]{}[[ ]{}]{})[w\_]{} + 2w\_R = - [1]{} [Py]{} \[hill3\] ( [t]{} + [ ]{}[[ ]{}]{})[w\_z]{} = - z\^2 - [1]{} [Pz]{} The “0” subscript has been dropped in the $2\Omega$ terms in equations (\[hill1\]) and (\[hill2\]), and in the derivative term on the right of equation (\[hill1\]). The time derivative is taken in the corotating frame, viz.: = [ t]{} + \_0 Equations (\[hill0\]–\[hill3\]) are well known, and have been used extensively in both numerical and analytical studies. The fundamental approach dates from nineteenth century treatments of the Earth-moon system (Hill 1878). Scale symmetry in the Hill Equations ==================================== The local equations of motion incorporate an important symmetry in their structure. Let [ ]{}([ ]{}, t), ([ ]{}, t), P([ ]{}, t), where ${ \mbox{\boldmath{$r$}} }=(x, y, z)$, be an exact solution to the Hill equations (\[hill0\]–\[hill3\]). Then, if $\alpha$ is an arbitrary constant, (1/)[ ]{}([ ]{}, t), ([ ]{}, t), (1/\^2) P([ ]{}, t) is also an exact solution to the same equations. The proof is a simple matter of direct substitution. An equivalent formulation of the scaling symmetry is [ ]{}([ ]{}/, t)(1/) [ ]{}([ ]{}, t) ([ ]{}/, t) ([ ]{}, t) P ([ ]{}/, t)(1/\^2) P ([ ]{}, t). In this form, with $\epsilon \ll 1$, we see that any solution of the Hill equations that involves very small length scales has a rescaled counterpart solution with exactly the same time dependence. In particular, any solution corresponding to a breakdown into turbulence must be present on both large and small scales. The implications of this scaling symmetry are of particular importance for understanding and testing the possible existence of local nonlinear instabilities in Keplerian disks. The key point is that any such instability would have to exist not just at small scales, but at all scales. Finite difference numerical codes would find such instabilities, if they existed. Indeed, a constant specific angular momentum profile is nonlinearly unstable, and is found to be so even at resolutions as crude as $32^3$. By way of contrast, local Keplerian profiles show no evidence of nonlinear instability at resolutions up to $256^3$, instead converge to the same stable solution in codes with completely different numerical diffusion properties (Hawley, Balbus, & Winters 1999). The argument that small scale flow structure is somehow being repressed is simply untenable. To see how the Reynolds number changes with scale, assume that a flow is characterized by an effective kinematic viscosity $\nu$. The scaling argument we have just given applies to inviscid equations, so we should not expect it to hold in the presence of viscosity. The Reynolds number associated with the small scale solution is Re\_[s]{} = [w l ]{} The Reynolds number associated with the large scale solution is Re\_[l]{} = [wl]{} where $w$ here means $w(l, t)$, the value of the velocity function evaluated at a fiducial value length $l$ and time $t$. $Re_l =Re_s/\epsilon^2 \ll Re_{s}$ because at larger scales both the velocity and the length scales increase by a factor of $1/\epsilon$. In a numerical simulation, strict scaling invariance is not obeyed. Instead, the large scale solutions approach the inviscid limit, while their sufficiently small scale counterparts are damped. But by behaving nearly inviscidly, the large scale solutions capture the behavior of the Hill system at all scales. What this Result Does Not Show ============================== Obviously, scale invariance does not constitute a proof of nonlinear stability in any Keplerian flow. There are several points we have not covered. First, the local approximation ignores boundary conditions. In laboratory flows, the fluid is always bounded by hard walls, and boundary layers form. A recent laboratory confirmation of the MRI also finds finite amplitude velocity fluctuations in a magnetically stable flow, for example. But the source of such disturbances are boundary layers (Sisan et al. 2004). The Hill equations emerge in the limit $R\rightarrow\infty$, and therefore curvature terms drop out of the analysis. Instabilities that depend, for example, upon inflection points or vorticity maxima in the background rotation profile would not appear in this limit. Nothing precludes them from forming in the $w$ velocity profile, however, and if such instabilities were present they should manifest on large scales as well as small. In any case, the criticism of the numerical simulations is that extremely small structure is being lost, and that high Reynolds number differential rotation is supposedly intrinsically unstable. It is very difficult to see how large scale curvature could play an essential destabilizing role here. In these equations, the curvature terms are nonsingular perturbations. Planar Couette and Poiseuille flows break down into turbulence without assistance from geometrical curvature. Our Hill analysis together with numerical simulations would also suggest that a non-Keplerian disk with, say, $\Omega \propto R^{-1.8}$ is nonlinearly stable. But an annulus supporting such a profile is in fact [*linearly*]{} unstable (Goldreich, Goodman, & Narayan 1986), transporting angular momentum outward even in its linear phase. The point is that the annulus supports edge modes that become unstable, and these global modes do not exist in the local approximation. The existence of a similar instabilities in disks found in nature cannot be ruled out, though to date none afflicting Keplerian disks have been found. The disk thermal structure could also be unstable, at least in principle. Nothing presented in this work bears on these types of instabilities. Finally, there are technical loopholes to the argument presented in this paper. What if the unstable solution required not just some small scales to be resolved, but very disparate scales? Why this should be so is far from clear, but this possibility cannot be ruled out [*a priori.*]{} Indeed, one could imagine that a fractal structure is required down to infinitesimal scales. Rescaling would not bring such a solution to larger characteristic length scales, by definition. This solution is obviously not characterized by a critical Reynolds number, above which it is necessary to be seen. The critical Reynolds number would be infinity! This is not the argument made by proponents of nonlinear high Reynolds number instability. Such a solution may remain a mathematical possibility, but not one that can be realized in nature. Conclusion ========== The local dynamics of Keplerian or other astrophysical disk profiles can be captured by a an established formalism known as the local, or Hill, approximation. The resulting system of equations has an exact scale invariance, so that any flow characterized by very small scales has an exact large scale counterpart with same stability properties. This feature of the Hill equations implies that finite difference codes at available resolutions are sufficient to explore the possibility of [*local*]{} nonlinear shear instabilities in astrophysical disks. If simulations accurately describe the large scale behavior of the Hill system, there is nothing more to uncover at small scales; it is simply renormalized large scale behavior. The absence of any observed instabilities in Keplerian numerical studies, coupled with the ready manifestation of such instabilities in local shear layers and constant specific angular momentum systems, suggests that any putative nonmagnetic disk instability would have to incorporate physics beyond simple differential rotation. I thank C. Gammie, J. Hawley, K. Menou, and C. Terquem for useful comments. This work is supported by NASA grants NAG5-13288 and NNG04GK77G. Balbus, S. A. 2003, , 41, 555 Balbus, S. A. & Hawley, J. F. 1991, ApJ, 376, 214 Balbus, S. A. & Hawley, J. F. 1998. Rev. Mod. Phys., 70, 1. Blaes, O. M. 1987, MNRAS, 227, 975 Crawford, J. A., & Kraft, R. P. 1956, ApJ, 123, 44 Dubrulle, B. 1993, Icarus, 106, 59 Fromang, S., Terquem, C., & Balbus, C. 2001, MNRAS, 329, 18 Gammie, C. F. 1996, ApJ, 457, 355 Gammie, C. F., & Menou, K. 1998, ApJ, 492, L75 Goldreich, P., Goodman, J., & Narayan, R. 1986, MNRAS, 221, 339 Hawley, J. F. 1991, ApJ, 381, 496 Hawley, J. F., Balbus, S. A., & Winters, W. F. 1999, ApJ, 518, 394 Hill, G. W. 1878, Am. J. Math., 1, 5 Richard, D. 2003, å, 408, 409 Richard, D., & Zahn, J.-P. 1999, A&A, 347, 734 Shakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 Sisan, D. R., Mujica, N., Tillotson, W. A., Huang, Y.-M., Dorland, W., Hassam, A., Antonsen, T. M., & Lathrop, D. P. 2004, Phys. Rev., in press (physics/0401125).

ArXiv

--- abstract: 'We present a comprehensive analysis of different techniques available for the spectroscopic analysis of FGK stars, and provide a recommended methodology which efficiently estimates accurate stellar atmospheric parameters for large samples of stars. Our analysis includes a simultaneous equivalent width analysis of [Fe]{} and [Fe]{} spectral lines, and for the first time, utilises on-the-fly NLTE corrections of individual [Fe]{} lines. We further investigate several temperature scales, finding that estimates from Balmer line measurements provide the most accurate effective temperatures at all metallicites. We apply our analysis to a large sample of both dwarf and giant stars selected from the RAVE survey. We then show that the difference between parameters determined by our method and that by standard 1D LTE excitation-ionisation balance of Fe reveals substantial systematic biases: up to $400$ K in effective temperature, $1.0$ dex in surface gravity, and $0.4$ dex in metallicity for stars with $\feh\sim-2.5$. This has large implications for the study of the stellar populations in the Milky Way.' author: - | Gregory R. Ruchti,$^{1}$[^1] Maria Bergemann,$^{1}$ Aldo Serenelli,$^{2}$ Luca Casagrande$^{3}$ and Karin Lind$^{1}$\ $^{1}$Max Planck Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany\ $^{2}$Instituto de Ciencias del Espacio (CSIC-IEEC), Facultad de Ciencias, Campus UAB, 08193 Bellaterra, Spain\ $^{3}$Research School of Astronomy & Astrophysics, Mount Stromlo Observatory, The Australian National University, ACT 2611, Australia\ date: 'Accepted 2012 October 30. Received 2012 September 26' title: 'Unveiling systematic biases in 1D LTE excitation-ionisation balance of Fe for FGK stars. A novel approach to determination of stellar parameters.' --- \[firstpage\] stars: abundances — stars: late-type — stars: Population II Introduction {#sec-intro} ============ The fundamental atmospheric (effective temperature, surface gravity, and metallicity) and physical (mass and age) parameters of stars provide the major observational foundation for chemo-dynamical studies of the Milky Way and other galaxies in the Local Group. With the dawn of large spectroscopic surveys to study individual stars, such as SEGUE [@yanny09], RAVE [@steinmetz06], Gaia-ESO [@gilmore12], and HERMES [@barden08], these parameters are used to infer the characteristics of different populations of stars that comprise the Milky Way. Stellar parameters determined by spectroscopic methods are of a key importance. The only way to accurately measure metallicity is through spectroscopy, which thus underlies photometric calibrations [e.g., @holmberg07; @an09; @arnadottir10; @casagrande11], while high-resolution spectroscopy is also used to correct the low-resolution results [e.g., @carollo10]. The atmospheric parameters can all be estimated from a spectrum in a consistent and efficient way. This also avoids the problem of reddening inherent in photometry since spectroscopic parameters are not sensitive to reddening. The spectroscopic parameters can then be used alone or in combination with photometric information to fit individual stars to theoretical isochrones or evolutionary tracks to determine the stellar mass, age, and distance of a star. A common method for deriving the spectroscopic atmospheric parameters is to use the information from [Fe]{} and [Fe]{} absorption lines under the assumption of hydrostatic equilibrium (HE) and local thermodynamic equilibrium (LTE). Many previous studies have used some variation of this technique (e.g., ionisation or excitation equilibrium) to determine the stellar atmospheric parameters and abundances, and henceforth distances and kinematics, of FGK stars in the Milky Way. For example, some have used this procedure to estimate the effective temperature, surface gravity, and metallicity of a star [e.g., @fulbright00; @prochaska00; @johnson02], while others use photometric estimates of effective temperature in combination with the ionisation equilibrium of the abundance of iron in LTE to estimate surface gravity and metallicity [e.g., @mcwilliam95; @francois03; @bai04; @allendep06; @lai08]. However, both observational [e.g., @fuhrmann98; @ivans01; @ruchti11; @bruntt12] and theoretical evidence [e.g., @thevenin99; @asplund05; @mashonkina11] suggest that systematic biases are present within such analyses due to the breakdown of the assumption of LTE. More recently, @bergemann12 and @lind12 quantified the effects of non-local thermodynamic equilibrium (NLTE) on the determination of surface gravity and metallicity, revealing very substantial systematic biases in the estimates at low metallicity and/or surface gravity. It is therefore extremely important to develop sophisticated methods, which reconcile these effects in order to derive accurate spectroscopic parameters. This is the first in a series of papers, in which we develop new, robust methods to determine the fundamental parameters of FGK stars and then apply these techniques to large stellar samples to study the chemical and dynamical properties of the different stellar populations of the Milky Way. In this work, we utilise the sample of stars selected from the RAVE survey originally published in @ruchti11 [hereafter R11] to formulate the methodology to derive very accurate atmospheric parameters. We consider several temperature scales and show that the Balmer line method is the most reliable among the different methods presently available. Further, we have developed the necessary tools to apply on-the-fly NLTE corrections[^2] to [Fe]{} lines, utilising the grid described in @lind12. We verify our method using a sample of standard stars with interferometric estimates of effective temperature and/or [*Hipparcos*]{} parallaxes. We then perform a comprehensive comparison to standard 1D, LTE techniques for the spectral analysis of stars, finding significant systematic biases. Sample Selection and Observations ================================= NLTE effects in iron are most prominent in low-metallicity stars [@lind12; @bergemann12]. We therefore chose the metal-poor sample from R11 for our study. These stars were originally selected for high-resolution observations based on data obtained by the RAVE survey in order to study the metal-poor thick disk of the Milky Way. Spectral data for these stars were obtained using high-resolution echelle spectrographs at several facilities around the world. Full details of the observations and data reduction of the spectra can be found in R11. Briefly, all spectrographs delivered a resolving power greater than 30,000 and covered the full optical wavelength range. Further, nearly all spectra had signal-to-noise ratios greater than $100:1$ per pixel. The equivalent widths (EWs) of both [Fe]{} and [Fe]{} lines, taken from the line lists of @fulbright00 and @johnson02, were measured using the ARES code [@sousa07]. However, during measurement quality checks, we found that the continuum was poorly estimated for some lines. We therefore determined EWs for these affected lines using hand measurements. Stellar Parameter Analyses {#sec-par} ========================== We computed the stellar parameters for each star using two different methods. In the first method, which is commonly used in the literature, we derived an effective temperature, $\tlte$, surface gravity, $\logglte$, metallicity, $\fehlte$, and microturbulence, $\vtlte$, from the ionisation and excitation equilibrium of Fe in LTE. This is hereafter denoted as the LTE-Fe method. We used an iterative procedure that utilised the `MOOG` analysis program [@sneden73] and 1D, plane-parallel `ATLAS-ODF` model atmospheres from Kurucz[^3] computed under the assumption of LTE and HE. In our procedure, the stellar effective temperature was set by minimising the magnitude of the slope of the relationship between the abundance of iron from [Fe]{} lines and the excitation potential of each line. Similarly, the microturbulent velocity was found by minimising the slope between the abundance of iron from [Fe]{} lines and the reduced EW of each line. The surface gravity was then estimated by minimising the difference between the abundance of iron measured from [Fe]{} and [Fe]{} lines. Iterations continued until all of the criteria above were satisfied. Finally, $\fehlte$ was chosen to equal the abundance of iron from the analysis. Our results for this method are described in Section \[sec-lte\]. The second method, denoted as the NLTE-Opt method, consists of two parts. First, we determined the optimal effective temperature estimate, $\tfin$, for each star (see Section \[sec-temp\] for more details). Then, we utilised `MOOG` to compute a new surface gravity, $\loggf$, metallicity, $\fehf$, and microturbulence, $\vtf$. This was done using the same iterative techniques as the LTE-Fe method, that is the ionisation balance of the abundance of iron from [Fe]{} and [Fe]{} lines. There are, however, three important differences. First, the stellar effective temperature was held fixed to the optimal value, $\tfin$. Second, we restricted the analysis to Fe lines with excitation potentials above 2 eV, since these lines are less sensitive to 3D effects as compared to the low-excitation lines [see the discussion in @bergemann12]. Third, the abundance of iron from each [Fe]{} line was adjusted according to the NLTE correction for that line at the stellar parameters of the current iteration in the procedure. The NLTE corrections were determined using the NLTE grid computed in @lind12 and applied on-the-fly via a wrapper program to `MOOG`. Note that the NLTE calculations presented in @lind12 were analogously calibrated using the ionisation equilibria of a handful of well-known stars. Our extended sample, including more stars with direct measurements of surface gravity and effective temperature, provide support for the realism of this calibration. The grid extends down to $\logg=1$. We imposed a routine which linearly extrapolated the NLTE corrections to below this value. The results of extrapolations were checked against NLTE grids presented in @bergemann12a and no significant differences were found. Further, @lind12 found very small NLTE corrections for [Fe]{} lines. We therefore do not apply any correction to the [Fe]{} lines. Iterations continued until the difference between the average abundance of iron from the [Fe]{} lines and the NLTE-adjusted [Fe]{} lines were in agreement (within $\pm0.05$ dex) and the slope of the relationship between the reduced EW of the [Fe]{} lines and their NLTE-adjusted iron abundance was minimised. Sections \[sec-temp\] and \[sec-nlte\] describe our final stellar parameter estimates for this method. Initial LTE-Fe Parameters {#sec-lte} ========================= The initial LTE-Fe stellar parameters for our sample stars are listed in Table \[tab-par\]. Residuals in the minimizations of this technique gave typical internal errors of 0.1 dex in both $\logglte$ and $\fehlte$ and $\sim55$ K in $\tlte$. As we show in the following sections, these small internal errors can be quite misleading as they are not representative of the actual accuracy of stellar parameter estimates. Often, especially in metal-poor stars, estimates of $\teff$, $\logg$, and $\feh$, that result from this method are far too low when compared to other, more accurate data (cf., R11). [@rrrrrrrrrrrrrrrrr@]{} & & & & & & & &\ \ Star & $\teff$ & $\eteff$ & $\logg$ & $\feh$ & $\vt$ & $E(B\mbox{-}V)$ & $\tirfm$ & $\etirfm$ & $\tr11$ & $\tbal$ & $\etbal$ & $\teff$ & $\eteff$ & $\logg$ & $\feh$ & $\vt$\ & (K) & (K) & ($\pm0.1$) & ($\pm0.1$) & & & (K) & (K) & ($\pm140$ K) & (K) & (K) & (K) & (K) & ($\pm0.1$) & ($\pm0.1$) &\ C0023306-163143 & 5128 & 58 & 2.40 & -2.63 & 1.3 & – & – & – & 5528 & 5400 & 100 & 5443 & 101 & 3.20 & -2.29 & 0.9\ C0315358-094743 & 4628 & 40 & 1.51 & -1.40 & 1.5 & – & – & – & 4722 & 4800 & 100 & 4774 & 89 & 2.06 & -1.31 & 1.6\ C0408404-462531 & 4466 & 40 & 0.25 & -2.25 & 2.2 & – & – & – & 4600 & – & – & 4600 & 90 & 1.03 & -2.10 & 2.1\ C0549576-334007 & 5151 & 50 & 2.53 & -1.94 & 1.3 & – & – & – & 5379 & 5400 & 100 & 5393 & 82 & 3.16 & -1.70 & 1.1\ C1141088-453528 & 4439 & 40 & 0.39 & -2.42 & 2.1 & – & – & – & 4592 & 4500 & 200 & 4562 & 123 & 1.10 & -2.28 & 1.9\ This table is published in its entirety in the electronic edition of the MNRAS. A portion is shown here for guidance regarding its form and content. Effective Temperature Optimisation {#sec-temp} ================================== It was found in @bergemann12 and @lind12 that taking into account NLTE in the solution of excitation equilibrium does not lead to a significant improvement of the stellar effective temperature. This was also supported by our test calculations for a sub-sample of stars. [Fe]{} lines formed in LTE or NLTE are still affected by convective surface inhomogeneities and overall different mean temperature/density stratifications, which are most prominent in strong low-excitation [Fe]{} lines [@shchukina05; @bergemann12]. Using 1D hydrostatic models with either LTE or NLTE radiative transfer thus leads to effective temperature estimates that are too low when the excitation balance of [Fe]{} lines is used (see below). It is therefore important that the stellar effective temperature be estimated by other means. Three Effective Temperature Scales ---------------------------------- We used three different methods to compute the effective temperature. The first estimate, $\tbal$, was derived from the wings of the Balmer lines, which is among the most reliable methods available for the effective temperature determination of FGK stars [e.g., @fuhrmann93; @fuhrmann98; @barklem02; @cowley02; @gehren06; @mashonkina08]. The only restriction of this method is that for stars cooler than $4500$ K, the wings of [H]{} lines become too weak to allow reliable determination of $\teff$. Profile fits of H$_\alpha$ and H$_\beta$ lines were performed by careful visual inspection of different portions of the observed spectrum in the near and far wings of the Balmer lines which were free of contaminant stellar lines. Figures \[fig-bal3\] and \[fig-bal5\] show two example fits to H$_{\alpha}$. Note that the Balmer lines were self-contained within a single order in each spectrum. Therefore, we did not use neighbouring orders for the continuum normalisation. Theoretical profiles were computed using the SIU code with `MAFAGS-ODF` model atmospheres [@fuhrmann98; @grupp04a]. Same as `ATLAS-ODF` (Section \[sec-par\]), the MAFAGS models were computed with Kurucz opacity distribution functions, thus the differences between the model atmosphere stratifications are expected to be minimal in our range of stellar parameters. For self-broadening of H lines, we used the @ali65 theory. As shown by @grupp04a this method successfully reproduces the Balmer line spectrum of the Sun within $20$ K, and provides accurate stellar parameters that agree very well with [*Hipparcos*]{} astrometry [@grupp04b]. The errors are obtained directly from profile fitting, and they are largely internal, $\pm50$ to 100 K. ![Example fit to H$_{\alpha}$ in the spectrum of the metal-poor ($\feh\sim-1.0$) dwarf, J142911.4-053131. The solid, red curve shows the best fit to the data (at 5700 K), while the blue, dashed curves represent $\pm100$ K around the fit.[]{data-label="fig-bal3"}](J142.eps){width="84mm"} ![Example fit to H$_{\alpha}$ in the spectrum of the metal-poor ($\feh\sim-2.3$) giant, J230222.8-683323. The solid, red curve shows the best fit to the data (at 5200 K), while the blue, dashed curves represent $\pm100$ K around the fit.[]{data-label="fig-bal5"}](J230.eps){width="84mm"} A key advantage of the Balmer lines is that they are insensitive to interstellar reddening, which affects photometric techniques (see below). However, the Balmer line effective temperature scale could be affected by systematic biases, caused by the physical limitation of the models. The influence of deviations from LTE in the [H]{} line formation in application to cool metal-poor stars was studied by @mashonkina08. Comparing our results to the NLTE estimates by @mashonkina08 for the stars in common, we obtain: $\Delta \teff$(our - M08) $= -70$ K (HD 122563 metal-poor giant), $\Delta \teff$(our - M08) $= 50$ K (HD 84937, metal-poor turn-off). The difference is clearly within the $\teff$ uncertainties. On the other side, it should be kept in mind that the atomic data for NLTE calculations for hydrogen are of insufficient quality and, at present, do not allow accurate quantitative assessment of NLTE effects in H, as elaborately discussed by @barklem07. Likewise, the influence of granulation is difficult to assess. @ludwig09 presented 3D effective temperature corrections for Balmer lines for a few points on the HRD, for which 3D radiative-hydrodynamics simulations of stellar convection are available. For the Sun[^4], they find $\Delta \teff (\rm{3D - 1D}) \approx 35$ K, and for a typical metal-poor subdwarf with \[Fe/H\] $=-2$, $\Delta \teff (\rm{3D - 1D})$ of the order $50$ to $80$ K (average over H$_\alpha$, H$_\beta$, and H$_\gamma$). However, in the absence of consistent 3D NLTE calculations, it is not possible to tell whether 3D and NLTE effects will amplify or cancel for FGK stars. Thus, we do not apply any theoretical corrections to our Balmer effective temperatures. Currently, the only way to understand whether our Balmer $\teff$ scale is affected by systematics is by comparing with independent methods, in particular interferometry. We, therefore, computed the Balmer $\teff$ for several nearby stars with direct and indirect interferometric angular diameter measurements. The results are listed in Table \[tab-inter\], while we plot the difference between our Balmer estimate and that from interferometry in Figure \[fig-int\]. Both $\teff$ scales show an agreement of $3\pm60$ K for stars with $\feh > -1$, while the Balmer estimate is $\sim50$ K warmer than $\tint$ at the lowest metallicities. These differences are well within the combined errors in the interferometric and Balmer measurements. This suggests that deviations from 1D HE *and* LTE are either minimal, or affect both interferometric and Balmer $\teff$ in exactly same way. Also note, for the stars in common with @cayrel11, our estimates are fully consistent. -------- --------- -------- --------- ------------------ --------- ------------------ ------ HD $\logg$ $\feh$ $\tint$ $\sigma_{\tint}$ $\tbal$ $\sigma_{\tbal}$ Ref. (K) (K) (K) (K) 6582 4.50 -0.70 5343 18 5295 100 a 10700 4.50 -0.50 5376 22 5320 100 a 22049 4.50 0.00 5107 21 5050 100 a 22879 4.23 -0.86 5786 16 5800 100 b\* 27697 2.70 0.00 4897 65 4900 100 c 28305 2.00 0.00 4843 62 4800 100 c 29139 1.22 -0.22 3871 48 4000 200 c 49933 4.21 -0.42 6635 18 6530 100 b 61421 3.90 -0.10 6555 17 6500 100 a 62509 2.88 0.12 4858 60 4870 100 c 84937 4.00 -2.00 6275 17 6315 100 b\* 85503 2.50 0.30 4433 51 4450 100 b 100407 2.87 -0.04 5044 33 5025 100 b 102870 4.00 0.20 6062 20 6075 100 a 121370 4.00 0.20 5964 18 5975 100 a 122563 1.65 -2.50 4598 42 4650 100 d 124897 1.60 -0.54 4226 53 4240 200 c 140283 3.70 -2.50 5720 29 5775 100 b\* 140573 2.00 0.00 4558 56 4610 100 c 150680 4.00 0.00 5728 24 5795 100 a 161797 4.00 0.20 5540 27 5550 100 a 215665 2.25 0.12 4699 71 4800 100 c -------- --------- -------- --------- ------------------ --------- ------------------ ------ : Effective temperatures determined from direct interferometric measurements of angular diameters.[]{data-label="tab-inter"} References for $\tint$: a - @cayrel11; b - [*Gaia*]{} calibration stars from U. Heiter (2012, priv. comm.) (Those marked with a \[\*\] have angular diameters determined using the surface-brightness relations from @kervella04); c - @mozurkewich03; d - @creevey12 ![Comparison of the Balmer effective temperature estimate to that from interferometry, as listed in Table \[tab-inter\]. The Balmer effective temperature estimates agree with the interferometric estimate to within $\sim100$ K.[]{data-label="fig-int"}](TbalvTint.eps){width="84mm"} For the second method, we utilised the Tycho-2 and 2MASS photometry of each star to compute effective temperature estimates, $\tirfm$, using the infrared flux method (IRFM), as presented in @casagrande10 [hereafter C10]. Note that in C10 the IRFM calculations were applied only to dwarfs and subgiants and validated by comparison with a large body of interferometric angular diameters. However, the same code can be safely applied to lower surface gravities, as shown by comparison with newly determined angular diameters for giants [@huber12]. The advantage of this method is that it is much less sensitive to model assumptions that are required for spectroscopic analyses. However, the quality of the photometric data used to compute IRFM effective temperatures, as well as interstellar reddening, can still largely affect the result. In our case reddening has been estimated using the @drimmel03 map, with distances derived from our spectroscopic $\logg$. Typical values of $E(B\mbox{-}V)$ are around 0.05 mag, although for some of the brightest giants the value can be considerably larger (see Table \[tab-par\]). Finally, we chose the effective temperature estimates from R11 (denoted as $\tr11$) as our third effective temperature scale. These estimates were based on the R11 calibration, which was derived from the the trend between $\fehlte$ and the difference between $\tlte$ and the 2MASS $\jks$ photometric effective temperature for several globular cluster stars, [*Hipparcos*]{} stars, and low-reddened stars in the R11 sample. In principle, this method should yield similar effective temperature estimates to that of IRFM, since it utilises the colour-temperature transformations presented in @ghernandez09, which were based upon their IRFM calculations. The advantage is that the R11 calibration relies on the $\jks$ colour, which is less sensitive to reddening ($E(\jks)\sim0.5E(B\mbox{-}V)$). However, note that the $\jks$ colour correlates only mildly with $\teff$, and thus calibrations involving that index exhibit a rather larger internal dispersion (in our case 139 K for dwarfs and 94 K for giants) when compared to IRFM effective temperature estimates for standard stars. Further, effective temperatures computed using the calibration in C10 are typically $\sim40$ K hotter than those computed in @ghernandez09. A detailed explanation for this discrepancy is given in C10. Comparisons ----------- We next applied each of the above methods to our sample stars, the values of which can be found in Table \[tab-par\]. Note that for several stars, the Balmer lines fell in the middle of the order of the spectrum. The continuum cannot be determined with sufficient accuracy in such regions. We therefore did not measure the Balmer lines for those stars. Further, not all stars in our sample have Tycho-2 photometry estimates. We were unable to compute an IRFM effective temperature estimate for these stars. Figure \[fig-tcomp\] shows the comparisons between the three effective temperature estimates when applied to our sample stars. The estimates from $\tbal$ and $\tr11$ show remarkable agreement, with a difference of only $1\pm 79$ K. The IRFM effective temperatures, however, are systematically higher than both $\tbal$ and $\tr11$ (by $119\pm215$ K and $113\pm186$ K, respectively), with an increasing dispersion towards hotter effective temperatures. ![Comparison of the three different $\teff$ scales. Both the estimate from R11 ($\tr11$) and the Balmer lines ($\tbal$) agree within $\sim80$ K, while that from IRFM ($\tirfm$) is systematically higher than both, and shows an increasing dispersion with increasing effective temperature. Error bars, which show the mean error in each $\teff$ scale, are displayed in the upper left corner.[]{data-label="fig-tcomp"}](Tcomp.eps){width="84mm"} It is possible that inherent NLTE and 3D effects could be influencing the Balmer effective temperature scale, however, we see excellent agreement with $\tr11$ and interferometric measurements. As stated previously, the effective temperature estimates computed in C10 are about 40 K warmer than those computed in @ghernandez09. Further, the $\jks$ calibration has a large internal dispersion. However, the difference between $\tirfm$ and $\tr11$ extends well beyond these limits. It is possible that the uncertainty in the interstellar reddening may be systematically affecting the IRFM estimates. In order to test the accuracy of the estimates of reddening from the @drimmel03 map, we also tried to measure $E(B-V)$ using the interstellar [Na]{} lines. However, the majority of the stars in our sample had multi-component interstellar [Na]{} features, or the feature was not discernible from the stellar Na lines. Note, for several of the stars in which we could measure single-component interstellar [Na]{} lines, the Drimmel et al. estimates and the [Na]{} estimates were on average different by $<0.02$ mag, which translates to a difference of $<100$ K in effective temperature. Given these differences, reddening alone cannot account for the very large differences ($\ge300$ K) for many stars. Instead, the large scatter mostly likely arises from the poor quality of the Tycho-2 magnitudes for stars fainter than $V_{\rm Tycho}\sim9$. For the kind of stars analysed in this work, $B$ and $V$ magnitudes are the dominant contributors to the bolometric flux, as compared to the infrared 2MASS magnitudes. Should a star be matched with a brighter (dimmer) source in $B$ and $V$, then the bolometric flux will be over-estimated (underestimated) by a very large amount, and IRFM will return a systematically higher (lower) $\tirfm$ estimate. Using the sample in C10, it is possible to compute $\teff$ using both the Tycho-2 $V_{\rm Tycho}\mbox{-}K_{\rm s}$ and Johnson-Cousins $V_{\rm JC}\mbox{-}K_{\rm s}$ calibrations. We plot the difference between $T(V_{\rm Tycho}\mbox{-}K_{\rm s})$ and $T(V_{\rm JC}\mbox{-}K_{\rm s})$ as a function of the Tycho-2 $V$-magnitude for the C10 sample (black points) in Figure \[fig-ty\]. In addition, we have over-plotted the difference between $\tirfm$ and $\tbal$ for the stars in our present sample (red triangles). As shown in the figure, both samples exhibit a large scatter in the difference in effective temperature estimates at $V_{\rm Tycho}\ge9$. Further, the hotter stars are on average among the faintest stars in our sample and thus have larger errors in $V_{\rm Tycho}$. This is a clear indicator that the poor quality of the Tycho-2 photometric measurements for our stars is responsible for the discrepancy between $\tirfm$ and $\tbal$. Final Effective Temperature Estimates ------------------------------------- From the comparisons above, Balmer line measurements provide the most reliable effective temperature estimates for all stars in our sample. In addition, $\tr11$ exhibits small differences with respect to $\tbal$ across all effective temperatures, as illustrated in Figure \[fig-tcomp\]. In contrast, IRFM effective temperatures appear to show a large dispersion, which we attribute to large errors in Tycho-2 photometry, as well as uncertainty in the reddening. We therefore adopted the mean of only $\tbal$ and $\tr11$, weighted according to the internal errors from each method, as our final $\tfin$ estimate. This also serves to reduce the internal error on the final optimal effective temperature estimate, which was typically $\simlt100$ K. As noted previously, we could not measure the Balmer lines in several of the stars in our sample. For those stars, we adopted the $\tr11$ estimate. The authors of R11 adopted an error of 140 K in their $\tr11$ estimate, which was derived from the residuals in their calibration. However, the comparison between $\tbal$ and $\tr11$ in Figure \[fig-tcomp\] suggests that this value was overestimated. The mean error in $\tfin$, for those stars with both a $\tbal$ and $\tr11$ estimate, was 90 K. We therefore adopted this value for stars with only a single $\tr11$ estimate. Our final $\tfin$ values and corresponding errors can be found in Table \[tab-par\]. Surface Gravity and \[Fe/H\] in NLTE {#sec-nlte} ==================================== Using the final values of $\tfin$ described in the previous section, we derived the remaining stellar parameters using the NLTE-Opt method described in Section \[sec-par\]. We first validated this methodology by applying our analysis to a sample of 18 “standard" stars, which have [*Hipparcos*]{} parallaxes. The spectra for these stars were obtained for the analysis in @fulbright00, and are of similar quality to our sample. Both the LTE-Fe and NLTE-Opt atmospheric parameters for each standard star are given in Table \[tab-hip\]. Using the [*Hipparcos*]{} parallax and an estimate of the bolometric correction derived from the bolometric flux relations presented in @ghernandez09, we also computed an “astrometric surface gravity" ($g_{\pi}=4\pi GM\sigma T^4/L$) for each star, which is listed as $\loggpi$ in Table \[tab-hip\]. Note, we computed an astrometric surface gravity using other various flux relations [@alonso95; @casagrande10; @torres10], finding results within $\simlt0.1$ dex of that computed using the relation in @ghernandez09. Further, we assumed a mass of $0.8~M_{\odot}$ for $\feh<-1$ and $0.9~M_{\odot}$ for $\feh\ge-1$. However, a difference of $0.1~M_{\odot}$ will only change the astrometric surface gravity by $\sim0.05$ dex. The NLTE-Opt surface gravity estimates show a remarkable $0.02\pm0.11$ dex agreement with the astrometric surface gravity, while the LTE-Fe estimates are too low by $-0.32\pm0.39$ dex. Given the agreement, we applied the above analysis to our sample stars. The final NLTE-Opt estimates for surface gravity and metallicity, as well as for the microturbulence, can be found in Table \[tab-par\]. We adopted 0.1 dex error in both the surface gravity and metallicity, based on our comparisons with the standard stars above. ----------- ----------------- ------------ ------------ --------------- ------------ ------------ ------------ HD $\tlte$ $\logglte$ $\fehlte$ $\tfin$ $\loggf$ $\fehf$ $\loggpi$ [*err*]{} ($\sim\pm60$ K) ($\pm0.1$) ($\pm0.1$) ($<\pm100$ K) ($\pm0.1$) ($\pm0.1$) ($\pm0.1$) 22879 5726 4.04 -0.92 5817 4.27 -0.89 4.33 24616 5084 3.34 -0.62 5071 3.40 -0.69 3.29 59374 5741 4.04 -0.96 5877 4.33 -0.88 4.49 84937 6137 3.58 -2.34 6374 4.18 -2.11 4.15 108317 4922 1.89 -2.58 5367 3.04 -2.14 3.14 111721 4956 2.52 -1.37 5091 2.93 -1.29 2.70 122956 4569 1.15 -1.75 4750 1.94 -1.61 2.03 134169 5868 4.03 -0.77 5924 4.20 -0.74 4.03 140283 5413 2.81 -2.79 5834 3.71 -2.41 3.73 157466 6070 4.41 -0.34 6002 4.37 -0.41 4.35 160693 5808 4.29 -0.47 5749 4.24 -0.55 4.31 184499 5740 4.11 -0.58 5766 4.23 -0.57 4.08 193901 5555 3.94 -1.18 5775 4.39 -1.01 4.57 194598 5814 4.02 -1.23 5991 4.39 -1.10 4.27 201891 5676 3.89 -1.21 5871 4.30 -1.06 4.30 204155 5696 3.94 -0.71 5733 4.08 -0.69 4.03 207978 6343 3.93 -0.62 6294 4.02 -0.62 3.96 222794 5588 3.99 -0.66 5604 4.08 -0.66 3.91 ----------- ----------------- ------------ ------------ --------------- ------------ ------------ ------------ NLTE-Opt vs. LTE-Fe =================== In Figure \[fig-lte\], we compare our final NLTE-Opt stellar parameters to those derived using the LTE-Fe method. The differences in the estimates of effective temperature, surface gravity, metallicity, and microturbulence all display clear trends with decreasing metallicity. The microturbulent velocity is underestimated by $\sim0.1\mbox{-}0.2$  until $\fehlte\sim-1.8$, where $\vtlte$ becomes larger than $\vtf$. The differences between $\teff$ range from $200$ to $400$ K for metal-poor giants, and $-50$ to $-100$ K for dwarfs. The differences for $\log g$ and \[Fe/H\] reach a factor of $30$ in surface gravity ($\Delta \log g = 1.5$ dex) and a factor of $3$ in metallicity ($\Delta$ \[Fe/H\] $=0.5$ dex) at \[Fe/H\] $\sim -2.5$. ![Comparison of stellar parameters derived using the LTE-Fe method and the NLTE-Opt stellar parameters vs. $\fehlte$. The difference in effective temperature, surface gravity, and metallicity show a large systematic increase with decreasing metallicity. The dual trends seen in $\Delta~\teff$, $\Delta~\logg$, and $\Delta~\feh$ are a result of the R11 effective temperature calibration, in which the authors found that stars with effective temperatures less than 4500 K only required a small correction to $\tlte$. Therefore, these stars stand out in the plots.[]{data-label="fig-lte"}](Sparam-comp.eps){width="84mm"} Figure \[fig-hrd\] illustrates how the different LTE-Fe and NLTE-Opt results can change the position of each star in the $\logg$ vs. $\log(\teff)$ plane. In addition, we have included several evolutionary tracks, computed using the GARSTEC code [@weiss08], for comparison. Generally, the NLTE-Opt estimates of surface gravity and effective temperature trace the morphology of the theoretical tracks much more accurately. Several features are most notable. The NLTE-Opt parameters lead to far less stars that lie on or above the tip of the red giant branch, and more stars occupy the middle or lower portion of the RGB. Also, stars at the turn-off and subgiant branch are now more consistent with stellar evolution calculations. Figures \[fig-lte\] and \[fig-hrd\] further prompted us to determine the relative importance of the effective temperature scale versus the NLTE corrections for gravities and metallicities in the NLTE-Opt method. We singled out the effect of the NLTE corrections by deriving additional, LTE-Opt surface gravity and metallicity estimates using LTE iron abundances combined with our $\tfin$ estimate. Note, as with the NLTE-Opt method, Fe lines which have an excitation potential below 2 eV were excluded. The comparison between these LTE-Opt estimates and the final NLTE-Opt estimates are shown in Figure \[fig-copt\]. As evident from this figure, solving for ionisation equilibrium in NLTE also leads to *systematic* changes in the $\log g$ and \[Fe/H\], such that LTE gravities are under-estimated by $0.1$ to $0.3$ dex, whereas the error in metallicity is about $0.05$ to $0.15$ dex. These effects are consistent with that seen in @lind12. We thus conclude that reliable effective temperatures are necessary to avoid substantial biases in a spectroscopic determination of $\log g$ and \[Fe/H\], such as displayed in Figure \[fig-lte\]. We have shown here that, at present, excitation balance of [Fe]{} lines with 1D hydrostatic model atmospheres in LTE does not provide the correct effective temperature scale, supporting the results by Bergemann et al. (2012). On the contrary, Balmer lines provide such a scale. Furthermore, NLTE effects on ionisation balance are necessary to eliminate the discrepancy between [Fe]{} and [Fe]{} lines, an effect that is present, regardless of the adopted $\teff$. Only in this way is it possible to determine accurate surface gravity and metallicity from Fe lines. Conclusion ========== In this work, we explore several available methods to determine effective temperature, surface gravity, and metallicity for late-type stars. The methods include excitation and ionization balance of Fe lines in LTE and NLTE, semi-empirically calibrated photometry (R11), and the Infra-Red flux method (IRFM). Applying these methods to the large set of high-resolution spectra of metal-poor FGK stars selected from the RAVE survey, we then devise a new efficient strategy which provides robust estimates of their atmospheric parameters. The principal components of our method are (i) Balmer lines to determine effective temperatures, (ii) NLTE ionization balance of Fe to determine $\logg$ and $\feh$, and (iii) restriction of the [Fe]{} lines to that with the lower level excitation potential greater than 2 eV to minimize the influence of 3D effects [@bergemann12]. A comparison of the new NLTE-Opt stellar parameters to that obtained from the widely-used method of 1D LTE excitation-ionization of Fe, LTE-Fe, reveals significant *systematic biases* in the latter. The difference between the NLTE-Opt and LTE-Fe parameters systematically increase with decreasing metallicity, and can be quite large for the metal-poor stars: from 200 to 400 K in $\teff$ , 0.5 to 1.5 dex in $\logg$, and 0.1 to 0.5 dex in $\feh$. These systematic trends are largely influenced by the difference in the estimate of the stellar effective temperature, and thus, a reliable effective temperature scale, such as the Balmer scale, is of critical importance in any spectral parameter analysis. However, a disparity between the abundance of iron from [Fe]{} and [Fe]{} lines still remains. It is therefore necessary to include the NLTE effects in [Fe]{} lines to eliminate this discrepancy. The implications of the very large differences between the NLTE-Opt and LTE-Fe estimates of atmospheric parameters extend beyond that of just the characterisation of stars by their surface parameters and abundance analyses. Spectroscopically derived parameters are often used to derive other fundamental stellar parameters such as mass, age and distance through comparison to stellar evolution models. The placement of a star along a given model will be largely influenced by the method used to determine the stellar parameters. For example, distance scales will change, which could affect the abundance gradients measured in the Milky Way (e.g., R11), as well as the controversial identification of different components in the MW halo [@schonrich11; @beers12]. We explore this in greater detail in the next paper of this series [@serenelli12]. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge valuable discussions with Martin Asplund, and are indebted to Ulrike Heiter for kindly providing interferometric temperatures for several [*Gaia*]{} calibration stars. We also acknowledge the staff members of Siding Spring Observatory, La Silla Observatory, Apache Point Observatory, and Las Campanas Observatory for their assistance in making the observations for this project possible. Greg Ruchti acknowledges support through grants from ESF EuroGenesis and Max Planck Society for the FirstStars collaboration. Aldo Serenelli is partially supported by the European Union International Reintegration Grant PIRG-GA-2009-247732, the MICINN grant AYA2011-24704, by the ESF EUROCORES Program EuroGENESIS (MICINN grant EUI2009-04170), by SGR grants of the Generalitat de Catalunya and by the EU-FEDER funds. 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ArXiv

--- abstract: 'The Gao-Wald theorem related to time delay [@gaowald] assumes that the Null Energy Condition and the Null Generic Condition are satisfied, and that the underlying gravity theory is General Relativity. In the present work it is shown that the Gao-Wald theorem is true if the space time is null geodesically complete, if the curvature satisfy some reasonable properties stated along the text, and if every null geodesic contains at least two conjugate points. This result may apply to modified theory of gravities and to violating Null Energy Condition models as well.' author: - 'Juliana Osorio Morales [^1] and Osvaldo P. Santillán [^2]' title: Variations of a theorem due to Gao and Wald --- Introduction ============ Since the introduction of the *Alcubierre bubble* [@alcubierre] or the *Krasnikov tube* [@krasnikov], there has been a growing interest in the concept of time delay in General Relativity as well as in modified theories of gravity. The definition of time delay is indeed very subtle [@olum]. The Alcubierre bubble is a space time in which it is possible to make a round trip from two stars $A$ and $B$ separated by a proper distance $D$ in such a way that a fixed observer at the star $A$ measures the proper time for the trip as less than $2D/c$. In fact, this time can be made arbitrary small. This fact does not indicate that the observers travel faster than light, as they are traveling inside their light cone. The Alcubierre constructions employ the fact that, for two comoving observers in an expanding universe, the rate of change of the proper distance to the proper time may be larger than $c$ or much more smaller, if there is contraction instead of expansion. The Alcubierre space time is Minkowski almost everywhere, except at a bubble around the traveler which endures only for a finite time, designed for making the round trip proper time measured by an observer at the star $A$ as small as possible. Details can be found in [@alcubierre]. The examples given above are interesting, however a more careful definition of time delay was introduced in [@olum]. In this reference, a space time which appears to allow time advance was constructed, but it was proven that it is in fact the flat Minkowski metric in unusual coordinates. This suggests that to analyze time advance by simple inspection of the metric may be misleading. However, given a space time that is Minkowski outside a tube or a bubble such that the Alcubierre or Krasnikov space times, the notion of time delay is well defined. By use of some results due to Tipler and Hawking [@tipler1]-[@tipler3], it can be shown that all these examples violate the Null Energy Conditions at least in some region of the manifold. Further issues related to time delay and quantum gravity can be found in the works [@otras]-[@otras4] and references therein. Recall that the Null Energy Condition states that the matter content energy momentum tensor satisfies $T_{\mu\nu}k^\mu k^\nu\geq 0$ for every null vector $k^\mu$ tangent to any null geodesic $\gamma$. This implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$ [@Wald]. On the other hand, the Null Generic Condition means that $k_{[\alpha} R_{\beta]\sigma\delta[\epsilon}k_{\gamma]}k^\sigma k^\delta\neq 0$ for some point in the geodesic $\gamma$. Both conditions automatically imply that any null geodesic $\gamma(\lambda)$ possesses at least a pair of conjugate points $p$ and $q$, if it is past and future inextendible, see [@Wald Proposition 9.3.7]. These results hold in the context of General Relativity, and should not be extrapolated to modified gravity theories without further analysis. The results just described raise the question of whether time delay could hold in theories which do not violate the Null Energy Conditions. In this context, a theorem due to Gao and Wald [@gaowald] may be relevant. Its statement is the following.\ *Gao-Wald theorem:* Consider a null geodesically complete space time ($M$, $g_{\mu\nu}$) such that the Null Energy and Null Generic Conditions are satisfied. Then, given a compact region $K$, there exists a compact $K'$ containing $K$ such that for any pairs of points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ connecting both points intersects $K$.\ The Gao-Wald theorem stated above is related to time advance hypothesis as follows. If there were possible to deform the geometry in a region $K$, similar perhaps to a bubble, in order to produce a time advance, then a fastest null geodesic would enter in the region $K$ in order to minimize this time. The theorem states that this is not possible if the Null Energy Conditions and Null Generic Conditions are satisfied in the space time in consideration. This may constitute a no go theorem. However, there is no control over the size of the region $K'$, thus this theorem should be considered only as a weak version of a time advance hypothesis. The aim of the present work is two folded. The first purpose is to show that the Null Energy and Null Generic conditions are not mandatory, neither is to work in the context of General Relativity, for the Gao-Wald theorem to be true. It will be shown that the Gao-Wald theorem holds when the following three requirements are satisfied.\ - *First requirement:* The space time ($M$, $g_{\mu\nu}$) is null geodesically complete.\ - *Second requirement:* Every null geodesic possesses at least two conjugate points.\ -*Third requirement:* Consider the set $S$ of pairs $\Lambda_0=$($p_0$, $k_0^\mu$) with $p$ a point in $M$ and $k_\mu$ a null vector in $TM_{p_0}$ properly normalized (see formula (\[norma\]) below) and defining a null geodesic $\gamma_0$. Then there exists an open set $O$ in $S$ containing $\Lambda_0$ for which the following two properties hold. For every pair $\Lambda=$($p$, $k^\mu$) in $O$, the corresponding geodesic $\gamma_\Lambda(\gamma)$ will posses a conjugate point $q$ to $p$, $q \in J_+(p)-I_+(p)$. Furthermore the map $h: O\to M$ such that $h(\Lambda)=q$ is continuous at $\Lambda_0$.\ The two properties described in the *third requirement* look a bit technical, but the intuition behind is the following. The first implies that, for a geodesic with two conjugate points $p_0$ and $q_0$, there exist an open set around $p_0$ such all the points $p$ in the open set will have a conjugate point $q$ with respect to some null geodesic emanating from them. The second part states that the conjugate point $q$ to $p$ will be very close to $q_0$ when $p$ is close to $p_0$ and when the geodesics are, in a very rough sense, “pointing in similar directions”. The second and main purpose of the present work is to prove that the *second requirement* implies the *third one* under some more or less reasonable hypotheses about the curvature of the space time. We feel that this statement may be relevant for extending the Gao-Wald results to more general gravity theories or to models violating the Null Energy Conditions. The organization of the present work is as follows. In section 2 some generalities about conjugate points in generic space times are discussed. In addition, certain topological issues related to the light cones in space times are also presented. The presentation is not exhaustive, but focused in the aspects more relevant for our purposes. At the end, a proof of the *third requirement* when the underlying model is General Relativity with Null Energy and Null Generic Conditions outlined. This is included by completeness, as this is one of the results to be generalized here. In section 3, some properties for the curvature of the space time are presented, which are not related neither to General Relativity nor to the weak and strong energy conditions, ensuring that the *second requirement* implies the *third requirement*. In section 4 the aforementioned implication is proved explicitly by the means of some propositions described in section 3. This section is rather technical. In section 5, the modified Gao-Wald theorem is proved explicitly, and the possible application of the obtained results is discussed. The *third requirement* in GR with Null Energy and Null Generic Conditions ========================================================================== As discussed above, the Gao-Wald theorem relies on the notion of conjugate points. Thus, it is convenient to recall some basic but important concepts about them, taking into account some standard references [@Wald]-[@penrose]. In addition, at the end of this section, a sketch of the proof of the *third requirement* in the context of GR and with Null Energy and Null Generic Conditions [@gaowald] is included. The next sections are devoted to generalize this proof to more general gravity models. Null geodesics and conjugate points ----------------------------------- In the present discussion, the space time ($M$, $g_{\mu\nu}$) is assumed to be null geodesically complete such that there exists a globally defined time like future pointing vector $t_\mu$ on it. Given a point $p$ in ($M$, $g_{\mu\nu}$), a point $q$ in $J_+(p)-I_+(p)$ is said to be conjugated to $p$ if the following holds. Consider a null geodesic $\gamma(\lambda)$ emanating from $p$, together with the associated differential equation =-R\^\_k\^k\^A\^\_, supplemented with the following initial conditions $$A^\mu_\nu|_p=0,\qquad \frac{dA^\mu_\nu}{d\lambda}\bigg|_p=\delta^\mu_\nu.$$ Here $\lambda$ is the affine parameter describing $\gamma(\lambda)$ and $k^\mu$ is a vector tangent to the curve $\gamma(\lambda)$, normalized by the following conditions k\^k\_=0, k\^t\_=-1. The point $q=\gamma(\lambda_0)$ is said to be conjugated to $p$ if and only if $$A_\mu^\nu(\lambda_0)=0.$$ The matrix $A^\mu_\nu(\lambda)$ has the following interpretation: the number $A^{\mu}_{\nu}$ are the coefficients of the Jacobi field $\eta^{\mu}$ along $\gamma$, i.e, $$\eta^\mu(\lambda)=A^\mu_\nu(\lambda) \frac{d\eta^\nu}{d\lambda}\bigg|_0,\qquad \eta(0)|_p=0,$$ then (\[smile\]) implies that $\eta(\lambda)$ satisfies the Jacobi equation (hence the name) on $\gamma$ given by =-R\^\_k\^k\^\^. The classical definition of a conjugate point $q$ to $p$ is the existence of a solution $\eta^\mu(\lambda)$ of the Jacobi equation such that $\eta^\mu(0)=0$ and $\eta^\mu(q)=0$. Clearly, the fact that $A^\mu_\nu(\lambda_0)=0$ implies that $\eta^\mu(q)=0$, thus $q$ is a conjugate point to $p$ in the usual sense. For further details see [@Wald Section 9.3]. There is no warrantee that there exists a point $q$ conjugate to a generic point $p$ for a given space time ($M$, $g_{\mu\nu}$). In addition, there might exist two or more different points $q$ and $s$ conjugate to $p$, joined to $p$ by different geodesics. The study of conjugate points has been proven to have many applications in Riemannian and Minkowski geometry. It is well known that, in Riemannian geometry, a geodesic $\gamma(\lambda)$ starting at a point $p=\gamma(0)$ and ending at a point $r=\gamma(\lambda_0)$ is not necessarily length minimizing if there is a conjugate point $q=\gamma(\lambda_1)$ to $p$ such that $\lambda_1<\lambda_0$. The presence of a conjugate point in the middle usually spoil the minimizing property. For time like geodesics in Minkowski geometries, the proper time elapsed to travel between $p$ and $r$ is not maximal if there is a conjugate point in the middle. For null geodesics, there is an important result which will be used below, see [@Wald Theorem 9.3.8]. Let $\gamma$ a smooth causal curve and let $p, r\in \gamma$. Then there does not exist a smooth one parameter family of causal curves $\gamma_s$ connecting both points, such that $\gamma_0=\gamma$ and such that $\gamma_s$ are time like for $s>0$ if and only if there is no conjugate point $q$ to $p$ in $\gamma$. By reading this statement as a positive affirmation, it is found that if a null curve connecting $p$ and $r$ can be deformed to a time like curve, then there is a pair of conjugate points in between and, conversely, if there is such pair, the curve can be deformed to a time like one. The matrix $A_\mu^\nu(\lambda)$ defined by equation (\[smile\]) takes values which depend on the choice of the null geodesic $\gamma$. For this reason it may be convenient to denote it as $(A_\gamma)^\mu_\nu$. The same follows for the quantity $$G_\gamma(\lambda)=\sqrt{\det A_\gamma(\lambda)},$$ which also vanish at both $p$ and $q$. Note that the initial conditions below (\[smile\]) imply that $\det A_\gamma>0$ until the point $q$ is reached, thus the square root in this definition does not pose a problem. The equation (\[smile\]) implies that $G_\gamma(\lambda)$ satisfies the following second order equation [@gaowald] =-\[\_\^+R\_ k\^k\^\]G\_, and that $G_\gamma(0)=0$ and $G_\gamma(\lambda_0)=0$. These two values correspond to the points $p$ and $q$. Here $\sigma_{\mu\nu}$ is the shear of the null geodesics emanating from $p$. The last is an equation of the form $$\frac{d^2 G_\gamma}{d\lambda^2}=-p_\gamma(\lambda) G_\gamma.$$ As near the point $p$ the initial conditions in (\[smile\]) imply that $A_\mu^\nu\sim \delta_\mu^\nu \lambda$ it follows that, at $\lambda=0$ one has that $$G_\gamma(0)=0, \qquad \frac{dG_\gamma(0)}{d\lambda}=0.$$ Then, if $p_\gamma(\lambda)$ is $C^{\infty}$, by taking derivatives of equation (\[smile2\]) with respect to $\lambda$ it may be shown that $$\frac{d^n G_\gamma(0)}{d\lambda^n}=0.$$ This suggest that $G_\gamma(\lambda)$ may not analytical at the point $\lambda=0$. Another typical equation appearing in the literature [@marolf] is given in terms of the expansion parameter $\theta_\gamma(\lambda)$, which is related to $G_\gamma(\lambda)$ by the formula G\_()=G\_i\_[\_i]{}\^\_() d, with $G_i=G(\lambda_i)$ the value of $\sqrt{\det A_\gamma(\lambda)}$ at generic parameter value $\lambda_i>0$. In terms of $\theta_\gamma$ the equation (\[smile2\]) becomes the well known Raychaudhuri equation +=-\_\^-R\_k\^k\^. The definition (\[folio\]) implies that \_= Thus $\theta_\gamma(\lambda)\to-\infty$ when $\lambda\to \lambda_0$, since $G_\gamma(\lambda)$ approaches to zero from positive values at $q$. Analogously, $\theta_\gamma(\lambda)\to\infty$ when $\lambda\to 0$, since $G_\gamma(\lambda)$ grows from the zero value when starting at $p$. On the other hand, the fact that $\theta_\gamma\to-\infty$ at $q$ itself does not imply that $G_\gamma(\lambda)\to 0$ when $\lambda\to\lambda_0$. This can be seen from (\[folio\]), as the integral of the divergent quantity $\theta_\gamma$ may be still convergent. By an elementary analysis of improper integrals it follows that, at the conjugate point $q=\gamma(\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ is divergent with degree \_()\~,0, up to multiplicative constant. The behavior (\[asin\]) will play an important role in the next sections. Some further remarks are in order. In a generic case, there is a possibility that $\theta=G'/2G$ may be divergent at a point $r$ non conjugate to $p$. In this case $G\neq 0$ but then $G'$ should be divergent. However, the equation (\[smile2\]) implies that $G''$ exists unless there is a singularity of $p_\gamma(\lambda)$. As an example, this may happen due to a some sort of singularity of the scalar $R_{\mu\nu}k^\mu k^\nu$ at $r$. The space times considered in the present work are assumed to be free of these pathologies. This means that $\theta_\gamma(\lambda)$ is well defined everywhere except at $p$ and $q$, where it takes the values $\pm \infty$. In other words, between the points $p$ and $q$ or, what is the same, when $\lambda$ varies in the interval $(0,\;\lambda_0)$, the expansion parameter $\theta_\gamma(\lambda)$ takes every real value. If instead $p$ does not have a conjugate point along $\gamma$, then $\theta_\gamma(\lambda)$ is expected to be finite and continuous for every finite value of $\lambda$. In addition, note that the quantity (\[folio\]) is not well defined when $\lambda_i \to 0$, that is, when the initial point is $p$. This reflects the expansion parameter is singular at $p$. Each of the equations (\[raychaudhuri\]) and (\[smile2\]) have their own advantages. In the following, both versions will play an important role, and will be employed in each situation by convenience. Future light cones in curved space times ---------------------------------------- In addition to conjugate points, another important concept is the future light cone emanating from a point $p$ in the space time ($M$, $g_{\mu\nu}$). Given the point $p$ one has to consider all the future directed null vectors $k^\mu$ in $TM_p$ which satisfy the normalization (\[norma\]). Far away from $p$ it is likely that these geodesics may form a congruence $\gamma_\sigma(\lambda)$, but for $\lambda=0$, the congruence is singular since $\gamma_\sigma(0)=p$ for every value of $\sigma$. In other words, $p$ is the tip of the cone. Close to the point $p$ there is an open set $U$ composed by points $p'$, with their respective set of future directed null vectors $k'^\mu$ in $TM_{p'}$ which satisfy the normalization (\[norma\]). When comparing geodesics emanating from different points $p$ and $p'$, one should compare not only both points but also the corresponding null vectors $k_\mu$ and $k'_\mu$. In some vague sense, two null geodesics $\gamma$ and $\gamma'$ are “close’ when $p$ and $p'$ are at close and the corresponding vectors $k_\mu$ and $k'_\mu$ ”point in similar directions". In order to put this comparison in more formal terms, it is convenient to introduce the set $S$ defined as follows [@gaowald] $$S=\{\Lambda=(p, k^\mu)~|~ p \in M, \quad k^\mu \in TM_p,\quad k^\mu k_\mu=0, \quad k^\mu t_\mu=-1\}.$$ This set has an appropriate topology which allows to compare a pair $\Lambda=(p, k^\mu)$ with another one $\Lambda'=(p', k'^\mu)$ and to determine if they are “close”. The definition implies that the vectors $k^\mu$ are all null and satisfying the normalization (\[norma\]). The null geodesic corresponding to the element $\Lambda=(p, k^\mu)$ will be denoted as $\gamma_\Lambda(\lambda)$ in the following. All the quantities depending on this curve such as $G_\gamma(\lambda)$ will be subsequently denoted as $G_\Lambda(\lambda)$ and so on. The reason for this notational change is the desire to study continuity properties of these quantities as functions on $S$. Proof of the *third requirement* for GR with Null Conditions ------------------------------------------------------------ The proof for the *third requirement* when the underlying theory is GR and the Null Energy and Null Generic Conditions are fulfilled is given in [@gaowald]. As this is one of the theorems to be generalized here, it is convenient to sketch the original argument. Consider a null geodesic $\gamma_0(\lambda)$ with $p_0=\gamma_0(0)$ and $q_0=\gamma_0(\lambda_0)$ conjugate points along it, with $\lambda_0>0$. Then $G_0(0)=G_0(\lambda_0)=0$ and $G_0(\lambda)>0$ for all $\lambda$ in the interval $0<\lambda<\lambda_0$. The Null Energy Condition $T_{\mu\nu}k^\mu k^\nu\geq 0$ implies, in the context of General Relativity, that $R_{\mu\nu}k^\mu k^\nu\geq 0$. This, together with (\[smile2\]) shows that $G_0''(\lambda)<0$ in the interval $0<\lambda<\lambda_0$. The mean value theorem applied to $G_0$ shows that $G_0'(\lambda_1)=-C^2$ for some value $\lambda_1$ in the interval and furthermore $G_0'(\lambda_1)<-C^2$ for $\lambda_1<\lambda<\lambda_0$, with $C^2$ a positive constant. By choosing $\lambda_0-\delta<\lambda<\lambda_0$ one has that $$\frac{G_0(\lambda_1)}{|G'_0(\lambda_1)|}<\delta,$$ since $|G_0'(\lambda)|$ is larger than $C^2$ if $\delta$ is small enough. Consider now a small open $O\subset S$ around the point $\Lambda_0=(p_0, k^\mu)$ generating $\gamma_0(\lambda)$. As $G_\Lambda(\lambda)$ and its derivatives are continuous when moving in this open, then $G'_\Lambda(\lambda)<0$ and $$\frac{G_\Lambda(\lambda_1)}{|G'_\Lambda(\lambda_1)|}<\delta,$$ for all the $\Lambda=(p, k^\mu)\in O$ if $O$ is small enough. As $G_\Lambda(\lambda)>0$ and $G_\Lambda''(\lambda)>0$ due to the Null Energy Condition, it can be shown that for all the points $\Lambda=(p, k^\mu)$ in $O$ one has $G_\Lambda(\lambda')=0$ for some point $\lambda'$ such that $|\lambda'-\lambda_0|<\delta$, see [@gaowald] for further details. This shows that there exists a conjugate point $q$ to $p$, which is close to $q_0$ when $O$ is small enough. This is basically the statement of the *third requirement*. For the purposes of the present work however, the Null Energy and Null Generic conditions are not assumed to hold. This means that it can not be assumed that $G_\Lambda''(\lambda)<0$ neither that mean value theorem applied to $G_\Lambda(\lambda)$, in the form presented above, is true. The following part is devoted to sort out the technical complications arising by relaxing these two important conditions. This effort may be useful, as it may allow to extend the Gao-Wald theorem to models with Averaged Null Energy Conditions or Quantum Null Energy Conditions [@averaged1]-[@averaged25] or to modified gravity theories [@odintsov]-[@odintsov3]. The interest in these conditions arises when considering quantum effects in gravitational models. The assumed properties for the space time $(M,\;g_{\mu\nu})$ ============================================================ The next step is to specify a new set of properties for the null geodesically complete space time ($M$, $g_{\mu\nu}$) which, together with the *second requirement*, lead directly to the *third requirement*. The implication is shown in the next sections. The postulated properties are the following.\ *Property 1:* $(M, \; g_{\mu\nu})$ is null geodesically complete. Every null geodesic $\gamma_\Lambda(\lambda)$ in $(M, \; g_{\mu\nu})$ will contain at least a pair of conjugate points $p$ and $q$, with $q$ in $J_+(p)-I_+(p)$.\ *Property 2:* For any constant $c>0$ the integral I\_(\_i)=\_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, corresponding to a generic geodesic $\gamma_\Lambda$, is always finite, for every finite value of the initial affine parameter $\lambda_i$. In addition, the integrand is continuous and derivable everywhere in the space time $(M, \;g_{\mu\nu})$.\ The *Property 1* is equivalent to the *first* and *second requirements* stated in the introduction. Note that space time manifold ($M$, $g_{\mu\nu}$) is *not* assumed to be a solution of General Relativity neither matter is assumed to satisfy the Null Energy and Null Generic Conditions. It should be emphasized that these properties are not necessarily the unique properties leading to a Gao-Wald theorem. The task of finding variation may be a relevant one. The *Property 2* looks a bit technical, but it may clarified as follows. It basically suggests that a light traveller measures the quantity $[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$, which involves the curvature, and does not find an asymptotic grow of an exponential type, with negative sign. In other words, the behavior $$[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda) \sim- e^{c\lambda},$$ when $\lambda\to \infty$ is forbidden. This is a ad hoc hypothesis, but in authors’s opinion, it is physically reasonable and may cover part of the spectrum of null energy violating models. The main task is to show that properties 1 and 2 imply that ($M$, $g_{\mu\nu}$) satisfy the three requirements described in the previous section and consequently that they lead to a Gao-Wald theorem. There is another reason for considering the quantity (\[tang\]), which is related to a mathematical property of the Raychaudhuri equation. This property is summarized in the propositions of the next subsection. Some mathematical consequences of the Properties 1 and 2 -------------------------------------------------------- The following proposition was used extensively in the works [@galloway1]-[@galloway2]. For the origins of the relation between Riccati inequality and Raychaudhuri equation see [@erlich]. \[prop1\] Given a non linear differential equation of the Riccati form +=p\_(), with an initial condition $\theta(0)=\theta_i$ then, if there exists a constant $c_\Lambda>0$ such that \_i+&lt; \_ \_[0]{}\^e\^[-c\_]{} p\_() d, it follows that a solution $\theta_\Lambda$ does not extend beyond a finite value of the affine parameter $\lambda=\lambda_e$. The equation (\[u\]) may be converted, by the change of variables y()=-(\_+c\_)e\^[-c\_]{},into the following one =+r(),y(0)=y\_0. Here r()=e\^[-c\_]{}(p\_()+),q()=e\^[-c\_]{},and $y_0=-\theta_i-c_\Lambda$. From the definition of $q(\lambda)$ given in (\[erre\]) it is clear that \_[+]{}\_0\^+. Assume that the initial condition is such that \_[+]{} \^\_0 r()d&gt;-y\_0, and that $y(\lambda)$ extend to the whole interval $[0,\infty)$. This will imply a contradiction which will show that this affirmation is false thus, $y(\lambda)$ does not extend beyond a finite affine parameter value $\lambda=\lambda_e$. The same observation will hold for $\theta_\Lambda$, as it is defined in terms of $y(\lambda)$ by (\[y\]). In order to show the aforementioned contradiction, note that (\[cudo\]) implies the existence of affine parameter $\lambda_1$ for which $$\int^\lambda_0 r(\xi)d\xi>-y_0, \qquad \textrm{for}\qquad \lambda>\lambda_1.$$ By integrating the equation (\[teor\]) and taking into account the last inequality, it follows that y()=\_0\^ d+\_0\^r() d+y\_0&gt;\_0\^ d. It is convenient to introduce the quantity given by R()=\_0\^ d. As $q(\lambda)$ is positive and $\lambda>0$, it can directly be seen that that $R(\lambda)\geq 0$, the equality holds only for $\lambda=0$. This definition and the inequality (\[teor2\]) shows that &lt;=, for $\lambda>\lambda_1$. From here it is concluded, for every $\lambda_2>\lambda_1$, that $$\int_{\lambda_2}^{\lambda}\frac{d\xi}{q(\xi)}<\int_{\lambda_2}^{\lambda}\frac{1}{R^2}\frac{dR}{d\lambda}d\xi=\frac{1}{R(\lambda_2)}-\frac{1}{R(\lambda)}<\frac{1}{R(\lambda_2)}.$$ However, by condition (\[cudon\]) it follows that the left hand is not bounded when $\lambda\to \infty$. Thus, the last inequality makes sense only for times $\lambda<\lambda_e$, with $\lambda_e$ a fixed time. This shows that $y(\lambda)$ can not extend to the whole interval $ [0,\infty)$, but only to an interval inside $[0, \lambda_e)$. The same applies for the quantity $\theta_\Lambda$ related to $y(\lambda)$ by $y(\lambda)=-(\theta_\Lambda+c_\Lambda)e^{-c_\Lambda\lambda}$. In terms of $\theta_\Lambda$ the condition (\[cudo\]) becomes $$\theta_0+c_\Lambda< \lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}\bigg(p_\Lambda(\lambda)+\frac{c_\Lambda^2}{2}\bigg)d\xi<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi+\frac{c_\Lambda}{2},$$ or, equivalently $$\theta_0+\frac{c_\Lambda}{2}<\lim_{\lambda\to\infty} \textrm{inf}\int^\lambda_0 e^{-c_\Lambda\lambda}p_\Lambda(\lambda)d\xi.$$ This is precisely the condition (\[tang2\]), which shows the desired result. Proposition \[prop1\] may apply to the Raychaudhuri equation for $\theta_\Lambda$ with $$p_\Lambda(\lambda)=-[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda).$$ In fact, the right hand of (\[tang2\]) becomes (\[tang\]) under this identification. This point will be elaborated in more detail below. \[theta\_i\] Consider again a null geodesically complete space-time $(M,g_{\mu\nu})$ satisfying Properties 1 and 2. There are a pair of conjugate points along any null geodesic $\gamma_\Lambda(\lambda)$, generically denoted as $p$ and $q$. Then for any positive real number $c_\Lambda>0$, there exists a point $s=\gamma_\Lambda(\lambda_i)$ such that $\theta_i:=\theta_\Lambda(\lambda_i)$ satisfies the following inequality \_i+&lt; \_[+]{}\_[\_i]{}\^e\^[-c\_]{} p\_(). d, Here $p=\gamma_\Lambda(0)$, the value of $\lambda_i$ in the integral in (\[tang3\]) is such that $0<\lambda_{i}<\lambda_{e}$, where $\lambda_e$ is the parameter defined by $q=\gamma_\Lambda(\lambda_e)$. This remark is a direct consequence of the fact that $\theta_{\Lambda}((0,\lambda_{e}))=\mathbb R$, the existence of such $\lambda_{i}$ is guaranteed by continuity. This can be seen as follows. Recall that the scalar expansion $\theta_\Lambda$, is such that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$, which corresponds to the point $q$. On the other hand, as the integrand function in is continuous and by the Property 2 it is seen that all the integrals $$I(\lambda_i)=\liminf_{\lambda\to+\infty} \int_{\lambda_i}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ are bounded from below for every $c_\Lambda>0$ and $\lambda_i$. By considering $I(\lambda_{i})$ as a function of $\lambda_{i}$ with $c_\Lambda>0$ fixed, it is seen that $I(\lambda_{i})$ attains a global minima $I_M$ when $\lambda_i$ varies in the compact interval $[0,\lambda_e]$. This follows from the fact that the function $I(\lambda_i)$ by Property 2 is bounded and the interval just defined is compact. Choose a special value of the parameter $\lambda_i$ such that $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}<I_M.$$ This parameter exists as $\theta_\Lambda(\lambda)$ takes every real value continuously in $(0, \lambda_e)$. By the minimality of $I_M$ it is seen that this condition implies (\[tang3\]), and this clarifies the Remark \[theta\_i\].[^3] The next task is to apply the content of Proposition \[prop1\] to the study of conjugate points in the null geodesically complete space time ($M$, $g_{\mu\nu}$). \[prop 2\] Let $(M,g_{\mu\nu})$ be a null geodesically complete space time satisfying the Property 1 and Property 2. Consider a geodesic $\gamma_\Lambda(\lambda)$. Let $q$ be conjugate to $p$ through $\gamma_{\Lambda}$ such that $q\in J_+(p)-I_+(p)$. Then the value of the affine parameter $\lambda_e$ such that $\gamma_\Lambda(\lambda_e)=q$ is given implicitly by the following formula: =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_1$ is any value of the affine parameter such that $\lambda_i<\lambda_1<\lambda_e$ and the quantity $R_\Lambda(\lambda)$ is defined in (\[dji\]). The definition of $R_\Lambda(\lambda)$ involves $c_\Lambda$ but the formula (\[explota\]) is universal, that is, does not depends on the choice of $c_\Lambda$, neither on the choice of $\lambda_1$. Without loss of generality, it may be assumed that $\gamma_\Lambda(0)=p$ and $\gamma_\Lambda(\lambda_e)=q$. First, it is convenient to re-write the Raychaudhuri equation in the form . This gives $$y(\lambda)=\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +\int_{\lambda_1}^\lambda r(\xi) d\xi+y_0,$$ with $y(\lambda)$ defined in (\[y\]). From here it is seen that $$\frac{e^{c_\Lambda \lambda}y^2(\lambda)}{2}=\bigg[\int_{\lambda_i}^\lambda \frac{y^2(\xi)}{q(\xi)} d\xi +I(\lambda)\bigg]^2\frac{ e^{c_\Lambda \lambda}}{2} ,$$ with $I(\lambda)=\int_{\lambda_i}^\lambda r(\xi) d\xi+y_0$. The last equation can be expressed in terms of the quantity $R_\Lambda(\lambda)$ defined in (\[dji\]), the result is $$\frac{dR}{d\lambda}=[R_\Lambda(\lambda)+I(\lambda)]^2\frac{e^{c_\Lambda \lambda}}{2}.$$ By dividing this result by $R^2_\Lambda(\lambda)$ and by integrating with respect to $\lambda$ leads to $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_\Lambda(\lambda_2)}=\int_{\lambda_1}^{\lambda_2}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_1<\lambda_2$ and both values are in $(0, \lambda_e)$ but otherwise arbitrary. Now, under the working hypothesis, the value $\theta_\Lambda(\lambda)\to-\infty$ as $\lambda\to \lambda_e$ signals the presence of a conjugate point. Then the asymptotic condition (\[asin\]) applies and it is seen from (\[dji\]) that $R_\Lambda(\lambda)\to \infty$ when $\lambda\to \lambda_e$. Then, at $\lambda_2=\lambda_e$, the last expression becomes $$\frac{1}{R_\Lambda(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg[1+\frac{I(\xi)}{R_\Lambda(\xi)}\bigg]^2\frac{e^{c_\Lambda \xi}}{2}d\xi.$$ Here $\lambda_i<\lambda_1<\lambda_e$ is arbitrary. By expressing the last formula in terms of $\theta_\Lambda(\lambda)$ from (\[y\]) and by taking into account the first (\[erre\]) the expression (\[explota\]) is obtained. Proof of the *third requirement* as a consequence of Properties 1 and 2 ======================================================================= The purpose of this section is to show that Property 1 and Property 2 given above imply the *third requirement*. This leads to the conclusion that the Gao-Wald theorem holds for space times satisfying these properties. Before making formal statements, it is worth to give the intuition behind the proof of *third requirement* given below. One considers a reference null geodesic $\gamma_0(\lambda)$ which, by Property 1, has two conjugate points let’s say $p_0$ and $q_0$. Remark \[theta\_i\] shows that (\[tang2\]) is satisfied for some point in the middle $s_0$. That is, for this point $s_0=\gamma_0(\lambda_i)$ one has $$\theta_0(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_0(\xi) d\xi.$$ The strategy is to show that, for some geodesics emanating from points $p$ close to $p_0$, the condition (\[tang2\]) is also satisfied for the points $s=\gamma_\Lambda(\lambda_i)$, which are close to $s_0$. That is $$\theta_\Lambda(\lambda_i)+\frac{c_\Lambda}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c_\Lambda \xi} p_\Lambda(\xi) d\xi,$$ for all these geodesics. The Proposition \[prop1\] will allow to conclude that $\theta_\Lambda(\lambda)$ also is going to tend to $-\infty$ at a finite value of the affine parameter $\lambda$. Therefore, these points $p$ will have a conjugate point $q$ joined by the null geodesic $\gamma_\Lambda(\lambda)$. This leads to the first part of the *third requirement*. The second part is more subtle. The point is that, by assumption, the reference geodesic $\gamma_0(\lambda)$ has two conjugate points $p_0$ and $q_0$. At these points the determinant $G_0(\lambda)$ defined in (\[smile2\]) vanishes. Thus $G_0(0)=0$ and $G_0(\lambda_e)=0$. By continuity in $S$, one may work by analogy with section 2.3 and try to show that $|G_\Lambda(\lambda_e)|<\epsilon$ if $\Lambda\in O$ with $O$ an open in $S$ small enough. But even taking into account that $G_\Lambda(\lambda)$ has a very small modulus, this does not ensure that it is going to vanish for $|\lambda-\lambda_e|<\delta$. In the present case, as the Null Energy Condition is not assumed valid, it follows that $G''_\Lambda(\lambda)$ is not always negative, and the mean value result of section 2.3 does not work. It may be the case that the value of $G_\Lambda(\lambda)$ is close to zero at $\lambda_e$, then grows very rapidly and only at a value $\lambda=\lambda'_e$ which is very far from $\lambda_e$ will vanish. This would spoil the continuity property of the *third requirement*. Below, it will be proven that this is not the case. The proof relies heavily on the formula (\[explota\]) for the explosion value of the affine parameter $\lambda=\lambda_e$, by considering how it varies when moving along “close geodesics”. In view of this discussion, it is convenient to divide the proof of the *third requirement* in two parts. Proof of the first part of the third requirement ------------------------------------------------ Consider an arbitrary null reference geodesic $\gamma_0$ in the space time ($M$, $g_{\mu\nu}$). The *first requirement* implies that it has two conjugate points $p_0$ and $q_0$, where $q_0$ belongs to $J_+(p_0)-I_+(p_0)$. Without losing generality one may assume that the affine parameter is such that $p_0=\gamma_0(0)$, the value at $q_0$ will be denoted as $\lambda=\lambda_0$. The expansion parameter $\theta_0\to \infty$ at $\lambda\to 0^+$ and $\theta_0\to -\infty$ at $q_0$. As stated before, this leads to the important conclusion that this expansion parameter $\theta_0$ takes every real value when $\lambda$ moves in the interval ($0$, $\lambda_0$), if $q$ is the first conjugate point to $p$. The reference null geodesic $\gamma_0$ is generated by a point $\Lambda_0=$($p_0$, $k_0^\mu$) in $S$. At a given value $\lambda_i$ such $0<\lambda_i<\lambda_0$, a generic value of $\theta(\lambda_i)=\theta_i$ is achieved. Denote the corresponding point in the space time by $s_0=\gamma_0(\lambda_i)$. Consider the map given by[^4] $H(\Lambda, \lambda)=\gamma_{\Lambda}(\lambda)$. For any small enough neighborhood $U$ in $M$ containing $s_0$ there exists an open $O$ in $S$ containing $\Lambda_0$ such that $H(\Lambda, \lambda)$ belongs to $U$ for every point $\Lambda=(p, \;k^\mu)$ in $O$ if $|\lambda-\lambda_i|<\delta$. Here $\gamma_\Lambda(0)=p$. In these terms, one may show the following proposition.\ \[1stpart3rdrequire\] Given a null geodesically complete space time ($M$, $g_{\mu\nu}$) with the Property 1 and 2, consider a geodesic $\gamma_0(\lambda)$ with pair of conjugate points $p_0$ and $q_0$, corresponding to a point $\Lambda_0=(p_0$, $k_0^\mu)$ in $S$. Then there exists an open $O$ in $S$ containing $\Lambda_0$ such that every geodesic $\gamma_\Lambda(\lambda)$ generated by points $\Lambda=(p$, $k^\mu)$ in $O$ possess a conjugate point $q$ to $p$. Before going to the proof, note that the Property 1 already states that every null geodesic contains a pair of conjugate points. But it does not specify where these points are located along the geodesic. The new information this proposition gives is that, once $p_0$ has a conjugate point, then all the points $p$ in a neighbourhood of $p_0$ will have a conjugate point along to some geodesic emanating from them. Choose a compact set $O'$ containing the point $\Lambda_0$ defined in the previous paragraph. Consider a curve generated by a point $\Lambda$ inside the compact set $O'$. By Property 2, the integral $$I(\lambda_s)=\lim_{\lambda\to\infty}\text{inf}\int_{\lambda_s}^\lambda e^{-c_\Lambda\xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\xi) d\xi,$$ is bounded from below for any fixed value $\lambda_s$. Vary the value of $\lambda_s$ in the interval $[0,\;\lambda_0]$, which is compact, and find the smallest value $I_i$, which will be achieved for a specific value $\lambda^\ast_s$. At this point, this procedure mimics the proof of Proposition 2. The strategy however is not doing this with one particular null geodesic, but with all the geodesics $\gamma_\Lambda(\lambda)$ generated by the points $\Lambda$ in $O'$. The resulting minimum for a given geodesic, denoted by $I_{\Lambda}$, is not necessarily a continuous function in $O'$ but, by Property 2, is bounded by below. As $O'$ is compact, there will exists an infimum value at a point $\Lambda_m\in O'$, with its corresponding smallest value $I_{\Lambda_m}$. By construction this value is smaller or equal than the corresponding to any other generic point $\Lambda$ in $O'$. Below, for notational simplicity, this value will be denoted by $I_m$ instead of $I_{\Lambda_m}$. Once the value $I_m$ has ben found, choose a value $\lambda_i$ such that $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$. The required value of $\lambda_i$ in $(0,\lambda_0)$ exists since, as discussed above, the expansion parameter $\theta_0(\lambda)$ for the reference geodesic $\gamma_0$ takes every real value when $\lambda$ varies in that closed interval. Then, from the minimality of $I_m$ it is clear that $\theta_0(\lambda_i)+c/2<I_m-\eta$ implies that $$\theta_0(\lambda_i)+\frac{c}{2}< \lim_{\lambda\to\infty} \text{inf}\int_{\lambda_i}^\lambda e^{-c \xi} [R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_0(\xi) d\xi.$$ In these terms, one may choose an open $O$ inside $O'$ containing $\Lambda_0$, such that for every $\theta_\Lambda(\lambda)$ determined by a point $\Lambda$ in $O$ the inequality $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ holds. The validity of this statement may be seen from the continuity of $G(\lambda, \Lambda)$ and $G'(\lambda, \Lambda)$ in $O$, which implies the continuity of $\theta_\Lambda(\lambda)=G'(\Lambda,\lambda)/G(\Lambda, \lambda)$ with respect to $\Lambda$ if $G(\lambda, \Lambda)\neq 0$, that is, outside a conjugate point. This continuity property follows from the fact that $(A_\Lambda)_\mu^\nu$ satisfies the ordinary equation (\[smile\]), and thus $(A_\Lambda)_\mu^\nu$ and $G_\Lambda(\lambda)$ vary continuously with respect to $\Lambda$ and $\lambda$. Now, for $\epsilon<\eta$, the minimality of $I_m$, together with the fact that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ and $\theta_0(\lambda_i)+c/2<I_m-\eta$, imply that \_(\_i)+&lt; \_ \_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for all the points $\Lambda$ of $O\subset S$. A direct application of this fact and the Proposition 1 implies that for any point $\Lambda=$($p$, $k^\mu$) in the open set $O$, if $\gamma_\Lambda(0)=p$, the solution $\theta_\Lambda(\lambda)$ does not extend beyond a finite value $\lambda_e$ if $\epsilon<\eta$. As discussed below formula (\[asin\]), for the space time in consideration the expansion parameter $\theta_\Lambda(\lambda)$ is continuous everywhere except at the conjugate point. Therefore, the only possibility is that $\theta_\Lambda(\lambda)\to -\infty$ when $\lambda\to \lambda_e$. This implies that at $\lambda_e$ there appears a conjugate point $q$ to $p$. This concludes the proof. Proof of the second part of the third requirement ------------------------------------------------- Before proving the second part, the following observation is needed. In the proof of Proposition \[1stpart3rdrequire\], the initial value $\theta_0(\lambda_i)+c/2<I_m-\eta$ with $\eta> C>0$ has been selected. Choose $\eta>c/2+\epsilon$. From the minimality $I_m$ and from (\[plat\]) it follows that \_(\_i)+c&lt;\_[\_i]{}\^e\^[-c ]{} \[R\_k\^k\^+\_\^\]\_()d, for every value of $\lambda\geq \lambda_i$ and $\Lambda$ in $O$. This particular choice will be useful below. In these terms, the proposition to be proved is the following. Consider a generic point $\Lambda=(p, k^\mu)$ in an open $O\subset S$ containing $\Lambda_0=(p_0, k_0^\mu)$. Under the conditions of the Proposition \[1stpart3rdrequire\], the map $h: O\to M$ defined by $h(\Lambda)=q$, with $q$ the first conjugate point to $p$, is continuous at $\Lambda_0$. As in the proof given before, the geodesic $\Lambda_0$ joins two conjugate points $p_0$ and $q_0$ and, by Proposition \[1stpart3rdrequire\], there is an open $O$ in $S$ containing $\Lambda_0$ such that, for all $\Lambda=(p, k^\mu)$ in $O$, there is a conjugate point $q$. By the proof of Proposition \[1stpart3rdrequire\] and the discussion above, one may find a value $\lambda_i$ such that for all the geodesics defined by every $\Lambda$ in $O$, the inequality (\[inicial\]) holds. Note that the value $\lambda_i$ is strictly larger than zero if the origin is defined such that $\gamma_\Lambda(0)=p$. The length parameter $\lambda_e$ that defines the conjugate point $q_0$ in the geodesic $\gamma_0$ is given by (\[explota\]) which, adapted to the present situation, is given by =\_[\_1]{}\^[\_e]{}{1+}\^2e\^[c]{}d. Here $\lambda_i<\lambda_1<\lambda_e$ is an initial parameter, and p\_0()=\[R\_k\^k\^+\_\^\](),R\_0(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_0()+c)\^2 d. The length parameter defining the conjugate point $q$ for another geodesic $\gamma_\Lambda(\lambda)$ is $\lambda_e+\Delta\lambda_e$, and is given by =\_[\_1]{}\^[\_e+\_e]{}{1+}\^2e\^[c]{}d, with p\_()=\[R\_k\^k\^+\_\^\]\_(),R\_(\_1)=\_[\_i]{}\^[\_1]{}e\^[c]{}(\_()+c)\^2 d. Note that $\Delta \lambda_e$ depends on the choice of the geodesic, that is, $\Delta \lambda_e=f(\Lambda)$. This dependence would be implicitly understood in the following reasoning. The task is to show that $|\Delta \lambda_e|=|f(\Lambda))|<\epsilon$ when $\Lambda$ is in an open $O$ of $S$ small enough, containing $\Lambda_0$. A point that might cause confusion is that, in principle, $\lambda_e+\Delta\lambda_e$ may be such that $\lambda_e+\Delta \lambda_e<\lambda_1$, as $\Delta\lambda_e$ may be negative. But the formula (\[mat2\]) is true for the opposite case. Thus, $\lambda_1$ should not be chosen so arbitrary. However, it has been mentioned that, from the continuity of $\theta_\Lambda(\lambda)$ in $S$ outside a conjugate point, one has that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small enough. In fact $|\theta_\Lambda(\lambda)-\theta_0(\lambda_i)|<\epsilon$ if $O$ is small and $|\lambda-\lambda_i|\leq \delta$. This means that, up to a point $\lambda=\lambda_i+\delta'$ with $\delta'<\delta$, the value of $\theta_\Lambda(\lambda)$ does not explode. Choose $\lambda_i<\lambda_1<\lambda_i+\delta'$, then $\lambda_e+\Delta\lambda_e>\lambda_1$ for all the curves parameterized by $\Lambda$. Then the mentioned problem does not arise. The actual value of $\lambda_1$ is not known, but it exists, and this is the only thing needed in the following reasoning. The subtraction of both expressions (\[mat\]) and (\[mat2\]) obtained above gives that $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e+\Delta \lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_1}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi)\bigg] \bigg\}^2e^{c\beta}d\beta.$$ The last expression can be cast in the following form $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta$$ +\_e {1+}\^2e\^[c’]{}, where in the last integral the mean value theorem has been employed and $\lambda'$ is some value in the interval $[\lambda_e, \; \lambda_e+\Delta \lambda_e]$. The last formula (\[artdeco\]) already gives an intuition about the intended proof. The condition (\[inicial\]) implies that the term multiplying $\Delta \lambda_e$ in is strictly positive (as $R_\Lambda>0$). It vanish for $\lambda=\lambda_i$ but $\lambda'>\lambda_1>\lambda_i$. Thus, $R_\Lambda(\lambda')$ is never vanishing. On the other hand, one may show by use of analysis methods that the remaining terms are as small as possible by restricting $O$ to be is small enough. Thus $\Delta \lambda_e$ will be also very small, and this will prove the continuity property stated in the Proposition. Roughly speaking, it will imply that a point $q$ conjugated to $p$ is close to a point $q_0$ conjugated to $p_0$. A method to prove this intuition goes as follows. First note that, from the definitions (\[prima\]) and (\[prima2\]) and by use of a mean value theorem, one has that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg|=\bigg|\frac{R_0(\lambda_1)-R_\Lambda(\lambda_1)}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|=\frac{1}{R_\Lambda(\lambda_1)R_0(\lambda_1)}\bigg|\int_{\lambda_{i}}^{\lambda_{1}}e^{c\xi}[(\theta_0(\xi)+c)^2-(\theta_\Lambda(\xi)+c)^2]\,d\xi\bigg|$$ =|\_0(’)-\_(’)||\_0(’)+\_(’)+2c|(\_1-\_i). where in the last expression the mean value theorem was applied and the resulting expression was factored. Here the value $\lambda'$ depends on the geodesic, that is, $\lambda'=g(\Lambda)$. The quantity (\[cito\]) can be as small as possible since $\delta\theta(\lambda')=\theta_0(\lambda')-\theta_\Lambda(\lambda')$ goes to zero and the other quantities are under control. To see this clearly, given the open $O'$ in $S$ consider a compact $O_c\subset O'$. Find the minimum value of $R_\Lambda(\lambda_1)$ in this compact, denoted as $R_m$, and the maximum value of $\theta_\Lambda(\lambda)$ in $O_c\times [\lambda_1,\lambda_i]$, denoted as $\theta_m$. Then choose another open $O\subset O_c$. The last expression implies that |-| |\_m||2\_m+2c|(\_1-\_i), for any $\Lambda$ in $O$. Here $\delta\theta_m$ is the maximum value of $\delta \theta(\lambda')$ in $O_c\times [\lambda_i,\lambda_1]$. As this set is compact, the Cantor-Heine theorem allows to conclude that $|\theta_0(\lambda)-\theta_\Lambda(\lambda')|$ is small when $|\lambda-\lambda'|<\delta$ and $O_c$ is small enough, independently on the value of $\lambda'(\Lambda)$. This implies in particular that, by making $O'$ and consequently $O_c$ and $O$ small enough one may chose $\delta \theta_m$ such that $$|\delta \theta_m|< \frac{R_m^2\epsilon}{(\lambda_1-\lambda_i)|2\theta_m+2c|}.$$ in $O$, which implies (\[socia\]) that |-| , for all $\Lambda \in O$. This prove that the left side of (\[artdeco\]) can be made arbitrarily small. On the other hand, the absolute value of the sum of those terms of the right side of (\[artdeco\]) which are independent on $\Delta \lambda_e$ can be converted by use of a mean value theorem into $$I=\bigg|\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_\Lambda(\beta)}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\beta}d\beta$$ $$-\int_{\lambda_1}^{\lambda_e}\bigg\{1+\frac{1}{R_0(\beta)}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \beta})+\int_{\lambda_i}^\beta e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\beta}d\beta\bigg|$$ $$=\bigg|(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]\bigg\}^2e^{c\lambda'}$$ $$-(\lambda_e-\lambda_1)\bigg\{1+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}^2e^{c\lambda'}\bigg|$$ $$=(\lambda_e-\lambda_1)e^{2c\lambda'}\bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$\times \bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|.$$ $$\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M \bigg|\bigg\{\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ -}| In the last expression $\lambda'$ is some value in the interval $[\lambda_1,\lambda_e]$, which comes from the mean value theorem for integrals, and $M$ denotes the maximum defined by $$M=\text{Max}\bigg|\bigg\{2+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}\bigg|_{O_1},$$ in an initial compact set $O_1\times [\lambda_1,\lambda_e]$. This maximum exists since the function $R_\lambda(\lambda)$ is never zero for $\lambda$ in $[\lambda_1,\lambda_e]$ and the integrals are convergent since the space time $(M, g_{\mu\nu})$ in consideration is such that $p_\Lambda(\lambda)=[R_{\mu\nu}k^\mu k^\nu+\sigma_{\mu\nu}\sigma^{\mu\nu}]_\Lambda(\lambda)$ is not divergent at finite values of $\lambda$. The expression inside the brackets in the last step in (\[last\]) is then $$D_\Lambda=\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg],$$ where the denomination $D_\Lambda$ is chosen in order to emphasize that it represents a difference. Write this quantity as $$D_\Lambda=\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$+\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg]$$ $$-\frac{1}{R_0(\lambda')}\bigg[-\theta_0(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]$$ By taking into account (\[andros\]), it is already seen that the last quantity is bounded since the integral is finite. In any case, by simplifying some terms, the last expression becomes $$D_\Lambda=\frac{1}{R_0(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)+\theta_0(\lambda_i)+\int_{\lambda_1}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi) \bigg]\bigg\}$$ $$+\bigg(\frac{1}{R_\Lambda(\lambda')}-\frac{1}{R_0(\lambda')}\bigg)\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi\bigg],$$ From the continuity of $\theta_\Lambda(\lambda)$ it follows that $|\theta_\Lambda(\lambda_i)-\theta_0(\lambda_i)|\leq \epsilon'/4$ by choosing a suitable open $O\subset O_1$ containing $\Lambda_0$. In addition, one has that $$\bigg|\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\xi)d\xi-\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_0(\xi)d\xi \bigg|\leq (\lambda_e-\lambda_i)\delta p^M_\Lambda,$$ where $\delta p^M_\Lambda$ denote the maximum value of $\delta p_\Lambda=e^{-c\lambda_i}(p_\Lambda(\lambda)-p_0(\lambda))$. Again, by a Cantor-Heine argument it can be seen that this maximum can be made arbitrarily small by making $O_1$ small enough. In particular, it can be made smaller than $\epsilon'/4(\lambda_e-\lambda_1)$. The minimum value of $R_0(\lambda')$ is $R_0(\lambda_1)$, as (\[prima2\]) shows that this quantity is monotone increasingly with $\lambda$. Thus by selecting $\epsilon'=\epsilon R_0(\lambda_1)$ one has that |D\_|+|(-)|. As the integrand in the last expression is finite for every curve $\gamma_\Lambda(\lambda)$ then it has a maximum $M'$ in the factors in parenthesis in the right side in the compact $C=O_1\times[\lambda_1,\lambda_e] $. By choosing $O_1$ and consequently $O$ small enough one may use (\[andros\]) in order to prove that $$\bigg|\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}\bigg| \leq\widetilde{\epsilon}.$$ By choosing $\widetilde{\epsilon}=\epsilon/2 M'$ it is seen from (\[last2\]) that |D\_|, for $\Lambda$ inside $O$. Now, the inequality (\[last\]) implies that $$I\leq (\lambda_e-\lambda_1)e^{2c\lambda_e} M |D_\Lambda|.$$ By use of (\[epsi\]) it is clear that, by choosing $O_1$ and thus $O$ small enough then $|D_\Lambda|\leq \epsilon/M(\lambda_e-\lambda_1)e^{2c\lambda_e} $, and this implies that I. In view of this discussion, consider again (\[artdeco\]). It is clear that it can be written as $$\frac{1}{R_\Lambda(\lambda_1)}-\frac{1}{R_0(\lambda_1)}=\pm I+\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'},$$ where the $\pm$ is to indicate that in the definition (\[last\]) of $I$, the absolute value has been taken. The sign of the quantity inside the modulus is not known, thus the identity holds with a plus or a minus sign, which will be not relevant in the following discussion. From the last expression and by taking into account (\[ep\]) and (\[andros\]), it is concluded that, by choosing a very small open $O$ around $\Lambda_0$, one has $$\bigg|\Delta \lambda_e \bigg\{1+\frac{1}{R_\Lambda(\lambda')}\bigg[-\theta_\Lambda(\lambda_i)-c+\frac{c}{2}(e^{-c\lambda_i}-e^{-c \lambda'})+\int_{\lambda_i}^{\lambda'} e^{-c\xi} p_\Lambda(\lambda')d\xi\bigg]\bigg\}^2e^{c\lambda'}\bigg|\leq \epsilon_1+\epsilon_2,$$ where both $\epsilon_i$ comes either from $I$ in (\[ep\]) or either from (\[andros\]). The quantity multiplying $\Delta \lambda_e$ is ensured to be positive by the choice (\[inicial\]) employed. In addition, it has a maximum in a compact $O'$ inside $O$ containing $\Lambda_0$, called $M_\Delta$. Find an open $O_2$ inside $O'$ containing $\Lambda_0$ such that $\epsilon_i\leq \epsilon/2 M_\Delta e^{c\lambda_e}$. Then $$|\Delta \lambda_e(\Lambda)|\leq \epsilon,$$ for every $\Lambda$ in $O_2$. Thus $\Delta \lambda_e(\Lambda)\to 0$ continuously as $\Lambda\to \Lambda_0$, which shows the desired result. A modified Gao-Wald theorem =========================== After proving that the Properties 1 and 2 described in section 3 imply that the *second requirement* and the *third requirement* stated in section 1 are satisfied, the next task is to show that the Gao-Wald theorem [@gaowald] is true when these properties are satisfied. Let $(M,g_{\mu\nu})$ be a null geodesically complete space time such that the Property 1 and Property 2 (given in section 3) are satisfied. Then, given a compact region $K$ in $M$ there exists a compact $K'$ containing $K$ such that, for any two points $p, q\notin K'$ and $q$ belonging to $J_+(p)-I_+(p)$, no causal curve $\gamma$ joining $p$ with $q$ can intersect $K$. As the space time manifold $M$ is assumed to be paracompact, it can be made into a Riemannian manifold with Riemannian metric $q_{\mu\nu}$. This metric can be assumed to be complete by multiplying it by a conformal factor if necessary [@Hicks]. Fix a point $r \in M$ and let $d_{r}:M\to\mathbb R$, $d_{r}(s)$ denotes the geodesic distance between $r$ and $s$ using the metric $q_{\mu\nu}$. This function is continuos in $M$ and for all $R>0$ the set $B_{R}=\{p\in M:~ d_{r}(p)\leq R\}$ is compact (see [@Hicks Theorem 15]). In these terms, given $\Lambda \in S$ let $\gamma_{\Lambda}$ a null geodesic determined by $\Lambda$, let’s define the function $f:S\to \mathbb R$ by: $$\begin{aligned} f(\Lambda) =& \inf_{R}\{B_{R}\text{ contains a connected segment of $\gamma_{\Lambda}$ that includes the initial point }\\ &\quad \quad \text{determined by $\Lambda$ together with a pair of conjugate points of $\gamma_{\Lambda}$}\}\end{aligned}$$ The function $f(\Lambda)$ is upper semicontinuous [@gaowald], when the *second requirement* and the *third requirement* are satisfied, the proof has been given in the reference [@gaowald]. Let $K\subset M$ be a compact set. Let $S_{K}=\{(p,k^{\mu}\in S \text{ with } p \in K)\}$, since the tangent bundle has the product topology, $K$ is compact and $k^{\mu}$ is of bounded norm, then $S_{K}$ is compact. Since $f$ is upper-semicontinuous, it must achieve a maximum in $S_{K}$, let’s denote it by $\bar R$. Let $K' = B_{\bar R}$. Let $p,q\notin K'$ and $q\in J_+(p)-I_+(p)$ and let $\gamma$ a causal curve joining $p$ with $q$, then $\gamma$ must be a null geodesic since $q\in J_+(p)-I_+(p)$. However, the Proposition \[temporal\] insures that $\gamma$ does not contain a pair of conjugate points between $p$ and $q$. If $\gamma\cap K\neq \emptyset$ then by the definition of $K'$, $\gamma$ must have a pair of conjugate points lying in $K'$ and in between $p$ and $q$. This contradiction completes the proof. Note that the theorem given above is true if Properties 1 and 2 are replaced by the *first*, *second* and *third requirements* given in the introduction, as the proof would be unchanged. In brief, in the present work it has been shown that the Gao-Wald theorem holds when the space time ($M$, $g_{\mu\nu}$) is null geodesically complete, every null geodesic posses at least a pair of conjugate points, and the curvature is such that a the quantity $R_{\mu\nu}k^\mu k^\nu\neq -e^{c\lambda}$ for large $\lambda$ values, when evaluated on a null geodesic. Therefore this result may apply to models which violate the Null Energy Condition [@averaged1]-[@averaged25] or to modified gravity theories such as the ones described in [@odintsov]-[@odintsov3]. This deserves further attention. In our opinion, the curvature condition introduced in the text may hold for several interesting solutions for these models. On the other hand, the condition that every null geodesic contains a pair of conjugate points sounds a bit stringent, and it may be of interest to relax it if possible. Another relevant task is to understand it is possible to control the size of the region $K'$ of the Gao-Wald theorem. We leave this for a future investigation. Acknowledgments {#acknowledgments .unnumbered} =============== Both authors are supported by CONICET, Argentina. [99]{} S. Gao and R. Wald Class. Quant. Grav 17 (2000) 4999. M. Alcubierre, Class. Quantum Grav. 11 (1994) L73. S. V. Krasnikov, Phys. Rev. D 57 (1998) 4760. K. Olum, Phys. Rev. Lett. 81 (1998) 3567. F. Tipler, Phys. Rev. Lett. 37 (1976) 879. F. Tipler, Ann. Phys. 108 (1977) 1. S. W. Hawking, Phys. Rev. D 46 (1992) 603. M. Visser, B. Bassett, and S. Liberati, Nucl. Phys. B88 (Proc. Supl) (2000) 267 M. Visser, B. Bassett, and S. Liberati, in General Relativity and Relativistic Astrophysics, Proceedings of the Eighth Canadian Conference, ed. by C.P Burgess and R.C. Meyers, (AIP Press, Melville, New York, 1999). X. Camanho, J. Edelstein, J. Maldacena and A. Zhiboedov JHEP 02 (2016) 20. G. Papallo and H. Reall JHEP 11 (2015) 109. X. Camanho, J. Edelstein and A. Zhiboedov Int. J. Mod. Phys. D 24 (2015) 1544031. J. Edelstein, G. Giribet, C. Gomez, E. Kilicarslan, M. Leoni and B. Tekin Phys. Rev. D 95 (2017) 104016. R.M. Wald, General Relativity, University of Chicago Press (Chicago, 1984). S.W. Hawking and G.F.R. Ellis, The Large Scale Structure of SpaceTime, Cambridge University Press (Cambridge, 1973). J. Beem, P. Ehrlich and K. Easley Global Lorentzian Geometry CRC press 1981. B. O Neill Semi-Riemannian Geometry with Applications to General Relativity Academic Press 1983. R. Penrose, in Essays in General Relativity, ed. by F.J. Tipler, Academic Press (New York, 1980). E. Flanagan, D. Marolf, and R. Wald Phys. 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Kontou and K. Olum Phys. Rev. D 90 (2014) 024031. D. Urban and K. Olum Phys. Rev. D 81 (2010) 024039. D. Urban and K. Olum Phys. Rev. D 81 (2010) 124004. M. Visser Phys. Lett. B 349 (1995) 443. M. Visser Phys. Rev. D 54 (1996) 5103. M. Visser Phys.Rev. D 54 (1996) 5116. M. Visser Phys. Rev. D 56 (1997) 936. R. Wald Phys. Rev. D 17 (1978) 1477. F. J. Tipler Phys. Rev. D17 (1978) 2521. A. Borde Class. Quant. Grav. 4 (1987) 343. R. Bousso, Z. Fisher, S. Leichenauer, and A. C. Wall Phys. Rev. D93 (2016) 064044. S. Balakrishnan, T. Faulkner, Z. U. Khandker, and H. Wang JHEP 09 (2019) 020. J. Koeller and S. Leichenauer Phys. Rev. D 94 (2016) 024026. F. Rosso JHEP 03 (2020) 186. S. Nojiri, S. Odintsov and V. Oikonomou Phys. Rept. 692 (2017) 1. K. S. Stelle, Phys. Rev. D 16 (1977) 953. K. S. Stelle, Gen. Rel. Grav. 9 (1978) 353. D. Lovelock J. Math. Phys. 12 (1971) 498. [^1]: Instituto de Matemática Luis Santaló (IMAS), UBA CONICET, Buenos Aires, Argentina [email protected]. [^2]: Instituto de Matemática Luis Santaló (IMAS), UBA CONICET, Buenos Aires, Argentina [email protected] and [email protected]. [^3]: Note that the remark \[theta\_i\] is in agreement with Proposition \[prop1\], as $\theta_\Lambda(\lambda)$ is expected to explode at affine parameter value $\lambda_e$. [^4]: This map is defined such that $H: O\times [0,\lambda_0]\to M$ where $O$ is an open in $S$.

ArXiv

--- abstract: 'A particular initial state for the construction of a perturbative QCD expansion is investigated. It is formed as a coherent superposition of zero momentum gluon pairs and shows Lorentz as well as global $SU(3)$ symmetries. The general form of the Wick theorem is discussed, and it follows that the gluon and ghost propagators determined by the proposed vacuum state, coincides with the ones used in an alternative of the usual perturbation theory proposed in a previous work, and reviewed here. Therefore, the ability of such a procedure of producing a finite gluon condensation parameter already in the first orders of perturbation theory is naturally explained. It also follows that this state satisfies the physicality condition of the BRST procedure in its Kugo and Ojima formulation. A brief review of the canonical quantization for gauge fields, developed by Kugo and Ojima, is done and the value of the gauge parameter $\alpha$ is fixed to $\alpha=1$ where the procedure is greatly simplified. Therefore, after assuming that the adiabatic connection of the interaction does not take out the state from the interacting physical space, the predictions of the perturbation expansion for the physical quantities, at the value $\alpha=1$, should have meaning. The validity of this conclusion solves the gauge dependence indeterminacy remained in the proposed perturbation expansion.' author: - '[**Author: Marcos Rigol Madrazo**]{}\' - '[**Advisor: Dr. Alejandro Cabo Montes de Oca**]{}' date: Havana 1999 title: | Instituto Superior de Ciencias y Tecnología Nucleares\ Dissertation Diploma Thesis\ --- Introduction ============ Quantum Chromodynamics (QCD) was discovered in the seventies and it has been considered as the fundamental theory for the strong interactions. A theory of such sort, showing a non abelian invariance group, was first suggested by Yang and Mills [@Yang]. The main idea in it is the principle of local gauge invariance, which for example, in Quantum Electrodynamics (QED) means that the phase of the wave function can be defined in an arbitrary way at any point of the space-time. In a non abelian theory, the arbitrary phase is generalized to an arbitrary transformation in an internal symmetry group, for QCD the internal symmetry group is $SU(3)$. The discovery of this theory generated radical changes in the character of the Modern Theoretical Physics and as a consequence it has been deeply investigated in the last years. It is believed at the present time that all the interactions in Nature are gauge invariant [@Green]. In one limit, the smallness of the coupling constant at high momentum (asymptotic freedom) made possible the theoretical investigation of the so-called hard processes by using the familiar perturbative language. This so called perturbative QCD (PQCD) was satisfactorily developed. However, relevant phenomena associated with the strong interactions can’t be described by the standard perturbative methods and the development of the so-called non-perturbative QCD is at the moment one of the great challenges of Theoretical Physics. One of the most peculiar characteristics of the strong interactions is the color confinement. According to this philosophy colored objects, like quarks and gluons, can’t be observed as free particles in contrast with hadrons that are colorless composite states and effectively detected. The physical nature of such phenomenon remains unclear. Qualitatively, it is compared with the Meissner effect in superconductors, in which the magnetic field is expelled from the bulk in which the Cooper pairs condensate exists. It is considered that the QCD vacuum expels the color fields from it. Numerous attempts to explain this property have been made, for example explicit calculations in which the theory is regularized in a spatial lattice [@Creutz], and also through the construction of phenomenological models. One of this models, the so called MIT Bag Model [@Chodos], assumes that a bag or bubble is formed around the objects having color in such a manner that they could not escape from it, because their effective mass is smaller inside the bag volume and very high outside. The dimensional quantity introduced in this model is $B$, the bag constant, which is the pressure that the vacuum makes on the color field. Another approach is the so called String Model [@Gervais] which is based in the assumption that the interaction forces between quarks and antiquarks grow with the distance, in such a way that the energy increases linearly with the string length $E(L)=kL$. The main parameter introduced in this theory is the string tension $k$ which determines the strength of the confining interaction potential. A fundamental problem in QCD is the nature of its ground state [@Shuryak]. This state is imagined as a very dense state of matter, composed of gluons and quarks interacting in a complicated way. Its properties are not easy accessible in experiments, because quarks and gluons fields can’t be directly observed, only the color neutral hadrons are detected. Furthermore, the interactions between quarks can’t be directly determine, because their scattering amplitudes can’t be measured. It is known, from the experience in solid-state physics, that a good understanding of the ground state structure implies a natural explanation of many of the phenomenological facts concerning to its excitations. The theory of superconductivity is a good example, up to the moment in which a good theory of the ground state was at hand the description of its excitations remained basically phenomenological. It is already accepted that in QCD the zero-point oscillations of the coupled modes produce a finite energy density, such effects are called non-perturbative ones. Obviously such an energy density can be subtracted by definition, however this procedure does not solve the problem, because soft modes are rearranged in the excited states and the variation of their energy should be unavoidable considered. This energy density is determined phenomenologically and its numerical estimate is [@Shuryak] $$E_{vac}\simeq -f\langle 0\mid g^2G^2\mid 0\rangle \simeq 0.5GeV/fm^3,$$ where the so-called non-perturbative gluonic condensate $\langle 0\mid g^2G^2\mid 0\rangle$ was introduced and phenomenologically evaluated by Shiftman, Vainshtein and Zakharov [@Zakharov]. The negative sign of $ E_{vac}$ means that the non-perturbative vacuum energy is lower that the one associated to the perturbative vacuum. Some of the QCD vacuum models, developed to explain the above-mentioned properties, are mentioned below. These models can be classified considering the dimension of the manifold in which the non-perturbative field fluctuations are concentrated. 1- The “instanton” model, in which it is assumed that the field is gathered in some localized regions of the space and time as instantaneous fluctuations. These are considered as fluctuations concentrated in zero dimensional manifolds. 2- The “soliton” model, in which it is assumed that the non linear gauge fields create some kind of stable particles or solitons (i.e. glueballs [@Hansson] or monopoles [@Mandelstam]) in the space. The space-time manifold to be considered for these models is one-dimensional. 3- The “string” model, in which closed strings (field created between color charges shows a form resembling a flux tube or string) are present in the vacuum. In space-time the history of these strings is a 2-dimensional surface, so in this picture the fluctuations are concentrated in closed surfaces. 4- The last model to be mentioned is the simplest one. It will be discussed here in more detail because it furnished the starting roots of the present discussion. This model is the “homogeneous” vacuum model, in which it is assumed that a magnetic field exist in the vacuum [@Savv1]. In the homogeneous vacuum field model, the existence of a constant magnetic abelian field $H$ is assumed. A simple calculation in the one loop approximation gives as result the following energy density [@Shuryak] $$E\left(H\right) =\frac{H^2}2\left(1+\frac{bg^2}{16\pi ^2}\ln \left(\frac H{\Lambda ^2}\right) \right).$$ This formula predicts negative energy values for small values of the field $ H $, so the usual perturbative ground state with $H=0$ is unstable with respect to the formation of a state with a non vanishing field intensity [@Shuryak] $$H_{vac}=\Lambda ^2\exp \left(-\frac{16\pi ^2}{bg^2}-\frac 12\right),$$ at which the energy $E\left(H\right)$ has a minimum. With the use of this model an extensive number of physical problems, related with the hadron structure, confinement, etc. have been investigated. Nevertheless, after some time its intense study was abandoned. The main reason were: 1\. The perturbative relation giving $E_{vac}$ would be only valid if the second order of the perturbative expansion is relatively small. 2\. The specific spatial and color directions of the magnetic field break the now seemingly indispensable Lorentz and $SU(3)$ invariance of the ground state. 3\. The magnetic moment of the vector particle (gluon) is such that its energy in the presence of the field has a negative eigenvalue, which also makes unstable the homogeneous magnetic field $H$. Before presenting the objectives of the present work it should be stressed that QCD quantization [@Faddeev-Popov] is realized in the same way as that in QED, and it can be shown that QCD is renormalizable. The quadratic field terms in the QCD Lagrangian ($L_{QCD}$), which depend on the quark and gluon fields, have the same form that the ones corresponding to the electrons and photons in QED. However, in connection with the interaction, there appears a substantial difference due to the coupling of the gluon to itself. In order to assure the unitarity of the quantum theory of gauge fields, it was necessary to introduce fictitious particles called the Faddeev-Popov ghosts, which carry color charge, behaves as fermions (their fields anticommute) in spite of their boson like propagation. These particles cancel out the contributions of the non-physical gauge field degrees of freedom, and in physical calculations only appear as internal lines of the Feynman diagrams. As it is well known, a perturbative expansion depends on the initial conditions at $t\rightarrow \pm \infty $ or what is the same on the states in which the expansion is based. The perturbation theory at finite orders differs in attention to the ground state selected, or from a functional point of view what boundary conditions are chosen. The perturbation theory in QED (PQED) is in excellent correspondence with the experimental facts. In this theory the expansion is based on a perturbative vacuum state that is the empty of the Fock space, excluding the presence of fermion and boson particles. This is a radical simplification of the exact perturbative ground state that should be a complex combination of states on the Fock basis. Formally the expansion around the Fock vacuum contains all the effects associated to the exact vacuum, but it would require from infinite orders of the expansion in the coupling constant for describing them. The rapid convergence of the perturbative series in PQED indicates that the higher excited states of the Fock basis expansion, in the real vacuum, have a short life and a small influence on physical observable. In QCD the color confinement indicates that the ground state has a non-trivial structure, which in terms of a Fock expansion could be represented with the formation of a gluon “condensate”. Therefore, it should not be surprising that the PQCD fails to describe the low energy physics where the propagator of gluons could be affected by the presence of the condensate, even under the validity of a modified perturbative expansion. Such a perturbative condensate could generate all the effects over the physical observable, which in the standard expansion could require an infinite number of terms of the series. In a previous work [@Cabo], following the above ideas, the construction of a modified perturbation theory for QCD was implemented. This construction retained the main invariance of the theory (the Lorentz and $SU(3)$ ones), and it was also able to reproduce some of the main physical predictions of the chromomagnetic field models. The central idea in that work was to modify the perturbative expansion in such a way that the effects of a gluon condensate could be incorporated. Such a modification is needed to be searched through the connection of the interaction on an alternative state in the Fock space designed to incorporate the presence of the gluon condensate. It is not excluded that this procedure could be also a crude approximation of the reality as in the case in which the connection is done on the Fock vacuum (QED). However, this procedure could produce a reasonable if not good description of the low energy physics. If such is the case the low and high-energy descriptions of QCD could be unified in a common unified perturbation theory. In particular, in that previous work [@Cabo] the results had the interesting outcome of producing a non vanishing mean value for the relevant quantity $G^2$. In addition the effective potential, in terms of the condensation parameter at a first order approximation, showed a minimum at non-vanishing values of that parameter. Therefore, the procedure was able to reproduce at least some central predictions of the chromomagnetic models and general QCD analysis. The main objective of the present work is to search the foundations of the mentioned perturbation theory. The concrete aim is to find a physical state in the Fock space of the non-interacting theory being able to generate that expansion. The canonical quantization formalism for gauge fields, developed by Kugo and Ojima is employed. The exposition will be organized as follows: The Chapter 2 is divided in three sections. In the first one a review of the former work [@Cabo] is done, by also establishing the needs for the present one and the objectives which are planned to be analyzed and solved. In the second section the operational quantization method for gauge fields developed by Kugo and Ojima is discussed. Starting from it, in the third section it is exposed the ansatz for the Fock space state that generates the desired form of the perturbative expansion. The proof that the state satisfies the physical state condition is also given in this section. The Chapter 3 is divided in three sections. In the first one an analysis for the general form of the generating functional in an arbitrary ground state is made. In the second section it is shown that the proposed state can generate the desired modification for the gluon propagator by a proper selection of the parameters at hand. In the third section the modification of the propagator for the ghost particles is investigated, such propagator was not modified in the work [@Cabo] and here this procedure is justified as compatible within the present description. Finally, two appendices are introduced for a detailed analysis of the most elaborated parts in the calculation of transverse, longitudinal and scalar modes contribution to the gluon propagator modification. Ground State Ansatz =================== The previous work [@Cabo] is reviewed, as motivation for the present discussion, and the objectives for the present work stated. It is also reviewed the canonical quantization method for gauge fields developed by Kugo and Ojima (K.O.). Finally the QCD modified vacuum state is proposed and it is shown that this state satisfies the BRST physicality conditions imposed by the K.O. formalism. Motivation ---------- In this section a review of a previous work [@Cabo] is made. The main properties of this approach, as was mentioned in the introduction, were: a\) The ability to produce a gluon-condensation parameter value $\left\langle G^2\right\rangle $ directly in the first approximation. b\) The prediction of a minimum of the effective action for non-vanishing values of the condensation parameter. The discussion in [@Cabo] opened the possibility of reproducing some interesting physical implications of the early chromomagnetic field models for the QCD vacuum [@Savv1; @Savv2] by also solving some of their main shortcoming: The breaking of Lorentz and $SU\left(3\right)$ invariance. However, the discussion in [@Cabo] had also a limitation; that is it was unknown if the state that generated the proposed modification to the gluon propagator was a physical state of the theory. This shortcoming, could be expressed in the gauge parameter dependence of the calculated gluon mass. Below it is reminded the main analysis in [@Cabo]. The exposition was referred to the Euclidean space and the followed conventions were used, $$\begin{aligned} \nabla _\mu ^{ab} &=&\delta ^{ab}\partial _\mu +gf^{abc}A_\mu ^c, \\ F_{\mu \nu }^a &=&\partial _\mu A_\nu ^a-\partial _\nu A_\mu ^a+f^{abc}A_\mu ^cA_\nu ^b,\end{aligned}$$ where $g$ is the coupling constant and $f^{abc}$ are the structure constant of $SU(3)$. The action for the problem, including the auxiliary sources for all the fields was taken as, $$\begin{aligned} S_T\left[ A,\overline{C},C\right] &=&\int d^4x\left\{ -\frac 14F_{\mu \nu }^aF_{\mu \nu }^a+\frac 1{2\alpha }\partial _\mu A_\mu ^a\partial _\nu A_\nu ^a+\overline{C}^a\nabla _\mu ^{ab}\partial _\mu C^b\right. \\ &&\text{ \qquad \qquad \qquad \qquad \qquad \qquad }\left. +J_\mu ^aA_\mu ^a+ \overline{\xi }^aC^a+\overline{C}^a\xi ^a\right\},\end{aligned}$$ where $A_\mu,$ $\overline{C},C$ are the gauge and ghost fields and $\alpha $ is the gauge fixing parameter [@Faddeev] for the Lorentz gauge. The generating functional for the Green functions was expressed in the form $$Z_T\left[ J,\xi,\overline{\xi }\right] =\frac 1N\int D\left( A,\overline{C},C\right) \exp \left\{ S_T\left[ A,\overline{C},C\right] \right\},$$ which through the usual Legendre transformation led to the effective action, $$\Gamma \left[ \Phi \right] =\ln Z\left[ J\right] -J_i\Phi _i,\text{ \quad with ~}\Phi _i=\frac{\delta \ln Z\left[ J\right] }{\delta J_i}, \label{efec}$$ $\Phi _i$ denoted the mean values of the fields, and the compact notation of DeWitt [@Daemi], $$\Phi _i\equiv \left(A,\overline{C},C\right) ;\text{ \qquad }J_i\equiv \left(J,\xi,\overline{\xi }\right),$$ was used. In which $\Phi _i$ and $J_i$ indicate all the fields and sources at a space-time point, respectively. Repeated indices imply space-time integration as well as summation over all the field types and over their Lorentz and color components. The one-loop effective action and the corresponding “quantum” Lagrange equations, in the compact notation, were considered as, $$\begin{aligned} \Gamma \left[ \Phi \right] &=&S\left[ \Phi \right] +\frac 12\ln DetD\left[ \Phi \right], \label{acc} \\ \Gamma _{,i}\left[ \Phi \right] &=&S_{,i}\left[ \Phi \right] +\frac 12S_{,ikj}D_{kj}=-J_i, \label{ecmo}\end{aligned}$$ the functional derivatives were denoted by $$L_{,i}\left[ \Phi \right] =\frac{\delta L\left[ \Phi \right] }{\delta \Phi _i }$$ and the action defined by $S_T=S+J_i\Phi _i$. The $\Phi$ dependent propagator $D$ was defined, as usual, through $$D_{ij}=-S_{,ij}^{-1}\left[ \Phi \right], \label{D}$$ After considering a null mean value for the vector field $\Phi$, as requires the $SO(4)$ invariance, the propagator relation (\[D\]) took the form $$D_{ij}=-S_{,ij}^{-1}\left[ 0\right].$$ In this case the only non vanishing second derivatives of the action were, $$\begin{aligned} \frac{\delta ^2S}{\delta A_\mu ^a\left(x\right) \delta A_\nu ^b\left(x^{\prime }\right) }\left[ 0\right] &=&\delta ^{ab}\left( \partial _{x}^2\delta _{\mu \nu }-\left(1+\frac 1\alpha \right) \partial _\mu ^{x}\partial _\nu ^{x}\right) \delta \left(x-x^{\prime }\right), \label{Sglu} \\ \frac{\delta ^2S}{\delta C^a\left(x\right) \delta \overline{C}^b\left( x^{\prime }\right) }\left[ 0\right] &=&\delta ^{ab}\partial _x^2\delta \left(x-x^{\prime }\right). \label{Sgho}\end{aligned}$$ The gluon and ghost propagators are the inverse kernels of (\[Sglu\]) and (\[Sgho\]). Here, the alternative for a perturbative description of gluon condensation appeared. As (\[Sglu\]) consist of derivatives only, the inverse kernel of the gluon propagator could include coordinate independent terms reflecting a sort of gluon condensation. It should be noticed that the propagator is a $SO\left(4\right)$ tensor (not a vector) then a constant term in it does not led necessary to a breaking of the $SO\left(4\right)$ invariance [@Cabo]. Accordingly with the above remark gluon and ghost propagators were selected as $$\begin{aligned} D_{\mu \nu }^{ab}\left(x\right) &=&\int \frac{dp}{\left(2\pi \right) ^4} \left[ C\delta ^{ab}\delta _{\mu \nu }\delta \left( p\right) +\frac{\delta ^{ab}}{p^2}\left(\delta _{\mu \nu }-\left( 1+\alpha \right) \frac{p_\mu p_\nu }{p^2}\right) \right] \exp \left(ipx\right), \label{Dglu} \\ D_G^{ab}\left(x\right) &=&\int \frac{dp}{\left(2\pi \right) ^4}\frac{ \delta ^{ab}}{p^2}\exp \left(ipx\right), \label{Dgho}\end{aligned}$$ and it was checked that the equations of motion (\[ecmo\]), considering (\[Dglu\]) and (\[Dgho\]) and taking vanishing gluon and ghost fields, were satisfied. After that some implications of the modified gluon propagator, in the standard perturbative calculations, were analyzed [@Cabo]. The first interesting result obtained was the standard one loop polarization tensor. It was modified by a massive term, depending on the condensate parameter, with the form $$m^2=\frac{3g^2}{\left(2\pi \right) ^4}C\left(1-\alpha \right). \label{mass}$$ This result had a dependence on the gauge parameter $\alpha$; which as was mentioned above is one of the shortcomings of the discussion [@Cabo] because it was unknown if this mass term was generated by a non-physical vacuum state. In the present work the idea is to solve this difficulty by explicitly constructing a perturbative state leading to the considered form of the propagator, but also satisfying the BRST physical state condition in the non-interacting limit. The mean value of the squared field intensity operator was also calculated [@Cabo], within the simplest approximation (the tree approximation), with the use of the proposed propagator. That is, it was evaluated the expression $$\langle 0\mid S_g\left[ \Phi \right] \mid 0\rangle \equiv \frac 1N\left[ \int D\left(\Phi \right) S_g\left[ \Phi \right] \exp S_T\left[ \Phi \right] \right] _{J_i=0},$$ with $$S_g\left[ \Phi \right] \equiv \int d^4x\left\{ -\frac 14F_{\mu \nu }^a\left(x\right) F_{\mu \nu }^a\left(x\right) \right\},$$ and the following result was obtained, $$\langle 0\mid S_g\left[ \Phi \right] \mid 0\rangle =-\frac{72g^2C^2}{\left(2\pi \right) ^8}\int d^4x.$$ Then the mean value of $G^2$ took the form $$G^2\equiv \langle 0\mid F_{\mu \nu }^a\left(x\right) F_{\mu \nu }^a\left(x\right) \mid 0\rangle =\frac{288g^2C^2}{\left(2\pi \right) ^8}. \label{G2}$$ The substitution of Eq. (\[G2\]) in Eq. (\[mass\]) gave a rough estimate of the gluon mass. It was selected a particular value of $\alpha =0$ and assumed the more or less accepted value of $g^2G^2$ in the physical vacuum $$g^2G^2\cong 0.5\left(\frac{GeV}{c^2}\right) ^4,$$ then the estimated value of the gluon mass became $$m=0.35\frac{GeV}{c^2}.$$ Finally, an evaluation for the contribution to the effective potential of all the one-loop graphs, having only mass term insertions in the polarization tensor, was done. The result, in terms of $G^2$ (\[G2\]), turned to be of the form [@Cabo], $$V\left(G^2\right) =\frac{G^2}4+\frac 3{16\pi ^2}g^2\frac{G^2}{32}\ln \frac{ g^2G^2}{\mu ^4}, \label{VG}$$ where $\mu $ is the dimensional parameter included by the renormalization procedure. As it can be noticed in (\[VG\]), the effective potential indicates the spontaneous generation of a $G^2$ condensate from the usual perturbative vacuum $\left(G^2=0\right) $. This occurred in close analogy with the chromomagnetic fields. Then from the reviewed functional treatment, there are some interesting features that allow believing that the above procedure could describe relevant phenomena of the low energy region through a perturbative expansion. However some questions needed to be answered and taken as objectives of the present work are: 1- To determine under what conditions the new gluon propagator (\[Dglu\]) corresponds to a modified vacuum satisfying the physical state condition. This could also help in the understanding of the $\alpha$ dependence in the gluonic mass term. 2- To investigate the form of the ghost propagator in the modified vacuum state, because in the previous work [@Cabo] it was taken the as same of the usual perturbative theory. Operational Quantization Formalism ---------------------------------- As it is well known the non-abelian character of Yang-Mills fields determines the asymptotic freedom property, and the quark-confinement problem of QCD. This character simultaneously makes difficult the quantization of such theories. The first approach to this quantization was made by Faddeev and Popov in the path integral formalism [@Faddeev-Popov], with the resulting correct Feynman rules including the Faddeev-Popov ghost fields and the renormalizability of the theory. But this approach has the problem of the absence of notions about the state vector space and the Heisemberg operators. In this case due the non-abelian character of the theory it is not possible to use the operators formalism developed by Gupta-Bleuler [@Gupta] or the more general Nakanishi-Lautrup version [@Nakanishi], which can be used only for the abelian case. This situation occurs because de S-Matrix calculated with those procedures is not unitary in the non abelian case, as it was first mentioned by Feynman [@Feynman]. In the present work the operator formalism developed by T. Kugo and I. Ojima [@Kugo], for the first consistent quantization of the Yang-Mills fields, is considered. This formulation uses the Lagrangian invariance under a global symmetry operation called the BRST transformation [@BRST]. In the following a brief review of the K.O. work is done and the following conventions are used. Let $G$ be a compact Lie group, and $\Lambda$ any matrix in the adjoin representation of its associated Lie Algebra. The matrix $\Lambda$ can be represented as a linear combination of the form $$\Lambda =\Lambda ^aT^a,$$ were $T^a$ are the generators $(a=1,...,$Dim$G=n)$, which can be chosen as Hermitian ones and satisfying $$\left[ T^a,T^b\right] =if^{abc}T^c.$$ The field variations under infinitesimal gauge transformations are given by $$\begin{aligned} \delta _\Lambda A_\mu ^a\left(x\right) &=&\partial _\mu \Lambda ^a\left(x\right) +gf^{acb}A_\mu ^c\left(x\right) \Lambda ^b\left( x\right) =D_\mu ^{ab}\left(x\right) \Lambda ^b, \\ D_\mu ^{ab}\left(x\right) &=&\partial _\mu \delta ^{ab}+gf^{acb}A_\mu ^c\left(x\right).\end{aligned}$$ The metric $g_{\mu \nu }$ is taken in the convention $$g_{00}=-g_{ii}=1\qquad \text{for}\quad i=1,2,3.$$ The complete G.D. Lagrangian to be considered is the one employed in the operator quantization approach [@OjimaTex]. Its explicit form is given by $$\begin{aligned} \mathcal{L} &=&\mathcal{L}_{YM}+\mathcal{L}_{GF}+\mathcal{L}_{FP} \label{Lag} \\ \mathcal{L}_{YM} &=&-\frac 14F_{\mu \nu }^a\left( x\right) F^{\mu \nu,a}\left(x\right), \label{YM} \\ \mathcal{L}_{GF} &=&-\partial ^\mu B^a\left(x\right) A_\mu ^a\left(x\right) +\frac \alpha 2B^a\left(x\right) B^a\left( x\right), \label{GF} \\ \mathcal{L}_{FP} &=&-i\partial ^\mu \overline{c}^a\left(x\right) D_\mu ^{ab}\left(x\right) c^b\left( x\right), \label{FP}\end{aligned}$$ where field intensity is $$F_{\mu \nu }^a\left(x\right) =\partial _\mu A_\nu ^a\left( x\right) -\partial _\nu A_\mu ^a\left(x\right) +gf^{abc}A_\mu ^b\left(x\right) A_\nu ^c\left(x\right).$$ Relation (\[YM\]) defines the standard Yang-Mills Lagrangian, Eq. (\[GF\]) defines the gauge fixing term which can be also rewritten in the form $$\mathcal{L}_{GF}=-\frac 1{2\alpha }\left(\partial ^\mu A_\mu ^a\left(x\right) \right) ^2+\frac \alpha 2\left(B^a\left(x\right) +\frac 1\alpha \partial ^\mu A_\mu ^a\left(x\right) \right) ^2-\partial ^\mu \left( B^a\left(x\right) A_\mu ^a\left(x\right) \right),$$ equivalent to the more familiar $-\frac 1{2\alpha }\left(\partial ^\mu A_\mu ^a\left(x\right) \right) ^2$, at the equations of motion level [@Faddeev] and Feynman diagram expansion. Finally, Eq. (\[FP\]) describes the non-physical Faddeev-Popov ghost sector. The definition for such fields in the Kugo and Ojima (K.O.) approach is satisfying $$\overline{c}^{\dagger }=\overline{c},\text{ \qquad }c^{\dagger }=c.$$ That is, the ghost fields are Hermitian. In the Faddeev-Popov formalism [@Faddeev] they satisfy $$C^{\dagger }=\overline{C},\text{ \qquad }\overline{C}^{\dagger }=C.$$ However, a simple change of variables is able to transform between the ghost fields satisfying both kind of conjugation conditions. The selected conjugation properties, for this sector, allowed Kugo and Ojima to solve various formal problems existing for the application of the BRST operator quantization method to QCD, for example the hermiticity of the Lagrangian, which guarantees the unitarity of the S-Matrix. The physical state conditions in the BRST procedure [@OjimaTex] are given by $$\begin{aligned} &&Q_B\mid phys\rangle =0, \nonumber \\ &&Q_C\mid phys\rangle =0,\end{aligned}$$ where $$Q_B=\int d^3x\left[ B^a\left(x\right) \overleftrightarrow{\partial _0} c^a\left(x\right) +gB^a\left(x\right) f^{abc}A_0^b\left( x\right) c^c\left(x\right) +\frac i2g\partial _0\left( \overline{c}^a\right) f^{abc}c^b\left(x\right) c^c\left(x\right) \right],$$ with $$f\left(x\right) \overleftrightarrow{\partial _0}g\left(x\right) \equiv f\left(x\right) \partial _0g\left(x\right) -\partial _0\left(f\left(x\right) \right) g\left(x\right).$$ The BRST charge is conserved as a consequence of the BRST symmetry of the Lagrangian (\[Lag\]). The also conserved charge $Q_C$ is given by $$Q_C=i\int d^3x\left[ \overline{c}^a\left(x\right) \overleftrightarrow{ \partial _0}c^a\left(x\right) +g\overline{c}^a\left(x\right) f^{abc}A_0^b\left(x\right) c^c\left(x\right) \right],$$ its conservation comes from the Noether theorem, due to the Lagrangian invariance (\[Lag\]) under the phase transformation $c\rightarrow e^\theta c,\ \overline{c}\rightarrow e^{-\theta }\overline{c}$. This charge defines the so called “ghost number” as the difference between the number of ghost $c$ and $\overline{c}$. The analysis here is restricted to the Yang-Mills Theory without spontaneous breaking of the gauge symmetry. The quantization for the theory defined by the Lagrangian (\[Lag\]), considering the interacting free limit $g\rightarrow 0$, leads to the following commutation relations between the free fields, $$\begin{aligned} \left[ A_\mu ^a\left(x\right),A_\nu ^b\left(y\right) \right] &=&\delta ^{ab}\left(-ig_{\mu \nu }D\left(x-y\right) +i\left( 1-\alpha \right) \partial _\mu \partial _\nu E\left(x-y\right) \right), \nonumber \\ \left[ A_\mu ^a\left(x\right),B^b\left( y\right) \right] &=&\delta ^{ab}\left(-i\partial _\mu D\left( x-y\right) \right), \nonumber \\ \left[ B^a\left( x\right),B^b\left(y\right) \right] &=&\left\{ \overline{c} ^a\left(x\right),\overline{c}^b\left(y\right) \right\} =\left\{ c^a\left(x\right),c^b\left(y\right) \right\} =0, \nonumber \\ \left\{ c^a\left(x\right),\overline{c}^b\left(y\right) \right\} &=&-D\left(x-y\right), \label{com}\end{aligned}$$ The $E$ functions are defined by [@OjimaTex] $$E_{\left(.\right) }\left(x\right) =\frac 12\left(\nabla ^2\right) ^{-1}\left(x_0\partial ^0-1\right) D_{\left(.\right) }\left( x\right).$$ The equations of motion for the non-interacting fields take the simple form $$\begin{aligned} \Box A_\mu ^a\left(x\right) -\left(1-\alpha \right) \partial _\mu B^a\left(x\right) &=&0, \\ \partial ^\mu A_\mu ^a\left(x\right) +\alpha B^a\left(x\right) &=&0, \label{liga1} \\ \Box B^a\left( x\right) =\Box c^a\left(x\right) =\Box \overline{c} ^a\left( x\right) &=&0.\end{aligned}$$ This equations can be solved for an arbitrary values of the $\alpha$ parameter. However, the discussion will be restricted to the case $\alpha =1$ which corresponds to the situation in which all the gluon components satisfy the D’Alambert equation. This selection, as considered in the framework of the usual perturbative expansion, implies that you are not able to check the $\alpha$ independence of the physical quantities. This simplification is a necessary requirement. In the present discussion, the aim is to construct a perturbative state that satisfies the BRST physical state condition, in order to connect adiabatically the interaction. Then, the physical character of all the prediction will follow whenever the former assumption that adiabatic connection do not take the state out of the physical subspace at any intermediate state. The consideration of different values of $\alpha $, would be also a convenient recourse for checking the $\alpha$ independent perturbative expansion. However, at this stage it is preferred to delay this more technical issue for future work. In that way the field equations for the $\alpha =1$ are $$\begin{aligned} \Box A_\mu ^a\left(x\right)=\Box B^a\left(x\right) =\Box c^a\left( x\right) =\Box \overline{c} ^a\left(x\right) &=&0, \label{movi1} \\ \partial ^\mu A_\mu ^a\left(x\right) +B^a\left(x\right) &=&0. \label{movi2}\end{aligned}$$ The solutions of the set (\[movi1\]), (\[movi2\]) can be written as $$\begin{aligned} A_\mu ^a\left(x\right) &=&\sum\limits_{\vec{k},\sigma }\left( A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k},\sigma }^{a+}f_{k,\mu }^{\sigma *}\left(x\right) \right), \nonumber \\ B^a\left(x\right) &=&\sum\limits_{\vec{k}}\left(B_{\vec{k}}^ag_k\left(x\right) +B_{\vec{k}}^{a+}g_k^{*}\left(x\right) \right), \nonumber \\ c^a\left(x\right) &=&\sum\limits_{\vec{k}}\left( c_{\vec{k}}^ag_k\left(x\right) +c_{\vec{k}}^{a+}g_k^{*}\left( x\right) \right), \nonumber \\ \overline{c}^a\left(x\right) &=&\sum\limits_{\vec{k}}\left(\overline{c}_{ \vec{k}}^ag_k\left( x\right) +\overline{c}_{\vec{k}}^{a+}g_k^{*}\left(x\right) \right).\end{aligned}$$ The wave packets system, for non-massive scalar and vector fields, are taken in the form $$\begin{aligned} g_k\left(x\right) &=&\frac 1{\sqrt{2Vk_0}}\exp \left(-ikx\right), \nonumber \\ f_{k,\mu }^\sigma \left(x\right) &=&\frac 1{\sqrt{2Vk_0}}\epsilon _\mu ^\sigma \left(k\right) \exp \left( -ikx\right). \label{pol}\end{aligned}$$ The polarization vectors, in Eq. (\[pol\]) are defined by $$\vec{k}\cdot \vec{\epsilon}_\sigma \left(k\right) =0,\ \epsilon _\sigma ^0\left(k\right) =0,$$ and satisfy $$\vec{\epsilon}_\sigma \left(k\right) \cdot \vec{\epsilon}_\tau \left(k\right) =\delta _{\sigma \tau },$$ where $\sigma,\tau =1,2$ are the transverse modes. For the longitudinal $L$ and scalar $S$ modes the definitions are $$\begin{aligned} \epsilon _L^\mu \left(k\right) &=&-ik^\mu =-i\left(\left| \vec{k}\right|, \vec{k}\right),\ \epsilon _L^{\mu *}\left( k\right) =-\epsilon _L^\mu \left(k\right), \\ \epsilon _S^\mu \left(k\right) &=&-i\frac{\overline{k}^\mu }{2\left| \vec{k} \right| ^2}=\frac{-i\left(\left| \vec{k}\right|,-\vec{k}\right) }{2\left| \vec{k}\right| ^2},\ \epsilon _S^{\mu *}\left(k\right) =-\epsilon _S^\mu \left(k\right),\end{aligned}$$ and satisfy $$\begin{aligned} \epsilon _L^{\mu *}\left(k\right) \cdot \epsilon _{L,\mu }\left( k\right) &=&\epsilon _S^{\mu *}\left(k\right) \cdot \epsilon _{S,\mu }\left(k\right) =0, \\ \epsilon _L^{\mu *}\left(k\right) \cdot \epsilon _{S,\mu }\left(k\right) &=&1.\end{aligned}$$ The scalar product of the defined polarizations define the metric matrix $$\widetilde{\eta }_{\sigma \tau }=\epsilon _\sigma ^{\mu *}\left(k\right) \cdot \epsilon _{\tau,\mu }\left(k\right)\equiv \left( \begin{array}{cccc} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array} \right).$$ Now it is possible to introduce the contravariant (in the polarization index) polarizations $$\epsilon ^{\sigma,\mu }\left(k\right) =\sum\limits_{1,2,L,S}\widetilde{ \eta }^{\sigma \tau }\cdot \epsilon _\tau ^\mu \left(k\right),$$ satisfying $$\sum\limits_\sigma \epsilon ^{\sigma,\mu }\left(k\right) \cdot \epsilon _\sigma ^{\nu *}\left(k\right) =\sum\limits_{\sigma,\tau }\widetilde{\eta } ^{\sigma \tau }\cdot \epsilon _\tau ^\mu \left( k\right) \cdot \epsilon _\sigma ^{\nu *}\left(k\right) =g^{\mu \nu }$$ and $$\begin{aligned} \epsilon ^{\sigma,\mu }\left(k\right) \cdot \epsilon _\mu ^{\tau *}\left(k\right) &=&\widetilde{\eta }^{\sigma \tau }, \\ \widetilde{\eta }^{\sigma \tau ^{\prime }}\cdot \widetilde{\eta }_{\tau ^{\prime }\tau } &=&\delta _\tau ^\sigma.\end{aligned}$$ After that, it follows for the vector functions $$\sum\limits_{\vec{k},\sigma }f_{k,\sigma }^\mu \left(x\right) \cdot f_k^{\sigma,\nu *}\left(y\right) =g^{\mu \nu }D_{+}\left( x-y\right).$$ As it can be seen from (\[movi2\]) the $A_{\vec{k},\sigma }^a$ and $B_{\vec{k}}^a$ modes are not all independent. Indeed, it follows from (\[movi2\]) that $$B_{\vec{k}}^a=A_{\vec{k}}^{S,a}=A_{\vec{k},L}^a.$$ Then excluding the scalar mode, the free Heisemberg fields expansion takes the form $$A_\mu ^a\left(x\right) =\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k}}^af_{k,S,\mu }\left(x\right) \right)+h.c.,$$ where $h.c.$ represents the Hermitian conjugate of the first term. In order to satisfy the commutations relations (\[com\]) the creation and annihilation operator, associated to the Fourier components of the field, should obey $$\begin{aligned} \left[ A_{\vec{k},\sigma }^a,A_{\vec{k}^{\prime },\sigma ^{\prime }}^{a^{\prime }+}\right] &=&-\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }}\eta _{\sigma \sigma ^{\prime }}, \nonumber \\ \left\{ c_{\vec{k}}^a,\overline{c}_{\vec{k}^{\prime }}^{a^{\prime }+}\right\} &=&i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }}, \nonumber \\ \left\{ \overline{c}_{\vec{k}}^a,c_{\vec{k}^{\prime }}^{a^{\prime }+}\right\} &=&-i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }}\end{aligned}$$ and all the other vanish. In a symbolic matrix form these relations can be arranged as follows $$\begin{array}{cccccc} & A_{\vec{k}^{\prime },\sigma ^{\prime }}^{a^{\prime }+} & A_{\vec{k} ^{\prime }}^{L,a^{\prime }+} & B_{\vec{k}^{\prime }}^{a^{\prime }+} & c_{ \vec{k}^{\prime }}^{a+} & \overline{c}_{\vec{k}^{\prime }}^{a+} \\ A_{\vec{k},\sigma }^a & \delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }}\delta _{\sigma \sigma ^{\prime }} & 0 & 0 & 0 & 0 \\ A_{\vec{k}}^{L,a} & 0 & 0 & -\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k} ^{\prime }} & 0 & 0 \\ B_{\vec{k}}^a & 0 & -\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k}^{\prime }} & 0 & 0 & 0 \\ c_{\vec{k}}^a & 0 & 0 & 0 & 0 & i\delta ^{aa^{\prime }}\delta _{\vec{k}\vec{k }^{\prime }} \\ \overline{c}_{\vec{k}}^a & 0 & 0 & 0 & -i\delta ^{aa^{\prime }}\delta _{\vec{ k}\vec{k}^{\prime }} & 0 \end{array} \label{commu}$$ The above commutation rules and equation of motions define the quantized non-interacting limit of G.D. Then, it is possible now to define the alternative interacting free ground state to be considered for the adiabatic connection of the interaction. As discussed before, the expectation is that the physics of the perturbation theory to be developed will be able to furnish good description of some low energy physical effects. It is interesting to comment now that one of the first tasks proposed for the present work was to construct a state, in quantum electrodynamics, able to generate a modification for the photon propagator similar to the one proposed in [@Cabo] for gluons. It was used the quantification operator method developed by Gupta and Bleuler (GB), however was impossible to find any state generating this covariant propagator modification and satisfying the physical state condition imposed by this formalism. In the GB formalism the physical state condition is given by $$\partial ^\mu A_\mu ^{+}\left(x\right) \mid \Phi \rangle =0,$$ or in terms of the annihilation operators [@Sokolov], by $$k_0\left(A_{\vec{k},3}-A_{\vec{k},0}\right) \mid \Phi \rangle =0.$$ The more general state satisfying this condition is [@GuptaTex] $$\mid \Phi \rangle =\sum\limits_{m,n_{1,}n_2}B_{n_{1,}n_2,m}\mid \Phi \left(n_{1,}n_2,m\right) \rangle,$$ with $$\mid \Phi \left(n_{1,}n_2,m\right) \rangle =\left(m!\right) ^{-\frac 12}\left(A_{\vec{k},3}^{+}-A_{\vec{k},0}^{+}\right) ^m\left(n_1!n_2!\right) ^{-\frac 12}\left( A_{\vec{k},1}^{+}\right) ^{n_1}\left(A_{ \vec{k},2}^{+}\right) ^{n_2}\mid 0\rangle,$$ where $B_{n_{1,}n_2,m}$ are arbitrary constants. This general form of the state is the one that disabled to find a covariant modification to the propagator. The alternative vacuum state ---------------------------- In the present section the construction of a relativistic invariant ground state in the non-interacting limit of QCD is considered. It will be required that the proposed state satisfies the BRST physical state conditions. Then this state will have an opportunity to furnish the gluodynamics ground state under the adiabatic connection of the interaction. After beginning to work in the K.O. formalism some indications were found, that the appropriate state obeying the physical state conditions in this procedure, and with possibilities for generating the modification to the gluon propagator proposed in the previous work, could have the general structure $$\mid \phi \rangle =\exp \sum\limits_a\left(\sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{ \vec{p},\sigma }^{a+}+C_3\left(\left| \vec{p}\right| \right) \left(B_{\vec{ p}}^{a+}A_{\vec{p}}^{L,a+}+i\overline{c}_{\vec{p}}^{a+}c_{\vec{p} }^{a+}\right) \right) \mid 0\rangle, \label{Vacuum}$$ where $\vec{p}$ is an auxiliary momentum chosen as one of the few smallest momenta of the quantized theory in a finite volume $V$. This value will be taken later in the limit $V\rightarrow \infty $ for recovering Lorentz invariance. From here the sum on the color index $a$ will be explicit. The parameters $C_i\left(\left| \vec{p}\right| \right)$, $i=1,2,3,$ will be fixed below from the condition that the free propagator associated to a state satisfying the BRST physical state condition, coincides with the one proposed in the previous work [@Cabo]. The solution of this problem, then would give a more solid foundation to the physical implications of the discussion in that work. It should also be noticed that the state defined by Eq. (\[Vacuum\]) has some similarity with the coherent states [@Itzykson]. However, in the present case, the creation operators appear in squares. Thus, the argument of the exponential creates pairs of physical and non-physical particles. An important property of this function is that its construction in terms of creation operator pairs determines that the mean value of an odd number of field operators vanishes. This at variance with the standard coherent state in which the mean values of the fields are nonzero. The vanishing of the mean fields is a property in common with the standard perturbative vacuum, in which Lorentz invariance could be broken by a non-vanishing expectation value of a 4-vector the gauge field. It should be also stressed that this state as formed by the superposition of gluons state pairs suggests a connection with some recent works in the literature that consider the formation gluons pairs due to the color interactions. Now it is checked that the state (\[Vacuum\]) satisfies the BRST physical state conditions $$\begin{aligned} &&Q_B\mid \Phi \rangle =0, \nonumber \\ &&Q_C\mid \Phi \rangle =0.\end{aligned}$$ The expressions for these charges in the interaction free limit [@OjimaTex] are $$\begin{aligned} &&Q_B=i\sum\limits_{\vec{k},a}\left( c_{\vec{k}}^{a+}B_{\vec{k}}^a-B_{\vec{k} }^{a+\ }c_{\vec{k}}^a\right), \nonumber \\ &&Q_C=i\sum\limits_{\vec{k},a}\left( \overline{c}_{\vec{k}}^{a+}c_{\vec{k} }^a+c_{\vec{k}}^{a+}\overline{c}_{\vec{k}}^a\right).\end{aligned}$$ Considering first the action of $Q_B$ on the proposed state, [ $$\begin{aligned} &&Q_B\mid \Phi \rangle =i\exp \left\{ \sum\limits_{\sigma,a}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \times \nonumber \\ &&\times \left(\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \sum\limits_{ \vec{k},b}c_{\vec{k}}^{b+}B_{\vec{k}}^b\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \right. \\ &&-\left. \exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \sum\limits_{\vec{k},b}B_{\vec{k} }^{b+}c_{\vec{k}}^b\exp \left\{ \sum\limits_aC_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \right) \mid 0\rangle =0, \nonumber\end{aligned}$$ ]{}where the identity $$\left[ B_{\vec{k}}^b,\exp \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right] =-C_3\left(\left| \vec{p} \right| \right) B_{\vec{p}}^{b+}\delta _{\vec{k},\vec{p}}\exp \sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{ p}}^{L,a+}, \label{ident1}$$ was used. For the action of $Q_C$ on the considered state it follows [ $$\begin{aligned} &&Q_C\mid \Phi \rangle =i\exp \left\{ \sum\limits_{\sigma,a}\frac 12C_\sigma \left(\left| \vec{p}\right| \right)A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}+\sum\limits_aC_3\left(\left| \vec{p}\right| \right) B_{\vec{p} }^{a+}A_{\vec{p}}^{L,a+}\right\} \\ &&\times \left[ \sum\limits_{\vec{k},b} \overline{c}_{\vec{k}}^{b+}c_{\vec{k} }^b\left( 1+\sum\limits_aiC_3\left(\left| \vec{p}\right| \right) \overline{ c }_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) +\sum\limits_{\vec{k},b}c_{\vec{k} }^{b+}\overline{c}_{\vec{k}}^b\left(1+\sum\limits_aiC_3\left( \left| \vec{p} \right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \right] \mid 0\rangle =0 \nonumber\end{aligned}$$ ]{}which vanishes due to the commutation rules of the ghost operators (\[commu\]). Next, the evaluation of norm of the proposed state is considered, which due to the commutation properties of the operator can be written as $$\begin{aligned} \langle \Phi \mid \Phi \rangle =\prod\limits_{a=1,..,8} &\prod\limits_{\sigma =1,2}&\langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^aA_{\vec{p},\sigma }^a\right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p} \right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \nonumber \\ &\times &\langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right\} \exp \left\{ C_3\left( \left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \nonumber \\ &\times &\langle 0\mid \left(1-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \left(1+iC_3\left( \left| \vec{ p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle.\end{aligned}$$ For the product of the factors associated with transverse modes and the eight values of the color index, after expanding the exponential in series, it follows that $$\begin{aligned} &&\left[ \langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left( \left| \vec{p }\right| \right) A_{\vec{p},\sigma }^aA_{\vec{p},\sigma }^a\right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \langle 0\mid \sum\limits_{m=0}^\infty \left| \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(A_{\vec{p},\sigma }^a\right) ^{2m}\left( A_{\vec{p},\sigma }^{a+}\right) ^{2m}}{\left(m!\right) ^2}\mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \sum\limits_{m=0}^\infty \left| \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(2m\right) !}{\left(m!\right) ^2}\right] ^8, \label{normT}\end{aligned}$$ where the identity $$\langle 0\mid \left(A_{\vec{p},\sigma }^a\right) ^{2m}\left( A_{\vec{p},\sigma }^{a+}\right) ^{2m}\mid 0\rangle =\left( 2m\right) !,$$ was used. The factors linked with the scalar and longitudinal modes can be transformed as follows $$\begin{aligned} &&\left[ \langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right\} \exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \langle 0\mid \sum\limits_{m=0}^\infty \left| C_3\left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(A_{\vec{p}}^{L,a}B_{\vec{p} }^a\right) ^m\left( B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right) ^m}{\left(m!\right) ^2}\mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ \sum\limits_{m=0}^\infty \left| C_3\left(\left| \vec{p}\right| \right) \right| ^{2m}\right] ^8=\left[ \frac 1{\left(1-\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right] ^8\text{ for}\quad \left| C_3\left(\left| \vec{p}\right| \right) \right| <1, \label{normLS}\end{aligned}$$ in which the identity $$\langle 0\mid \left(A_{\vec{p}}^{L,a}B_{\vec{p}}^a\right) ^m\left( B_{\vec{p }}^{a+}A_{\vec{p}}^{L,a+}\right) ^m\mid 0\rangle =\left( m!\right) ^2,$$ was employed. Finally the factor connected with the ghost fields can be calculated as follows $$\begin{aligned} &&\left[ \langle 0\mid \left(1-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \left( 1+iC_3\left(\left| \vec{ p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle \right] ^8 \nonumber \\ &&=\left[ 1+\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\langle 0\mid c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\overline{c}_{\vec{p}}^{a+}c_{ \vec{p}}^{a+}\mid 0\rangle \right] =\left[ 1-\left| C_3\left( \left| \vec{p} \right| \right) \right| ^2\right] ^8. \label{normG}\end{aligned}$$ After substituting all the calculated factors, the norm of the state can be written as $$N=\langle \Phi \mid \Phi \rangle =\prod\limits_{\sigma =1,2}\left[ \sum\limits_{m=0}^\infty \left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^{2m}\frac{\left(2m\right) !}{\left(m!\right) ^2}\right] ^8.$$ Therefore, it is possible to define the normalized state $$\mid \widetilde{\Phi }\rangle =\frac 1{\sqrt{N}}\mid \Phi \rangle.$$ Note that, as it should be expected, the norm is not dependent on the $ C_3\left(\left| \vec{p}\right| \right) $ parameter which defines the non-physical particle operators entering in the definition of the proposed vacuum state. Propagator Modifications ======================== The general form for generating functionals and propagators, for boson and fermion particles in an arbitrary vacuum state, are analyzed. The modification for the gluon and ghost propagators, introduced by the vacuum state defined in the previous chapter, are calculated. General Form of the Propagator ------------------------------ As it is well known in the Quantum Field Theory to calculate any element of the S-Matrix, after applying the reduction formulas, it is necessary to obtain the vacuum expectation value of the temporal ordering of Heisemberg operators [@Gasiorowicz]. That is it is needed to calculate $$\langle \Psi \mid T\left(\hat{A}_H\left(x_1\right) \hat{A}_H\left( x_2\right) \hat{A}_H\left(x_3\right)...\right) \mid \Psi \rangle, \label{orden1}$$ where $\mid \Psi \rangle $ is the real vacuum of the interacting theory. For simplifying the exposition it is considered a scalar field, the generalization for vector fields is straightforward. Using the relations between the operators in the Interaction and Heisemberg representations $$\begin{aligned} &&\hat{A}_H\left(x\right) =\hat{U}\left(0,t\right) \hat{A}_I\left( x\right) \hat{U}\left(t,0\right), \\ &&\hat{U}\left( t_1,t_2\right) \hat{U}\left(t_2,t_3\right) =\hat{U}\left( t_1,t_3\right)\end{aligned}$$ and assuming that the real vacuum interacting state can be obtained from the non-interacting one under the adiabatic connection of the interaction. The expression (\[orden1\]) takes the form [@Gasiorowicz] $$\frac{\langle \Phi \mid T\left\{ \hat{A}_I\left(x_1\right) \hat{A}_I\left(x_2\right) \hat{A}_I\left(x_3\right)...\exp \left( -\int\limits_{-\infty }^\infty H_i\left(t\right) dt\right) \right\} \mid \Phi \rangle }{\langle \Phi \mid T\left\{ \exp \left(-\int\limits_{-\infty }^\infty H_i\left(t\right) dt\right) \right\} \mid \Phi \rangle }, \label{orden2}$$ where $\Phi $ is the non interacting vacuum of the theory. To evaluate these quantities it is needed to develop the exponential in series of perturbation theory and calculate the vacuum expectation values of the temporal ordering of fields in the interaction representation ($ \hat{A}_I\left(x\right) $), but in this representation the field operators are like free fields ($\hat{A}^0\left(x\right))$ about which much is known. $$\hat{A}_I\left(x\right) =\hat{A}^0\left(x\right).$$ And it is necessary to evaluate terms of the form $$\langle \Phi \mid T\left(\hat{A}^0\left(x_1\right) \hat{A}^0\left( x_2\right) \hat{A}^0\left(x_3\right)...\right) \mid \Phi \rangle.$$ Introducing the auxiliary generating functional $$Z\left[ J\right] \equiv \langle \Phi \mid T\left(\exp \left\{ i\int d^4xJ\left(x\right) A^0\left(x\right) \right\} \right) \mid \Phi \rangle, \label{genfun}$$ it is possible to write for the relevant expectation values the expression $$\langle \Phi \mid T\left(\hat{A}^0\left(x_1\right) \hat{A}^0\left( x_2\right) \hat{A}^0\left(x_3\right)...\right) \mid \Phi \rangle =\left(\frac 1i\frac \delta {\delta J\left(x_1\right) }\frac 1i\frac \delta {\delta J\left(x_1\right) }\frac 1i\frac \delta {\delta J\left(x_1\right) }...Z\left[ J\right] \right) _{J=0}.$$ Considering now the auxiliary functional $$Z\left[ J;t\right]\equiv \langle \Phi \mid T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^0\left( x\right) \right\} \right) \mid \Phi \rangle$$ and defining $W\left(t\right)$ through the relation $$\begin{aligned} &&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^0\left(x\right) \right\} \right)\nonumber \\ && =T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right), \label{T11}\end{aligned}$$ where $A^{0-}\left(x\right)$ and $A^{0+}\left( x\right)$ are the negative (creation) and positive (annihilation) frequency parts, respectively. The $t$ differentiation on the expression (\[T11\]), takes the form $$\begin{aligned} &i&\int\limits_{x_0=t}d^3xJ\left(x\right) A^0\left(x\right) T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right) \nonumber \\ &=&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left(x\right) \right\} \right) \frac{dW\left(t\right) }{dt}+ \nonumber \\ &&+i\int\limits_{x_0=t}d^3xJ\left(x\right) A^{0-}\left(x\right) T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left( x\right) A^{0-}\left(x\right) \right\} \right) W\left(t\right).\end{aligned}$$ Keeping in mind that the free field creation operators commute, for all times, the following relation holds $$\left[ A^{0-}\left(x\right),A^{0-}\left(y\right) \right] =0,$$ then the $T$ instruction can be eliminated and after some algebra is obtained [ $$\begin{aligned} \frac{dW\left(t\right) }{dt} &=&i\exp \left\{ -i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left( x\right) \right\} \int\limits_{x_0=t}d^3xJ\left(x\right) A^{0+}\left(x\right) \times \nonumber \\ &&\times \exp \left\{ i\int\limits_{-\infty }^tdt\int d^3xJ\left(x\right) A^{0-}\left( x\right) \right\} W\left(t\right) \nonumber \\ &=&i\int\limits_{y_0=t}d^3yJ\left(y\right) \left\{ A^{0+}\left( y\right) -i\int\limits_{-\infty }^td^4xJ\left(x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\} W\left( t\right). \label{dif1}\end{aligned}$$]{} The initial condition on $W\left(t\right)$ is $$W\left(-\infty \right) =1.$$ Then the solution of (\[dif1\]) is $$\begin{aligned} W\left(t\right) &=&\exp \left\{ i\int\limits_{-\infty }^td^4yJ\left(y\right) A^{0+}\left(y\right) \right\}\nonumber \\&& \times \exp \left\{ \int\limits_{-\infty }^td^4y \int\limits_{-\infty }^{y_0}d^4xJ\left(y\right) J\left(x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\},\end{aligned}$$ when $t\rightarrow \infty $ this expression takes the form $$\begin{aligned} W\left(\infty \right) &=&\exp \left\{ i\int d^4yJ\left(y\right) A^{0+}\left(y\right) \right\} \times \\ &&\times \exp \left\{ \int d^4xd^4y\theta \left(y_0-x_0\right) J\left(y\right) J\left( x\right) \left[ A^{0-}\left(x\right),A^{0+}\left(y\right) \right] \right\}.\end{aligned}$$ Therefore, the generating functional (\[genfun\]) can be written in the following way [@Gasiorowicz] $$\begin{aligned} Z\left[ J\right] &\equiv &\langle \Phi \mid \exp \left\{ i\int d^4xJ\left(x\right) A^{0-}\left(x\right) \right\} \exp \left\{ i\int d^4yJ\left(y\right) A^{0+}\left(y\right) \right\} \mid \Phi \rangle \nonumber \\ &&\times \exp \left\{ \frac i2\int d^4xd^4yJ\left(x\right) D(x-y)J\left(y\right) \right\}, \label{bosones}\end{aligned}$$ where $D(x-y)$ is the usual propagator for an scalar particle. In case that is needed to calculate a similar matrix element for fermions the following functional is defined $$Z\left[ \eta,\bar{\eta}\right] \equiv \langle \Phi \mid T\left( \exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^0\left(x\right) + \bar{\psi}^0\left(x\right) \eta \left(x\right) \right] \right\} \right) \mid \Phi \rangle.$$ Because of the anticommuting properties of $\bar{\psi},\ \psi$ fields the introduced sources $\bar{\eta},\ \eta$ satisfy anticommuting relations between then and with the field operators. Here is assumed the left differentiation convention, then the S-Matrix element can be calculate by the following expression $$\begin{aligned} &&\langle \Phi \mid T\left(\psi ^0\left(y_1\right) \bar{\psi}^0\left(z_1\right) \psi ^0\left(y_2\right) ...\bar{\psi}^0\left(z_k\right) \right) \mid \Phi \rangle \nonumber \\ &&=\left(\frac 1i\frac \delta {\delta \eta \left( z_k\right) }...\frac 1i\frac \delta {\delta \bar{\eta}\left( y_2\right) }\frac 1i\frac \delta {\delta \eta \left(z_1\right) }\frac 1i\frac \delta {\delta \bar{\eta} \left(y_1\right) }Z\left[ \eta,\bar{\eta}\right] \right) _{\eta,\bar{\eta} =0}.\end{aligned}$$ Now, in the same way that for the bosons, the following auxiliary functional is defined by $$Z\left[ \eta,\bar{\eta};t\right] \equiv \langle \Phi \mid T\left( \exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta}\left(x\right) \psi ^0\left(x\right) +\bar{\psi}^0\left( x\right) \eta \left(x\right) \right] \right\} \right) \mid \Phi \rangle$$ and the corresponding $G\left(t\right)$ functional by $$\begin{aligned} &&T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta} \left(x\right) \psi ^0\left(x\right) +\bar{\psi}^0\left(x\right) \eta \left(x\right) \right] \right\} \right) \nonumber \\ &&=T\left(\exp \left\{ i\int\limits_{-\infty }^tdt\int d^3x\left[ \bar{\eta} \left(x\right) \psi ^{0-}\left( x\right) +\bar{\psi}^{0-}\left(x\right) \eta \left(x\right) \right] \right\} \right) G\left(t\right). \label{ferm1}\end{aligned}$$ Manipulations completely parallel to those leading to (\[dif1\]) give [ $$\begin{aligned} \frac{dG\left(t\right) }{dt}=i &&\left[ \int\limits_{y_0=t}d^3y\left(\bar{ \eta}\left(y\right) \psi ^{0+}\left(y\right) +\bar{\psi}^{0+}\left(y\right) \eta \left( y\right) \right) \right. \nonumber \\ &&+i\int\limits_{-\infty }^td^4x\int\limits_{y_0=t}d^3y\bar{\eta}\left(y\right) \left\{ \psi ^{0+}\left(y\right),\bar{\psi}^{0-}\left(x\right) \right\} \eta \left(x\right) \nonumber \\ &&\left. -i\int\limits_{-\infty }^td^4x\int\limits_{y_0=t}d^3y\bar{\eta} \left(x\right) \left\{ \psi ^{0-}\left(x\right),\bar{\psi}^{0+}\left(y\right) \right\} \eta \left(y\right) \right] G\left(t\right),\end{aligned}$$]{} This equation is easily integrated to obtain the solution $$\begin{aligned} G\left(t\right) &=&\exp \left\{ i\int\limits_{-\infty }^td^4y\left(\bar{ \eta}\left(y\right) \psi ^{0+}\left(y\right) +\bar{\psi}^{0+}\left(y\right) \eta \left(y\right) \right) \right\} \nonumber \\ &&\times \exp \left\{ -\int\limits_{-\infty }^td^4y\int\limits_{-\infty }^{y_0}d^4x\bar{\eta}\left(y\right) \left\{ \psi ^{0+}\left(y\right),\bar{ \psi}^{0-}\left(x\right) \right\} \eta \left(x\right) \right\} \nonumber \\ &&\times \exp \left\{ \int\limits_{-\infty }^td^4x\int\limits_{-\infty }^{x_0}d^4y\bar{\eta}\left(y\right) \left\{ \psi ^{0-}\left(y\right),\bar{ \psi}^{0+}\left(x\right) \right\} \eta \left(x\right) \right\},\end{aligned}$$ where in the last term the dummy variables $x$ and $y$ were interchanged. Consequently the following expression for the generating functional arise [@Gasiorowicz] [ $$\begin{aligned} Z\left[ \eta,\bar{\eta}\right] &\equiv &\langle \Phi \mid \exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^{0-}\left( x\right) +\bar{ \psi}^{0-}\left(x\right) \eta \left(x\right) \right] \right\} \nonumber \\ &&\quad\times\exp \left\{ i\int d^4x\left[ \bar{\eta}\left(x\right) \psi ^{0+}\left(x\right) +\bar{\psi}^{0+}\left(x\right) \eta \left(x\right) \right] \right\} \mid \Phi \rangle \nonumber \\ &&\times \exp \left\{ i\int d^4xd^4y\bar{\eta}\left(x\right) S\left(x-y\right) \eta \left(y\right) \right\} \label{fermiones}\end{aligned}$$]{} where $S\left(x-y\right)$ is the standard fermion propagator. As much for the case of bosons as for fermions the term related with the vacuum expectation value for the usual vacuum is one. This is so because the annihilation operators are located to the right and to the left those of creation. However in the present work the vacuum expectation values generate the propagator modifications, because the vacuum state considered is not the trivial one. The other term in the generating functional expression, that is expressed by a simple exponential of c numbers, gives the usual propagator and it has the same form when is calculated by this operational method or alternatively by the functional method. Then, starting from the analysis in the present section it can be concluded that from an operation formalism point of view any modification to the usual propagators is only generated by a change in the vacuum state of the theory. And these modifications can be determined through the vacuum expectation values in (\[bosones\]) and (\[fermiones\]). From a functional formalism point of view, the propagator modifications are generated by changes in the boundary conditions. Modified Gluon Propagator ------------------------- As it follows from the general form of the Wick Theorem, analyzed in the previous section, the modification of the gluon propagator introduced by the modified vacuum state (\[Vacuum\]) is defined by the expression $$\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4xJ^{\mu ,a}\left(x\right) A_\mu ^{a-}\left(x\right) \right\} \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \widetilde{\Phi } \rangle, \label{mod}$$ for each value of the color index $a$. All the different colors can be worked out independently because of the commutation relations between the annihilation and creation operators for the free theory. At the necessary point of the analysis all the color contributions will be included. The annihilation and creation fields in (\[mod\]) are given by $$\begin{aligned} A_\mu ^{a+}\left(x\right) &=&\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k}}^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k} }^af_{k,S,\mu }\left(x\right) \right), \\ A_\mu ^{a-}\left(x\right) &=&\sum\limits_{\vec{k}}\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^{a+}f_{k,\mu }^{\sigma *}\left(x\right) +A_{\vec{k}}^{L,a+}f_{k,L,\mu }^{*}\left(x\right) +B_{\vec{k} }^{a+}f_{k,S,\mu }^{*}\left( x\right) \right).\end{aligned}$$ In what follows it is calculated explicitly, for each color, the action of the exponential operators $$\begin{aligned} &&\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left( x\right) \right\} \mid \Phi \rangle \nonumber \\ &&=\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} }\left( \sum\limits_{\sigma =1,2}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) +A_{\vec{k}}^{L,a}f_{k,L,\mu }\left(x\right) +B_{\vec{k} }^af_{k,S,\mu }\left(x\right) \right) \right\} \nonumber \\ &&\quad\times \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}+C_3\left(\left| \vec{p}\right| \right) \left(B_{\vec{p}}^{a+}A_{ \vec{p}}^{L,a+}+i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \right\} \mid 0\rangle. \label{expd}\end{aligned}$$ After a systematic use of the commutation relations among the annihilation and creation operators, the exponential operators can be decomposed in products of exponential for each space-time mode. This fact allows to perform the calculation for each mode independently. Then the expression (\[expd\]) takes the form [ $$\begin{aligned} &\prod\limits_{\sigma =1,2}&\exp \left\{ i\int d^4xJ^{\mu ,a}\left(x\right) \sum\limits_{\vec{k}}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) \right\} \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \nonumber \\ &\times &\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k }}\left(B_{\vec{k}}^af_{k,S,\mu }\left( x\right)+A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right)\right) \right\} \nonumber \\ &\times &\exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p} }^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \exp \left\{ C_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right\} \mid 0\rangle.\end{aligned}$$]{} For a transverse component, it is necessary to calculate $$\exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k}}A_{ \vec{k},\sigma }^af_{k,\mu }^\sigma \left(x\right) \right\} \exp \left\{ \frac 12C_\sigma \left( \left| \vec{p}\right| \right) A_{\vec{p},\sigma }^{a+}A_{\vec{p},\sigma }^{a+}\right\} \mid 0\rangle \ \text{\qquad for } \sigma =1,2 \label{modT}$$ The following recourse is used to calculate this expression; calling $U$ the first exponential in (\[modT\]) this expression can be written as $$\exp \left\{ \frac 12C_\sigma\left(p\right) \left( UA_{\vec{p},\sigma }^{a+}U^{-1}\right) \left(UA_{\vec{p},\sigma }^{a+}U^{-1}\right) \right\} \mid 0\rangle, \label{modTt}$$ since $$U^{-1}\mid 0\rangle =\mid 0\rangle.$$ The inverse $U^{-1}$ is the same $U$ when in the exponential argument the sign is changed. Using the Baker-Hausdorf formula $$\exp [\hat{F}]\hat{G}\exp [-\hat{F}]=\exp \left\{ [\hat{F},\ ]\right\} \hat{G }=\sum \frac 1{n!}\left[ \hat{F},\left[ \hat{F},....,\left[ \hat{F},\hat{G} \right].....\right] \right]$$ and noticing that only the first and the second term in the expansion are non-vanishing when $\hat{F}$ and $\hat{G}$ are linear functions of annihilation and creation operators, it follows $$\exp [\hat{F}]\hat{G}\exp [-\hat{F}]=\hat{G}+\left[ \hat{F},\hat{G}\right].$$ Therefore, for the relevant commutators appearing in (\[modTt\]) it follows $$\left[ i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k}}A_{\vec{k},\sigma }^af_{k,\mu }^\sigma \left( x\right),A_{\vec{p},\sigma }^{a+}\right] =i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right).$$ Then for the expression (\[modT\]) the following result is obtained $$\exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left(A_{ \vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \ \label{T}$$ For the longitudinal and scalar modes, following the above procedure, the result obtained is $$\begin{aligned} &&\exp \left\{i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} }\left(B_{\vec{k}}^af_{k,S,\mu }\left( x\right) +A_{\vec{k} }^{L,a}f_{k,L,\mu }\left(x\right) \right) \right\}\nonumber \\ && \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) B_{\vec{p}}^{a+}A_{\vec{p}}^{L,a+}\right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left( B_{\vec{p} }^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\&& \qquad\qquad\qquad\qquad \times \left. \left(A_{\vec{p}}^{L,a+} -i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle.\label{L}\end{aligned}$$ where the expressions below were used $$\begin{aligned} &&\left[ \left(i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} } A_{\vec{k} }^{L,a}f_{k,L,\mu }\left( x\right) \right),B_{\vec{p}}^{a+}\right] =-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \\ &&\left[ \left(i\int d^4xJ^{\mu,a}\left(x\right) \sum\limits_{\vec{k} } B_{\vec{k}}^af_{k,S,\mu }\left(x\right) \right),A_{\vec{p} }^{L,a+}\right] =-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right).\end{aligned}$$ For the full modification calculation (\[mod\]), it is necessary to evaluate $$\langle \Phi \mid \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a-}\left(x\right) \right\} =\left(\exp \left\{ -i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \Phi \rangle \right) ^{\dagger }, \label{left}$$ which can be easily obtained by conjugating the result for the right hand side, through (\[T\]) and (\[L\]). Then, substituting (\[T\]), (\[L\]) and (\[left\]) in (\[mod\]), the following expression should be calculated $$\begin{aligned} &&\frac 1N\langle 0\mid \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma ^{*}\left(\left| \vec{p}\right| \right) \left(A_{\vec{p},\sigma }^a+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right) \right) ^2\right\} \nonumber \\ &&\qquad \times \exp \left\{ \sum\limits_{\sigma =1,2}\frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left( A_{\vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \nonumber \\ &&\times\langle 0\mid \exp \left\{ C_3^{*}\left( \left| \vec{p}\right| \right) \left(B_{\vec{p}}^a-i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,L,\mu }^{*}\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left(x\right) \right) \right\} \nonumber \\ &&\qquad \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left(B_{ \vec{p}}^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle \nonumber \\ &&\times\langle 0\mid \exp \left(-iC_3^{*}\left(\left| \vec{p}\right| \right) c_{\vec{p}}^a\overline{c}_{\vec{p}}^a\right) \exp \left( iC_3\left(\left| \vec{p}\right| \right) \overline{c}_{\vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle \label{mod1}\end{aligned}$$ In the expression (\[mod1\]) the calculated contribution, for each transverse mode, is $$\exp \left\{ -\int \frac{d^4xd^4y}{2Vp_0}J^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) \epsilon _\mu ^\sigma \left(p\right) \epsilon _\nu ^\sigma \left(p\right) \frac{\left(C_\sigma \left( \left| \vec{p}\right| \right) +C_\sigma ^{*}\left(\left| \vec{p}\right| \right) +2\left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^2\right) }{2\left(1-\left| C_\sigma \left(\left| \vec{p}\right| \right) \right| ^2\right) }\right\}, \label{T1}$$ and the longitudinal and scalar mode contribution is $$\exp \left\{ -\int \frac{d^4xd^4y}{2Vp_0}J^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) \epsilon _{S,\mu }\left(p\right) \epsilon _{L,\nu }\left(p\right) \frac{\left(C_3\left(\left| \vec{p}\right| \right) +C_3^{*}\left(\left| \vec{p}\right| \right) +2\left| C_3\left(\left| \vec{p} \right| \right) \right| ^2\right) }{\left( 1-\left| C_3\left(\left| \vec{p} \right| \right) \right| ^2\right) }\right\}, \label{L1}$$ The detailed analysis of these calculations can be found in the Appendixes 1 and 2. Therefore, after collecting the contributions of all the modes, assuming $ C_1\left(\left| \vec{p}\right| \right) =C_2\left( \left| \vec{p}\right| \right) =C_3\left(\left| \vec{p}\right| \right) $ (which follows necessarily in order to obtain Lorentz invariance) and using the properties of the defined vectors basis, the modification to the propagator becomes $$\exp \left\{ \frac 12\int \frac{d^4xd^4y}{2p_0V}J^{\mu,a}\left( x\right) J^{\nu,a}\left(y\right) g_{\mu \nu }\left[ \frac{\left( C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }{\left(1-\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right] \right\}. \label{totmodF}$$ In the expression (\[totmodF\]), the combination of the $C_1\left(\left| \vec{p}\right| \right) $ constant is always real and nonnegative, for all $ \left| C_1\left(\left| \vec{p}\right| \right) \right| <1$. Now it is possible to perform the limit process $\vec{p}\rightarrow 0$. In doing this limit, it is considered that each component of the linear momentum $\vec{p}$ is related with the quantization volume by $$p_x\sim \frac 1a,\ p_y\sim \frac 1b,\ p_z\sim \frac 1c,\ V=abc\sim \frac 1{\left| \vec{p}\right| ^3},$$ $\ \ $ And it is necessary to calculate $$\lim_{\vec{p}\rightarrow 0}\frac{\left(C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) }{4p_0V\left(1-\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) },\label{Lim1}$$ Then, after fixing a dependence of the arbitrary constant $C_1$ of the form $ \left| C_1\left(\left| \vec{p}\right| \right) \right| \sim \left(1-\kappa \left| \vec{p}\right| ^2\right),\kappa >0$, and $C_1\left(0\right) \neq -1$ the limit (\[Lim1\]) becomes $$\lim_{\vec{p}\rightarrow 0}\frac{\left(C_1\left(\left| \vec{p}\right| \right) +C_1^{*}\left(\left| \vec{p}\right| \right) +2\left| C_1\left(\left| \vec{p}\right| \right) \right| ^2\right) \left| \vec{p}\right| ^3\frac 1{\left(1-\left(1-\kappa \left| \vec{p}\right| ^2\right) ^2\right) }}{4p_0}=\frac C{2\left(2\pi \right) ^4}$$ where $C$ is an arbitrary real and nonnegative constant, determined by the also real and nonnegative constant $\kappa$. Therefore, the total modification to the propagator including all color values turns to be [ $$\begin{aligned} &\prod\limits_{a=1,..,8}&\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a-}\left( x\right) \right\} \exp \left\{ i\int d^4xJ^{\mu,a}\left(x\right) A_\mu ^{a+}\left(x\right) \right\} \mid \widetilde{\Phi }\rangle \nonumber \\ &=&\exp \left\{ \sum\limits_{a=1,..8}\int d^4xd^4yJ^{\mu,a}\left(x\right) J^{\nu,a}\left(y\right) g_{\mu \nu }\frac C{2\left(2\pi \right) ^4}\right\}.\end{aligned}$$]{} The generating functional associated to the proposed initial state, including the usual perturbative piece for $\alpha =1$, can be written in the form $$Z[J]=\exp \left\{ \frac i2\sum\limits_{a,b=1,..8}\int d^4xd^4yJ^{\mu,a}\left(x\right) \widetilde{D}_{\mu \nu }^{ab}(x-y)J^{\nu,b}\left(y\right) \right\},$$ where $$\widetilde{D}_{\mu \nu }^{ab}(x-y)=\int \frac{d^4k}{\left(2\pi \right) ^4} \delta ^{ab}g_{\mu \nu }\left[ \frac 1{k^2}-iC\delta \left(k\right) \right] \exp \left\{ -ik\left(x-y\right) \right\} \label{propag}$$ which shows that the gluon propagator has the same form proposed in [@Cabo], for the selected gauge parameter value $\alpha =1$ (which corresponds to $\alpha =-1$ in that reference). Modified Ghost Propagator ------------------------- In the present section the possible modification to the ghost propagator will be analyzed. As was shown in Sec. 3.1 for fermionic particles the expression for the modification, introduced by a nontrivial vacuum state, is $$\begin{aligned} &&\langle \widetilde{\Phi } \mid \exp \left\{ i\int d^4x\left( \overline{\xi }^a\left(x\right) c^{a-}\left(x\right) +\overline{c}^{a-}\left(x\right) \xi ^a\left(x\right) \right) \right\} \nonumber \\&&\qquad \times \exp \left\{ i\int d^4x\left( \overline{\xi }^a\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^a\left(x\right) \right) \right\} \mid \widetilde{\Phi }\rangle, \label{ini}\end{aligned}$$ where $$\begin{aligned} c^{a+}\left(x\right) &=&\sum\limits_{\vec{k}}c_{\vec{k}}^ag_k\left(x\right),\qquad c^{a-}\left(x\right) =\sum\limits_{\vec{k}}c_{\vec{k} }^{a+}g_k^{*}\left(x\right), \nonumber \\ \overline{c}^{a+}\left( x\right) &=&\sum\limits_{\vec{k}}\overline{c}_{\vec{k }}^ag_k\left(x\right),\qquad \overline{c}^{a-}\left(x\right) =\sum\limits_{\vec{k}}\overline{c}_{\vec{k}}^{a+}g_k^{*}\left( x\right).\end{aligned}$$ Now it is calculated explicitly the action of the exponential operator $$\begin{aligned} &&\exp \left\{ i\int d^4x\left(\overline{\xi }^a\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^a\left( x\right) \right) \right\} \exp \left\{ C_3\left(\left| \vec{p}\right| \right) i\overline{c}_{\vec{p}}^{a+} c_{\vec{p}}^{a+}\right\} \mid 0\rangle \nonumber \\ &&=\left( 1+i\int d^4y\overline{\xi }^a\left(y\right) \sum\limits_{\vec{k} ^{\prime }}c_{\vec{k}^{\prime }}^ag_{k^{\prime }}\left(y\right) \right) \left(1+i\int d^4x\sum\limits_{\vec{k}} \overline{c}_{\vec{k}}^ag_k\left(x\right) \xi ^a\left(x\right) \right) \times \nonumber \\ &&\qquad \times \left(1+C_3\left( \left| \vec{p}\right| \right) i\overline{c}_{ \vec{p}}^{a+}c_{\vec{p}}^{a+}\right) \mid 0\rangle. \label{Ghost}\end{aligned}$$ The grassman character of the field and sources allowed expanding the exponential retaining only the first two terms in the expansion. With the use of the following relations [$$\begin{aligned} \overline{\xi }^a\left(y\right) c_{\vec{k}^{\prime }}^a\overline{c}_{\vec{p} }^{a+}c_{\vec{p}}^{a+}\mid 0\rangle &=&i\delta _{\vec{k}^{\prime },\vec{p}} \overline{\xi }^a\left( y\right) c_{\vec{p}}^{a+}\mid 0\rangle, \nonumber \\ \overline{c}_{\vec{k}}^a\xi ^a\left(x\right) \overline{c}_{\vec{p}}^{a+}c_{ \vec{p}}^{a+}\mid 0\rangle &=&i\delta _{\vec{k},\vec{p}}\overline{c}_{\vec{p} }^{a+}\xi ^a\left(x\right) \mid 0\rangle, \nonumber \\ \overline{\xi }^a\left(y\right) c_{\vec{k}^{\prime }}^ai\delta _{\vec{k}, \vec{p}}\overline{c}_{\vec{p}}^{a+}\xi ^a\left(x\right) \mid 0\rangle &=&-\delta _{\vec{k},\vec{p}}\delta _{\vec{k}^{\prime },\vec{p}}\overline{ \xi }^a\left(y\right) \xi ^a\left(x\right) \mid 0\rangle,\end{aligned}$$ ]{}the expression (\[Ghost\]) can be written as $$\begin{aligned} &&\left[ 1+C_3\left(\left| \vec{p}\right| \right) \left( i\overline{c}_{ \vec{p}}^{a+}c_{\vec{p}}^{a+}-i\int d^4xg_p\left( x\right) \left(\overline{ \xi }^a\left(x\right) c_{\vec{p}}^{a+}+\overline{c}_{\vec{p}}^{a+}\xi ^a\left(x\right) \right) \right. \right. + \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p\left(y\right) g_p\left(x\right) \overline{\xi }^a\left(y\right) \xi ^a\left(x\right) \right) \right] \mid 0\rangle. \label{rhs}\end{aligned}$$ In addition the formula $$\begin{aligned} &&\langle \widetilde{\Phi }\mid \exp \left\{ i\int d^4x\left( \overline{\xi } ^a\left(x\right) c^{a-}\left(x\right) +\overline{c}^{a-}\left(x\right) \xi ^a\left(x\right) \right) \right\} \nonumber \\ &&=\left[ \exp \left\{ i\int d^4x\left( \overline{\xi }^{a\dagger }\left(x\right) c^{a+}\left(x\right) +\overline{c}^{a+}\left(x\right) \xi ^{a\dagger }\left(x\right) \right) \right\} \mid \widetilde{\Phi }\rangle \right] ^{\dagger }, \label{rsh1}\end{aligned}$$ allows to calculate the left hand side of (\[ini\]) using (\[rhs\]). Then the expression (\[ini\]), substituting (\[rhs\]) and (\[rsh1\]), takes the form [ $$\begin{aligned} &\langle 0\mid &\left[ 1-C_3^{*}\left(\left| \vec{p}\right| \right) \left(ic_{\vec{p}}^a\overline{c}_{\vec{p}}^a-i\int d^4xg_p^{*}\left(x\right) \left(c_{\vec{p}}^a\overline{\xi }^a\left(x\right) +\xi ^a\left(x\right) \overline{c}_{\vec{p}}^a\right) \right. \right. \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p^{*}\left(y\right) g_p^{*}\left(x\right) \xi ^a\left(y\right) \bar{\xi}^a\left( x\right) \right) \right] \nonumber \\ &\times &\left[ 1+C_3\left( \left| \vec{p}\right| \right) \left(i\overline{c }_{\vec{p}}^{a+}c_{\vec{p}}^{a+}-i\int d^4xg_p\left(x\right) \left(\overline{\xi }^a\left(x\right) c_{\vec{p}}^{a+}+\overline{c}_{\vec{p} }^{a+}\xi ^a\left(x\right) \right) \right. \right. \nonumber \\ &&\left. \left. +i\int d^4y\int d^4xg_p\left(y\right) g_p\left(x\right) \overline{\xi }^a\left(y\right) \xi ^a\left(x\right) \right) \right] \mid 0\rangle. \label{ghomod}\end{aligned}$$]{} In this case, the expression (\[ghomod\]) calculus is easier than the one realized for gluons. And the result of its contribution, canceling out the normalization factor, is $$\exp \left[ \frac{i\int d^4xd^4y\overline{\xi }^a\left(x\right) \xi ^a\left(y\right) \left(C_3\left(\left| \vec{p}\right| \right) +C_3^{*}\left(\left| \vec{p}\right| \right) -2\left| C_3\left( \left| \vec{p }\right| \right) \right| ^2\right) }{2Vp_0\left( 1-\left| C_3\left(\left| \vec{p}\right| \right) \right| ^2\right) }\right],$$ which in the limit $\vec{p}\rightarrow 0$, under the same condition considered for the gluon modification limit, takes the form $$\exp \left\{ -\sum\limits_{a=1,..8}i\int d^4xd^4y\overline{\xi }^a\left(x\right) \xi ^a\left(y\right) \frac{C_G}{\left(2\pi \right) ^4}\right\}.$$ In this expression $C_G$ is a real and nonnegative constant. It is interesting to note that choosing $C_3\left(0\right) =1$, then $C_G=0$ and there is no modification to the ghost propagator as was chosen in the previous work [@Cabo]. The ghost generating functional associated to the proposed initial state, including the usual perturbative piece for $\alpha =1$, can be written in the form $$Z_G[\overline{\xi },\xi ]=\exp \left\{ i\sum\limits_{a,b=1,..8}\int d^4xd^4y \overline{\xi }^a\left( x\right) \widetilde{D}_G^{ab}(x-y)\xi ^b\left(y\right) \right\},$$ where $$\widetilde{D}_G^{ab}(x-y)=\int \frac{d^4k}{\left(2\pi \right) ^4}\delta ^{ab}\left[ \frac{\left(-i\right) }{k^2}-C_G\delta \left(k\right) \right] \exp \left\{ -ik\left(x-y\right) \right\}.$$ Summary ======= By using the operational formulation for Quantum Gauge Fields Theory developed by Kugo and Ojima, a particular state vector for QCD in the non-interacting limit, that obeys the BRST physical state condition, was constructed. The general motivation for looking this wave function is to search for a reasonably good description of low energy QCD properties, through giving foundation to the perturbative expansion proposed in [@Cabo]. The high energy QCD description should not be affected by the modified perturbative initial state. In addition it can be expected that the adiabatic connection of the color interaction starting with it as an initial condition, generate at the end the true QCD interacting ground state. In case of having the above properties, the analysis would allow to understand the real vacuum as a superposition of infinite number of soft gluon pairs. It has been checked that properly fixing the free parameters in the constructed state, the perturbation expansion proposed in the former work [@Cabo] is reproduced for the special value $\alpha =1$ of the gauge constant. Therefore, the appropriate gauge is determined for which the expansion introduced in that work is produced by an initial state, satisfying the physical state condition for the BRST quantization procedure. The fact that the non-interacting initial state is a physical one, lead to expect that the final wave-function after the adiabatic connection of the color interaction will also satisfy the physical state condition for the interacting theory. If this assumption is correct, the results for calculations of transition amplitudes and the values of physical quantities should be also physically meaningful. In future, a quantization procedure for arbitrary values of $\alpha$ will be also considered. It is expected that with its help the gauge parameter independence of the physical quantities could be implemented. It seems possible that the results of this generalization will lead to $\alpha $ dependent polarizations for gluons and ghosts and their respective propagators, which however could produce $\alpha $ independent results for the physical quantities. However, this discussion will be delayed for future consideration. It is important to mention now a result obtained during the calculation of the gluon propagator modification, in the chosen construction. It is that the arbitrary constant $C$ is determined here to be real and nonnegative. This outcome restricts an existing arbitrariness in the discussion given in the previous work. As this quantity $C$ is also determining the square of the generated gluon mass as positive or negative, real or imaginary, therefore it seems very congruent to arrive to a definite prediction of $C$ as real and positive. The modification to the standard free ghost propagator introduced by the proposed initial state, was also calculated. Moreover, after considering the free parameter in the proposed trial state as real, which it seems the most natural assumption, the ghost propagator is not be modified, as it was assumed in [@Cabo]. Some tasks which can be addressed in future works are: The study of the applicability of the Gell-Mann and Low theorem with respect to the adiabatic connection of the interaction, starting from the here proposed initial state. The investigation of zero modes quantization, that is gluon states with exact vanishing four momentum. The ability to consider them with success would allow a formally cleaner definition of the proposed state, by excluding the auxiliary momentum $\vec{p}$ recursively used in the construction carry out. Finally, the application of the proposed perturbation theory in the study of some problems related with confinement and the hadron structure. Transverse Mode Contribution ============================ The transverse mode contribution is determined by the expression $$\begin{aligned} &&\langle 0\mid \exp \left\{ \frac 12C_\sigma ^{*}\left(\left| \vec{p} \right| \right) \left(A_{\vec{p},\sigma }^a+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right) \right) ^2\right\} \nonumber \\ &&\quad\times \exp \left\{ \frac 12C_\sigma \left(\left| \vec{p}\right| \right) \left(A_{\vec{p},\sigma }^{a+}+i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right) \right) ^2\right\} \mid 0\rangle \label{A1}\end{aligned}$$ For simplifying the exposition, the following notation is introduced $$\begin{aligned} C^{*} &\equiv &C_\sigma ^{*}\left(\left| \vec{p}\right| \right),\ C\equiv C_\sigma \left(\left| \vec{p}\right| \right),\text{ \qquad }\hat{A} ^{+}\equiv A_{\vec{p},\sigma }^{a+},\ \hat{A}\equiv A_{\vec{p},\sigma }^a \text{\ }, \nonumber \\ a_1 &\equiv &i\int d^4xJ^{\mu,a}\left(x\right) f_{p,\mu }^{\sigma *}\left(x\right),\text{ \qquad }a_2\equiv i\int d^4xJ^{\mu ,a}\left(x\right) f_{p,\mu }^\sigma \left(x\right). \label{nota1}\end{aligned}$$ Then the expression (\[A1\]) takes the form $$\begin{aligned} &&\langle 0 \mid \exp \left\{ \frac 12C^{*}\left( \hat{A}+a_1\right) ^2\right\} \exp \left\{ \frac 12C\left( \hat{A}^{+}+a_2\right) ^2\right\} \mid 0\rangle \label{A2} \\ &&=\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2a_2^2\right\} \langle 0\mid \exp \left\{ \frac{C^{*}}2\hat{A}^2 +C^{*}a_1\hat{A}\right\} \exp \left\{ \frac C2 \hat{A}^{+2}+Ca_2\hat{A}^{+}\right\} \mid 0\rangle. \nonumber\end{aligned}$$ For the action of the exponential, linear in the annihilation operator, at the left on the right the result obtained is $$\begin{aligned} &&\exp \left\{ C^{*}a_1\hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}+Ca_2\hat{A}^{+}\right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ \frac C2\left(\hat{A}^{+}+C^{*}a_1\right) ^2+Ca_2\left(\hat{A}^{+}+C^{*}a_1\right) \right\} \mid 0\rangle, \label{A3}\end{aligned}$$ where the same procedure used for calculating (\[modT\]) is considered. The expression (\[A2\]), considering (\[A3\]), can be written in the form $$\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\right\} \langle 0\mid \exp \left\{ \frac{C^{*}}2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}+C\hat{A}^{+}\left(C^{*}a_{1}+a_2\right) \right\} \mid 0\rangle \label{A4}$$ It is possible in (\[A4\]) to act with the exponential linear in the creation operator at the right on the left and the result is $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\left(1+\left| C\right| ^2\right) \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*}C\left(C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{A5}\end{aligned}$$ In such a way after n-steps it is possible to arrive to a recurrence relation, which can be proven by mathematical induction. This recurrence relation has the form $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left(2m+1\right) }\right] \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*n+1}C^{n+1} \left( C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{rec1}\end{aligned}$$ Lets probe it, acting with the exponential linear in the annihilation operator at the left on the right the result is $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\left(\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] +\left| C\right| ^{4\left(n+1\right) }\right) \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}+C^{*n+1}C^{n+2}\left(C^{*}a_{1}+a_2\right) \hat{A}^{+}\right\} \mid 0\rangle,\end{aligned}$$ now acting on the left with the exponential linear in the creation operator is obtained the relation $$\begin{aligned} &&\exp \left\{ \frac{C^{*}}2a_1^2+\frac C2\left( C^{*}a_{1}+a_2\right) ^2\sum\limits_{m=0}^{n+1}\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] \right\} \nonumber \\ &&\times \langle 0 \mid \exp \left\{ \frac{C^{*}}2\hat{A}^2+C^{*n+2}C^{n+2} \left( C^{*}a_{1}+a_2\right) \hat{A}\right\} \exp \left\{ \frac C2\hat{A} ^{+2}\right\} \mid 0\rangle \label{A6}\end{aligned}$$ which probe the recurrence relation (\[rec1\]). At this point the limit $n\rightarrow \infty$ is taken, considering $\left| C\right| <1$ which implies that $$\begin{aligned} &&\lim_{n\rightarrow \infty }\left| C\right| ^{2n}=0, \nonumber \\ &&\lim_{n\rightarrow \infty }\sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left( 2m+1\right) }\right] =\frac 1{\left(1-\left| C\right| ^2\right) }, \label{lim}\end{aligned}$$ and the expression (\[A6\]) in this limit has the form, $$\exp \left\{ \frac{\left(C^{*}a_{1}^2+Ca_2^2+2C^{*}Ca_1a_2\right) }{2\left(1-\left| C\right| ^2\right) }\right\} \langle 0\mid \exp \left\{ \frac{C^{*}} 2\hat{A}^2\right\} \exp \left\{ \frac C2\hat{A}^{+2}\right\} \mid 0\rangle \label{A7}$$ Finally, the notation (\[nota1\]) is substituted in (\[A7\]). After that, the functions of $\vec{p}$ are expanded in the vicinity of $\vec{p}=0$, keeping in mind that the sources are located in a space finite region it is necessary to consider only the first terms in the expansion. Then for the expression (\[A7\]) it is obtained the result (\[T1\]), the renormalization factors cancel out. Longitudinal and Scalar Modes Contribution ========================================== The longitudinal and scalar modes contribution is determined by the expression $$\begin{aligned} &&\langle 0\mid \exp \left\{ C_3^{*}\left(\left| \vec{p}\right| \right) \left(B_{\vec{p}}^a-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }^{*}\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left(x\right) \right) \right\} \nonumber \\ &&\qquad \times \exp \left\{ C_3\left(\left| \vec{p}\right| \right) \left(B_{ \vec{p}}^{a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }\left(x\right) \right) \right. \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad \times \left. \left( A_{\vec{p}}^{L,a+}-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }\left(x\right) \right) \right\} \mid 0\rangle \label{B1}\end{aligned}$$ introducing the following notation, $$\begin{aligned} C^{*} &\equiv &C_3^{*}\left(\left| \vec{p}\right| \right),\ C\equiv C_3\left(\left| \vec{p}\right| \right),\text{ \ \ }\hat{A}^{+}\equiv A_{ \vec{p}}^{L,a+},\ \hat{A}\equiv A_{\vec{p}}^{L,a},\text{ \ \ }\hat{B} ^{+}\equiv B_{\vec{p}}^{a+},\ \hat{B}\equiv B_{\vec{p}}^a, \nonumber \\ a_1 &\equiv &-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,S,\mu }^{*}\left( x\right),\text{ \qquad }a_2\equiv -i\int d^4xJ^{\mu,a}\left( x\right) f_{p,S,\mu }\left(x\right), \nonumber \\ b_1 &\equiv &-i\int d^4xJ^{\mu,a}\left(x\right) f_{p,L,\mu }^{*}\left( x\right),\text{ \qquad }b_2\equiv -i\int d^4xJ^{\mu,a}\left( x\right) f_{p,L,\mu }\left(x\right), \label{nota2}\end{aligned}$$ the expression (\[B1\]) takes the form $$\begin{aligned} &&\langle 0\mid \exp \left\{ C^{*}\left(\hat{A}+a_1\right) \left( \hat{B} +b_1\right) \right\} \exp \left\{ C\left( \hat{B}^{+}+b_2\right) \left(\hat{A}^{+}+a_2\right) \right\} \mid 0\rangle \nonumber \\ &&=\exp \left\{ C^{*}a_1b_1+Ca_2b_2\right\} \langle 0\mid \exp \left\{ C^{*}\left(\hat{A}\hat{B} +b_1\hat{A}+a_1\hat{B}\right) \right\} \nonumber \\ &&\text{ \qquad \qquad \qquad \qquad \qquad \qquad }\times \exp \left\{ C\left(\hat{B}^{+}\hat{A}^{+}+ b_2\hat{A}^{+}+a_2\hat{B}^{+}\right) \right\} \mid 0\rangle \label{B2}\end{aligned}$$ For the action of the exponential linear in the annihilation operator at the left on the right, is obtained $$\begin{aligned} &&\exp \left\{ C^{*}\left(b_1\hat{A}+a_1\hat{B}\right) \right\} \exp \left\{ C\left( \hat{B}^{+}\hat{A}^{+}+b_2\hat{A}^{+}+a_2\hat{B}^{+}\right) \right\} \mid 0\rangle \label{B3} \\ &&=\exp \left\{ C\left[ \left(\hat{B}^{+}-C^{*}b_1\right) \left(\hat{A} ^{+}-C^{*}a_1\right) +b_2\left(\hat{A}^{+}-C^{*}a_1\right) +a_2\left(\hat{B }^{+}-C^{*}b_1\right) \right] \right\} \mid 0\rangle, \nonumber\end{aligned}$$ the same procedure used for calculating (\[L\]) is considered. Following the same steps described in the previous appended for transverse modes, in this case the recurrence relation obtained for longitudinal and scalar modes is [ $$\begin{aligned} &&\exp \left\{ C^{*}a_1b_1+C\left(C^{*}a_1-a_2\right) \left( C^{*}b_1-b_2\right) \sum\limits_{m=0}^n\left[ \left| C\right| ^{2\left(2m\right) }+\left| C\right| ^{2\left(2m+1\right) }\right] \right\}\label{B4} \\ &&\langle 0\mid \exp \left\{ C^{*}\hat{A}\hat{B}+C^{*n+1}C^{n+1}\left(\left( C^{*}b_1-b_2\right) \hat{A}+\left(C^{*}a_1-a_2\right) \hat{B}\right) \right\} \exp \left\{ C\hat{B}^{+}\hat{A}^{+}\right\} \mid 0\rangle.\nonumber\end{aligned}$$]{} For the expression (\[B4\]), in the limit $n\rightarrow \infty $ considering $\left| C\right| <1$, the following relation is obtained $$\begin{aligned} &&\exp \left\{ C^{*}a_1b_1+C\left(C^{*}a_1-a_2\right) \left( C^{*}b_1-b_2\right) \frac 1{\left(1-\left| C\right| ^2\right) }\right\} \nonumber \\ &&\quad \langle 0\mid \exp \left\{ C^{*}\hat{A}\hat{B}\right\} \exp \left\{ C \hat{B}^{+}\hat{A}^{+}\right\} \mid 0\rangle. \label{B5}\end{aligned}$$ Finally, the notation (\[nota2\]) is substituted in (\[B5\]), the functions of $\vec{p}$ are expanded in the vicinity of $\vec{p}=0$, and the result (\[L1\]) is obtained. 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ArXiv

--- abstract: 'We consider a radiation from a uniformly accelerating harmonic oscillator whose minimal coupling to the scalar field changes suddenly. The exact time evolutions of the quantum operators are given in terms of a classical solution of a forced harmonic oscillator. After the jumping of the coupling constant there occurs a fast absorption of energy into the oscillator, and then a slow emission follows. Here the absorbed energy is independent of the acceleration and proportional to the log of a high momentum cutoff of the field. The emitted energy depends on the acceleration and also proportional to the log of the cutoff. Especially, if the coupling is comparable to the natural frequency of the detector ($e^2/(4m) \sim \omega_0$) enormous energies are radiated away from the oscillator.' address: - ' Dept. of Physics, Sungkyunkwan Univ., SUWON 440-746, KOREA ' - ' Dept. of Physics, KAIST , Taejon 305-701, KOREA ' author: - 'Hyeong-Chan Kim[^1]' - 'Jae Kwan Kim[^2]' date: 'April 8, 1997' title: Radiation from a uniformly accelerating harmonic oscillator --- [      pacs number 04.60.+n, 03.70.+k ]{} Introduction ============ It is well known that the concept of a particle depends on the motion of an observer [@birrell]. Especially, the Minkowski vacuum is a canonical ensemble with the temperature $a/2\pi$ from the point of view of a uniformly accelerated observer with the acceleration $a$ (the thermalization theorem) [@unruh]. This observer dependence is most easily shown if one use a particle detector model invented by Unruh [@unruh] and DeWitt [@dewitt79]. It consists of an idealized point particle with internal energy levels labeled by $E$, coupled via a monopole interaction with a scalar field $\phi$ (Unruh-DeWitt model). Following these, many works emerged in the literature. Letaw [@Letaw] exhibited the stationary world lines, on which the detectors in a vacuum have a time-independent excitation spectra. Grove and Ottewill [@Grove:Otte] studied the problem of a non-extended detector, and clarified the radiation effect arising both from the walls of the detector and from the interaction with the field. Several authors [@hinton; @takagi] discussed the anisotropic nature of the thermal radiation of the accelerated detector. A full review for this thermal character was given by Takagi [@takagi86]. The vacuum noise seen by a uniformly accelerated observer in flat space-times of arbitrary dimensions was investigated and was shown to exhibit the phenomenon of the apparent inversion of statistics in odd dimensions, which was discussed precisely by Unruh [@unruh86] and Fukazawa [@fukazawa]. A few years ago, the excitation rate associated with a uniformly accelerated finite-time detector interacting with the Minkowski vacuum has been analyzed in an inertial frame by Svaiter and Svaiter [@svaiter]. They found a logarithmic ultraviolet divergences on the transition amplitude, which was due to the instantaneous switching of the detector [@higuchi]. This UV divergence does not occur in lower dimensions. Grove argue that a macroscopic constantly accelerating object will emit negative energy radiation until equilibrium with the Minkowski vacuum is achieved [@grove]. Several years ago a new particle detector model–a harmonic oscillator coupled to a scalar field in $1+1$ dimensions–was introduced by Raine, Sciama, and Grove(RSG) [@raine]. Several aspects of this model was discussed in connection with the ‘open quantum system’ [@unruhZurek; @unruhWald; @anglin]. Hinterleitner [@hin] and Massar, Parentani, and Brout [@massar] shown that there is a polarization cloud which surrounds the detector at all times and energy is exchanged with it locally. Audretsch and Müller [@aud] explored nonlocal pair correlations in accelerated detector. Recently stochastic aspects of this detector were discussed by Raval, Hu, and Anglin [@raval]. These works mainly interested on the asymptotic states with time independent coupling. In this paper we consider the intermediate region during the equilibrium achieved between the detector and the field. We show that this is not a simple energy absorption process but there are two main stages after the two systems in contact with. First stage is a fast absorption of energy of the oscillator from the field. This occurs shortly after the change of the coupling in a time which is much smaller than the inverse of the characteristic frequency of the oscillator. The total energy absorbed during this period is independent of the acceleration and depends on the log of a high momentum cutoff. Second stage is slow emission of radiation which exponentially decrease in a time scale of the coupling constant. The total radiated energy during this period depends on the acceleration of the oscillator. If the coupling constant is small then the total radiation is smaller than the inertial one. But if the coupling is comparable to the characteristic frequency, enormous energies are radiated away from the oscillator. In the case of a weakly coupled system, the absorbed energy during the first stage is larger than the emitted one during the second stage. In Sec. II-A, we describe the model in Minkowski space and give the general form of the solution for the field and the oscillator. These evolutions of the operators are given by use of the inhomogeneous solution $G(\omega,t)$ of a forced harmonic oscillator. Similarly, all physical quantities like the correlation function or the stress tensor can be expressed with this single function. In Sec. II-B, the model is generalized to incorporate the uniformly accelerating oscillators. Sec. III is devoted to present two solvable models. $G(\omega,t)$ is obtained in the asymptotic region. We obtain the stress tensor in Sec. IV when the detector is turned on suddenly. Sec. V is summary and discussions. There are two appendices where we describe the details of the calculation of the stress tensor. Models for the particle detector ================================ Let us consider a minimally coupled system of a massless real scalar field $\phi(t,x)$ in two dimensions and a detector of a harmonic oscillator $q(t)$ with mass $m$. The action for this system is $$\begin{aligned} \label{ac} S &=& \int \mbox{d}x \mbox{d}t \frac{1}{2}\left\{ \left(\frac{\partial}{\partial t} \phi(t,x) \right)^2 - \left( \frac{\partial}{\partial x} \phi(t,x)\right)^2 \right\} \\ &+& \int d\tau \left\{ \frac{1}{2} m \left(\frac{d q(\tau)}{d \tau} \right)^2 -\frac{1}{2} m \omega_0^2 q^2(\tau) -e(\tau)q(\tau) \frac{ d\phi}{d \tau} \left(t(\tau),x(\tau)\right) \right\} . \nonumber\end{aligned}$$ The oscillator follows the explicitly given path $(t(\tau),x(\tau))$ where $\tau$ is the proper time of the oscillator along the path. In this paper, we select two paths through which the oscillator moves: the inertial and the uniformly accelerated. Varying Eq. (\[ac\]) with respect to $\phi(t,x)$ and $q(\tau)$ we get the Heisenberg equation of motion for the field and the oscillator $$\begin{aligned} \Box \phi(t,x) &=& \frac{de(\tau)q(\tau)}{d \tau} \delta (\rho), \label{2}\\ m \left( \frac{d}{d \tau}\right)^2 q(\tau) &+& m \omega_0^2 q(\tau) = - e(\tau)\frac{d \phi}{d \tau}(t(\tau),x(\tau)), \label{3}\end{aligned}$$ where $\rho$ is an appropriate space coordinate which is orthogonal to $\tau$ and the path of the oscillator can be represented as $\rho=0$. Eq. (\[2\]) can be integrated to give $$\begin{aligned} \label{phi:uv} \phi(t,x) = \phi^0(t,x) + \frac{e(\tau_{ret})}{2} q(\tau_{ret}),\end{aligned}$$ where $\tau_{ret}$ is the value of $\tau$ at the intersection of the past lightcone of $(t,x)$ and the detector trajectory. where we have used the explicit retarded propagator of a massless field $$\begin{aligned} G_{\mbox{ret}}(t,x;0,0) = \frac{1}{2} \theta(t+x) \theta(t-x).\end{aligned}$$ Substituting the solution (\[phi:uv\]) into Eq. (\[3\]), one get $$\begin{aligned} \label{qeq} m \ddot{q}(\tau) + \frac{1}{2} e^2(\tau) \dot{q}(\tau) + m \left( \omega_0^2 + \frac{\dot{e}^2(\tau)}{4 m}\right) q(\tau) = -e(\tau) \dot{\phi}^0(t(\tau),x(\tau)).\end{aligned}$$ The redefinitions $$\begin{aligned} M(\tau) &=& m \exp\left(\int_{\tau_0}^{\tau} \mbox{d}\tau \frac{e^2(\tau)}{2m } \right), \label{M:t} \\ \omega^2(\tau) &=& \omega_0^2 + \frac{\dot{e}^2(\tau)}{4m}, \\ F(\tau) &=& - \frac{e(\tau)}{m} \frac{d\phi^0}{d\tau} (t(\tau),x(\tau)).\end{aligned}$$ make Eq. (\[qeq\]) into the equation of motion of the forced harmonic oscillator with the effective mass $M(t)$, and the frequency $\omega^2(t)$ $$\begin{aligned} \label{q'':F} \ddot{q}(\tau) + \frac{d\ln M(\tau)}{d \tau}\dot{q}(\tau) + \omega^2(\tau) q(\tau) = F(\tau).\end{aligned}$$ Here $F(\tau)$ is the force density per unit effective mass. We take the normalization of the effective mass as $M(\tau_0)= m$ at some initial time $\tau_0$. As one see from Eq. (\[q”:F\]), we can arbitrarily choose the normalization of the effective mass. Note that we can rewrite this equation into quadratic form: $$\begin{aligned} \label{quad} \left[ \left(\frac{d}{d\tau}\right)^2 + \Omega^2(\tau) \right] \sqrt{M(\tau)} q(\tau) = \sqrt{M(\tau)} F(\tau),\end{aligned}$$ where $$\begin{aligned} \label{Omega} \Omega^2(\tau) = \omega_0^2- \left( \frac{e^2(\tau)}{4m}\right)^2.\end{aligned}$$ The behavior of a homogeneous solution of Eq. (\[quad\]) changes from oscillatory to exponential decay according to the value of $\Omega^2(t)$. We restrict our discussion into $\Omega^2(t)$ greater than zero. If $\epsilon(\tau) \ll \omega_0$ then $\Omega$ is natural positive frequency mode of the oscillator. The behavior of the homogeneous solution, in this case, is $$\begin{aligned} f(t) = \frac{1}{\sqrt{M(\tau)}} exp\left[{\pm i \int^\tau \Omega(\tau') d\tau'}\right].\end{aligned}$$ Let the initial Heisenberg operators for the oscillator to be $q(\tau_0)$ and $p(\tau_0)=m \dot{q}(\tau_0)$. Then the exact quantum motion of $q(\tau)$ which is subjected to the external force $F(\tau)$ in the Heisenberg picture is given by [@kim] $$\begin{aligned} \label{q:0A} q(\tau) &=& q_O(\tau)+ q_F(\tau) \nonumber \\ &=& q(\tau_0) \frac{\sqrt{g_-(\tau) g_+(\tau_0) }}{ \omega_I} \cos \left[\Theta(\tau) - \chi(\tau_0)\right] + p(\tau_0) \frac{ \sqrt{g_-(\tau)g_-(\tau_0)}}{\omega_I} \sin\Theta(\tau) \\ &+& A_F(\tau) + A_F^{\dagger}(\tau). \nonumber\end{aligned}$$ In this equations we use the following definitions: $$\begin{aligned} g_-(\tau) &=& f(\tau) f^*(\tau), \label{g_-(t)} \\ g_0(\tau) &=& - \frac{M(\tau)}{2} \dot{g}_-(\tau) , \nonumber \\ g_+(\tau) &=& M^2(\tau)\left|\dot{f}(\tau)\right|^2, \nonumber \\ \label{phase} \Theta(\tau) &=& \int^\tau_{\tau_0}\mbox{d}\tau \frac{\omega_I}{M(\tau) g_{-}(\tau)}, \end{aligned}$$ where $f(\tau)$ is a homogeneous solution of Eq. (\[q”:F\]) and $\omega_I = \sqrt{g_{+}(\tau) g_{-}(\tau) - g_{0}^2(\tau)}$ is invariant under the time evolution. For the definition of $g_i(\tau)$ $(i= \pm,0)$ see [@kim] and references therein. If $\tau$ is Killing time, we can expand the free field solution into its positive solutions and negative solutions. Let us classify its solution by $\omega$ and set the positive solution as $u_\omega$. Therefore $$\begin{aligned} \frac{\partial}{\partial \tau} u_\omega = -i |\omega| u_\omega\end{aligned}$$ and The free field solution in two dimension can be written as $$\begin{aligned} \phi^0(t,x) &=& \int_{-\infty}^{\infty} \mbox{d}k [ a_k u_k(\tau, \rho) + a_k^{\dagger} u_k^*(\tau,\rho) ] \end{aligned}$$ where $a_k$ and $a_k^\dagger$ is the corresponding creation and annihilation operators and $u_k$ is proportional to $ e^{-i\omega \tau}$ . With this choice, we can write the annihilation part of the inhomogeneous solution $q_F(\tau)= A_F(\tau)+ A_F^{\dagger}(\tau)$ as $$\begin{aligned} A_F(\tau) = \int_0^{\infty} \mbox{d}\omega \omega G(\omega,\tau)( a_{\omega} + a_{-\omega}).\end{aligned}$$ Where $G(\omega,\tau)$ is the classical inhomogeneous solution of the forced harmonic oscillator equation $$\begin{aligned} \ddot{G}(\omega,\tau) + \frac{d \ln M(\tau)}{d\tau} \dot{G}(\omega,\tau) + \omega^2(\tau) G(\omega,\tau) = -i \frac{e(\tau)}{m} u_\omega\end{aligned}$$ with the initial condition $$\begin{aligned} \label{condition1} G(\omega,0)=0 \hspace{1cm} \dot{G}(\omega,0) = 0.\end{aligned}$$ If we analyze $G(\omega,\tau)$, we can know all time evolutions of the operators in principle. The general solution for $G(\omega,\tau)$ can be written as $$\begin{aligned} \label{G:g} G(\omega,\tau) = g(\omega, \tau) - g^*(-\omega,\tau),\end{aligned}$$ where $$\begin{aligned} \label{g:tau} g(\omega,\tau) = e^{i \Theta(\tau)}\frac{\sqrt{g_{-}(\tau)}}{2 m \omega_I} \int_{\tau_0}^\tau \mbox{d}\tau' \sqrt{g_-(\tau')} M(\tau') e(\tau') e^{-i \Theta(\tau')} u_\omega (x(\tau'),t(\tau')).\end{aligned}$$ One can show that Eq. (\[q:0A\]) satisfies (\[q”:F\]) by direct substitution. The high momentum behavior of $G(\omega,t)$ is $O(1/\omega^{5/2})$ except some special case like the sudden jumping of the coupling constant, which we consider in Sec. IV. In the case of large $\omega$ the integral of (\[g:tau\]) is approximately given by the contributions around $\tau_0$, which makes the arguments of the exponential of $u_\omega(\tau)$ vanishes. Therefore the first approximation of $g(\omega,\tau)$ is of the form $\int d\tau' f(\tau_0) e^{\pm i \omega \tau}$. But this term is canceled in $G(\omega,\tau)= g(\omega,\tau) -g^*(-\omega,\tau)$, and leaves only the $O(1/\omega^{5/2})$ and higher terms. The inertial oscillator {#sec:II-1} ----------------------- At first let us consider the simplest inertial path: $x= 0$ and $t = \tau$. Moreover the mode solution is $u_k = 1/\sqrt{4 \pi |k|} e^{-i (|k| t -k x)}$. Let the initial state of the combined system to be $$\begin{aligned} \label{instate} \left|i\right> =\left|n \right> \left|0\right>_M,\end{aligned}$$ the $n$th excited state for the oscillator and the Minkowski vacuum state for the field. The correlation functions of $q(t)$ for state (\[instate\]) is $$\begin{aligned} \left<q_O(t) q_O(t')\right> &=& (2n+1) \frac{\sqrt{g_{-}(t)g_{-}(t')}}{2 \omega_I} \exp\left\{-i[ \Theta(t)- \Theta(t')] \right\},\\ \left<q_F(t) q_F(t')\right> &=& 2 \int \mbox{d}\omega \omega^2 G(\omega,t) G^*(\omega, t'),\\ \left<q_O(t) q_F(t')\right> &=& 0. \end{aligned}$$ The correlation of the homogeneous part decrease because $M(t)$ increase monotonically. Therefore for a large enough time the correlation is governed by the inhomogeneous term. If there is absent of $1/(\omega)^{3/2}$ term in $G(\omega,t)$ there is no UV contribution to the correlation function. As we see in the previous section, this is normally true. Therefore in the case of a slowly varying coupling, the main contribution to the correlation comes from the frequency region around the resonance frequence $\Omega(t)$ (See Eq. (\[quad\]) ). The system is symmetric about $x=0$. Therefore, it is enough to obtain the correlations of the field in the area $x,x'<0$. In this area Eq. (\[phi:uv\]) becomes $$\begin{aligned} \phi(t,x) = \phi^0_R(u) + \phi^0_L(v) + \frac{1}{2} e(v) [q_O(v)+q_F(v)].\end{aligned}$$ Therefore the correlations of the field and the oscillator is for the state (\[instate\]) are $$\begin{aligned} \left<\phi^0_R(u) q_F(v')\right> &=& \left<q_F(v') \phi^0_R(u)\right>^* = \int \mbox{d}\omega \omega G^*(\omega, v') u_{\omega}(u), \\ \left<\phi^0_L(v) q_F(v')\right> &=& \left<q_F(v') \phi^0_L(v)\right>^* = \int \mbox{d}\omega \omega G^*(\omega, v') u_{\omega}(v).\end{aligned}$$ From these, one get the renormalized correlation function $$\begin{aligned} \label{correlation} &&\left<\phi(t,x) \phi(t',x')\right> - \left<\phi^0(t,x)\phi^0(t',x')\right> = \frac{e(v)}{2} \left\{ \left<\phi^0_R(u) q_F(v')\right> + \left<\phi^0_L(v) q_F(v')\right> \right\} \nonumber \\ &&~~~+ \frac{e(v')}{2}\left\{\left<q_F(v)\phi^0_R(u')\right> + \left<q_F(v)\phi^0_L(v')\right> \right\} + \frac{e(v)e(v')}{4} \left\{\left<q_O(v) q_O(v')\right> + \left<q_F(v)q_F(v')\right> \right\} \nonumber \\ &&~=(2n+1) \frac{e(v)e(v')}{4} \frac{ \sqrt{g_{-}(v)g_{-}(v')}}{2 \omega_I} \exp\left\{-i[ \Theta(v)- \Theta(v')] \right\} \\ &&~~+ \frac{e(v)e(v')}{2} \int \mbox{d}\omega \omega^2 G(\omega,v) G^*(\omega, v') \nonumber \\ &&~~+ \frac{e(v)}{2} \int \mbox{d}\omega\omega G^*(\omega,v') \left[ u_\omega(v) +u_\omega(u)\right] + \frac{e(v')}{2} \int \mbox{d}\omega\omega G(\omega,v) \left[ u_\omega^*(v') +u_\omega^*(u') \right]. \nonumber \end{aligned}$$ The uniformly accelerated oscillator ------------------------------------- Now let us consider a uniformly accelerating trajectory $x = \frac{1}{a} \cosh a\tau, t= \frac{1}{a}\sinh a \tau $. Rindler space $(\tau,\rho)$ is given by $$\begin{aligned} x = \frac{1}{a} ~e^{a\rho} \cosh a\tau, ~~ t = \frac{1}{a} ~e^{a\rho} \sinh a \tau.\end{aligned}$$ In this system the retarded time is $$\begin{aligned} \tau_{ret} &=&\tau-\rho ~~~\mbox{for} ~~ \rho > 0,\\ &=&\tau+\rho ~~~\mbox{for} ~~ \rho < 0 .\nonumber \end{aligned}$$ At the right Rindler wedge, the free field $\phi^0(t,x)$ can be expanded with the normal modes of Rindler space-time as $$\begin{aligned} \phi^0(t,x) &=& \sum_{k=-\infty}^{\infty} [b_k \xi_k + H.C.] \\ &=& \sum_{\lambda =0}^{\infty}[b_\lambda \xi_\lambda(U) + b_{-\lambda} \xi_{\lambda}(V) + H.C.], \nonumber\end{aligned}$$ where $U = \tau - \rho=-\mbox{ln}(-a u)/a $, $V = \tau + \rho= \mbox{ln}(av)/a$, and $\xi_\lambda = 1/\sqrt{4 \pi |\lambda|} e^{-i \lambda U} = 1/\sqrt{4 \pi |\lambda|} (-a u)^{i \lambda/a}$, and $b_\lambda, b_\lambda^\dagger$ is the creation and annihilation operator in the Rindler spacetime. Therefore we can set $u_\omega \rightarrow \xi_\lambda$ and $\omega \rightarrow \lambda$ in Sec. II. Let us consider the initial state to be (\[instate\]). The expectation value of $q(\tau)$ for $|i>$ is zero. The correlation functions for $|i>$ are $$\begin{aligned} \left<q_O(\tau)\right.&&\left. q_O(\tau')\right> = (2n+1) \frac{\sqrt{g_{-}(\tau)g_{-}(\tau')}}{2 \omega_I} \exp\left\{-i[ \Theta(\tau)- \Theta(\tau')] \right\},\\ \left<q_F(\tau)\right.&&\left. q_F(\tau')\right> = 2 \int \mbox{d}\lambda \lambda^2 \left[ \left\{1 + N(\lambda/a)\right\} G(\lambda, \tau) G^*( \lambda, \tau') \right. \\ +&& \left. N(\lambda/a) G^*(\lambda, \tau)G(\lambda, \tau') \right], \nonumber \\ \left<\phi^0_R(U)\right.&&\left. q_F(\tau'_{ret})\right> = \left<q_F(\tau'_{ret}) \phi^0_R(U) \right>^* \\ =&& \int \mbox{d}\lambda \lambda \left[ \xi_{\lambda}(U) G^*(\lambda, \tau') \left\{ 1+N(\lambda/a) \right\} + \xi^*_{\lambda} (U) G(\lambda, \tau'_{ret}) N(\lambda/a) \right], \nonumber \\ \left<\phi^0_L(V)\right.&& \left.q_F(\tau'_{ret})\right> = \left<q_F(\tau'_{ret}) \phi^0_L(V)\right>^* \\ =&& \int \mbox{d}\lambda \lambda \left[\xi_\lambda(V) G^*(\lambda, \tau_{ret}') \left\{ 1+N(\lambda/a) \right\} + \xi_\lambda^*(V) G(\lambda, \tau'_{ret}) N(\lambda/a) \right], \nonumber\end{aligned}$$ where $N(\Omega) = 1/(e^{2\pi \Omega}-1)$. We use the fact that the Minkowski vacuum is FDU thermal bath with temperature $a/(2\pi)$ to the accelerating observer. The renormalized correlation function can be obtained with the same method of inertial case. These correlation functions can be devided into two classes: First is those of zero temperature and second is thermal contributions. The high momentum behavior of the second is cut off by the presence of the exponential in the denominator. Therefore it is evident that the first term dominate the correlation if there is UV divergences due to the existence of the $1/\omega$ term in $G(\lambda,\tau)$. The sudden jump case of the coupling is exactly that case. Exactly solvable models {#sec:solvable} ======================= Constant coupling ----------------- The most easiest problem is, of course, the case of constant coupling ($e(\tau)=e$). In this case $$\begin{aligned} M(\tau) &=& \frac{e^2}{2} (\tau-\tau_0) \\ f(\tau) &=& \frac{1}{\sqrt{m}} e^{\pm (i \Omega +e^2/(2m))(\tau-\tau_0)} \end{aligned}$$ where $\Omega ^2 = \omega_0^2 - \epsilon^2$. $G(\omega, \tau)$ satisfies the following equation: $$\begin{aligned} \ddot{G}(\omega,\tau) + \frac{e^2}{2m} \dot{G}(\omega,\tau) + \omega_0^2 G(\omega,\tau) = -i \frac{e}{m} u_\omega\end{aligned}$$ The general solution to this equation is given by $$\begin{aligned} \label{sol1} G(\omega,\tau) &=& a \exp(-\epsilon \tau) \exp(i\sqrt{\omega_0^2-\epsilon^2} \tau+ \alpha) + i e \chi(\omega) u_\omega (\tau,0) \end{aligned}$$ where $$\begin{aligned} \chi(\omega) = \frac{1}{m \left[\omega_0^2 -\omega^2 -2i \epsilon \omega \right]} \end{aligned}$$ and $\epsilon = e^2/4m$. The coefficients $a$ and $\alpha$ must be chosen $G(\omega,\tau)$ to satisfy the initial condition (\[condition1\]). The first term exponentially decay therefore there remain only the second term asymptotically. This is exactly the same result with Massar, Parentani, and Brout [@massar]. Turn on of the coupling {#sec:turnon} ----------------------- The next example is given by the coupling $$\begin{aligned} \label{cha} 2\epsilon(\tau) =\frac{e^2(\tau)}{2 m} &=& \frac{e_-^2}{4m} \left(1- \tanh \frac{\tau}{d} \right)+ \frac{e_+^2}{4m} \left(1+ \tanh \frac{\tau}{d}\right) \\ & =& \epsilon_{-}\left(1- \tanh \frac{\tau}{d}\right)+\epsilon_{+} \left(1+ \tanh \frac{\tau}{d} \right). \nonumber\end{aligned}$$ The limit $d \rightarrow 0$ corresponds to the sudden jump and $d \rightarrow \infty$ to the adiabatic one. Eq. (\[Omega\]) $$\begin{aligned} \Omega^2(\tau) = \omega_0^2 - \epsilon^2(\tau) = \frac{\omega_-^2}{2}\left(1- \tanh \frac{\tau}{d} \right) + \frac{\omega_+^2}{2}\left(1+ \tanh \frac{\tau}{d} \right) + \frac{(\epsilon_+- \epsilon_-)^2/4}{\cosh^2(\tau/d)}\end{aligned}$$ has two limiting values $\omega_\pm^2 = \omega_0^2 -\epsilon_\pm^2$ at the positive and negative infinity. These two values define a natural positive frequency modes of the oscillator in the past and the future asymptotic region. The effective mass (\[M:t\]) becomes $$\begin{aligned} \label{mass} M(\tau) = m \exp \int 2 \epsilon(\tau) \mbox{d}\tau = m \left( \frac{\cosh \tau/d}{ \cosh \tau_0/d}\right)^{(\epsilon_+-\epsilon_-)d} \exp \left[ (\epsilon_+ +\epsilon_-)(\tau-\tau_0) \right].\end{aligned}$$ From these we get the homogeneous solution for the classical equation of motion (\[q”:F\]) $$\begin{aligned} \label{f:t} f(\tau) = \frac{1}{\sqrt{M(\tau)}} e^{-i(\omega_+ + \omega_-)\tau/2} \left(\cosh \frac{\tau}{d} \right)^{-i(\omega_+-\omega_-)d/2} ~_2F_1(\alpha_-, \alpha_+;1-i \omega_- d; y),\end{aligned}$$ where $$\begin{aligned} y &=& \frac{1+ \tanh \tau/d}{2}, \\ \alpha_\pm &=& \frac{1 \pm \sqrt{1+ (\epsilon_+-\epsilon_-)^2d^2}}{2} + i \frac{(\omega_+ - \omega_-)d}{2},\end{aligned}$$ and $_2F_1$ is the hypergeometric function [@morse]. We choose Eq. (\[f:t\]) to be pure positive frequency mode at the past infinity. On the other hand, it becomes generally mixture of the positive and negative modes at the future: $$\begin{aligned} \lim_{\tau \rightarrow -\infty} f(\tau) &=& \frac{2^{i(\omega_+-\omega_-)d/2}}{ \sqrt{M(\tau)}} e^{-i \omega_-\tau} , \label{f:-} \\ \lim_{\tau \rightarrow \infty} f(\tau) &=& \frac{2^{i(\omega_+-\omega_-)d/2}}{ \sqrt{M(\tau)}} \left( \alpha e^{-i \omega_+ \tau} + \beta e^{i \omega_+ \tau} \right). \label{f:+}\end{aligned}$$ where $$\begin{aligned} \alpha &=&\frac{\Gamma(1-i\omega_- d) \Gamma(1-i\omega_-d -\alpha_- -\alpha_+) }{\Gamma(1-i\omega_-d -\alpha_-) \Gamma(1-i\omega_-d -\alpha_+)} , \\ \beta &=&\frac{\Gamma(1- i\omega_-d) \Gamma(\alpha_- +\alpha_+ -1+i\omega_-d) }{\Gamma(\alpha_-)\Gamma(\alpha_+)}. \end{aligned}$$ The absolute squares of $\alpha$ and $\beta$ $$\begin{aligned} |\alpha|^2 &=& \frac{1}{2}\frac{\omega_-}{\omega_+} \frac{ \cosh\pi(\omega_+ + \omega_-)d + \cos 2\pi x}{\sinh \pi \omega_- d \sinh \pi \omega_+ d}, \\ |\beta|^2 &=& \frac{1}{2}\frac{\omega_-}{\omega_+} \frac{ \cosh\pi(\omega_+ - \omega_-)d + \cos 2\pi x}{\sinh \pi \omega_- d \sinh \pi \omega_+ d}\end{aligned}$$ satisfy a Bogolubov type relation $$\begin{aligned} |\alpha|^2 -|\beta|^2 =\frac{\omega_-}{\omega_+},\end{aligned}$$ where $x = \sqrt{1 + (\epsilon_+ -\epsilon_-)^2 d^2}/2$. The factor $\omega_-/\omega_+$ comes from the change of the natural frequency of the oscillator [@jyji2]. At the present problem the initial homogeneous solution for the oscillator decays by $1/\sqrt{M(\tau)}$ factor so the asymptotic form for large $\tau$ is given by $q_F$. Therefore the particle creation or other related topics must be discussed with the inhomogeneous solution $G(\omega,\tau)$ with respect to the positive frequency mode at the future asymptotic region. Since our primary purpose is not the oscillator state but the radiation from the oscillator, we do not discuss it further. In the adiabatic limit $\beta$ vanishes, on the other hand, in the sudden jump limit it becomes $(1-\omega_-/\omega_+)/2$. From (\[f:t\]) one get $$\begin{aligned} \label{g_-:on} g_-(\tau) = f(\tau) f^*(\tau) = \frac{|_2F_1(\alpha_-, \alpha_+;1-i \omega_-d;y)|^2}{M(\tau)},\end{aligned}$$ and the invariant frequency $\omega_I= \omega_-$. The integral of generalized frequency (\[phase\]) is $$\begin{aligned} \label{phase1} \Theta(\tau) &=& \omega_I \int^\tau \mbox{d}\tau' \frac{1}{|_2F_1(\alpha_-, \alpha_+; 1-i \omega_-d;y)|^2} \nonumber \\ &=& \omega_I \int^\tau \mbox{d}\tau'\frac{1}{R^2(\tau)} - \omega_I \int^\tau \mbox{d}\tau'\frac{1}{R^2(\tau)} \left(1- \frac{R^2(\tau)}{|_2F_1(\alpha_-, \alpha_+;1-i \omega_-d;y)|^2} \right) \\ &=& \theta(\tau) -\theta(\tau_0) - \theta_f(\tau), \nonumber\end{aligned}$$ where $R(\tau)$ and $\theta(\tau)$ are the absolute value and the real phase of $$\begin{aligned} R e^{-i \theta(\tau)}=\alpha e^{-i \omega_+ \tau} +\beta e^{+i \omega_+\tau},\end{aligned}$$ and $\theta_f(\tau)$ approaches to some finite value as $t \rightarrow \infty$. Eqs. (\[q:0A\] and (\[g\_-:on\]) shows $q_O(t)$ decreases exponentially for $\tau \gg d$. Using these and Eq. (\[G:g\]) we get the asymptotic form $$\begin{aligned} \label{Ginfty:chi} G(\omega,\tau) &=& i e_+ \chi(\omega) u_\omega(\tau,0) \\ &-& \frac{1}{2 m \omega_I \sqrt{M(\tau)}} \left[\left\{\alpha \chi_f(-\omega) - \beta^* \chi_f(\omega)\right\} e^{-i \omega_+ \tau} + \left\{\beta \chi_f(-\omega) -\alpha^* \chi_f(\omega) \right\} e^{i \omega_+ \tau} \right]\nonumber\end{aligned}$$ where $$\begin{aligned} \chi(\omega) &=& \frac{1}{m [\omega_0^2 - \omega^2 -2i \epsilon_+\omega]},\end{aligned}$$ and $\chi_f(\omega) = \lim_{\tau\rightarrow \infty} \chi_f(\omega,\tau)$. $$\begin{aligned} \label{chif} \chi_f(\omega,\tau) &=& -e_+ \int_{\tau_0}^\tau d\tau'\sqrt{M(\tau')} R(\tau') e^{-i\omega_+\tau'} u_\omega(\tau' ,0) \\ & & \cdot \left[1- \frac{\sqrt{M(\tau')g_-(\tau')}}{R(\tau')} \frac{e(\tau')}{e_+} e^{i[\omega_+\tau' - \theta(\tau')]} \right]. \nonumber\end{aligned}$$ This result is similar with that of the constant coupling except $\chi_f(\omega)$ is determined by the integral. Stress Energy tensor in the Sudden Jump Limit {#sec:jump} ============================================= Now let us study the stress tensor of the scalar field in the presence of the oscillator. We consider instant switching process ($d \rightarrow 0$ limit of Sec. \[sec:turnon\]). We solve this problem up to zeroth order on $d$ or $e^{-2|\tau|/d}$, where $\tau$ is the proper time seen by the oscillator. We calculate $G(\omega,\tau)$ without explicit choice of coordinates system because it is common both the inertial and the uniformly accelerating oscillator. Let us set $\tau_0=-\infty$ and rescale the mass to be $M(0) =m$. Similarly we also set $\Theta(0) =0$. After carrying out the integral (\[chif\]) explicitly in the limit $|t| \gg d$ we get $$\begin{aligned} \label{G:-+} G(\omega, \tau) &=& i e_- \chi_-(\omega) u_\omega(\tau,0), \hspace{5.9cm} ~~~~~ \mbox{for} ~~ \tau \ll -d, \\ &=& G_\infty(\omega,\tau) + \frac{i e_+}{2\sqrt{4 \pi \omega}} e^{-\epsilon_+ \tau} \left\{ \chi_+^*(-\omega) e^{i\omega_+\tau}+\chi_+(\omega)e^{-i\omega_+\tau} \right\}, \mbox{ for}~~ \tau \gg d, \nonumber \end{aligned}$$ where $$\begin{aligned} \label{chip:chida} \chi_+(\omega) = \chi_d(\omega)+\frac{e_-}{e_+} \chi_-(\omega),\end{aligned}$$ and $$\begin{aligned} \chi_d(\omega) &=& \frac{1}{m \omega_+} \left[\frac{1}{\omega- \omega_+ +i \epsilon_+} - \frac{1- e_-/e_+ }{ + \omega-\omega_++i(\epsilon_+ - 2/d)} \right. \\ &-&\left. \frac{e_-}{2e_+} \left(\frac{1}{\omega-\omega_-+i\epsilon_- }+ \frac{1}{\omega +\omega_- +i \epsilon_-} \right)\right], \label{chi_d} \nonumber \\ \chi_-(\omega) &=& \chi_-^*(-\omega)= \frac{1}{m}\frac{1}{\omega_0^2-\omega^2- 2i \omega \epsilon_-}. \label{chi_a}\end{aligned}$$ Here needs some remarks. All physical quantities like the coupling, classical solution, and effective mass must be continuous at $\tau=0$. This constraints demands the second term in $\chi_d$, which makes $G(\omega,\tau)$ to be quadratically decrease for large $\omega$. $G_\infty(\omega,\tau) = i e_+ \chi(\omega,\tau) u_\omega(\tau,0) $ dominates the asymptotic form of $G(\omega,\tau)$ and the second term, which is the effect of the change of the coupling, decrease exponentially on the time scale of $1/\epsilon_+$. At $\tau <0$ the inhomogeneous solution $G(\omega,\tau)$ is that of the equilibrium. Therefore our system represent a system which is in equilibrium at $\tau<0$ become dynamic due to the change of the coupling at $\tau=0$. The solution $G(\omega,\tau) $describe this dynamic approaching process to equilibrium. If we restrict the region of $\omega$ as $0 < \omega < \Gamma \ll 1/d$, we can ignore the $d$ dependent term in $\chi_d(\omega)$. In this limit $\chi_d(\omega)$ becomes $O(1/\omega)$ and gives cutoff dependent UV behaviors. On the other hands in the case of $\Gamma \gg 1/d$, $\chi_d(\omega)$ is $O(1/\omega^2)$ which makes the theory UV finite. But we cannot get a sensible theory because the asymptotic form of the stress tensor crucially depends on $1/d$, which is unphysical. Therefore we restrict the cutoff $\Gamma \ll 1/d$ and also restrict our attention to $|\tau| > 1/\Gamma$. The Stress tensor in the presence of a inertial oscillator {#sec:stressiner} ---------------------------------------------------------- In Sec. (\[sec:II-1\]) we obtain the renormalized correlation function (Eq. (27)) in the region $x , x'<0$. In this region there is no $u, u'$ dependent terms. Therefore $T_{uu}$ component of the stress tensor vanishes. Moreover, $\left<T_{uv}\right> = tr T/4 = 0$ since we are dealing with a massless field in two dimension and there is no trace anomaly because the curvature is zero. The stress tensor vanishes in the region $v<0$ since $G(\omega,t)$ has the same form with the asymptotic case (\[Ginfty:chi\]) and there is no radiation asymptotically. In the region $t > 0$ we must calculate the stress tensor explicitly. We restrict our attention to $v > 1/\Gamma \gg d$ since we are interested in the radiation after turn on the coupling. After taking differentiation of the correlation function with respect to $v$ and $v'$ followed by the limit $v' \rightarrow v$ we get $$\begin{aligned} \label{Tvv:G} T_{vv} &=& T_1 + \frac{e_+}{2} T_2 + \frac{e_+^2}{2} T_3,\end{aligned}$$ where $$\begin{aligned} T_1 &=& (2n+1) \frac{e_+^2}{8 \omega_I} \lim_{v'\rightarrow v} \left(\partial_v \partial_v'\sqrt{g_-(v)g_-(v')} \exp \left[-i\left\{ \Theta(v) - \Theta(v')\right\}\right] \right) \\ T_2 &=& \int d \omega \omega \left[ \partial_v G^*(\omega,v) \partial_v u_\omega(v) + \partial_v G(\omega,v)\partial_v u_\omega^*(v) \right], \\ T_3 &=& \int d \omega \omega^2 \partial_v G(\omega,v) \partial_v G^*(\omega,v).\end{aligned}$$ Where we ignore terms related with $\dot{e}(v)$ which is important for $t \sim d$. Since we consider only the region $t > 1/\Gamma \gg d$, it is safe to ignore such terms. Now let us write down only the dominant terms of the stress tensor. (For details see appendix.) For small $v$, it is dominant the interference ($T_2$) between the oscillator and the field. $$\begin{aligned} T_{vv}= -\frac{e^2_+ \omega_0}{8 \pi m \omega_+} e^{-\epsilon_+ v} \left(1- \frac{e_-}{e_+} \right) \frac{1}{v} \cos (\omega_+v + \theta) , \hspace{0.5cm} \mbox{for} \hspace{0.3cm} \frac{1}{\omega_0} \gg v > \frac{1}{\Gamma}.\end{aligned}$$ In this region the energy is absorbed into the oscillator from the field with the amount $$\begin{aligned} \label{Eabs} E_{absorbed} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right) \ln\frac{\Gamma}{\omega_0} + \mbox{smaller terms}.\end{aligned}$$ For $v \gg 1/\Gamma$, energy is radiated away from the oscillator and $T_3$ is dominant. $$\begin{aligned} \label{Tvvasym} T_{vv} &=& \frac{e_+^4}{8 \pi} \left(\frac{\omega_0}{m \omega_+} \right)^2 e^{-2\epsilon_+ v} \cos^2(\omega_+ v + \theta) \left(1-\frac{e_-}{e_+}\right)^2 \mbox{ln}\left(\Gamma/\omega_0\right) \hspace{0.5cm} \mbox{for} \hspace{.3cm} v \gg \frac{1}{\Gamma}.\end{aligned}$$ where $\tan \theta = \epsilon_+/\omega_+$. The total radiated energy in this region is $$\begin{aligned} \label{Erad} E_{radiated} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right)^2 \ln\frac{\Gamma}{\omega_0}\end{aligned}$$ Therefore we can conclude that in general the absorbed energy into the oscillator is greater than the radiated one. The Stress tensor in the presence of a uniformly accelerating oscillator {#sec:stressacc} ------------------------------------------------------------------------ With the same reason given at the previous subsection, $T^A_{uu}$ and $T^A_{uv}$ are zero. Similarly, the stress tensor vanishes for $ V < 0$. In the region $V>0$, we also restrict to $V > 1/\Gamma \gg 1/d$, and we ignored $\dot{e}(\tau)$ related terms. In this region, $T^A_{vv}$ components can be written as follows: $$\begin{aligned} T^A_{vv} = T^A_{1}+ \frac{e_+}{2} T^A_{2} + \frac{e_+^2}{2} T^A_{3},\end{aligned}$$ where $$\begin{aligned} T^A_{1} &=& (2n+1)\frac{e_+^2}{8 \omega_I} \lim_{v' \rightarrow v} \left( e^{-a (V + V')} \partial_V \partial_V' \sqrt{g_-(V)g_-(V')} \exp \left[-i\left\{ \Theta(V) - \Theta(V')\right\}\right] \right) \\ T^A_{2} &=& \int d \lambda \lambda \left\{1 + 2 N\left( \frac{\lambda}{a} \right) \right\} \left[\partial_V \xi_{\lambda}(V) \partial_V G^*(\lambda,V) + \partial_V \xi_{\lambda}^*(V) \partial_V G(\lambda,v) \right] e^{-2a V}, \\ T^A_{3} &=& \int d \lambda \lambda^2 \left\{1 + 2 N\left( \frac{\lambda}{a} \right) \right\} \partial_V G(\lambda,V) \partial_V G^*(\lambda, V) e^{-2 a V}.\end{aligned}$$ For small $V$ the stress tensor is dominated by $T_2^A$ which is given by $$\begin{aligned} T_{vv} ^A &=& \frac{e^2_+ }{8 \pi m \omega_+} \left(1- \frac{e_-}{e_+} \right) \frac{a}{\ln av } \\ & &\cdot \left\{ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right\} (av)^{-2- \epsilon_+/a} \hspace{.5cm} \mbox{for} \hspace{.2cm} \frac{1}{\omega_0} \gg V > \frac{1}{\Gamma}, \nonumber\end{aligned}$$ This is exactly the same form with the inertial oscillator except the retardation factor $(a v)^{-2}$ due to the acceleration and the mere coordinate change $v \rightarrow \ln av/a$. The total absorbed energy is given by $$\begin{aligned} E^A_{absorbed} = \frac{e^2_+}{8 \pi m } \left(1- \frac{e_-}{e_+}\right) \ln\frac{\Gamma}{\omega_0} + \mbox{smaller terms}.\end{aligned}$$ This is exactly the same with Eq. (\[Eabs\]). Physically, it is natural because there is no enough time the acceleration to act on the short time interference. For $V \gg 1/\Gamma$, $T_3^A$ is dominant. $$\begin{aligned} T_{vv}^A = \frac{e_+^2}{8 \pi m \omega_+} \left( 1- \frac{e_-}{e_+}\right)^2 \ln \frac{\Gamma}{ \omega_0} \left\{ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right\}^2 (av)^{-2- 2\epsilon_+/a} \hspace{.2cm} \mbox{for} \hspace{.2cm} V \gg \frac{1}{\Gamma},\end{aligned}$$ This equation is quite similar to Eq.(\[Tvvasym\]) except the retardation effect and the coordinate change ($v \rightarrow \ln{a v}/a $) due to acceleration. This is because the main effect to the radiation comes from the high momentum region. The total energy radiated away from the oscillator is $$\begin{aligned} E^A_{radiated} &=& \frac{e_+^2}{8 \pi m} \left(1-\frac{e_-}{e_+} \right)^2 \ln \left(\frac{\Gamma}{\omega_0} \right) \tan \theta \left[\frac{1}{(x+ \sin \theta) \cos \theta} + \frac{\cos 2 \theta x - \tan \theta}{x^2 + 2 \sin \theta x + 1} \right] ,\end{aligned}$$ where $x = a/(2 \omega_0)$. Summary and Discussion ====================== We have discussed the influence of a harmonic oscillator on a scalar quantum field in $1+ 1$ dimensions. These are illustrated by calculating the radiation of the scalar field from the oscillator. The first step to do this is to express the time evolutions with the classical inhomogeneous solution $G(\omega,t)$ of a damping forced harmonic oscillator. Then we applied this result to the sudden jumping limit of the coupling and obtain the change of the stress tensor in the presence of the oscillator. There are two main effects on the radiation. The first is due to sudden change of the coupling which is described by the correlation between the oscillator and the field. This effect rapidly die out but the oscillator absorbs large energy from the field through this correlation. Moreover this absorption is independent of the acceleration. Subsequently, slow radiation from the oscillator take place. In case of an inertial oscillator this radiation is smaller than the absorbed energy through the first stage. The behavior of the total radiated energy become nontrivial if the detector is accelerated. In the case of a small coupling constant ( $\epsilon_+ \ll \omega_0$), the radiated energy is maximized by $a=0$. But there is a peak of the radiated energy at a non-zero acceleration if $\theta $ greater than some value $\theta_0 \sim 1.07702$ or $$\begin{aligned} \left(\frac{\epsilon_+}{\omega_0}\right)^2 > \frac{- \omega_+/\omega_0 + \sqrt{(\omega_+/\omega_0)^2 - 8 \omega_+/\omega_0 +8}}{4(1-\omega_+/\omega_0)}\end{aligned}$$ In this case the radiated energy can greater than the absorbed one during the first stage. Especially, if $\epsilon \rightarrow \omega_0$ then the radiated energy becomes extremely large for non-zero acceleration. If the acceleration is large enough, the radiation decrease according to the inverse of the acceleration. The following two points can help to understand this phenomena. First, the radiation is not due to the acceleration but due to the change of the coupling. Second, as acceleration grows the unit proper time of an accelerating oscillator correspond to a larger coordinate time to the Minkowski observer. Therefore the coordinate time which takes to vary the coupling becomes larger for the larger acceleration. [Fig. 1. Total radiated energy\ Total radiated energy is plotted according to the acceleration. The acceleration of each time is given by $ a/(2\omega_0)= \{\pi/6, \pi/3, \pi/2-0.3, \pi/2-0.2, \pi/2-0.1\}$ from the below. The unit for energy is $\frac{e_+^2}{8 \pi m} \left(1-\frac{e_-}{e_+} \right)^2 \ln \left(\frac{\Gamma}{\omega_0} \right)$. ]{}  \  \   acknowledgements ================ This work was supported by the Korea Science and Engineering Foundation (KOSEF). One of the authors thanks to Min-Ho Lee for his helpful discussions. Appendix {#appendix .unnumbered} ======== The Stress Energy Tensor of the field in the sudden jump of the coupling – Inertial Case ---------------------------------------------------------------------------------------- In this appendix we obtain the stress tensor for the model of Sec. \[sec:turnon\] in the sudden jump limit. It is easy to know that the stress tensor simply vanishes for $v<0$, from (\[G:-+\]). Therefore we calculate it only for $v \geq 0$ in the left hand side of the oscillator. The stress tensor (\[Tvv:G\]) is composed of three terms. The first term can be evaluated easily to become $$\begin{aligned} \label{T1} T_1 &=&\frac{\epsilon_+}{4 \omega_{-}\omega_{+}^2} e^{-2 \epsilon_+ v} \left[(\omega_+^2+ \omega_-^2)\omega_0^2 +(\epsilon_+^2-\omega_+^2 )(\omega_+^2-\omega_-^2) \cos 2\omega_+ v \right. \\ &&+ \left. 2\epsilon_+ \omega_+ ( \omega_+^2- \omega_-^2) \sin2 \omega_+ v \right]. \nonumber\end{aligned}$$ $T_2$ is sum of two terms which are mutually complex conjugate. One of these is $$\begin{aligned} \label{Gu} \int d \omega \omega \partial_v G(\omega,v) \partial_{v} u_\omega^*(v) = \int d \omega \omega \partial_v G_\infty(\omega,v) \partial_{v} u_\omega^*(v) +\frac{i e_+}{8 \pi}e^{-\epsilon_+ v} T_{2a},\end{aligned}$$ where $$\begin{aligned} \label{T2a} T_{2a} = -\beta_+ e^{i\omega_+v} \int d \omega \omega \chi_+^*(-\omega) e^{i\omega v} + \beta_+^* e^{-i \omega_+v} \int d\omega \omega \chi_+(\omega) e^{i \omega v} ,\end{aligned}$$ and we define the constant $$\begin{aligned} \beta_\pm = \omega_\pm + i \epsilon_\pm.\end{aligned}$$ Therefore $$\begin{aligned} T_{2a}-T_{2a}^* &=& -\frac{1}{m \omega_+} \beta_+ e^{i \omega_+ v} \left[ \left(1-\frac{e_-}{e_+} \right) \frac{2i}{v} + 2 \beta_+ e^{-i\beta_+ v} Ei(i \beta_+ v) \right. \\ &-& \left. \frac{e_-}{e_+}\left\{ \left(1+ \frac{\omega_+}{\omega_-} \right) \beta_- e^{-i \beta_- v} Ei(i \beta_- v) - \left(1+ \frac{\omega_+}{\omega_-} \right) \beta_-^* e^{i \beta_-^*v} Ei(-i \beta_-^*v) \right\} \right] \nonumber \\ &-& C.C. \nonumber\end{aligned}$$ Rather than use this complex form, lets us extract only its limiting form for small and large $v$ . In the region $ d \ll 1/\Gamma < v \ll 1/\omega_0 $ $$\begin{aligned} T_2 &=& \frac{e_+}{4 \pi m \omega_+} e^{-\epsilon_+ v}\left(1- \frac{e_-}{e_+} \right) \frac{ \omega_0}{v} \cos (\omega_+v+ \theta)\end{aligned}$$ and for large $v \gg 1/\omega_0$, $T_2 = O(e^{-2 \epsilon_+ v})$. Where $ \tan \theta = \epsilon_+/\omega_+$. Finally, let us evaluate $T_3$. If we define the following integrals $$\begin{aligned} I_1(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi_d(-\omega) e^{-i \omega v}, \\ I_2(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi^*_d (\omega) e^{-i \omega v}, \\ J(v) &=& \int_0^\Gamma d \omega \omega^2 \chi(\omega) \chi_-(-\omega) e^{-i \omega v} ,\end{aligned}$$ then $T_3$ becomes $$\begin{aligned} T_3 = && \lim_{v' \rightarrow v} \int d \omega \omega^2 \partial_v G_\infty(\omega,v) \partial_{v'} G_\infty^*(\omega,v') \\ &+& \frac{e_+^2}{8 \pi} \left( e^{-i\beta_+^* v} \beta_+^* T_{3a} + e^{i \beta_+ v} \beta_+ T_{3a}^* \right)+ \frac{e_+^2}{16 \pi} e^{-2\epsilon_+ v} T_{3b}. \label{T3:T3ab} \nonumber\end{aligned}$$ where $$\begin{aligned} T_{3a} &=& -I_1(v) + I_2^*(v)+ \frac{e_-}{e_+}[-J(v)+ J^*(v)] \label{T3a:int} \\ T_{3b} &=& \omega_0^2 \int _0^\Gamma d \omega \omega \left( |\chi_+(\omega)|^2+|\chi_+(-\omega)|^2 \right) \label{T3b:int} \\ &&-\beta_+^2 e^{2i\omega_+ v} \int _0 ^\Gamma d \omega \omega\chi_+^*(-\omega)\chi_+^*( \omega)-\beta_+^{*2} e^{-2i\omega_+ v} \int_0^\Gamma d \omega \omega \chi_+(-\omega)\chi_+(\omega). \nonumber \end{aligned}$$ where we have introduced explicit high momentum cut-off $\Gamma$ to regularize the UV behaviors. The first term of $T_3$ is canceled by the $\lim_{v\rightarrow v'}\int d \omega \omega (\partial_vG_\infty \partial_{v'}u^* +\partial_v u^* \partial_{v'}G_\infty)$ term of $T_2$ (The detail of the calculation can be consulted in Ref. [@massar].) As one can see in $T_{3b} $ major contribution to the stress tensor comes from the ultra-violet region. As one can easily see $T_{3b}$ don’t have UV contribution. Therefore the major contribution comes from $T_{3b}$. Let us examine $T_{3b}$ in detail. $\chi_-$ is of order $O(1/\omega^2)$ for large $\omega$, therefore only the first term of the integral $$\begin{aligned} \label{intchip} \int d&& \omega \omega |\chi_+(\omega)|^2 \\ &&= \int d \omega \omega |\chi_d|^2 +\frac{e_-}{e_+} \int d \omega \omega \left( \chi_d \chi_-^* + \chi_-\chi_d^* \right) +\frac{e_-^2}{e_+^2} \int d \omega \omega |\chi_-|^2 \nonumber\end{aligned}$$ can have important ultra-violet contribution. If one try to extract only the high momentum part it is ease to show that $$\begin{aligned} \int d \omega \omega |\chi_d|^2 &\cong & \frac{1}{m^2 \omega_+^2} \left(1-\frac{e_-}{e_+} \right)^2 \int^\Gamma d\omega \frac{1}{\omega} \\ &=& \frac{1}{m^2 \omega_+^2} \left(1-\frac{e_-}{e_+} \right)^2 \mbox{ln}\left(\Gamma/\omega_0\right). \nonumber\end{aligned}$$ Therefore $$\begin{aligned} T_3 = \frac{e_+^2}{4 \pi} \left(\frac{\omega_0}{m \omega_+} \right)^2 e^{-2\epsilon_+ v} \cos^2(\omega_+ v + \theta) \left(1-\frac{e_-}{e_+}\right)^2 \mbox{ln}\left(\Gamma/\omega_0\right).\end{aligned}$$ where $\tan \theta = \epsilon_+/\omega_+$. The stress tensor of the field in the sudden jump limit of the coupling – Accelerating Case ------------------------------------------------------------------------------------------- The stress tensor for $V<0$ vanishes. In the region $V>0$, the term $T^A_{1}$ is $$\begin{aligned} T^A_1 (v) &=& \frac{\epsilon_+}{4 \omega_- \omega_+^2} \left[ (\omega_+^2+ \omega_-^2)\omega_0^2 \right.\\ &-& \left. \frac{1}{2} (\omega_+^2-\omega_-^2) \left(\beta_+^{*2} (av)^{-2i\omega_+/a} + \beta_+^2 (av)^{2i \omega_+/a} \right) \right] (av)^{-2(1+\epsilon_+/a)}. \nonumber\end{aligned}$$ The integral for $T^A_{2,3}$ are of the form $\int d \lambda \lambda (1 + 2 N(\lambda/a) )f(\lambda,V)$. One can separate the ultra-violet (UV) $2\int d \lambda \lambda f(\lambda, V)$ from its thermal contributions $ \int d \lambda \lambda N(\lambda/a) f(\lambda,V)$. Let us look at each terms more closely. The UV term of $T^A_3$ is $$\begin{aligned} T^A_{3UV} = && \int^\Gamma d \lambda \lambda^2 \partial_v G_\infty(\lambda,V) \partial_{v} G_\infty^*(\lambda,V) \\ &+& \frac{e_+^2}{8 \pi} e^{-2aV} \left( e^{-i\beta_+^* V} \beta_+^* T^A_{3a} + e^{i \beta_+ V} \beta_+ T^{A*}_{3a} \right)+ \frac{e_+^2}{16 \pi} e^{-2(\epsilon_++a) V} T^A_{3b}. \label{T^A_3},\nonumber\end{aligned}$$ where $T^A_{3a}$ and $T^A_{3b}$ are given by Eqs. (\[T3a:int\]) and (\[T3b:int\]) if one replace $\omega \rightarrow \lambda $, $v \rightarrow V$. The first term of $T^A_3$ is canceled by the $\int d \lambda \lambda \left\{\partial_vG_\infty(\lambda,V) \partial_{v} \xi^*_\lambda(V) +\partial_v \xi _\lambda(V) \partial_{v}G_\infty^*(\lambda,V)\right\}$ term of $T^A_2$ [@massar]. The dominant term for this UV contribution is $$\begin{aligned} \label{T_3^AUV} \frac{e_+^2}{4 \pi}&& \left(\frac{1-e_-/e_+}{m \omega_+} \right)^2 \ln \frac{\Gamma}{\omega_0} \omega_0^2 \cos^2(\omega_+ V + \theta) e^{-2(\epsilon_++a) V} \\ &&=\frac{e_+^2}{16 \pi}\left(\frac{1-e_-/e_+}{m \omega_+} \right)^2 \ln \frac{ \Gamma}{\omega_0} \left[ 2 \omega_0^2 + \beta_+^2 (av)^{2i \frac{\omega_+}{ a}} + \beta_+^{*2} (av)^{-2i \frac{\omega_+}{a}} \right] (av)^{-2(1+ \epsilon/a)} \nonumber\end{aligned}$$ There are thermal contributions in $T^A_3$ but we can argue that it does not give comparable contribution to the UV term. The general form of the integral of the thermal part is $$\begin{aligned} \int d \lambda \frac{2\lambda^2}{e^{\lambda/a}-1} \partial _V G(\lambda, V) \partial_V G^*(\lambda, V).\end{aligned}$$ As one can easily see, there is no UV divergence because of the thermal factor in the denominator. Moreover there is no IR contributions which comes from $\lambda \sim 0$. Therefore it do not give terms depends on the cutoff $\Gamma$ which is the main contribution of the $T^A_{3UV}$. In case of $T^A_2$ the situation is much different to $T^A_3$ because there are no UV contributions and the main contribution of it is only for small $V$. So we must calculate it exactly in the region $\frac{1}{\Gamma} < V \ll 1/\omega_0, 2/a$. $T_2^A$ is sum of two terms which are mutually complex conjugate. One of these is $$\begin{aligned} &&\int d \lambda \lambda \coth (2\pi \lambda/a) \partial_V G(\lambda, V) \partial_V \xi_{\lambda}^*(V) e^{-2a V} \\ &&= \int d \lambda \lambda \coth (2\pi \lambda/a) \partial_V G_\infty(\lambda, V) \partial_V \xi_{\lambda}^*(V) e^{-2aV} + \frac{i e_+}{8 \pi} e^{-(\epsilon_+ + 2 a)v} T^A_{2a}, \nonumber\end{aligned}$$ where $$\begin{aligned} \label{T^A_{2a}} T^A_{2a} &=& -\beta_+ e^{i\omega_+V} \int d \lambda \lambda \coth{2 \pi \lambda/a }\chi_+^*(-\lambda) e^{i\lambda V} \\ &+& \beta_+^* e^{-i \omega_+V} \int d\lambda \lambda \coth{2 \pi \lambda /a} \chi_+(\lambda) e^{i \lambda V} . \nonumber\end{aligned}$$ Therefore we must calculate $T^A_{2a}- T^{A*}_{2a}$. After change of variable and using the fact $\lambda \coth{2 \pi \lambda/a} $ is even function on $\lambda$, we get $$\begin{aligned} T^A_{2a} &-& T^{A*}_{2a} = -\beta_+ e^{-i \omega_+ V} \int_{-\infty}^{\infty} d \lambda \left[ \lambda \coth{2 \pi \lambda/a} \right] \xi_+^*(-\lambda)e^{i \lambda V} \\ &-& C.C. \nonumber\end{aligned}$$ Now we use $$\begin{aligned} \coth \pi x = \frac{1}{\pi x} + \frac{2 x}{\pi}\sum_{k=1}^{\infty} \frac{1}{x^2+ k^2}\end{aligned}$$ and do the residue integral along the upper half plane of the $\lambda$ plane, then we get $$\begin{aligned} T^A_{2a} &-& T^{A*}_{2a} = ia \beta_+ e^{-i \omega_+ V} \left[ S(-2i \beta_+) - \frac{e_-}{2 e_+} \left\{ \left(1+ \frac{\omega_+}{\omega_-} \right) S(-2i \beta_-) + \left(1- \frac{\omega_+}{\omega_-}\right) S(2i \beta_-^*) \right\} \right] \nonumber \\ &-& C.C\end{aligned}$$ where $$\begin{aligned} S(\beta)= \sum_{k=1}^{\infty} \frac{ak e^{-akV/2}}{ak + \beta}.\end{aligned}$$ If we restrict $V$ to $V \ll 2/a, 1/\omega_0$, we get $$\begin{aligned} S(\beta) \cong \frac{2}{a V} + \frac{\beta}{a} \ln\left(\frac{V}{\omega_0}\right).\end{aligned}$$ The second term is much smaller than the first. Therefore we can write $$\begin{aligned} T^A_2 &=& - \frac{e_+}{4 \pi} \left(1- \frac{e_-}{e_+} \right) e^{-(\epsilon_+ + 2a)V } \frac{\omega_0}{V} \cos(\omega_+ V+ \theta) \\ &=& - \frac{e_+}{4 \pi m \omega_+} \left( 1- \frac{e_-}{e_+} \right) \frac{ a}{ \ln av} \left[ \beta_+ (av)^{i \frac{ \omega_+}{a}} + \beta_+^* ( av)^{-i \frac{\omega_+}{a}} \right] (av)^{-(2+ \epsilon_+/a)}. \nonumber\end{aligned}$$ [8]{} N. D. Birrell and P. C. W. Davis, [*Quantum Fields in Curved Space*]{} (Cambridge Univ. Press, Cambridge, 1984). W. G. Unruh, Phys. Rev. D [**14**]{}, 870, (1976). B. S. DeWitt, in General Relativity, eds. S. W. Hawking and W. Israel (Cambridge: Cambridge University Press) (1979). J. R. Letaw, Phys. Rev. D [**23**]{} 1709 (1981). J. Phys. A: Math. Gen. [**16**]{} 3905 (1983). K. Hinton, P. C. W. Davis, and J. Pfautsch, Phys. Lett. B [**120**]{} 88 (1983). S. Takagi, Phys. Lett. B [**148**]{} 116 (1984). S. Takagi, Prog. Theor. Phys. 88 (1986). W. G. Unruh, Phys. Rev. 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--- author: - | CARLOS GERSHENSON\ Vrije Universiteit Brussel title: 'A General Methodology for Designing Self-Organizing Systems' --- Author’s address: Krijgskundestraat 33 B-1160 Brussel, Belgium [email protected] http://homepages.vub.ac.be/cgershen Introduction ============ Over the last half a century, much research in different areas has employed self-organizing systems to solve complex problems, e.g. [Ashby1956,Beer1966,BonabeauEtAl1999,EngineeringSOS2004,ZambonelliRana2005]{}. Recently, particular methodologies using the concepts of self-organization have been proposed in different areas, such as software engineering [WooldridgeEtAl2000,ZambonelliEtAl2003]{}, electrical engineering [RamamoorthyEtAl1993]{}, and collaborative support [@JonesEtAl1994]. However, there is as yet no general framework for constructing self-organizing systems. Different vocabularies are used in different areas, and with different goals. In this paper, I present an attempt to develop a general methodology that will be useful for designing and controlling *complex* systems [@Bar-Yam1997]. The proposed methodology, as with any methodology, does not provide ready-made solutions to problems. Rather, it provides a *conceptual framework*, a *language,* to assist the solution of problems. Also, many current problem solutions can be *described* as proposed. I am not suggesting new solutions, but an alternative way of thinking about them. As an example, many standardization efforts have been advanced in recent years, such as ontologies required for the Semantic Web [@Berners-LeeEtAl2001], or FIPA standards. I am not insinuating that standards are not necessary. Without them engineering would be chaos. But as they are now, they cannot predict future requirements. They are developed with a static frame of mind. They are not adaptive. What this work suggests is a way of introducing the expectation of change into the development process to be able to cope with the unexpected beforehand, in problem domains where this is desired. The paper is organized as follows: in the next section, notions of complexity and self-organization are discussed. In Section [secConceptualFw]{}, original concepts are presented. These will be used in the Methodology, exposed in Section \[secMethodology\]. In Section [secSOTL]{}, a case study concerning self-organizing traffic lights is used to illustrate the steps of the Methodology. Discussion and conclusions follow in Sections \[secDiscussion\] and \[secConclusions\]. Complexity and Self-organization ================================ There is no general definition of *complexity*, since the concept achieves different meanings in different contexts [@Edmonds1999]. Still, we can say that a system is complex if it consists of several *interacting* elements [@Simon1996], so that the behavior of the system will be difficult to deduce from the behavior of the parts. This occurs when there are many parts, and/or when there are many interactions between the parts. Typical examples of complex systems are a living cell, a society, an economy, an ecosystem, the Internet, the weather, a brain, and a city. These all consist of numerous elements whose interactions produce a global behavior that cannot be reduced to the behavior of their separate components [@GershensonHeylighen2005]. For example, a cell is considered a living system, but the elements that conform it are not alive. The properties of life arise from the complex dynamical *interactions* of the components. The properties of a system that are not present at the lower level (such as life), but are a product of the interactions of elements, are sometimes called *emergent* [@Anderson1972]. Another example can be seen with gold: it has properties, such as temperature, malleability, conductivity, and color, that emerge from the interactions of the gold atoms, since atoms do not have these properties. Even when there is no general definition or measure of complexity, a relative *notion* of complexity can be useful: \[theorem\][Notion]{} The complexity of a system scales with the number of its elements, the number of interactions between them, the complexities of the elements, and the complexities of the interactions [@Gershenson2002a]:[^1] $$C_{sys} \sim \#\overline{E} \#\overline{I} \sum_{j=0}^{\#\overline{E}}C_{e_{j}} \sum_{k=0}^{\#\overline{I}}C_{i_{k}} \label{eqComplexity}$$ The complexity on an interaction $C_{i}$ can be measured as the number of different possible interactions two elements can have.[^2] The problem of a strict definition of complexity lies in the fact that there is no way of drawing a line between simple and complex systems independently of a context. For example, the *dynamics* of a system can be simple (ordered), complex, or chaotic, having a complex structure. Cellular automata and random Boolean networks are a clear example of this, where moreover, the interactions of their components are quite simple. On the other hand, a *structurally* simple system can have complex and chaotic dynamics. For this case, the damped pendulum is a common example. Nevertheless, for practical purposes, the above notion will suffice, since it allows the comparison of the complexity of one system with another under a common frame of reference. Notice that the notion is recursive, so a basic level needs to be set contextually for comparing two systems. The term *self-organization* has been used in different areas with different meanings, as is cybernetics [@vonFoerster1960; @Ashby1962], thermodynamics [@NicolisPrigogine1977], biology [@CamazineEtAl2003], mathematics [@Lendaris1964], computing [@HeylighenGershenson2003], information theory [@Shalizi2001], synergetics [@Haken1981], and others [@SkarCoveney2003] (for a general overview, see [Heylighen2003sos]{}). However, the use of the term is subtle, since any dynamical system can be said to be self-organizing or not, depending partly on the observer [@GershensonHeylighen2003a; @Ashby1962]: If we decide to call a “preferred" state or set of states (i.e. attractor) of a system “organized", then the dynamics will lead to a self-organization of the system. It is not necessary to enter into a philosophical debate on the theoretical aspects of self-organization to work with it, so a practical notion will suffice: \[theorem\][Notion]{} A system *described* as self-organizing is one in which elements *interact* in order to achieve *dynamically* a global function or behavior. This function or behavior is not imposed by one single or a few elements, nor determined hierarchically. It is achieved *autonomously* as the elements interact with one another. These interactions produce feedbacks that regulate the system. All the previously mentioned examples of complex systems fulfill the definition of self-organization. More precisely, the question can be formulated as follows: *when is it useful to describe a system as self-organizing?* This will be when the system or environment is very dynamic and/or unpredictable. If we want the system to solve a problem, it is useful to describe a complex system as self-organizing when the “solution" is not known beforehand and/or is changing constantly. Then, the solution is dynamically strived for by the elements of the system. In this way, systems can adapt quickly to unforeseen changes as elements interact locally. In theory, a centralized approach could also solve the problem, but in practice such an approach would require too much time to compute the solution and would not be able to keep the pace with the changes in the system and its environment. In engineering, a self-organizing system would be one in which elements are designed in order to solve *dynamically* a problem or perform a function at the system level. Thus, the elements need to divide, but also integrate, the problem. For example, the parts of a car are designed to perform a function at the system level: to drive. However, the parts of a (normal) car do not change their behavior in time, so it might be redundant to call a car self-organizing. On the other hand, a swarm of robots [@DorigoEtAl2004] will be conveniently described as self-organizing, since each element of the swarm can change its behavior depending on the current situation. It should be noted that all engineered self-organizing systems are to a certain degree *autonomous*, since part of their actual behavior will not be determined by a designer. In order to understand self-organizing systems, two or more *levels of abstraction* [@Gershenson2002a] should be considered: elements (lower level) organize in a system (higher level), which can in turn organize with other systems to form a larger system (even higher level). The understanding of the system’s behavior will come from the relations observed between the descriptions at different levels. Note that the levels, and therefore also the terminology, can change according to the interests of the observer. For example, in some circumstances, it might be useful to refer to cells as elements (e.g. bacterial colonies); in others, as systems (e.g. genetic regulation); and in others still, as systems coordinating with other systems (e.g. morphogenesis). A system can cope with an unpredictable environment *autonomously* using different but closely related approaches: - **Adaptation** (learning, evolution) [@Holland1995]. The system changes its behavior to cope with the change. - **Anticipation** (cognition) [@Rosen1985]. The system predicts a change to cope with, and adjusts its behavior accordingly. This is a special case of adaptation, where the system does not require to experience a situation before responding to it. - **Robustness** [@vonNeumann1956; @Jen2005]. A system is robust if it continues to function in the face of perturbations [@Wagner2005]. This can be achieved with modularity [@Simon1996; @Watson2002], degeneracy [@FernandezSole2003], distributed robustness [@Wagner2004], or redundancy [@GershensonEtAl2006]. Successful self-organizing systems will use combinations of the these approaches to maintain their integrity in a changing and unexpected environment. Adaptation will enable the system to modify itself to “fit" better within the environment. Robustness will allow the system to withstand changes without losing its function or purpose, and thus allowing it to adapt. Anticipation will prepare the system for changes before these occur, adapting the system without it being perturbed. We can see that all of them should be taken into account while engineering self-organizing systems. In the following section, further concepts will be introduced that will be necessary to apply the methodology . The Conceptual Framework {#secConceptualFw} ======================== Elements of a complex system interact with each other. The actions of one element therefore affect other elements, directly or indirectly. For example, an animal can kill another animal directly, or indirectly cause its starvation by consuming its resources. These interactions can have negative, neutral, or positive effects on the system [@HeylighenCampbell1995]. Now, intuitively thinking, it may be that the “smoothening" of local interactions, i.e. the minimization of “interferences" or “friction" will lead to global improvement. But is this always the case? To answer this question, the terminology of multi-agent systems [Maes1994,WooldridgeJennings1995,Wooldridge2002,Schweitzer2003]{} can be used. We can say that: \[theorem\][Notion]{} An agent is a description of an entity that *acts* on its environment. Examples of this can be a trader acting on a market, a school of fish acting on a coral reef, or a computer acting on a network. Thus, every element, and every system, can be seen as agents with *goals* and behaviors thriving to reach those goals. The behavior of agents can affect (positively, negatively, or neutrally) the fulfillment of the goals of other agents, thereby establishing a relation. The *satisfaction* or fulfillment of the goals of an agent can be represented using a variable $% \sigma \in \lbrack 0,1]$.[^3] Relating this to the higher level, the satisfaction of a system $\sigma _{sys}$ can be recursively represented as a function $f: \mathbb{R} \rightarrow \lbrack 0..1]$ of the satisfaction of the $n$ elements conforming it: $$\sigma _{sys}=f\left( \sigma _{1},\sigma _{2},...,\sigma _{n},w_{0},w_{1},w_{2},...,w_{n}\right) \label{eqSigmaSys}$$ where $w_{0}$ is a bias and the other weights determine the importance given to each $\sigma _{i}$. If the system is homogeneous, then $f$ will be the weighted sum of $\sigma _{i}$, $w_{i}=\frac{1}{n}\forall i\neq 0$, $w_{0}=0$. Note that this would be very similar to the activation function used in many artificial neural networks [@Rojas1996]. For heterogenous systems, $f$ may be a nonlinear function. Nevertheless, the weights $w_{i}$’s are determined *tautologically* by the importance of the $\sigma $ of each element to the satisfaction of the system. Thus, it is a useful tautology to say that maximizing individual $\sigma $’s, adjusting individual behaviors (and thus relations), will maximize $\sigma _{sys}$. If several elements decrease $% \sigma _{sys}$ as they increase their $\sigma $, we would not consider them as part of the system. It is important to note that this is independent of the potential nonlinearity of $f$. An example can be seen with the immune system. It categorizes molecules and micro-organisms as akin or alien [VazVarela1978]{}. If they are considered as alien, they are attacked. Auto-immune diseases arise when this categorization is erroneous, and the immune system attacks vital elements of the organism. On the other hand, if pathogens are considered as part of the body, they are not attacked. Another example is provided by cancer. Carcinogenic cells can be seen as “rebel", and no longer part of the body, since their goals differ from the goal of the organism. Healthy cells are described easily as part of an organism. But when they turn carcinogenic, they can better be described as parasitic. The tautology is also useful because it gives a general mathematical representation for system satisfaction, which is independent of a particular system. A reductionist approach would assume that maximizing the satisfaction of the elements of a system would also maximize the satisfaction of the system. However, this is not always the case, since some elements can “take advantage" of other elements. Thus, we need to concentrate *also* on the interactions of the elements. If the model of a system considers more than two levels, then the $\sigma $ of higher levels will be recursively determined by the $\sigma $’s of lower levels. However, the $f$’s most probably will be very different on each level. Certainly, an important question remains: how do we determine the function $% f $ and the weights $w_{i}$’s? To this question there is no complete answer. One option would be to approximate $f$ numerically [@DeWolfEtAl2005]. An explicit $f$ may be difficult to find, but an approximation can be very useful. Another method consists of *lesioning* the system[^4]: removing or altering elements of the system, and observing the effect on $\sigma _{sys}$. Through analyzing the effects of different lesions, the function $f$ can be reconstructed and the weights $w_{i}$’s obtained. If a small change $\Delta \sigma _{i}$ in any $\sigma _{i}$ produces a change $\Delta \sigma _{sys}\geq \Delta \sigma _{i}$, the system can be said to be *fragile*. What could then be done to maximize $\sigma _{sys}$? How can we relate the $% \sigma _{i}$’s and avoid conflicts between elements? This is not an obvious task, for it implies bounding the agents’ behaviors that reduce other $% \sigma _{i}$’s, while preserving their functionality. Not only should the interference or friction between elements be minimized, but the synergy [Haken1981]{} or “positive interference" should also be promoted. Dealing with complex systems, it is not feasible to tell each element what to do or how to do it, but their behaviors need to be constrained or modified so that their goals will be reached, blocking the goals of other elements as little as possible. These constraints can be called *mediators* [Michod2003,Heylighen2003]{}. They can be imposed from the top down, developed from the bottom up, be part of the environment, or be embedded as an *aspect* [@tenHaafEtAl2002 Ch. 3] of the system. An example can be found in city traffic: traffic lights, signals and rules mediate among drivers, trying to minimize their conflicts, which result from the competition for limited resources, i.e. space to drive through. The role of a mediator is to arbitrate among the elements of a system, to minimize interferences and frictions and maximize synergy. Therefore, the efficiency of the mediator can be measured directly using $\sigma _{sys}$. Individually, we can measure the “friction" $\phi _{i}\in \lbrack -1,1]$ that agent $i$ causes in the rest of the system, relating the change in satisfaction $\Delta \sigma _{i}$ of element $i$ and the change in satisfaction of the system $\Delta \sigma _{sys}$: $$\phi _{i}=\frac{-\Delta \sigma _{i}-\Delta \sigma _{sys}\left( n-1\right) }{n% }. \label{eqFriction}$$ Friction occurs when the increase of satisfaction of one element causes a decrease in the satisfaction of some other elements that is greater than the increase. Note that $\phi _{i}=0$ does imply that there is no conflict, since one agent can “get" the satisfaction proportionally to the “loss" of satisfaction of (an)other agent(s). Negative friction would imply synergy, e.g. when $\Delta \sigma _{i}\geq 0$ while other elements also increase their $\sigma $. The role of a mediator would be to maximize $\sigma _{sys}$ by minimizing $\phi _{i}$’s. With this approach, friction can be seen as a type of *interaction* between elements. Thus, the problem can be put in a different way: how can we find/develop/evolve efficient mediators for a given system? One answer to this question is the methodology proposed in this paper. The answer will not be complete, since we cannot have precise knowledge of $f$ for large evolving complex systems. This is because the evolution of the system will change its own $f$ [@Kauffman2000], and the relationships among different $\sigma _{i}$’s. Therefore, predictions cannot be complete. However, the methodology proposes to follow steps to increase the understanding (and consequently the control) of the system and the relations between its elements. The goal is to identify conflicts and diminish them without creating new ones. This will increase the $\sigma _{i}$’s and thus $% \sigma _{sys}$. The precision of $f$ is not so relevant if this is achieved. It should be noted that the timescale chosen for measuring $\Delta \sigma _{i}$ is very important, since at short timescales the satisfaction can decrease, while on the long run it will increase. In other words, there can be a short term “sacrifice" to harvest a long term “reward". If the timescale is too small, a system might get stuck in a “local optimum", since all possible actions would decrease its satisfaction on the short term. But in some cases the long term benefit should be considered for maximization. A way of measuring the slow change of $\sigma _{i}$ would be with its integral over time for a certain interval $\Delta t$: $$\int_{t}^{t+\Delta t}\sigma _{i}dt. \label{eqIntegralSigma}$$ Another way of dealing with the local optima is to use neutral changes to explore alternative solutions [@Kimura1983]. Before going into further detail, it is worth noting that this is not a reductionist approach. Smoothing out local interactions will not provide straightforward clues as to what will occur at the higher level. Therefore, the system should be observed at both levels: making local and global changes, observing local and global behaviors, and analyzing how one affects the other. Concurrently, the *dependence* $\epsilon $ $\in \lbrack -1,1]$ of an element to the system can be measured by calculating the difference of the satisfaction $\sigma _{i}$ when the element interacts within the system and its satisfaction $\widetilde{\sigma _{i}}$ when the element is isolated. $$\epsilon =\sigma _{i}-\widetilde{\sigma _{i}}. \label{eqDependence}$$ In this way, full dependence is given when the satisfaction of the element within the system $\sigma _{i}$ is maximal and its satisfaction $\widetilde{\sigma _{i}}$ is minimal when the element is isolated. A negative $\epsilon $ would imply that the element would be more satisfied on its own and is actually “enslaved" by the system. Now we can use the dependences of the elements to a system to measure the *integration* $\tau $ $\in \lbrack -1,1]$ of a system, which can be seen also as a gradual measure of a meta-system transition (MST) [@Turchin1977]. $$\tau =\frac{1}{n}\sum\limits_{i=1}^{n}\epsilon _{i}. \label{eqIntegration-MST}$$ A MST is a gradual process, but it will be complete when elements are not able to reach their goals on their own, i.e. $\overline{\sigma _{i}}% \rightarrow 0$. Examples include cells in multi-cellular organisms and mitochondria in eukaryotes. In an evolutionary process, natural (multilevel [@Michod1997; @Lenaerts2003]) selection will tend to increase $\tau $ because this implies higher satisfaction both for the system and its elements (systems with a negative $\tau $ are not viable). Relations and mediators that contribute to this process will be selected, since higher $% \sigma $’s imply more chances of survival and reproduction. Human designers and engineers also select relations and mediators that increase the $\sigma $’s of elements and systems. Therefore, we can see that evolution will tend, in the long run, towards synergetic relationships [@Corning2003], even if resources are scarce. In the next section, the steps suggested for developing a self-organizing system are presented, using the concepts described in this section. The Methodology {#secMethodology} =============== The proposed methodology meets the requirements of a system, i.e. what the system should do, and enables the designer to produce a system that fulfills the requirements. The methodology includes the following steps: Representation, Modeling, Simulation, Application, and Evaluation, which will be exposed in the following subsections. Figure \[diagram\] presents these steps. These steps should not necessarily be followed one by one, since the stages merge with each other. There is also backtracking, when the designer needs to return to an earlier stage for reconsideration before finishing a cycle. This methodology should not be seen as a recipe that provides ready-made solutions, but rather as a guideline to direct the search for them. The stages proposed are not new, and similar to those proposed by iterative and incremental development methodologies. Still, it should be noted that the active feedback between stages within each iteration can help in the design of systems ready to face uncertainties in complex problem domains. The novelty of the methodology lies in the *vocabulary* used to describe self-organizing systems. Representation -------------- The goal of this step is to develop a *specification* (which might be tentative) of the components of the system. The designer should always remember the distinction between model and modeled. A model is an abstraction/description of a “real" system. Still, there can be several descriptions of the same system [Gershenson2002a,GershensonHeylighen2005]{}, and we cannot say that one is better than another independently of a context. There are many possible representations of a system. According to the *constraints* and *requirements*, which may be incomplete, the designer should choose an appropriate vocabulary (metaphors to speak about the system), abstraction levels, granularity, variables, and interactions that need to be taken into account. Certainly, these will also depend on the experience of the designer. The choice between different approaches can depend more on the expertise of the designer than on the benefits of the approaches. Even when there is a wide diversity of possible systems, a general approach for developing a Representation can be abstracted. The designer should try to divide a system into elements by identifying semi-independent modules, with internal goals and dynamics, and with few interactions with their environment. Since interactions in a model will increase the complexity of the model, we should group “clusters" of interacting variables into elements, and then study a minimal number of interactions between elements. The first constraints that help us are space and time. It is useful to group variables that are close to each other (i.e. interacting constantly) and consider them as elements that relate to other elements in occasional interactions. Multiscale analysis [@Bar-Yam2005] is a promising method for identifying levels and variables useful in a Representation. Since the proposed methodology considers elements as agents, another useful criterion for delimiting them is the identification of goals. These will be useful in the Modeling to measure the satisfaction $\sigma $ of the elements. We can look at genes as an example: groups of nucleotides co-occur and interact with other groups and with proteins. Genes are identified by observing nucleotides that keep close together and act together to perform a function. The fulfillment of this function can be seen as a goal of the gene. Dividing the system into modules also divides the problem it needs to solve, so a complex task will be able to be processed in parallel by different modules. Certainly, the integration of the “solutions" given by each module arises as a new problem. Nevertheless, modularity in a system also increases its robustness and adaptability [@Simon1996; @Watson2002; @FernandezSole2003]. The representation should consider at least two levels of abstraction, but if there are many variables and interactions in the system, more levels can be contemplated. Since elements and systems can be seen as agents, we can refer to all of them as $x$-agents, where $x$ denotes the level of abstraction relative to the simplest elements. For example, a three-layered abstraction would contemplate elements (0-agents) forming systems that are elements (subsystems, 1-agents) of a greater system (meta-system, 2-agents). If we are interested in modeling a research institute, 0-agents would be researchers, 1-agents would be research groups, and the research institute would be a 2-agent. Each of these have goals and satisfactions ($\sigma ^{x}$) that can be described and interrelated. For engineering purposes, the satisfaction of the highest level, i.e. the satisfaction of the system that is being designed, will be determined by the tasks expected from it. If these are fulfilled, then it can be said that the system is “satisfied". Thus, the designer should concentrate on engineering elements that will strive to reach this satisfaction. If there are few elements or interactions in the Representation, there will be low complexity, and therefore stable dynamics. The system might be better described using traditional approaches, since the current approach might prove redundant. A large variety of elements and/or interactions might imply a high complexity. Then, the Representation should be revised before entering the Modeling stage. Modeling -------- In science, models should ideally be as simple as possible, and predict as much as possible [@Shalizi2001]. These models will provide a better understanding of a phenomenon than complicated models. Therefore, a good model requires a good Representation. The “elegance" of the model will depend very much on the metaphors we use to speak about the system. If the model turns out to be cumbersome, the Representation should be revised. The Modeling should specify a Control* *mechanism that will ensure that the system does what it is required to do. Since we are interested in self-organizing systems, the Control will be *internal* and *distributed*. If the problem is too complex, it can be divided into different subproblems. The Modeling should also consider different trade-offs for the system. ### Control mechanism The Control mechanism can be seen as a *mediator* [@Heylighen2003] ensuring the proper interaction of the elements of the system, and one that should produce the desired performance. However, one cannot have a strict control over a self-organizing system. Rather, the system should be *steered* [@Wiener1948]. In a sense, self-organizing systems are like teenagers: they cannot be tightly controlled since they have their own goals. We can only attempt to steer their actions, trying to keep their internal variables under certain boundaries, so that the systems/teenagers do not “break" (in Ashby’s sense [@Ashby1947]). To develop a Control, the designer should find aspect systems, subsystems, or constraints that will prevent the negative interferences between elements (friction) and promote positive interferences (synergy). In other words, the designer should search for ways of minimizing frictions $\phi _{i}$’s that will result in maximization of the global satisfaction $\sigma _{sys}$. The performance of different mediators can be measured using equation ([eqSigmaSys]{}). The Control mechanism should be *adaptive*. Since the system is dynamic and there are several interactions within the system and with its environment, the Control mechanism should be able to cope with the changes within and outside the system, in other words, *robust*. An adaptive Control will be efficient in more contexts than a static one. In other words, the Control should be *active* in the search of solutions. A static Control will not be able to cope with the complexity of the system. There are several methods for developing an adaptive Control, e.g. [@SastryBodson1994]. But these should be applied in a distributed way, in an attempt to reduce friction and promote synergy. Different methods for reducing friction in a system can be identified. In the following cases, an agent A negatively affected by the behavior of an agent B will be considered[^5]: - **Tolerance**. This can be seen as the acceptance of others and their goals. A can tolerate B by modifying itself to reduce the friction caused by B, and therefore increase $\sigma _{A}$. This can be done by moving to another location, finding more resources, or making internal changes. - **Courtesy**. This would be the opposite case to Tolerance. B should modify its behavior not to reduce $\sigma _{A}$. - **Compromise**. A combination of Courtesy and Tolerance: both agents A and B should modify their behaviors to reduce the friction. This is a good alternative when both elements cause friction to each other. This will be common when A and B are similar, as in homogeneous systems. - **Imposition**. This could be seen as forced Courtesy. The behavior of B could be changed by force. The Control could achieve this by constraining B or imposing internal changes. - **Eradication**. As a special case of Imposition, B can be eradicated. This certainly would decrease $\sigma _{B}$, but can be an alternative when either $\sigma _{B}$ does not contribute much to $\sigma _{sys}$, or when the friction caused by B in the rest of the system is very high. - **Apoptosis**. B can “commit suicide". This would be a special case of Courtesy, where B would destroy itself for the sake of the system. The difference between Compromise/Apoptosis and Imposition/Eradication is that in the former cases, change is triggered by the agent itself, whereas in the latter the change is imposed from the “outside" by a mediator. Tolerance and Compromise could be generated by an agent or by a mediator. Different methods for reducing friction can be used for different problems. A good Control will select those in which other $\sigma $’s are not reduced more than the gain in $\sigma $’s. The choice of the method will also depend on the importance of different elements for the system. Since more important elements contribute more to $\sigma _{sys}$, these elements can be given preference by the Control in some cases. Different methods for increasing synergy can also be identified. These will consist of taking measures to increase $\sigma _{sys}$, even if some $\sigma $’s are reduced: - **Cooperation**. Two or more agents adapt their behavior for the benefit of the whole. This might or might not reduce some $\sigma $’s. - **Individualism**. An agent can choose to increase its $\sigma $ if it increases $\sigma _{sys}$. A mediator should prevent increases in $% \sigma $’s if these reduce $\sigma _{sys}$, i.e. friction. - **Altruism**. An agent can choose to sacrifice an increase of its $\sigma $ or to reduce its $\sigma $ in order to increase $\sigma _{sys}$. This would make sense only if the *relative* increase of $\sigma _{sys}$ is greater than the decrease of the $\sigma $ of the altruistic agent. A mediator should prevent wasted altruism. - **Exploitation**. This would be forced Altruism: an agent is driven to reduce its $\sigma $ to increase $\sigma _{sys}$. A common way of reducing friction is to enable agents to learn via reinforcement [@KaelblingEtAl1996]. With this method, agents tend to repeat actions that bring them satisfaction and avoid the ones that reduce it. Evolutionary approaches, such as genetic algorithms [@Mitchell1996], can also reduce friction and promote synergy. However, these tend to be “blind", in the sense that variations are made randomly, and only their effects are evaluated. With the criteria presented above, the search for solutions can be guided with a certain aim. However, if the relationship between the satisfaction of the elements and the satisfaction of the system is too obscure, “blind" methods remain a good alternative. In general, the Control should explore different alternatives, trying to constantly increase $\sigma _{sys}$ by minimizing friction and maximizing synergy. This is a constant process, since a self-organizing system is in a dynamic environment, producing “solutions" for the current situation. Note that a mediator will not necessarily maximize the satisfaction of the agents. However, it should try to do so for the system. ### Dividing the problem If the system is to deal with many parameters, then it can be seen as a *cognitive* system [@Gershenson2004]. It must “know" or “anticipate" what to do according to the current situation and previous history. Thus, the main problem, i.e. what the elements should do, could be divided into the problems of communication, cooperation, and coordination [GershensonHeylighen2004]{}. For a system to self-organize, its elements need to *communicate*: they need to “understand" what other elements, or mediators, “want" to tell them. This is easy if the interactions are simple: sensors can give *meaning* to the behaviors of other elements. But as interactions turn more complex, the *cognition* [@Gershenson2004] required by the elements should also be increased. New meanings can be learned [@Steels1998; @DeJong2000]to adapt to the changing conditions. These can be represented as “concepts" [@Gardenfors2000], or encoded, e.g., in the weights of a learning neural network [@Rojas1996]. The precise implementation and philosophical interpretations are not relevant if the outcome is the desired one. The problem of *cooperation* has been widely studied using game theory [@Axelrod1984]. There are several ways of promoting cooperation, especially if the system is designed. To mention mention only two of them: the use of tags [@RioloEtAl2001; @HalesEdmonds2003] and multiple levels of selection [@Michod1997; @Lenaerts2003] have proven to yield cooperative behavior. This will certainly reduce friction and therefore increase $\sigma _{sys}$. Elements of a system should *coordinate* while reducing friction, not to obstruct each other. An important aspect of coordination is the *division of labour*. This can promote synergy, since different elements can specialize in what they are good at and *trust[^6]* others to do what they are good at [@Gaines1994; @DiMarzoSerugendo2004]. This process will yield a higher $\sigma _{sys}$ compared to the case when every element is meant to perform all functions independently of how well each element performs each function. A good Control will promote division of labour by mediating a balance between *specialization* and *integration*: elements should devote more time doing what they are best at, but should still take into account the rest of the system. Another aspect of coordination is the *workflow*: if some tasks are prerequisites of other tasks, a mediator should synchronize the agents to minimize waiting times. ### Trade-offs A system needs to be able to cope with the complexity of its domain to achieve its goals. There are several trade-offs that can be identified to reach a balance and cope better with this complexity: - **Complexity of Elements/Interactions**. The complexity of the system required to cope with the complexity of its domain can be tackled at one end of the spectrum by complex elements with few/simple interactions, and at the other by simple elements with several/complex interactions. - **Quality/Quantity**. A system can consist at one extreme of a few complex elements, and at the other of several simple elements. - **Economy/Redundancy**. Solving a problem with as few elements as possible is economical. But a minimal system is very fragile. Redundancy is one way of favoring the *robustness* of the system [vonNeumann1966,FernandezSole2003,Wagner2004,GershensonEtAl2006]{}. Still, too much redundancy can also reduce the speed of adaptation and increase costs for maintaining the system. - **Homogeneity/Heterogeneity**. A homogeneous system will be easier to understand and control. A heterogenous system will be able to cope with more complexity with less elements, and will be able to adapt more quickly to sudden changes. If there is a system of ten agents each able to solve ten tasks, a homogeneous system will be able to solve more than ten tasks robustly. A fully heterogeneous system would be able to solve more than a hundred tasks, but it would be fragile if one agent failed. Heterogeneity also brings diversity, that can accelerate the speed of exploration, adaptation, and evolution, since different solutions can be sought in parallel. The diversity is also related to the amount of variety of perturbations that the system can cope with [@Ashby1956], i.e. robustness. - **System/Context**. The processing and storage of information can be carried out internally by the system, or the system can exploit its environment “throwing" complexity into it, i.e. allow it to store or process information [@GershensonEtAl2003a]. - **Ability/Clarity**. A powerful system will solve a number of complex problems, but it will not be very useful if the functioning of the system cannot be understood. Designers should be able to understand the system in order to be able to control it [@Schweitzer2003]. - **Generality/Particularity**. An abstract Modeling will enable the designer to apply the Modeling in different contexts. However, particular details should be considered to make the implementation feasible. There are only very relative ways of measuring the above mentioned trade-offs. However, they should be kept in mind during the development of the system. In a particular system, the trade-offs will become clearer once the Simulation is underway. They can then be reconsidered and the Modeling updated. Simulation ---------- The aim here is to build computer simulation(s) that implement the model(s) developed in the Modeling stage, and test different scenarios and mediator strategies. The Simulation development should proceed in stages: from abstract to particular. First, an abstract scenario should be used to test the main concepts developed during the Modeling. Only when these are tested and refined, should details be included in the Simulation. This is because particular details take time to develop, and there is no sense in investing before knowing wether the Modeling is on the right track. Details can also influence the result of the Simulation, so they should be put off until a time when the main mechanisms are understood. The Simulation should compare the proposed solutions with traditional approaches. This is to be sure that applying self-organization in the system brings any benefit. Ideally, the designer should develop more than one Control to test in the simulation. A rock-scissors-paper situation could arise, where no Control is best in all situations (also considering classic controls). The designer can then adjust or combine the Controls, and then compare again in the Simulation. Experiments conducted with the aid of the Simulation should go from simple to extensive. Simple experiments will show proof of concepts, and their results can be used to improve the Modeling. Once this is robust, extensive studies should be made to be certain of the performance of the system under different conditions. Based on the Simulation results and insights, the Modeling and Representation can be improved. A Simulation should be mature before taking the implementation into the real world. Application ----------- The role of this stage is basically to use the developed and tested model(s) in a real system. If this is a software system, the transition will not be so difficult. On the other hand, the transition to a real system can expose artifacts of a naive Simulation. A useful way to develop robust Simulations consists in adding some noise into the system [@Jakobi1997]. Good theoretical solutions can be very difficult/expensive/impossible to implement (e.g. if they involve instantaneous access to information, mind reading, teleportation, etc.). The feasibility of the Application should be taken into account during the whole design process. In other words, the designer should have an *implementation bias* in all the Methodology stages. If the proposed system turned out to be too expensive or complicated, all the designer’s efforts would be fruitless. If the system is developed for a client, there should be feedback between developers and clients during the whole process [@Cotton1996] to avoid client dissatisfaction once the system is implemented. The *legacy* of previous systems should also be considered for the design [ValckenaersEtAl2003]{}: if the current implementation is to be modified but not completely replaced, the designer is limited by the capabilities of the old system. Constraints permitting, a pilot study should be made before engaging in a full Application, to detect incongruences and unexpected issues between the Simulation or Modeling stages and the Application. With the results of this pilot study, the Simulation, Modeling, and Representation can be fine-tuned. Evaluation ---------- Once the Application is underway, the performance of new system should be measured and compared with the performances of the previous system(s). Constraints permitting, efforts should be continued to try to improve the system, since the requirements it has to meet will certainly change with time (e.g. changes of demand, capacity, etc.). The system will be more adaptive if it does not consider its solution as the best once and for all, and is able to change itself according to its performance and the changing requirements. Notes on the methodology ------------------------ - All returning arrows in the Figure \[diagram\] are given because it is not possible to predict the outcome of strategies before they have been tried out. All information and eventualities cannot be abstracted, nor emergent properties predicted before they have been observed. Emergent properties are *a posteriori*. - The proposed Methodology will be useful for unpredictable problem domains, where all the possible system’s situations cannot be considered beforehand. It could also be useful for creative tasks, where a self-organizing system can explore the design space in an alternative way. - Most methodologies in the literature apply to software systems, e.g. [@JacobsonEtAl1999; @Jennings2000]. This one is more general, since it is domain independent. The principles presented are such that can be applied to any domain for developing a functioning self-organizing system: Any system can be modeled as a group of agents, with satisfactions depending on their goals. The question is *when* is it useful to use this Methodology. Only application of the Methodology will provide an answer to this question. It should be noted that several approaches have been proposed in parallel, e.g. [@CaperaEtAl2003; @DeWolfHolvoet2005], that, as the present work, and are part of the ongoing effort by the research community to understand self-organizing systems. - The proposed Methodology is not quite a spiral model [@Boehm1988], because the last stage does not need to be reached to return to the first. This is, there is no need to deploy a working version (finish a cycle/iteration) before revisiting a previous stage, as for example in extreme programming. Rather, the Methodology is a *backtracking* model, where the designer can always return to previous stages. - It is not necessary to understand a solution before testing it. In many cases understanding can come only after testing, i.e., the global behavior of the system is irreducible. Certainly, understanding the causes of a phenomenon will allow the modelers to have a greater control over it. A detailed diagram of the different substeps of the Methodology can be appreciated in Figure \[diagram-detailed\]. Case Study: Self-organizing Traffic Lights {#secSOTL} ========================================== Recent work on self-organizing traffic lights [@Gershenson2005] will be used to illustrate the flow through the different steps of the Methodology. These traffic lights are called self-organizing because the global performance is given by the local rules followed by each traffic light: they are unaware of the state of other intersections and still manage to achieve global coordination. Traffic modeling has increased the understanding of this complex phenomenon [PrigogineHerman1971,Traffic95,Traffic97,Traffic99,Helbing1997,HelbingHuberman1998]{}. Even when vehicles can follow simple rules, their local interactions generate global patterns that cannot be reduced to individual behaviors. Controlling traffic lights in a city is not an easy task: it requires the coordination of a multitude of components; the components affect one another; furthermore, these components do not operate at the same pace over time. Traffic flows and densities change constantly. Therefore, this problem is suitable to be tackled by self-organization. A centralized system could also perform the task, but in practice the amount of computation required to process all the data from a city is too great to be able to respond in real time. Thus, a self-organizing system seems to be a promising alternative. **Requirements**. The goal is to develop a feasible and efficient traffic light control system. **Representation**. The traffic light system can be modelled on two levels: the vehicle level and the city level. These are easy to identify because vehicles are objects that move through the city, establishing clear spatiotemporal distinctions. The goal of the vehicles is to flow as fast as possible, so their “satisfaction" $\sigma $ can be measured in terms of their average speed and average waiting time at a red light. Cars will have a maximum $\sigma $ if they go as fast as they are allowed, and do not stop at intersections. $\sigma $ would be zero if a car stops indefinitely. The goal of the traffic light system on the city level is to enable vehicles to flow as fast as possible, while mediating their conflicts for space and time at intersections. This would minimize fuel consumption, noise, pollution, and stress in the population. The satisfaction of the city can be measured in terms of the average speeds and average waiting times of all vehicles (i.e. average of $\sigma _{i},\ \forall i$), and with the average percentage of stationary cars. $\sigma _{sys}$ will be maximum if all cars go as fast as possible, and are able to flow through the city without stopping. If a traffic jam occurs and all the vehicles stop, then $\sigma _{sys}$ would be minimal. **Modeling**. Now the problem for the Control can be formulated: find a mechanism that will coordinate traffic lights so that these will mediate between vehicles to reduce their friction (i.e. try to prevent them from arriving at the same time at crossings). This will maximize the satisfactions of the vehicles and of the city ($\sigma _{i}$’s and $\sigma _{sys}$). Since all vehicles contribute equally to $\sigma _{sys}$, ideally the Control should minimize frictions via Compromise. **Simulation**. A simple simulation was developed in NetLogo [Wilensky1999]{}, a multi-agent modeling environment. The “Gridlock" model [@WilenskyStroup2002] was extended to implement different traffic control strategies. It consists of an abstract traffic grid with intersections between cyclic single-lane arteries of two types: vertical or horizontal (similar to the scenarios of [@BML1992; @BrockfeldEtAl2001]). Cars only flow in a straight line, either eastbound or southbound. Each crossroad has traffic lights that allow traffic flow in only one of the intersecting arteries with a green light. Yellow or red lights stop the traffic. The light sequence for a given artery is green-yellow-red-green. Cars simply try to go at a maximum speed of 1 “patch" per timestep, but stop when a car or a red or yellow light is in front of them. Time is discrete, but not space. A “patch" is a square of the environment the size of a car. The simulation can be tested at the URL http://homepages.vub.ac.be/cgershen/sos/SOTL/SOTL.html . At first, a tentative model was implemented. The idea was unsuccessful. However, after refining the model, an efficient method was discovered, named *sotl-request*. **Modeling**. In the *sotl-request* method, each traffic light keeps a count $\kappa _{i}$ of the number of cars times time steps ($c\ast ts $) approaching *only* the red light, independently of the status or speed of the cars (i.e. moving or stopped). $\kappa _{i}$ can be seen as the integral of waiting/approaching cars over time. When $\kappa _{i}$ reaches a threshold $\theta $, the opposing green light turns yellow, and the following time step it turns red with $\kappa _{i}=0$ , while the red light which counted turns green. In this way, if there are more cars approaching or waiting before a red light, the light will turn green faster than if there are only few cars. This simple mechanism achieves self-organization as follows: if there is a single or just a few cars, these will be made to stop for a longer period before a red light. This gives time for other cars to join them. As more cars join the group, cars will be made to wait shorter periods before a red light. Once there are enough cars, the red light will turn green even before the oncoming cars reach the intersection, thereby generating “green waves". Having “platoons" or “convoys" of cars moving together improves traffic flow, compared to a homogeneous distribution of cars, since there are large empty areas between platoons, which can be used by crossing platoons with few interferences. The *sotl-request* method has no phase or internal clock. Traffic lights change only when the above conditions are met. If no cars are approaching a red light, the complementary light can remain green. **Representation**. It becomes clear now that it would be useful to consider traffic lights as agents as well. Their goal is to “get rid" of cars as quickly as possible. To do so, they should avoid having green lights on empty streets and red lights on streets with high traffic density. Since the satisfactions of the traffic lights and vehicles are complementary, they should interact via Cooperation to achieve synergy. Also, $\sigma _{sys}$ could be formulated in terms of the satisfactions of traffic lights, vehicles, or both. **Modeling**. Two classic methods were implemented to compare their performance with *sotl-request*: *marching* and *optim*. *Marching* is a very simple method. All traffic lights “march in step": all green lights are either southbound or eastbound, synchronized in time. Intersections have a phase $\varphi _{i}$, which counts time steps. $\varphi _{i}$ is reset to zero when the phase reaches a period value $p$. When $% \varphi _{i}==0$, red lights turn green, and yellow lights turn red. Green lights turn yellow one time step earlier, i.e. when $\varphi ==p-1$. A full cycle of an intersection consists of $2p$ time steps. “Marching" intersections are such that $\varphi _{i}==\varphi _{j},\forall i,j$. The *optim* method is implemented trying to set phases $\varphi _{i}$ of traffic lights so that, as soon as a red light turns green, a car that was made to stop would find the following traffic lights green. In other words, a fixed solution is obtained so that *green waves* flow to the southeast. The simulation environment has a radius of $r$ square patches, so that these can be identified with coordinates $(x_{i},y_{i}),$ $% x_{i},y_{i}\in \lbrack -r,r]$. Therefore, each artery consists of $2r+1$ patches. In order to synchronize all the intersections, red lights should turn green and yellow lights should turn red when $$\varphi _{i}==round(\frac{2r+x_{i}-y_{i}}{4})$$ and green lights should turn to yellow the previous time step. The period should be $p=r+3$. The three is added as an extra margin for the reaction and acceleration times of cars (found to be best, for low densities, by trial and error). These two methods are *non-adaptive*, in the sense that their behavior is dictated beforehand, and they do not consider the actual state of the traffic. Therefore, there cannot be Cooperation between vehicles and traffic lights, since the latter ones have fixed behaviors. On the other hand, traffic lights under the *sotl-request* method are sensitive to the current traffic condition, and can therefore respond to the needs of the incoming vehicles. **Simulation**. Preliminary experiments have shown that *sotl-request*, compared with the two traditional methods, achieves very good results for low traffic densities, but very poor results for high traffic densities. This is because depending on the value of $\theta $, high traffic densities can cause the traffic lights to change too fast. This obstructs traffic flow. A new model was developed, taking this factor into account. **Modeling**. The *sotl-phase* method takes *sotl-request* and only adds the following constraint: a traffic light will not be changed if the time passed since the last light change is less than a minimum phase, i.e. $\varphi _{i}<\varphi _{\min }$. Once $\varphi _{i}\geq \varphi _{\min } $, the lights will change if/when $\kappa _{i}$ $\geq $ $\theta $. This prevents the fast changing of lights[^7]. **Simulation**. *Sotl-phase* performed a bit less effectively than *sotl-request* for very low traffic densities, but still much better than the classic methods. However, *sotl-phase* outperformed them also for high densities. An unexpected phenomenon was observed: for certain traffic densities, *sotl-phase* achieved *full synchronization*, in the sense that no car stopped. Therefore, speeds were maximal and there were no waiting times nor sopped cars. Thus, satisfaction was maximal for vehicles, traffic lights, and the city. Still, this is not a realistic situation, because full synchronization is achieved due to the toroidal topology of the simulation environment. The full synchronization is achieved because platoons are promoted by the traffic lights, and platoons can move faster through the city modulating traffic lights. If two platoons are approaching an intersection, *sotl-phase* will stop one of them, and allow the other to pass without stopping. The latter platoon keeps its phase as it goes around the torus, and the former adjusts its speed until it finds a phase that does not cause a conflict with another platoon. **Modeling**. Understanding the behavior of the platoons, it can be seen that there is a favorable system/context trade-off. There is no need of direct communication between traffic lights, since information can actually be sent via platoons of vehicles. The traffic lights communicate *stigmergically* [@TheraulazBonabeau1999], i.e. via their environment, where the vehicles are conceptualized as the environment of traffic lights. **Simulation**. With encouraging results, changes were made to the Simulation to make it more realistic. Thus, a scenario similar to the one of [@FaietaHuberman1993] was developed. Traffic flow in four directions was introduced, alternating streets. This is, arteries still consist of one lane, but the directions alternate: southbound-northbound in vertical roads, and eastbound-westbound in horizontal roads. Also, the possibility of having more cars flowing in particular directions was introduced. Peak hour traffic can be simulated like this, regulating the percentages of cars that will flow in different roads. An option to switch off the torus in the simulation was added. Finally, a probability of turning at an intersection $ P_{turn}$ was included. Therefore, when a car reaches an intersection, it will have a probability $P_{turn}$ of reducing its speed and turning in the direction of the crossing street. This can cause cars to leave platoons, which are more stable when $P_{turn}=0$. The results of experiments in the more realistic Simulation confirmed the previous ones: self-organizing methods outperform classic ones. There can still be full synchronization with alternating streets, but not without a torus or with $P_{turn}>0$. **Modeling**. Another method was developed, *sotl-platoon*, adding two restrictions to *sotl-phase* for regulating the size of platoons. Before changing a red light to green, *sotl-platoon* checks if a platoon is not crossing through, not to break it. More precisely, a red light is not changed to green if on the crossing street there is at least one car approaching at $\omega $ patches from the intersection. This keeps platoons together. For high densities, this restriction alone would cause havoc, since large platoons would block the traffic flow of intersecting streets. To avoid this, a second restriction is introduced. The first restriction is not taken into account if there are more than $\mu $ cars approaching the intersection. Like this, long platoons can be broken, and the restriction only comes into place if a platoon will soon be through an intersection. **Simulation**. *Sotl-platoon* manages to keep platoons together, achieving full synchronization commonly for a wide density range, more effectively than *sotl-phase* (when the torus is active). This is because the restrictions of *sotl-platoon* prevent the breaking of platoons when these would leave few cars behind, with a small time cost for waiting vehicles. Still, this cost is much lower than breaking a platoon and waiting for separated vehicles to join back again so that they can switch red lights to green before reaching an intersection. However, for high traffic densities platoons aggregate too much, making traffic jams more probable. The *sotl-platoon* method fails when a platoon waiting to cross a street is long enough to reach the previous intersection, but not long enough to cut its tail. This will prevent waiting cars from advancing, until more cars join the long platoon. This failure could probably be avoided introducing further restrictions. In more realistic experiments (four directions, no torus, $P_{turn}=0.1$), *sotl-platoon* gives on average 30% (up to 40%) more average speed, half the stopped cars, and seven times less average waiting times than non-responsive methods. Complete results and graphics of the experiments discussed here can be found in [@Gershenson2005]. **Representation**. If priority is to be given to certain vehicles (e.g. public transport, emergency), weights can be added to give more importance to some $\sigma _{i}$’s. A meso-level might be considered, where properties of platoons can be observed: their behaviors, performance, and satisfaction and the relationships of these with the vehicle and city levels could enhance the understanding of the self-organizing traffic lights and even improve them. **Simulation**. Streets of varying distances between crossings were tested, and all the self-organizing methods maintained their good performance. Still more realistic simulations should be made before moving to the Implementation, because of the cost of such a system. At least, multiple-street intersections, multiple-lane streets, lane changing, different driving behaviors, and non homogeneous streets should be considered. **Application**. The proposed system has not been implemented yet. Still, it is feasible to do so, since there is the sensor technology to implement the discussed methods in an affordable way. Currently, a more realistic simulation is being developed in cooperation with the Brussels Ministry of Mobility and Public Works to study its potential application in the city of Brussels. A pilot study should be made before applying it widely, to fine tune different parameters and methods. External factors, e.g. pedestrians and cyclists, could also affect the performance of the system. Pedestrians could be taken into account considering them as cars approaching a red light. For example, a button could be used to inform the intersection of their presence, and this would contribute to the count $\kappa _{i}$. A mixed strategy between different methods could be considered, e.g. *sotl-platoon* for low and medium densities, and *sotl-phase* or *marching* for high densities. **Evaluation**. If a city deploys a self-organizing traffic light system, it should be monitored and compared with previous systems. This will help to improve the system. If the system would be an affordable success, its implementation in other cities would be promoted. Discussion {#secDiscussion} ========== As could be seen in the case study, the backtracking between different steps in the Methodology is necessary because the behavior of the system cannot be predicted from the Modeling, i.e. it is not reducible. It might be possible to reason about all possible outcomes of simple systems, and then to implement the solution. But when complexity needs to be dealt with, a mutual feedback between experience and reasoning needs to be established, since reasoning alone cannot process all the information required to predict the behavior of a complex system [@Edmonds2005]. For this same reason, it would be preferable for the Control to be distributed. Even when a central supercomputer could possibly solve a problem in real time, the information delay caused by data transmission and integration can reduce the efficiency of the system. Also, a distributed Control will be more robust, in as much as if a module malfunctions, the rest of the system can still provide reliable solutions. If a central Control fails, the whole system will stop working. The Simulation and Experiments are strictly necessary in the design of self-organizing systems [@Edmonds2005]. This is because their performance cannot be evaluated by purely formal methods [@EdmondsBryson2004]. Still, formal methods are necessary in the first stages of the Methodology. I am not suggesting a trial-and-error search. But since the behavior of a complex system in a complex environment cannot be predicted completely, the models need to be contrasted with experimentation before they can be validated. This Methodology suggests one possible path for finding solutions. Now the reader might wonder whether the proposed Methodology is a *top-down* or a *bottom-up* approach. And the answer is: it is both and neither, since (at least) higher and lower levels of abstraction need to be considered simultaneously. The approach tests different local behaviors, and observes local and global (and meso) performances, for local and global (and meso) requirements. Thus, the Methodology can be seen as a *multi-level* approach. Since “conflicts" between agents need to be solved at more than one level, the Control strategies should be carefully chosen and tested. A situation as in the prisoner’s dilemma [@Axelrod1984] might easily arise, when the “best" solution on one level/timescale is not the best solution on another level/timescale. Many frictions between agents are due to faulty communication, especially in social and political relations. If agents do not know the goals of others, it will be much more difficult to coordinate and increase $\sigma _{sys}$. For example, in a social system, knowing what people or corporations need to fulfill their goals is not so obvious. Still, with emerging technologies, social systems perform better in this respect. Already in the early 1970s, the project Cybersin in Chile followed this path [@MillerMedina2005]: it kept a daily log of productions and requirements from all over the country (e.g. mines, factories, etc.), in order to distribute products where they were needed most; and as quickly as possible. Another step towards providing faster response to the needs of both individuals and social systems can be found in e-government [@LayneLee2001]. A company should also follow these principles to be able to adapt as quickly as possible. It needs to develop “sensors” to perceive the satisfactions and conflicts of agents at different levels of abstraction, and needs to develop fast ways of adapting to emerging conflicts, as well as to changing economic environment. A tempting solution might be to develop a homogeneous system since, e.g., homogeneous societies have fewer conflicts [@Durkheim1893]. This is because all the elements of a homogeneous system pursue the same goals. Thus, less diversity is easier to control. However, less diversity will be less able to adapt to sudden changes. Nevertheless, societies cannot be *made* homogeneous without generating conflicts since people are already diverse, and therefore already have a diversity of goals. The legacy [@ValckenaersEtAl2003] of social systems gives less freedom to a designer, since some goals are already within the system. A social Control/mediator needs to satisfy these while trying to satisfy those of the social system. Conclusions {#secConclusions} =========== This paper suggests a conceptual framework and a general methodology for designing and controlling self-organizing systems. The Methodology proposes the exploration for proper Control mechanisms/mediators/constraints that will reduce frictions and promote synergy so that elements will dynamically reach a robust and efficient solution. The proposed Methodology is general, but certainly it is not the only way to *describe* self-organizing systems. Even if this paper is aimed mainly at engineers, it is rather philosophical. It presents no concrete results, but *ideas* that can be exploited to produce them. Certainly, these ideas have their roots in current practices, and many of them are not novel. Still, the aim of this work is not for novelty but for synthesis. The Methodology strives to build artificial systems. Still, these could be used to understand natural systems using the synthetic method [@Steels1993]. Therefore, the ideas presented here are potentially useful not only for engineering, but also for science. The backtracking ideology is also applicable to this Methodology: it will be improved once applied, through learning from experience. This Methodology is not final, but evolving. The more this Methodology is used, and in a wider variety of areas, the more potentially useful its abstractions will be. For example, would it be a good strategy to minimize the standard deviation of $\sigma $’s? This might possibly increase stability and reduce the probability of conflict, but this strategy, as any other, needs to be tested before it can be properly understood. It is worth noting that apart from self-organizing traffic lights [@Gershenson2005], the Methodology is currently being used to develop Ambient Intelligence protocols [@GershensonHeylighen2004] and to study self-organizing bureaucracies. Any system is liable to make mistakes (and *will* make them in an unpredictable environment). But a good system will *learn* from its mistakes. This is the basis for adaptation. It is pointless to attempt to build a “perfect" system, since it is not possible to predict future interactions with its environment. What should be done is to build systems that can adapt to their unexpected future and are robust enough not to be destroyed in the attempt. Self-organization provides one way to achieve this, but there is still much to be done to harness its full potential. I should like to thank Hugues Bersini, Marco Dorigo, Erden Göktepe, Dirk Helbing, Francis Heylighen, Diana Mangalagiu, Peter McBurney, Juan Julián Merelo, Marko Rodriguez, Frank Schweitzer, Sorin Solomon, and Franco Zambonelli for interesting discussions and comments. I also wish to thank Michael Whitburn for proof-reading an earlier version of the manuscript. This research was partly supported by the Consejo Nacional de Ciencia y Teconolgía (CONACyT) of México. [^1]: This can be confirmed mathematically in certain systems. As a general example, random Boolean networks [Kauffman1969,Kauffman1993,Gershenson2004c]{} show clearly that the complexity of the network increases with the number of elements and the number of interactions. [^2]: Certainly, the number of possible interactions for certain elements is impossible to enumerate or measure. [^3]: In some cases, $\sigma $ could be seen as a “fitness" [HeylighenCampbell1995]{}. However, in most genetic algorithms [Mitchell1996]{} a fitness function is imposed from the outside, whereas $% \sigma $ is a property of the agents, that can change with time. [^4]: This method has been used widely to detect functions in complex systems such as genetic regulatory networks and nervous systems. [^5]: Even when equation \[eqFriction\] relates the satisfaction of an element to the satisfaction of the system, this can also be used for the relation between satisfactions of two elements, when $\Delta \sigma _{i}=0$ for all other elements. [^6]: Trust is also important for communication and cooperation. [^7]: A similar method has been used successfully in the United Kingdom for some time, but for isolated intersections [@VincentYoung1986].

ArXiv

\#1 \#1 = harvmac =[ ]{} \#1\#2[\#1\#2==]{}\#1 **Abstract** \#1 =cmr10 =cmr7 =cmr5 =cmmi10 =cmmi7 =cmmi5 =cmsy10 =cmsy7 =cmsy5 =cmti10 =’177 =’177 =’177 =’60 =’60 =’60 \#1[\#1\^]{} \#1[[e]{}\^[\^[\#1]{}]{}]{} \#1\#2[\^[\#1]{}\_[\#2]{}]{} /\#1[/]{} \#1 \#1\#2 \#1[\_[[\#1]{}]{}]{} \#1\#2[\_[[\#1]{}]{}\_[[\#2]{}]{}]{} \#1\#2[[\#1\#2]{}]{} \#1\#2\#3[[\^2 \#1\#2 \#3]{}]{} |\#1 \#1[\#1 ]{} \#1[\#1|]{} \#1[| \#1]{} \#1[| \#1|]{} \#1 \#1[1.5ex-16.5mu \#1]{} \#1\#2 \#1[.3ex]{} \#1\#2\#3[Nucl. Phys. B[\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Lett. [\#1]{}B (\#2) \#3]{} \#1\#2\#3[Phys. Rev. Lett. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Rev. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Ann. Phys. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Phys. Rep. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Rev. Mod. Phys. [\#1]{} (\#2) \#3]{} \#1\#2\#3[Comm. Math. Phys. [\#1]{} (\#2) \#3]{} Zachary Guralnik [*Department of Physics, University of California at San Diego, La Jolla, CA 92093*]{} The divergence of lepton and baryon currents in the Standard Model is independent of the fermion masses. For a single family, the baryon and lepton number anomaly is where $W^{\mu\nu}$ is the $SU(2)$ field strength and $B^{\mu\nu}$ is the $U(1)$ field strength. This differs greatly from the axial current equations of Q.E.D. because in Q.E.D. the production of axial charge depends critically on whether or not the electron is massive. I will begin by reviewing the reasons for this sensitivity. Then I will show why these reasons are not applicable to a spontaneously broken theory with a vector current anomaly, such as the standard model. The results give some insight into the production of baryon number in the standard model by sphalerons, which has been of much recent interest. The divergence of the axial current in Q.E.D. \[\] is In a background gauge field the matrix element of the last term is The remaining terms are higher dimension functions of the gauge fields and vanish in an adiabatic aproximation. If the electron is massive then there is no axial charge violation in an adiabatic approximation because the first and last terms in equation   cancel. This cancellation is obvious from the start if one calculates the anomaly using a Pauli Villars regulator field. Then the regulated axial current satisfies where $\chi$ is the regulator field and $\Lambda$ is its mass. $\chi$ is bosonic, so $\chi$ loops have the opposite sign from $\psi$ loops. Therefore there can be no mass independent terms in the matrix element of $\del_{\mu}J^{5 \mu}_r$ in a background gauge field. This cancellation also has a simple spectral interpretation. An explanation of the Q.E.D. axial anomaly based upon the spectrum of a massless electron in a background magnetic field has been given by Nielson and Ninomiya \[\]. Their arguments are briefly summarized below. Consider a uniform background magnetic field in the z direction. In the massless case, positive and negative chirality fermions decouple, so there are two sets of Landau levels. The positive and negative chirality Landau levels contain zero-modes with $E=-p_z$ and $E=+p_z$ respectively. Suppose one turns on a positive uniform electric field ${\cal E}$ in the $z$ direction. In an adiabatic approximation, solutions flow along spectral lines according to the Lorentz force law ${dp\over dt}=e{\cal E}$. Thus right chiral zero-modes slide out of the Dirac sea while left chiral zero-modes slide deeper into the Dirac sea (). This motion produces a net axial charge but no electric charge. By a careful counting of states one reproduces the global form of the anomaly where V is the volume of space. Now consider the same background fields but suppose the electron is massive. In this case, there are no zero-modes among the Landau levels. In the absence of zero-modes adiabatic evolution just maps the Dirac sea into itself, so axial charge can not be adiabatically generated. The discussion above is not applicable to the standard model because standard model fermions can be given masses without changing the baryon or lepton number violation in fixed gauge field background. Dirac mass terms do not carry vector charge, so they do not effect the divergence of a vector current. Yet in an adiabatic limit it seems that presence or absence of mass terms $\it must$ effect the divergence of a current. In the following, this paradox will be resolved by solving the equations of motion for certain background fields which, according to the anomaly equation, should generate charge. I will demonstrate that spatially uniform backgrounds which generate vector charge have no adiabatic limit. Such backgrounds produce the anomaly by causing hopping between energy levels. On the other hand, localized instanton-like backgrounds do possess an adiabatic limit. Backgrounds of this type will be shown to produce the anomaly via fermionic bound states whose energies traverse the gap between $E=-m$ to $E=m$. This give a better understanding of the mechanism of baryon number production in the standard model by sphalerons. The sphaleron configuration corresponds to the half-way point with a zero energy bound state. Because of the chiral couplings, the standard model Landau levels are quite complicated. To avoid calculating Landau levels in $3+1$ dimensions, I will instead consider a spontaneously broken $U(1)$ axial gauge theory in $1+1$ dimensions. While the details of the computation are different, many of the results obtained in $1+1$ dimensions are expected to hold in $3+1$ dimensions. The lagrangian of this theory is This simplified model possesses the two traits whose consistency I wish to demonstrate; a massive spectrum and a mass independent vector current divergence, For the moment I will not consider the full dynamical theory, but only that given by where $\rho(x) =v$ asymptotically. It should be possible to demonstrate the anomaly by considering the momentum space equations of motion, as was done for massless Q.E.D. by Nielsen and Ninomiya using the Lorentz force law. A few remarks are in order about how to do this. Let the Dirac field in a background be expanded as follows: where $u_{p,i}$ are free massive spinors normalized to 1, and the index $i$ distinguishes between positive and negative frequency solutions when the backgrounds vanish. All the background dependance is contained in the time evolution of $c_{p,i}(t)$ When the backgrounds vanish, where Given a knowledge of which states are occupied at an initial time, one can determine which states are occupied at a final time by looking at the evolution of the coeficients $c_{p,i}$. At this point however, the use of this expansion to determine the vector charge or the particle number is very ambiguous. One can make transformations of $\psi$, corresponding to certain transformations of the background fields, which change the $c_{p,i}$. For example transformations exist which map something that looks like the Dirac sea into something that looks like an excited state with non zero vector charge. An invariant definition of charge is needed. Such a definition must depend on the background fields as well as the Fourier coefficients. In order to make the computation of the charge simple, I will only consider processes in which local gauge invariant functions of the background fields vanish at asymptotic times. This means that the initial and final $\theta$ and $A^{\mu}$ are gauge equivalent to $\theta =0$ and $A^{\mu} =0$. In this case the proper definition of charge at asymptotic times is simple. In Dirac sea language, one subtracts the number of vacant negative frequency states from the number of occupied positive frequency states. The occupation number of a positive or negative frequency state of momentum $p$ is proportional to $|c_{p,\pm}|^2$ in the gauge in which the backgrounds vanish. Equivalently, in second quantized language one can adopt a normal ordered definition of charge at asymptotic times. The change in the charge can then be written in terms of Bogolubov coefficients relating the operators $\hat c_{p,i}$ in the asymptotic past to those in the asymptotic future, where these operators are defined in the gauge in which the backgrounds vanish. Note that at intermediate times the gauge invariant backgrounds do not vanish so a well defined Bogolubov transformation between asymptotic past and intermediate times does not exist. Normal ordering is no longer sensible at intermediate times because solutions can not be classified as positive or negative frequency. However, I will never explicitly calculate the charge at intermediate times [^1]. In the spirit of the anomaly calculations done by Nielsen and Ninomiya, I will first consider a process in which a spatially uniform axial electric field is turned on and then off. I will also choose a uniform (spatially parallel transported) Higgs field background. The particular background to be considered is where ${\cal E}(t)={\cal E}$ for $0<t<T$ and $0$ at all other times. In this gauge, with the initial and final backgrounds vanishing, the coefficients $c_{p,i}$ have an immediate interpretation in terms of particle and charge production. Due to the axial electric field, vector charge generation is expected, and should be evident in the time evolution of these coeficients. The equations of motion for $c_{p,i}(t)$ are complicated at low p, but simplify greatly at large $|p|$. The simplification occurs because, as one would expect, the fermion mass can be neglected at large $|p|$. A straightforward calculation gives the equation describing behavior deep in the Dirac sea: This equation is not complete, but the neglected terms are all supressed by factors of ${m\over |p|}$. The solution is where which is easily recognized as an axial version of the Lorentz force law. Therefore states along the negative frequency spectral lines at large $|p|$ flow inward towards small $|p|$. Because of unitarity and Fermi statistics, solutions can not pile up at small $|p|$. Therefore there must be level hopping at small $|p|$. Positive frequency states must appear at a rate matching the inward flow of negative frequency states across some large $|p|$ cutoff (). I thus arrive at the result that the backgrounds of have no adiabatic limit. Therefore the absence of zeromodes has no effect on charge production. Putting the system on a line of length L with periodic boundary boundary conditions on the Fermi field, one finds that the number of states crossing the cutoff per unit time is ${L\over\pi}gE$. This yields the expected anomaly ${1\over L}{dQ\over dt}= {gE\over \pi}$. There is actually no reason to expect adiabatic behavior with uniform backgrounds. The backgrounds of   have singular time dependence when the electric field is turned on or off. One can make the time dependence of these backgrounds nonsingular either by smoothly switching the electric field on and off, or by going to $A^0=0$ gauge. If one does the former, one can try to make the backgrounds vary slowly in time by having the electric field ${\cal E}(t)$ vary slowly in time. However, no matter how slowly the electric field varies, $A^0$ will vary rapidly at large distances since $A^0=-{\cal E}(t)x$. In $A^0=0$ gauge, the backgrounds of  become One can try to make these backgrounds vary slowly in time by making ${\cal E}$ small. Yet, no matter how small ${\cal E}$ is, the Higgs phase $\theta$ winds wildly with time at large distances. Therefore the non adiabatic nature of uniform charge producing backgrounds is an infinite volume effect. It is actually easy to see the low momentum level hopping explicitly without invoking Fermi statistics. In $1+1$ dimensions $\gamma^{\mu}\gamma^5 =\epsilon^{\mu\nu}\gamma_{\nu}$. One can use this fortuitous fact to solve the equations of motion at all momenta. For $0<t<T$ the background fields of    are equivalent to a a background vector gauge field [^2] with $V^0=0$ and $V^1=-{\cal E}x$. The vector field strength vansishes, so the time evolution of $\psi$ at intermediate times is trivial. $\psi^{\prime}\equiv e^{{i\over 2}{\cal E}x^2}\psi$ evolves as a free field: One only has to transform back from $c^{\prime}$ to $c$ to get $c_{p,i}(t)$ as a function of the initial coefficients $c_{r,l}(0)$. The result is that where The quantity within the brackets can be written where $\phi$ is a massive free scalar field in $1+1$ dimensions, and $(z^0,z^1) \equiv (t,{p-r\over E})$. The “light cone” singularity in  gives the leading term of  : Let us rewrite this in a form which is easier to interpret: where I have used the fact that $\gamma^0\gamma^{\pm}=1 \pm \gamma^5$ in $1+1$ dimensions. The axial Lorentz force law is clearly visible in the delta functions and the associated left or right chiral projectors. The low momentum level hopping is also manifest. The hopping of negative frequency to positive frequency states is described by $T_{p,+:r,-}$. At large $p$ and $r$ of the same sign, the spinors $u_{p,+}$ and $u_{r,-}$ have opposite chirality so that $u^{\dagger}_{p,+} {1\pm\gamma^5\over 2}u_{r,-}$ vanishes. Thus in the limit of large momenta at fixed time, $T_{p,+;r,-}$ vanishes. However at small $|p|$ the spinors have mixed chirality so that  does not vanish when $p-r=\pm {\cal E}t$, and the predicted level hopping occurs. It is interesting to note that factor   is almost the transformation function associated with an axial transformation of $\psi$: is equivalent to where This is not to be confused with an axial $\it gauge$ transformation because the initial and final background fields are the same; $A^{\mu}=0$ and $\theta=0$. An axial gauge transformation does nothing, but an axial transformation which leaves the Higgs and gauge potentials unchanged can produce particles and vector charge. This should be no surprise given the bosonization rules \[\] for an axial gauge theory in $1+1$ dimensions. The vector charge density in bosonized form is where $\chi$ is the bosonic counterpart to $\psi$. An axial transformation corresponds to Therefore an axial transformation of the type  above produces a net vector charge. The uniform backgrounds of  are interesting but perverse because the gauge invariant objects built from the Higgs and gauge fields do not fall off at large spatial distances. Furthermore these configurations can exist only in an infinite volume because they are inconsistent with periodic boundary conditions. Therefore let us instead consider localized, charge producing backgrounds. By localized, I mean that the energy density carried by the backgrounds is at its minimum outside a spacetime disc of finite radius. At fixed $\Delta Q$ one can always make such backgrounds vary arbitrarily slowly in time, so that there is no argument against the existence of an adiabatic limit. We are again confronted with the puzzle of how vector charge can be produced by a weak electric field in a theory with a gap. The clue to the puzzle is that one can not go to unitary $(\theta=0)$ gauge from localized backgrounds which produce charge. For such backgrounds $D_{\mu}\phi=0$ asymptotically. Therefore If $\Delta Q$ is not zero, then $\phi^{\ast}\phi$ must vanish somewhere due to the non vanishing Higgs winding number. In the presence of such a defect there may be a bound state as well as the continuum of “scattering” solutions with $E=\pm\sqrt{p^2+m^2}$. In an adiabatic limit the only way charge can appear is if a bound state traverses the mass gap. As the defect is created and destroyed in a process with $\Delta Q=1$, the bound state energy should change continuously from $-m$ to $m$. I will show that this is indeed the case. The sphaleron corresponds to a bound state at the half-way point and has charge one half \[\]. An example of a localized configuration giving $\Delta Q =1$ is where the phase $\alpha (t)$ rotates by a total angle of $-\pi$ from $\alpha (-\infty)=0$ to $\alpha (\infty)=-\pi$. In an adiabatic limit $\alpha(t)$ varies slowly and the gauge fields can be neglected. The defect at $x=0$ is spatially pointlike for convenience; For a fixed $\alpha$, finding the spectrum is a trivial matching problem. (A less singular version of this background is drawn in ) One finds a set of scattering solutions with $E=\pm\sqrt{p^2+m^2}$, but there is also a bound state solution with $E^2<m^2$. Continuity of the solution across $x=0$ requires This yields a bound state with energy $E=-m\cos\alpha$. As $\alpha$ varies adiabatically from $0$ to $-\pi$, a single bound charge is carried across the gap. Note that this alone does not guarantee the net production of charge. A bound state could travel across the gap and leave a negative energy hole. The axial Lorentz force law causes negative frequency states to slide inwards towards zero momentum, which prevents the appearance of a hole. In an adiabatic approximation, the gauge fields are negligible pertubations on the spectrum, but drive the spectral flows needed to produce the anomaly. For more general localized backgrounds, an index theorem enables one to count the number of time dependent energy eigenvalues which travel across the gap. Consider spinor functions $f(x,\tau)$ anihilated by the operator where by varying the parameter $\tau$ from $-\infty$ to $\infty$ one goes slowly through the same cycle of Dirac hamiltonians $\hat H$ that occur in real time. I will write the energy eigenvalues as $E_n(\tau)$ and the energy eigenfunctions as $\chi_n (x,\tau)$. Since $\hat H (\tau)$ is a slowly varying function of $\tau$, the solutions of equation   can be written as where there is no sum on $n$ and This solution is only normalizable if $E_n(\tau)$ has a negative value at $\tau=-\infty$ and a positive value at $\tau=+\infty$. Now consider the adjoint operator A function $a_n(\tau) \chi_n(x,\tau)$ annihilated by $\hat D^{\dagger}$ is only normalizable if $E_n(\tau)$ has a positive value at $\tau=-\infty$ and a negative value at $\tau=+\infty$. Hence the total charge generated by bound states crossing the gap is equal to the difference in the number of normalizable modes annihilated by $\hat D$ and the number of normalizable modes annihilated by $\hat D^\dagger$. This quantity is known as the index of $\hat D$. The operator whose index I wish to calculate is where asymptotically $A^0$ is absent from $\hat D$ because it is negligible in an adiabatic approximation. One can take the adiabatic limit of a process with fixed $\Delta Q$ by making the following gauge invariant rescaling of the fields: In the large $\lambda$ limit $A^0$ vanishes. $A^1$ is a nonvanishing adiabatic parameter, but one can gauge it to zero. Doing so effects only the eigenfunctions of $\hat H (\tau)$ but not the eigenvalues. A straight-forward method to calulate the index of Dirac operators on $R_n$ has been constructed by Weinberg\[\]. Using these methods, the index of $\hat D$ with $A^1=0$ is found to be [^3] which is gauge invariant. This is just as one expects given equation . The relation of this index theorem to charge production can also be understood in terms of the euclidean path integral using methods due to Fujikawa \[\] and ’t Hooft \[\]. The fermionic portion of the partition function is where Let $\psi$ and $\bar\psi$ be expanded as where and $f_n(x)$ and $g_l(x)$ are normalized to one. There is a one to one mapping between eigenfunctions of $\hat K ^{\dagger} \hat K$ and $\hat K \hat K ^{\dagger}$ provided that the eigenvalue is not zero. $\hat K$ maps eigenfunctions of $\kr$ into eigenfunctions of $\kl$ with the same non zero eigenvalue, while $\hat K ^{\dagger}$ does the inverse mapping. However if $\kr f(x) =0$ or $\kl g(x) =0$, then there is no mapping because $\kl f(x)=0$ implies that $\hat K f(x) =0$, and $\kr g(x)=0$ implies that $\hat K^{\dagger} g(x)=0$. The difference between the number of zeromodes of $\kr$ and $\kl$ is given by the index of $\hat K$. A zeromode of either $\kr$ or $\kl$ contributes nothing to the euclidean action. Therefore the integral over the grassman coefficient of a zeromode will vanish unless the coefficient appears in the expansion of an operator in a Green’s function. It is easy to see from this that the contributions of a given Higgs and gauge field background to a Green’s function vanishes except when the number of $\psi$’s in the Green’s function differs from the number $\bar\psi$ ’s by the index of $\hat K$. For example, if $\kr$ has one zeromode $f_0(x)$ and $\kl$ has no zeromode, then In general the net vector charge produced is given by the index of $\hat K$, which in an adiabatic limit is the same as the index of $\hat D$ because the two operators differ only by a factor of $\gamma^0$. The connection between the spectral and path integral approaches to the anomaly is now clear [^4]. An interesting feature of the index theorem for a spontaneously broken axial theory is that it permits Higgs and gauge field backgrounds to create single fermions and not just pairs. The Euclidean equations of motion possess a symetry $\psi\rightarrow\gamma^0\psi^{\ast}$. In the absence of the Higgs coupling to fermions, $\hat K$ anticommutes with $\gamma^5$, so zeromodes can be chosen to be chiral. Therefore in the massless axial theory zeromodes occur in pairs of opposite chirality which are related by the above symetry. This pairing is a reflection of $Q_5$ conservation. However in the spontaneously broken axial theory, $Q_5$ has a Higgs component as well as a fermionic component, and only the sum is conserved. It is no longer true that $\lbrace\hat K , \gamma^5\rbrace = 0$. Therefore zeromodes can no longer be chosen to be chiral. In fact, in an adiabatic approximation one can prove that the mapping $\psi\rightarrow\gamma^0\psi^{\ast}$ does not yield independent solutions. This is done in appendix B. The production of single fermions by a background is not a violation of gauge or Lorentz invariance. For example a single fermion can not get a vacuum expectation value because the path integral over gauge and Higgs fields in the one instanton sector vanishes, even if the fermionic integral does not. So far it has only been demonstrated how charge violation proceeds independently of the fermion masses in the case of background Higgs and gauge fields. I will now show how this works in the dynamical case. This will be done by demonstrating the consistency of the Ward identities with a massive spectrum. Similiar results should hold for three current correlation functions in $3+1$ dimensions. The current equations are and A simple path integral manipulation relates the current equations to Ward idendities for $\JJ$ . One finds that and If it were not for the last term in  , the two Ward identities   and   would ensure the existence of a massless pole in the current correlator \[\]??[A.D. Dolgov and V.I. Zakharov, Nucl. Phys. B27 (1971) 525Y. Frishman, A. Schwimmer, T. Banks and S. Yankielowicz, Nucl. Phys. B177 (1981) 157S. Coleman and B. Grossman, Nucl. Phys. B203 (1982) 205.]{}. Naively one might expect the last term in  to give at most ${\cal O}(g)$ perturbative corrections to this pole or its residue. We are thus confronted with the same dilemma as before. The massive spectrum of a spontaneously broken U(1) axial gauge theory appears to be inconsistent with its vector current anomaly. The resolution of the puzzle lies in the fact that the gauge boson mass is proportional to $g$. It turns out that the last term in   contains an order zero piece which exactly cancels the first term at small $p^2$. The last term in   can be rewritten as where $\phi=\rho\exp{i\theta\gamma^5}$, $\bra{0}\rho\ket{0}=v$, and terms which do not give a zeroth order contribution have been dropped. In t’Hooft $\xi$-gauge there is no mixing between $\theta$ and $A^{\mu}$, so in momentum space the leading term of   is At small $p^2$ this is just $-{1\over \pi}\epsilon^{\mu\alpha}p_{\mu}$, giving the stated cancellation. An almost identical cancellation occurs in the Schwinger model \[\] with no fermion mass term. This model also has a massive spectrum. Furthermore the Ward identities are like those of the axial Higgs model, except that axial and vector labels are swapped: and In bosonized form \[\] the last term of the latter ward identity can be written as where $\phi$ is a scalar field with mass ${e\over\sqrt\pi}$. At momentum small compared to the coupling $e$, this becomes ${1\over\pi}\epsilon^{\alpha\nu}p_{\nu}$ which cancels against the first (anomalous commutator) term of  . Thus the anomaly equation does not imply a massless pole. The apparent paradox of an anomaly equation which is insensitive to particle masses has been resolved in $1+1$ dimensions. The Higgs mechanism creates a gap, but also provides a means to cross the gap. In the presence of a localized background with Pontryagin number one, there is a bound fermion due to the winding Higgs background. This bound fermion acts as an “elevator” which carries charge across the gap. For uniform charge generating backgrounds, the Higgs degree of freedom prevents the existence of an adiabatic limit. In the dynamical case, the gauge boson becomes massive due to the Higgs. The gauge boson mass alters the anomalous ward identities in such a way that they do not imply the existence of a massless state. I believe the mechanisms described here should extend readily to $3+1$ dimensions and the standard model. [**Acknowledgements**]{} The author would like to thank David Kaplan, Aneesh Manohar and Jan Smit for useful discussions. This work was supported in part by the Department of Energy under grant number DOE-FG03-90ER40546, the Texas National Research Laboratory Commission under grant RGFY93-206, and by the National Science Foundation under grant PHY-8958081. In $1+1$ dimensions $\gamma^{\mu}\gamma^5= \epsilon_{\mu\nu}\gamma^{\nu}$ Therefore the $1+1$ dimensional Dirac equation with an axial gauge field $A^{\mu}$ is equivalent to the Dirac equation with a background vector gauge field $V^{\mu}$ where $V^{\mu}=\epsilon^{\mu\nu}A_{\nu}$. Thus it naively appears that if an axial gauge theory does not conserve vector charge, then neither does a vector gauge theory. Conversly if a vector theory does not conserve axial charge, it seems that an axial theory does not conserve axial charge either. Fortunately both these statements are not true. The reason they are not true in a finite volume is that there is an ambiguity in doing Bogoliubov transformations. This ambiguity is removed by choosing either axial or vector gauge invariance. Consider the massless axial gauge theory in an $S_1\otimes R_1$ space-time, and suppose charge is produced by a field strengh which vanishes at asymptotic times. The change in vector charge is equal to minus the change in the Chern-Simons number: Therefore the gauge can be chosen so that $A^{\mu}$ vanishes in either the asymptotic past or the asymptotic future, but not both. I will call the Fermi field $\psi^{in}$ or $\psi^{out}$ depending on whether $A^{\mu}$ vanishes in the past or future. $\psi^{in}$ can be expanded in terms of spinors which have definite momentum and frequency in the asymptotic past. Similiarly $\psi^{out}$ can be expanded in terms of spinors which have definite momentum and frequency in the asymptotic future. Particle production is then determined from the Bogoliubov transformation relating the two sets of expansion coefficients. Now suppose we were to consider the vector gauge theory with the backgrounds $V^{\mu}=\epsilon^{\mu\nu}A_{\nu}$. Suppose also that both the axial and vector field strenghths vanish at past and future times. If both field strengths vanish then $\epsilon^{\mu\nu}\del_{\mu}A_{\nu}$ and $\del_{\mu}A^{\mu}$ vanish and $A^{\mu}$ must be a constant. Consider a configuration with $A^{\mu}$=0 in that past and $A^{\mu}=a^{\mu}$ in the future. The difference between an axial gauge theory and a vector gauge theory lies in the relation between $\psi^{in}$ and $\psi^{out}$. For the axial theory while for the vector theory In light-cone coordinates, the two $\psi^{out}$ fields are related by the transformation This transformation changes the vector charge by an amount $g(a_+ - a_-){L\over 2\pi}$ and the axial charge by an amount proportional to $g(a_+ + a_-){L\over 2\pi}$, where $L$ is circumference of $S_1$. Thus in a finite volume one finds the desired result that the axial theory produces only vector charge and the vector theory produces only axial charge. The arguments above are not sufficient to show this result in an infinite volume. This is because in an infinite volume one can always find a gauge in which the vector potential vanishes in both the asymptotic past and asymptotic future [^5]. For these gauges there is no difference between the out fields in the axial theory and the out fields in the vector theory: both are equal to the in field. However there is no equivalence between localized gauge invariant backgrounds in the axial theory and localized gauge invariant backgrounds in the vector theory provided that either vector charge or axial charge respectively are produced. If the axial and vector field strenghs are both localized, then $\epsilon^{\mu\nu}\del_{\mu}A^{\nu}$ and $\del_{\mu}A^{\mu}$ vanish outside some finite region of space-time. This means that $A^{\mu}$ must be a constant outside this region. The Pontryagin index for both the axial and the vector theory therefore vanishes. Note also that for the massive axial theory, a winding Higgs background has no Q.E.D. counterpart. The Euclidean equations of motion for the fermions of a spontaneously broken axial gauge theory possess the symetry $\psi\rightarrow\ \gamma^0\psi^*$. In this appendix I show that, in an adiabatic limit, this symetry does not yield independent solutions. To be precise, a solution of $\hat K f_0(x,\tau)=0$ has the property that $\gamma^0f_0^*(x,\tau)=\exp(i\alpha)f_0(x,\tau)$, where the phase $\alpha$ is a constant. The same is true for spinors annihilated by the adjoint operator $\hat K ^{\dagger}$. Recall that the solution of $\hat K f_0(x,\tau)=0$ in an adiabatic limit is where $\chi_0(x,\tau)$ is an eigenfunction of the time dependent Hamiltonian for which the energy $E_0(\tau)$ crosses the gap. The Berry’s phase $\beta(\tau)$ will turn out to be important to prevent pairing of zeromodes. At asymptotic positive $x$ the magnitude of the Higgs field is $v$, and one can always choose the gauge so that the phase of the Higgs field is independent of $x$. With this choice the bound state eigenfunctions of $H(\tau)$ at large $x$ are of the form where $c(\tau)$ is the phase of Higgs, and $a(\tau)$ is an arbitrary phase. Therefore at large positive $x$ It is easy show that the above relation holds at all $x$ without knowing the exact form of the solution. If $\chi$ is an solution of then so is $\gamma^0\chi^{\ast}$, because Furthermore the eigenvalue equation   is linear and first order in $x$. Therefore if the relation   is true at any $x$, then it must be true at all $x$. We thus arrive at the result that It appears that there is a time dependent phase relation, but in fact the product of all the phases above is independent of $\tau$. The Euclidean equations of motion are linear and first order in $\tau$, and possess the symetry $\psi\rightarrow\gamma^0\psi^{\ast}$. Therefore if at some fixed $\tau$ then this relation must hold at all $\tau$. The symetry which gives pairs of zeromodes in the massless theory fails to give pairs in the spontaneously broken theory. [^1]: At intermediate times the charge is defined by axial gauge invariance and charge conjugation symetry. For example one can use an axially gauge invariant point split charge which is odd under charge conjugation. When the gauge fields vanish this is equivalent to the usual normal ordered definition of charge. [^2]: This method of solving the Dirac equation brings up a troubling question. If an axial gauge field background can generate vector charge, then apparently a vector gauge field background can also generate vector charge. I discuss why this last statement is not true in appendix A. [^3]: Weinberg applied his methods to count the number of zero energy modes of a vortex-fermion system in 2 spatial dimensions. This system was previously considered by Jackiw and Rossi \[\] who suggested the existence of an index theorem equating the number of fermion zero energy modes to the vortex number. The index theorem for their model is very similiar to the one considered in this paper. [^4]: This connection is not novel. The relation between modes annihilated by the Euclidean Dirac operator and spectral flows which take states in and out of the Dirac sea was discussed by Nielsen and Ninomiya in the context of massless fermions . [^5]: In a finite volume one is prevented from doing this by the gauge invariance of $\exp(ig\oint dx^1A_1)$

ArXiv

--- abstract: 'We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.' author: - 'Tian-Jun Li and Cheuk Yu Mak[^1]' title: Symplectic Divisorial Capping in Dimension 4 --- Introduction ============ In this paper, a [*symplectic divisor*]{} refers to a connected configuration of finitely many closed embedded symplectic surfaces $D=C_1 \cup \dots \cup C_k$ in a symplectic 4 dimensional manifold (possibly with boundary or non-compact) $(W, \omega)$. $D$ is further required to have the following properties: $D$ has empty intersection with $ \partial W$, no three $C_i$ intersect at a point, and any intersection between two surfaces is transversal and positive. The orientation of each $C_i$ is chosen to be positive with respect to $\omega$. Since we are interested in the germ of a symplectic divisor, $W$ is sometimes omitted in the writing and $(D,\omega)$, or simply $D$, is used to denote a symplectic divisor. A closed regular neighborhood of $D$ is called a plumbing of $D$. The plumbings are well defined up to orientation preserving diffeomorphism, so we can introduce topological invariants of $D$ using any of its plumbings. In particular, $b_2^{\pm}(D)$ is defined as $b_2^{\pm}$ of a plumbing. Similarly, we define the [*boundary*]{} of the divisor $D$, and we call the fundamental group of the boundary [*boundary fundamental group*]{} of $D$. In the same vein, when $\omega$ is exact on the boundary of a plumbing, we say that $\omega$ is exact on the boundary of $D$. A plumbing $P(D)$ of $D$ is called a [*concave (resp. convex) neigborhood*]{} if $P(D)$ is a strong concave (resp. convex) filling of its boundary. A symplectic divisor $D$ is called [*concave*]{} (resp. [*convex*]{}) if for any neighborhood $N$ of $D$, there is a concave (resp. convex) neighborhood $P(D) \subset N$ for the divisor. Through out this paper, all concave (resp. convex) fillings are symplectic strong concave (resp. strong convex) fillings and we simply call it cappings or concave fillings (resp. fillings or convex fillings). Suppose that $D$ is a concave (resp. convex) divisor. If a symplectic gluing ([@Et98]) can be performed for a concave (resp. convex) neighborhood of $D$ and a symplectic manifold $Y$ with convex (resp. concave) boundary to obtain a closed symplecitc manifold, then we call $D$ a [**capping**]{} (resp. [**filling**]{}) divisor. In both cases, we call $D$ a [**compactifying**]{} divisor of $Y$. Motivation ---------- We provide some motivation from two typical families of examples in algebraic geometry together with some general symplectic compactification phenomena. Suppose $Y$ is a smooth affine algebraic variety over $\mathbb{C}$. Then $Y$ can be compactified by a divisor $D$ to a projective variety $X$. By Hironaka’s resolution of singularities theorem, we could assume that $X$ is smooth and $D$ is a simple normal crossing divisor. In this case, $Y$ is a Stein manifold and $D$ has a concave neighborhood induced by a plurisubharmonic function on $Y$ ([@ElGr91]). Moreover, $Y$ is symplectomorphic to the completion of a suitably chosen Stein domain $\overline{Y}\subset Y$ (See e.g. [@McL12]). Therefore, compactifying $Y$ by $D$ in the algebro geometric situation is analogous to gluing $\overline{Y}$ with a concave neighborhood of $D$ along their contact boundaries [@Et98]. On the other hand, suppose we have a compact complex surface with an isolated normal singularity. We can resolve the isolated normal singularity and obtain a pair $(W,D)$, where $W$ is a smooth compact complex surface and $D$ is a simple normal crossing resolution divisor. In this case, we can define a Kähler form near $D$ such that $D$ has a convex neighborhood $P(D)$. If the Kähler form can be extended to $W$, then the Kähler compactification of $W-D$ by $D$ is analogous to gluing the symplectic manifold $W-Int(P(D))$ with $P(D)$ along their contact boundaries. From the symplectic point of view, there are both flexibility and constraints for capping a symplectic 4 manifold $Y$ with convex boundary. For flexibility, there are infinitely many ways to embed $Y$ in closed symplectic 4-manifolds (Theorem 1.3 of [@EtHo02]). This still holds even when $Y$ has only weak convex boundary (See [@El04] and [@Et04]). For constraints, it is well-known that (e.g. [@Hu13]) $Y$ does not have any exact capping. From these perspectives, divisor cappings might provide a suitable capping model to study (See also [@Ga03] and [@Ga03c]). On the other hand, divisor fillings have been studied by several authors. For instance, it is known that they are the maximal fillings for the canonical contact structures on Lens spaces (See [@Li08] and [@BhOz14]). In this setting, the following questions are natural: Suppose $D$ is a symplectic divisor. \(i) When is $D$ also a compactifying divisor? \(ii) What symplectic manifolds can be compactified by $D$? A Flowchart {#A Flowchart} ----------- Regarding the first question, observe that a divisor is a capping (resp. filling) divisor if it is concave (resp. convex), and embeddable in the following sense: If a symplectic divisor $D$ admits a symplectic embedding into a closed symplectic manifold $W$, then we call $D$ an [**embeddable**]{} divisor. We recall some results from the literature for the filling side. It is proved in [@GaSt09] that when the graph of a symplectic divisor is negative definite, it can always be perturbed to be a convex divisor. Moreover, a convex divisor is always embeddable, by [@EtHo02], hence a filling divisor. However, a concave divisor is not necessarily embeddable. An obstruction is provided by [@Mc90] (See Theorem \[McDuff\]). Our first main result: \[MAIN\] Let $D \subset (W,\omega_0)$ be a symplectic divisor. If the intersection form of $D$ is not negative definite and $\omega_0$ restricted to the boundary of $D$ is exact, then $\omega_0$ can be deformed through a family of symplectic forms $\omega_t$ on $W$ keeping $D$ symplectic and such that $(D,\omega_1)$ is a concave divisor. In particular, if $D$ is also an embeddable divisor, then it is a capping divisor after a deformation. It is convenient to associate an augmented graph $(\Gamma,a)$ to a symplectic divisor $(D,\omega)$, where $\Gamma$ is the graph of $D$ and $a$ is the area vector for the embedded symplectic surfaces (See Section \[Preliminary\] for details). The intersection form of $\Gamma$ is denoted by $Q_{\Gamma}$. Suppose $(\Gamma,a)$ is an augmented graph with $k$ vertices. Then, we say that $(\Gamma,a)$ satisfies the positive (resp. negative) [**GS criterion**]{} if there exists $z \in (0,\infty)^k$ (resp $(-\infty,0]^k$) such that $Q_{\Gamma}z=a$. A symplectic divisor is said to satisfy the positive (resp. negative) GS criterion if its associated augmented graph does. One important ingredient for the proof of Theorem \[MAIN\] is the following result. \[MAIN2\] Let $(D,\omega)$ be a symplectic divisor with $\omega$-orthogonal intersections. Then, $(D,\omega)$ has a concave (resp. convex) neighborhood inside any regular neighborhood of $D$ if $(D,\omega)$ satisfies the positive (resp. negative) GS criterion. The construction is essentially due to Gay and Stipsicz in [@GaSt09], which we call the GS construction. We remark that GS criteria can be verified easily. They are conditions on wrapping numbers in disguise. Therefore, by a recent result of Mark McLean [@McL14], Proposition \[MAIN2\] can be generalized to higher dimensions with GS criteria being replaced accordingly. Moreover, using techniques in [@McL14], we establish the necessity of the GS criterion and answer the uniqueness question in [@GaSt09]. \[obstruction-GS\] Let $D \subset (W,\omega)$ be an $\omega$-orthogonal symplectic divisor. If $(D,\omega)$ does not satisfy the positive (resp. negative) GS criterion. Then, there is a neighborhood $N$ of $D$ such that any plumbing $P(D) \subset N$ of $D$ is not a concave (resp. convex) neighborhood. \[uniqueness-GS\] Let $(D,\omega_i)$ be $\omega_i$-orthogonal symplectic divisors for $i=0,1$ such that both satisfy the positive (resp. negative) GS criterion. Then the concave (resp. convex) structures on the boundary of $(D,\omega_0)$ and $(D,\omega_1)$ via the GS construction are contactomorphic. In particular, when $\omega_0=\omega_1$, the contact structure constructed via GS construction is independent of choices, up to contactomorphism. Summarizing Theorem \[MAIN\] and Proposition \[MAIN2\], we have \[QHS\] Let $(D,\omega)$ be a symplectic divisor with $\omega$ exact on the boundary of $D$. Then $D$ is either a concave divisor or a convex divisor, possibly after a symplectic deformation. More precise information is illustrated by the following schematic flowchart. (exact) \[startstop\] [$\omega|_{\partial P(D)}$ exact?]{}; (not exact) \[startstop2, below of=exact\] [No concave nor convex neighborhood]{}; (definite) \[startstop, right of=exact\] [$Q_D$ negative definite?]{}; (convex) \[startstop2, below of=definite\] [Admits a convex neighborhood]{}; (GS criterion) \[startstop, right of=definite\] [$(D,\omega)$ satisfies positive GS criterion?]{}; (concave) \[startstop2, right of=GS criterion\] [Admits a concave neighborhood]{}; (deformation) \[startstop3, below of=GS criterion\] [No small concave neighborhood, but admits one after a deformation]{}; (exact) – node\[right\][no]{}(not exact); (exact) – node\[above\][yes]{}(definite); (definite) – node\[right\][yes]{}(convex); (definite) – node\[above\][no]{}(GS criterion); (GS criterion) – node\[above\][yes]{}(concave); (GS criterion) – node\[right\][no]{}(deformation); For a general divisor $(D,\omega)$, which is not necessarily $\omega$-orthogonal, the corresponding results for Proposition \[MAIN2\] and Theorem \[uniqueness-GS\] are still valid (See Proposition \[McLean0\] and Proposition \[McLean\]), by McLean’s construction. The generalization of Theorem \[obstruction-GS\] is a bit subtle. For an embeddable divisor $(D,\omega)$, we obtain in Theorem \[obstruction-closed case\] that if it does not satisfy the positive GS criterion then there is a neighborhood $N$ of $D$ such that any plumbing $P(D) \subset N$ is not a concave neighborhood. Divisors with Finite Boundary $\pi_1$ ------------------------------------- Using Theorem \[MAIN\], Theorem \[obstruction-closed case\], Proposition \[McLean0\] and Proposition \[McLean\], we classify, in our second main results Theorem \[main classification theorem\] and Theorem \[complete classification\], capping divisors (not necessarily $\omega$-orthogonal) with finite boundary fundamental group. This allows us to answer the second question if $D$ is a capping divisor with finite boundary fundamental group. As a consequence, we show that only finitely many minimal symplectic manifolds can be compactified by $D$, up to diffeomorphism. More details are described in Section \[Finiteness\]. Moreover, we also investigate special kinds of symplectic filling. In Section \[Non-Conjugate Phenomena\], we study pairs of symplectic divisors that compactify each other. \[Conjugate Definition\_Divisor\] For symplectic divisors $D_1$ and $D_2$, we say that they are [**conjugate**]{} to each other if there exists plumbings $P(D_1)$ and $P(D_2)$ for $D_1$ and $D_2$, respectively, such that $D_1$ is a capping divisor of $P(D_2)$ and $D_2$ is a filling divisor of $P(D_1)$. On the other hand, it is also interesting to investigate the category of symplectic manifolds having symplectic divisorial compactifications. Affine surfaces are certainly in this category. In this regard, symplectic cohomology could play an important role. Growth rate of symplectic cohomology has been used in [@Se08] and [@McL12] to distinguish a family of cotangent bundles from affine varieties. The proof actually applies to any Liouville domain which admits a divisor cap, so certain boundedness on the growth rate is necessary for a Liouville domain to be in this category. Finally, for symplectic manifolds in this category we would like to define invariants in terms of the divisorial compactifications (See [@LiZh11] for a related invariant). The remaining of this article is organized as follows. In Section \[Preliminary\], we give the proof of Proposition \[MAIN2\], Theorem \[obstruction-GS\] and Theorem \[uniqueness-GS\]. Section \[Operation on Divisors\] is mainly devoted to the proof of Theorem \[MAIN\]. We give the statement and proof of the classification of compactifying divisors with finite boundary fundamental group in Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors would like to thank Mark McLean for many helpful discussions, in particular for explaining the canonical contact structure in [@McL14]. They would also like to thank David Gay, Ko Honda, Burak Ozbagci, Andras Stipsicz, Weiwei Wu and Weiyi Zhang for their interest in this work. They are also grateful to Laura Starkston for stimulating discussions. Contact Structures on the Boundary {#Preliminary} ================================== Essential topological information of a symplectic divisor can be encoded by its [*graph*]{}. The graph is a weighted finite graph with vertices representing the surfaces and each edge joining two vertices representing an intersection between the two surfaces corresponding to the two vertices. Moreover, each vertex is weighted by its genus (a non-negative integer) and its self-intersection number (an integer). If each vertex is also weighted by its symplectic area (a positive real number), then we call it an [*augmented graph*]{}. Sometimes, the genera (and the symplectic area) are not explicitly stated. For simplicity, we would like to assume the symplectic divisors are connected. In what follow, we call a finite graph weighted by its self-intersection number and its genus (resp. and its area) with no edge coming from a vertex back to itself a graph (resp. an augmented graph). For a graph (resp. an augmented graph) $\Gamma$ (resp. $(\Gamma,a)$), we use $Q_{\Gamma}$ to denote the intersection matrix for $\Gamma$ (resp. and $a$ to denote the area weights for $\Gamma$). We denote the determinant of $Q_{\Gamma}$ as $\delta_{\Gamma}$. Moreover, $v_1,\dots,v_k$ are used to denote the vertices of $\Gamma$ and $s_i$, $g_i$ and $a_i$ are self-intersection, genus and area of $v_i$, respectively. Notice that, $\omega$ being exact on the boundary of a plumbing is equivalent to $[\omega]$ being able to be lifted to a relative cohomological class. Using Lefschetz duality, this is in turn equivalent to $[\omega]$ being able to be expressed as a linear combination $\sum\limits_{i=1}^k z_i[C_i]$, where $z_i \in \mathbb{R}$ and $D=C_1 \cup \dots \cup C_k$. As a result, $\omega$ is exact on the boundary of a plumbing if and only if there exist a solution $z$ for the equation $Q_{\Gamma}z=a$ (See Subsection \[wrapping numbers\] for a more detailed discussion). We also remark that the germ of a symplectic divisor $(D,\omega)$ with $\omega$-orthogonal intersections is uniquely determined by its augmented graph $(\Gamma,a)$ (See [@McR05] and Theorem 3.1 of [@GaSt09]) and a symplectic divisor can always be made $\omega$-orthogonal after a perturbation (See [@Go95]). \[2-1\] The graph $$\xymatrix{ \bullet^{2}_{v_1} \ar@{-}[r] & \bullet^{1}_{v_2}\\ }$$ where both vertices are of genus zero, represents a symplectic divisor of two spheres with self-intersection $2$ and $1$ and intersecting positively transversally at a point. \[realizable definition\] A graph $\Gamma$ is called [**realizable**]{} (resp. [**strongly realizable**]{}) if there is an embeddable (resp. compactifying) symplectic divisor $D$ such that its graph is the same as $\Gamma$. In this case, $D$ is called a realization (resp. strongly realization) of $\Gamma$. Similar to Definition \[realizable definition\], we can define realizability and strongly realizability for an augmented graph. If the area weights attached to $\Gamma$ is too arbitrary, it is possible that $(\Gamma,a)$ is not strongly realizable but $\Gamma$ is strongly realizable. Existence --------- In this subsection, Proposition \[MAIN2\] is given via two different approaches, namely, GS construction and McLean’s construction. ### Existence via the GS construction [@GaSt09] $(X,\omega,D,f,V)$ is said to be an [**orthogonal neighborhood 5-tuple**]{} if $(X,\omega)$ is a symplectic 4-manifold with $D$ being a collection of closed symplectic surfaces in $X$ intersecting $\omega$-orthogonally such that $f:X \to [0,\infty)$ is a smooth function with no critical value in $(0,\infty)$ and with $f^{-1}(0)=D$, and $V$ is a Liouville vector field on $X-D$. Moreover, if $df(V)>0$ (resp $<0$), then $(X,\omega,D,f,V)$ is called a convex (resp concave) neighborhood 5-tuple. \[fig GS construction\] (0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{}; (1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.4,1.8) coordinate (d\_1)– (5.4,1.8) coordinate (d\_2); (2.4,1.8) coordinate (e\_1)– (2.4,3) coordinate (e\_2); (2.5,3.3) node \[left\] [$R_{e_{\alpha \beta},v_{\alpha}} $]{}; (2.4,3) coordinate (j\_1)– (2.4,4.3) coordinate (j\_2); (1.5,3.9) coordinate (f\_1)– (2.4,4.3) coordinate (f\_2); (1.5,2.4) coordinate (g\_1)– (2.4,2.8) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (3.9,1.8) coordinate (h\_2); (3.8,1.3) node \[right\] [$R_{e_{\alpha \beta},v_{\beta}} $]{}; (4.5,0.9) coordinate (i\_1)– (5.4,1.8) coordinate (i\_2); (c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{\beta}'$]{} -| (xaxis -| c) node\[below\] [$z_{\alpha}'$]{}; (1.5,2.4) – (0,2.4) node\[left\] [$z_{\beta}'+\epsilon$]{}; (1.5,3.9) – (0,3.9) node\[left\] [$z_{\beta}'+2\epsilon$]{}; (3,0.9) – (3,0) node\[below\] [$z_{\alpha}'+\epsilon$]{}; (4.5,0.9) – (4.5,0) node\[below\] [$z_{\alpha}'+2\epsilon$]{}; Figure \[fig GS construction\] In [@GaSt09], Gay and Stipsicz constructed a convex orthogonal neighborhood 5-tuple $(X,\omega,D,f,V)$ when the augmented graph $(\Gamma,a)$ of $D$ satisfies the negative GS criterion. We first review their construction and an immediate consequence will be Proposition \[MAIN2\]. Let $z$ be a vector solving $Q_{\Gamma}z=a$ with $z \in (-\infty,0]^k$. Then $z'=(z_1',\dots,z_n')^T=\frac{-1}{2\pi}z$ has all entries being non-negative. We remark that the $z'$ we use corresponds to the $z$ in [@GaSt09]. For each vertex $v$ and each edge $e$ meeting the chosen $v$, we set $s_{v,e}$ to be an integer. These integers $s_{v,e}$ are chosen such that $\sum\limits_{\text{e meeting v}} s_{v,e}=s_v$ for all $v$, where $s_v$ is the self-intersection number of the vertex $v$. Also, set $x_{v,e}=s_{v,e}z_v'+z_{v'}'$, where $v'$ is the other vertex of the edge $e$. For each edge $e_{\alpha \beta}$ of $\Gamma$ joining vertices $v_\alpha$ and $v_\beta$, we construct a local model $N_{e_{\alpha \beta}}$ as follows. Let $\mu:\mathbb{S}^2 \times \mathbb{S}^2 \to [z_{\alpha}',z_{\alpha}'+1] \times [z_{\beta}',z_{\beta}'+1]$ be the moment map of $\mathbb{S}^2 \times \mathbb{S}^2$ onto its image. We use $p_1$ for coordinate in $[z_{\alpha}',z_{\alpha}'+1]$, $p_2$ for coordinate in $[z_{\beta}',z_{\beta}'+1]$ and $q_i \in \mathbb{R}/2\pi$ be the corresponding fibre coordinates so $\theta=p_1dq_1+p_2dq_2$ gives a primitive of the symplectic form $dp_1 \wedge dq_1 + dp_2 \wedge dq_2$ on the preimage of the interior of the moment image. Fix a small $\epsilon >0$ and let $D_1=\mu^{-1}(\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon])$ be a symplectic disc. Let also $D_2=\mu^{-1}([z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \})$ be another symplectic disc meeting $D_1$ $\omega$-orthogonal at the point $\mu^{-1}(\{z_{\alpha}'\} \times \{z_{\beta}' \})$. Our local model $N_{e_{\alpha \beta}}$ is going to be the preimage under $\mu$ of a region containing $\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon] \cup [z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \}$. A sufficiently small $\delta$ will be chosen. For this $\delta$, let $R_{e_{\alpha \beta},v_{\alpha}}$ be the closed parallelogram with vertices $(z_{\alpha}',z_{\beta}'+\epsilon), (z_{\alpha}',z_{\beta}'+2\epsilon), (z_{\alpha}'+\delta,z_{\beta}'+2\epsilon-s_{v_{\alpha},e_{\alpha \beta}} \delta), (z_{\alpha}'+\delta,z_{\beta}'+\epsilon-s_{v_{\alpha},e_{\alpha \beta}} \delta)$. Also, $R_{e_{\alpha \beta},v_{\beta}}$ is defined similarly as the closed parallelogram with vertices $(z_{\alpha}'+\epsilon,z_{\beta}'), (z_{\alpha}'+2 \epsilon,z_{\beta}'), (z_{\alpha}'+2\epsilon-s_{v_{\beta},e_{\alpha \beta}} \delta,z_{\beta}'+\delta), (z_{\alpha}'+\epsilon-s_{v_{\beta},e_{\alpha \beta}} \delta,z_{\beta}'+\delta)$. We extend the right vertical edge of $R_{e_{\alpha \beta},v_{\alpha}}$ downward and extend the top horizontal edge of $R_{e_{\alpha \beta},v_{\beta}}$ to the left until they meet at the point $(z_{\alpha}'+\delta,z_{\beta}'+\delta)$. Then, the top edge of $R_{e_{\alpha \beta},v_{\alpha}}$, the right edge of $R_{e_{\alpha \beta},v_{\beta}}$, the extension of right edge of $R_{e_{\alpha \beta},v_{\alpha}}$, the extension of top edge of $R_{e_{\alpha \beta},v_{\beta}}$, $\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon]$ and $[z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \}$ enclose a region. After rounding the corner symmetrically at $(z_{\alpha}'+\delta,z_{\beta}'+\delta)$, we call this closed region $R$. Now, we set $N_{e_{\alpha \beta}}$ to be the preimage of $R$ under $\mu$. See Figure \[fig GS construction\]. On the other hand, for each vertex $v_{\alpha}$, we also need to construct a local model $N_{v_{\alpha}}$. Let $g_{\alpha}$ be the genus of $v_{\alpha}$. We can form a genus $g_{\alpha}$ compact Riemann surface $\Sigma_{v_{\alpha}}$ such that the boundary components one to one correspond to the edges meeting $v_{\alpha}$. We denote the boundary component corresponding to $e_{\alpha \beta}$ by $\partial_{e_{\alpha \beta}} \Sigma_{v_{\alpha}}$. There exists a symplectic form $\bar{\omega}_{v_{\alpha}}$ and a Liouville vector field $\bar{X}_{v_{\alpha}}$ on $\Sigma_{v_{\alpha}}$ such that when we give the local coordinates $(t,\vartheta_1) \in (x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z}$ to the neighborhood of the boundary component $\partial_{e_{\alpha \beta}} \Sigma_{v_{\alpha}}$, we have that $\bar{\omega}_{v_{\alpha}}=dt \wedge d\vartheta_1$ and $\bar{X}_{v_{\alpha}}=t \partial_t$. Now, we form the local model $N_{v_{\alpha}}=\Sigma_{v_{\alpha}} \times \mathbb{D}^2_{\sqrt{2\delta}}$ with product symplectic form $\omega_{v_{\alpha}}=\bar{\omega}_{v_{\alpha}}+rdr \wedge d\vartheta_2$ and Liouville vector field $X_{v_{\alpha}}=\bar{X}_{v_{\alpha}}+(\frac{r}{2}+\frac{z_{v_{\alpha}}'}{r})\partial_r$, where $(r,\vartheta_2)$ is the standard polar coordinates on $\mathbb{D}^2_{\sqrt{2\delta}}$. Finally, the GS construction is done by gluing these local models appropriately. To be more precise, the preimage of $R_{e_{\alpha \beta},v_{\alpha}}$ of $N_{e_{\alpha \beta}}$ is glued via a symplectomorphism preserving the Liouville vector field to $[x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ of $N_{v_{\alpha}}$ and other matching pieces are glued similarly. When $\delta >0$ is chosen sufficiently small, this glued manifold give our desired convex orthogonal neighborhood 5-tuple with the symplectic divisor having graph $\Gamma$. We remark the whole construction works exactly the same if all entries of $z'$ are negative. In this case, all entries of $z$ are positive and we get the desired concave orthogonal neighborhood 5-tuple if $(\Gamma,a)$ satisfies the positive GS criterion. Now, if we have an $\omega'$-orthogonal divisor $(D',\omega')$ with augmented graph $(\Gamma,a)$, which is the same as that of the concave orthogonal neighborhood 5-tuple $(X,\omega,D,f,V)$, then there exist neighborhood $N'$ of $D'$ symplectomorphic to a neighborhood of $D$ and sending $D'$ to $D$ (See [@McR05] and [@GaSt09]). Therefore, a concave neighborhood of $D$ in $N$ give rise to a concave neighborhood of $D'$ in $N'$. This finishes the proof of Proposition \[MAIN2\]. ### Existence in Higher Dimensions via Wrapping Numbers {#wrapping numbers} To understand the geometrical meaning of the GS criteria, we recall wrapping numbers from [@McL14] and [@McL12]. Then, another construction for Proposition \[MAIN2\] is given. Let $(D,P(D),\omega)$ be a plumbing of a symplectic divisor. If $\omega$ is not exact on the boundary of $D$, then there is no Liouville flow $X$ near $\partial P(D)$ such that $\alpha=i_X \omega$ and $d\alpha=\omega$. Therefore, $D$ does not have concave nor convex neighborhood. When $\omega$ is exact on the boundary, let $\alpha$ be a $1$-form on $P(D)-D$ such that $d \alpha=\omega$. Let $\alpha_c$ be a $1$-form on $P(D)$ such that it is $0$ near $D$ and it equals $\alpha$ near $\partial P(D)$. Note that $[\omega-d\alpha_c] \in H^2(P(D), \partial P(D);\mathbb{R})$. Let its Lefschetz dual be $-\sum\limits_{i=1}^k \lambda_i [C_i] \in H_2(P(D); \mathbb{R})$. We call $\lambda_i$ the wrapping number of $\alpha$ around $C_i$. Also, there is another equivalent interpretation of wrapping numbers. If we symplectically embed a small disc to $P(D)$ meeting $C_i$ positively transversally at the origin of the disc, then the pull-back of $\alpha$ equals $ \frac{r^2}{2}d\vartheta + \frac{\lambda_i}{2\pi} d\vartheta +df$, where $(r,\vartheta)$ is the polar coordinates of the disc and $f$ is some function defined on the punctured disc. (See the paragraph before Lemma 5.17 of [@McL12]). From this point of view, we can see that the $z_i$’s in the GS criteria are minus of the wrapping numbers $-\lambda_i$’s for a lift of the symplectic class $[\omega] \in H^2(P(D);\mathbb{R})$ to $H^2(P(D),\partial P(D);\mathbb{R})$. In particular, $Q$ being non-degenerate is equivalent to lifting of symplectic class being unique, which is in turn equivalent to the connecting homomorphism $H^1(\partial P(D); \mathbb{R}) \to H^2(P(D),\partial P(D);\mathbb{R})$ is zero. When $Q$ is degenerate and for a fixed $\omega$, the equation $Qz=a$ having no solution for $z$ is equivalent to $\omega|_{\partial P(D)}$ being not exact. Similarly, when $Qz=a$ has a solution for $z$, then the solution is unique up to the kernel of $Q$, which corresponds to the unique lift of $\omega$ up to the image of the connecting homomorphism $H^1(\partial P(D); \mathbb{R}) \to H^2(P(D),\partial P(D);\mathbb{R})$. To summarize, we have Let $(D,\omega)$ be a symplectic divisor. Then, lifts $[\omega-d\alpha_c] \in H^2(P(D),\partial P(D);\mathbb{R})$ of the symplectic class $[\omega]$ are in one-to-one correspondence to the solution $z$ of $Q_Dz=a$ via the minus of Lefschetz dual $PD([\omega-d\alpha_c])= -\sum\limits_{i=1}^k \lambda_i [C_i]$ and $z_i=-\lambda_i$. Proposition \[MAIN2\] can be generalized to arbitrary dimension if we apply the constructions in the recent paper of McLean [@McL14]. We first recall an appropriate definition of a symplectic divisor in higher dimension (See [@McL14] or [@McL12]). Let $(W^{2n},\omega)$ be a symplectic manifold with or without boundary. Let $C_1, \dots, C_k$ be real codimension $2$ symplectic submanifolds of $W$ that intersect $\partial W$ trivially (if any). Assume all intersections among $C_i$ are transversal and positive, where positive is defined in the following sense. \(i) For each $I \subset \{1, \dots,k \}$, $C_I=\cap_{i \in I}C_i$ is a symplectic submanifold. \(ii) For each $I, J \subset \{1, \dots, k\}$ with $C_{I \cup J} \neq \emptyset$, we let $N_1$ be the symplectic normal bundle of $C_{I \cup J}$ in $C_I$ and $N_2$ be the symplectic normal bundle of $C_{I \cup J}$ in $C_J$. Then, it is required that the orientation of $N_1 \oplus N_2 \oplus TC_{I \cup J}$ is compatible with the orientation of $TW|_{C_{I \cup J}}$. We remark that the condition (ii) above guarantees that no three distinct $C_i$ intersect at a common point when $W$ is four dimensions. Therefore, this higher dimension definition coincides with the one we use in four dimension. To make our paper more consistent, in higher dimension, we call $D= C_1 \cup \dots \cup C_k$ a symplectic divisor if $D$ is moreover connected and the orientation of each $C_i$ is induced from $\omega^{n-1}|_{C_i}$. Now, for each $i$, let $N_i$ be a neighborhood of $C_i$ such that we have a smooth projection $p_i: N_i \to C_i$ with a connection rotating the disc fibers. Hence, for each $i$, we have a well-defined radial coordinate $r_i$ with respect to the fibration $p_i$ such that $C_i$ corresponds to $r_i=0$. Let $\bar{\rho}: [0,\delta) \to [0,1]$ be a smooth function such that $\bar{\rho}(x)=x^2$ near $x=0$ and $\bar{\rho}(x)=1$ when $x$ is close to $\delta$. Moreover, we require $\bar{\rho}'(x) \ge 0$. A smooth function $f: W-D \to \mathbb{R}$ is called compatible with $D$ if $f=\sum\limits_{i=1}^k \log(\bar{\rho}(r_i))+\bar{\tau}$ for some smooth $\bar{\tau}: W \to \mathbb{R}$ and choice of $\bar{\rho}(r_i)$ as above. Here is the analogue of Proposition \[MAIN2\] in arbitrary even dimension. \[McLean0\] Suppose $f: W^{2n}-D \to \mathbb{R}$ is compatible with $D$ and $D$ is a symplectic divisor with respect to $\omega$. Suppose $\theta \in \Omega^1(W^{2n}-D)$ is a primitive of $\omega$ on $W^{2n}-D$ such that it has positive (resp. negative) wrapping numbers for all $i=1, \dots, k$. Then, there exist $g: W^{2n}-D \to \mathbb{R}$ such that $df(X_{\theta+dg}) > 0$ (resp. $df(-X_{\theta+dg}) > 0$) near $D$, where $X_{\theta+dg}$ is the dual of $\theta+dg$ with respect to $\omega$. In particular, $D$ is a convex (resp. concave) divisor. This is essentially contained in Propositon 4.1 of [@McL14]–the only new statement is the last sentence. And Proposition 4.1 in [@McL14] is stated only for the case in which wrapping numbers are all positive, however, the proof there goes through without additional difficulty for the other case. We give here the most technical lemma adapted to the case of negative wrapping numbers and ambient manifold being dimension four for the sake of completeness. We remark that the $\omega$-orthogonal intersection condition is not required in his construction. Given $D=D_1 \cup D_2 \subset (U,\omega)$, where $D_1$ and $D_2$ are symplectic $2$-discs intersecting each other positively and transversally at a point $p$. Suppose $\theta \in \Omega^1(U-D)$ is a primitive of $\omega$ on $U-D$ such that it has negative wrapping numbers with respect to both $D_1$ and $D_2$. Then there exists $g$ such that for all smooth functions $f:U-D \to \mathbb{R}$ compatible wtih $D$, we have that $df(-X_{\theta+dg}) > c_f \|\theta+dg\| \| df\|$ near $D$, where $c_f>0$ is a constant depending on $f$. Also $c_1 \| db\| < \| \theta+dg \| < c_2 \| db\|$ near $D$ for some smooth function $b$ compatible with $D$, where $c_1$ and $c_2$ are some constants. By possibly shrinking $U$, we give a symplectic coordinate system at the intersection point $p$ such that $D_1=\{x_1=y_1=0\}$ and $0$ corresponds to $p$. Let $\pi_1$ be the projection to the $x_2,y_2$ coordinates. Write $x_1=r \cos \vartheta$ and $y_1=r \sin \vartheta$ and let $\tau=\frac{r^2}{2}$. Let $U_1=U-D_1$ and $\tilde{U_1}'$ be the universal cover of $U_1$ with covering map $\alpha$. Give $\tilde{U_1}'$ the coordinates $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})$ coming from pulling back the coordinates of $(\tau,\vartheta,x_2,y_2)$ by the covering map. Then, the pulled back symplectic form on $\tilde{U_1}'$ is given by $d\tilde{x_1} \wedge d\tilde{y_1} + d\tilde{x_2} \wedge d\tilde{y_2}$. Hence, we can enlarge $\tilde{U_1}'$ across $\{ \alpha^*\tau=\tilde{x_1}=0 \}$ to $\tilde{U_1}$ by identifying $\tilde{U_1}'$ as an open subset of $\mathbb{R}^4$ with standard symplectic form. Let $L_{\vartheta_0}=\{(\tau,\vartheta_0,x_2,y_2) \in \tilde{U_1}| \tau,x_2,y_2 \in \mathbb{R}\}$, which is a $3$-manifold depending on the choice of $\vartheta_0$. Let $T$ be the tangent space of $D_2$ at $0$ and identify it as a $2$ dimensional linear subspace in $(x_1,y_1,x_2,y_2)$ coordinates. Then, $l_{\vartheta_0}=\alpha(L_{\vartheta_0} \cap \tilde{U_1}') \cap T$ is an open ray starting from $0$ in $U$ because $D_1$ and $D_2$ are assumed to be transversal. If we pull back the tangent space of $l_{\vartheta_0}$ to the $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})$ coordinates in $\tilde{U_1}'$, it is spanned by a vector of the form $(1,0,a_{\vartheta_0},b_{\vartheta_0})$ for some $a_{\vartheta_0},b_{\vartheta_0}$. We identify this vector as a vector at $(0,\vartheta_0,0,0)$ and call it $v_{\vartheta_0}$. Notice that the $\omega$-dual of $v_{\vartheta_0}$ is $d\tilde{y_1}-b_{\vartheta_0}d\tilde{x_2}+a_{\vartheta_0}d\tilde{y_2}$, for all $\vartheta_0 \in [0,2\pi]$. Let $X_1$ be a vector field on $\tilde{U_1}$ such that $X_1=\frac{\lambda_1}{2\pi}v_{\vartheta_0}$ at $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})=(0,\vartheta_0,0,0)$ for all $\vartheta_0 \in [0,2\pi]$, where $\lambda_1$ is the wrapping number of $\theta$ with respect to $D_1$. We also require the $\omega$-dual of $X_1$ to be a closed form on $\tilde{U_1}$. This can be done because the $\omega$-dual of $X_1$ restricted to $\{\tilde{x_1}=\tilde{x_2}=\tilde{y_2}=0\}$ is closed. Furthermore, we can also assume $X_1$ is invariant under the $2\pi \mathbb{Z}$ action on $\tilde{y_1}$ coordinate. Note that $d\tilde{x_1}(X_1)=\frac{\lambda_1}{2\pi} < 0$ at $(0,\vartheta_0,0,0)$ for all $\vartheta_0$ so we have $d\tilde{x_1}(X_1) < 0$ near $\{\tilde{x_1}=\tilde{x_2}=\tilde{y_2}=0\}$. Let the $\omega$ dual of $X_1$ be $\tilde{q_1}$, which is exact as it is closed in $\tilde{U_1}$. Now, $\tilde{q_1}$ can be descended to a closed 1-form $q_1$ in $U_1$ under $\alpha$ with wrapping numbers $\lambda_1$ and $0$ with respect to $D_1$ and $D_2$, respectively. We can construct another closed 1-form $q_2$ in $U_2$ in the same way as $q_1$ with $D_1$ and $D_2$ swapped around. Notice that $q_1+q_2$ is a well-defined closed 1-form in $U-D$ with same wrapping numbers as that of $\theta$. Let $\theta'=\theta_1+q_1+q_2$ be such that $d(\theta')=\omega$ and $\theta_1$ has bounded norm. Since $\theta'$ has the same wrapping numbers as that of $\theta$, we can find a function $g: U-D \to \mathbb{R}$ such that $\theta'=\theta+dg=\theta_1+q_1+q_2$. We want to show that $df(-X_{\theta+dg}) > c_f \|\theta+dg\| \| df\|$ near $D$. It suffices to show that $df(-X_{q_1+q_2}) > c_f \|q_1+q_2\| \| df\|$ near $D$ as $\| \theta_1 \|$ is bounded. Since $f=\sum\limits_{i=1}^n \log(\rho(r_i))+\bar{\tau}$ for some smooth $\bar{\tau}: M \to \mathbb{R}$, it suffices to show that $\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2)) (-X_{q_1+q_2}) > c_f \|q_1+q_2\| \| \sum_{i=1}^2 (d\log(x_i'^2+y_i'^2))\|$, where $(x_1',y_1',x_2',y_2')$ are smooth coordinates adapted to the fibrations used to define compatibility. To do this, we pick a sequence of points $p_k \in U-D$ converging to $0$. Then $\frac{X_{q_1}}{\| q_1 \|}$ at $p_k$ converges (after passing to a subsequence) to a vector transversal to $D_1$ but tangential to $D_2$. The analogous statement is true for $\frac{X_{q_2}}{\| q_2 \|}$. Hence we have $\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2)) (-X_{q_1+q_2}) > c_f \sum\limits_{i=1}^2 \|q_i\| \| (d\log(x_i'^2+y_i'^2))\|$ and thus get the desired estimate (See [@McL14] for details). On the other hand, $c_1 \| db\| < \| \theta+dg \| < c_2 \| db\|$ near $D$ for some smooth function $b$ compatible with $D$ is easy to achieve by taking $b=C\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2))$ near $D$. Careful readers will find that when constructing a convex neighborhood, the GS construction works when wrapping numbers are all non-negative while McLean’s constructions work only when wrapping numbers are all positive. We end this subsection with a lemma saying that the GS construction is not really more powerful than McLean’s construction in dimension four. \[non-negative wrapping numbers\] Let $(D^{2n-2},\omega)$ be a symplectic divisor with $n>1$. Suppose $\omega$ is exact on the boundary with $\alpha$ being a primitive on $P(D)-D$. If the wrapping numbers of $\alpha$ are all non-negative, then all are positive. Suppose the wrapping numbers $\lambda_i$ of $\alpha$ are all non-negative and $\lambda_1=0$. Then, $\alpha$ can be extend over $C_1-\cup_{1 \in I, |I| \ge 2}C_I$, where we recall $C_I$ with $1 \in I$ are the symplectic submanifold of $C_1$ induced from intersection with other $C_i$. Therefore, $$\int_{C_1} \omega^{n-1} = \int_{P(\cup_{1 \in I, |I| \ge 2} C_I)} \omega^{n-1} - \int_{\partial P(\cup_{1 \in I, |I| \ge 2} C_I)} \alpha \wedge \omega^{n-2},$$ where $P(\cup_{1 \in I, |I| \ge 2} C_I)$ is a regular neighborhood of $\cup_{1 \in I, |I| \ge 2} C_I$ in $C_1$. We claim that $\int_{P(\cup_{1 \in I, |I| \ge 2} C_I)} \omega^{n-1} - \int_{\partial P(\cup_{1 \in I, |I| \ge 2} C_I)} \alpha \wedge \omega^{n-2} \le 0$ so we will arrive at a contradiction. We first assume that if $1 \in I$, then $C_I = \emptyset$ except $C_1$ and $C_{ \{1,2 \}}$. As a submanifold of $C_1$, $P(\cup_{1 \in I, |I| \ge 2} C_I)=P(C_{\{1,2 \}})$ can be symplectically identified with a closed $2$-disc bundle over $C_{ \{1,2 \}}$. For each fibre, $\alpha|_{\text{fibre}}= \frac{r^2}{2}d\vartheta + \frac{\lambda_2}{2\pi} d\vartheta +df$, where $(r,\vartheta)$ is the polar coordinates of the disc and $f$ is a smooth function defined on the punctured disc. Without loss of generality, we can assume $P(C_{\{1,2\}})$ is taken such that symplectic connection rotates the fibre and we have a well defined one form $\frac{\lambda_2}{2\pi} d\vartheta$ on $P(C_{\{1,2\}})-C_{\{1,2\}}$. Then, $\alpha- \frac{\lambda_2}{2\pi} d\vartheta -df$ can be defined over $P(C_{\{1,2\}})$ for some $f$ defined on $P(C_{\{1,2\}})-C_{\{1,2\}}$ and $$\begin{aligned} \int_{\partial P(C_{\{1,2\}})} (\alpha -\frac{\lambda_2}{2\pi} d\vartheta) \wedge \omega^{n-2} &=&\int_{\partial P(C_{\{1,2\}})} (\alpha -\frac{\lambda_2}{2\pi} d\vartheta-df) \wedge \omega^{n-2}\\ &=&\int_{P(C_{\{1,2\}})} rdr \wedge d\vartheta \wedge \omega^{n-2}\\ &=&\int_{P(C_{\{1,2\}})} \omega^{n-1}.\end{aligned}$$ Therefore, $$\begin{aligned} \int_{P(C_{\{1,2\}})} \omega^{n-1}- \int_{\partial P(C_{\{1,2\}})} \alpha \wedge \omega^{n-2} &=& -\int_{\partial P(C_{\{1,2\}})} \frac{\lambda_2}{2\pi} d\vartheta \wedge \omega^{n-2} \\ &=& -\lambda_2 \int_{C_{\{1,2\}}} \omega^{n-2} \le 0\end{aligned}$$ It is not hard to see that this argument can be generalized to more than two $C_I$ being non-empty, where $1 \in I$. This completes the proof. Obstruction ----------- In this subsection we prove Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\]. We first prove Theorem \[obstruction-GS\], in which $(D,\omega)$ is assumed to be $\omega$-orthogonal. Then the proof for Theorem \[obstruction-closed case\], which is similar, is sketched. ### Energy Lower Bound Given an $\omega$-orthogonal symplectic divisor $D =C_1 \cup \dots \cup C_k$ in a 4-manifold $(W,\omega)$, for each $i$, let $N_i$ be a neighborhood of $C_i$ together with a symplectic open disk fibration $p_i: N_i \to C_i$ such that the symplectic connection induced by $\omega$-orthogonal subspace of the fibers rotates the symplectic disc fibres. Hence, for each $i$, we have a well-defined radial coordinate $r_i$ with respect to the fibration $p_i$ such that $C_i$ corresponds to $r_i=0$. Also, $N_i$ are chosen such that the disk fibers are symplectomorphic to the standard open symplectic disk with radius $\epsilon_i$. We also assume $\min_{i=1}^k {r_i}=r_1$ (or simply $r_1=r_2=\dots=r_k$). Moreover, we require $p_{ij}: N_i \cap N_j \to C_{ij}$ to be a symplectic $\mathbb{D}^2 \times \mathbb{D}^2$ fibration such that $p_i|_{N_i \cap N_j}$ is the projection to the first factor and $p_j|_{N_i \cap N_j}$ is the projection to the second factor. Such choice of $p_i$ and $N_i$ exist (See Lemma 5.14 of [@McL12]). Let $(D=C_1 \cup \dots \cup C_k ,\omega)$ be a symplectic divisor with $p_i$ and $N_i$ as above. There exist an $\omega$-compatible almost complex structure $J_N$ on $N=\cup_{i=1}^k N_i$ such that $C_i$ are $J_N$-holomorphic, the projections $p_i$ are $J_N$-holomorphic and the fibers are $J_N$-holomorphic. Using $p_{ij}$, we can define a product complex structure on $\mathbb{D}^2 \times \mathbb{D}^2 = N_i \cap N_j$. Since $p_{ij}$ are compatible with $p_i$ and $p_j$, we can extend this almost complex structure such that $J|_{C_i}$ and $J|_{C_j}$ are complex structures, $(p_l)_*J=J|_{C_l}$ and $J(r_l\partial_{r_l})=\partial_{\vartheta_l}$ for $l=i,j$, where $(r_i,\vartheta_i)$ and $(r_j,\vartheta_j)$ are polar coordinates of the disk fiber for $p_i$ and $p_j$, respectively. Although $\vartheta_i$ and $\vartheta_j$ are not well-defined if the disk bundle has non-trivial Euler class, $\partial_{\vartheta_l}$ are well-defined for $k=i,j$. Since the almost complex structure $J$ is ‘product-like’, $J$ is compatible with the symplectic form $\omega$. We call this desired almost complex structure $J_N$. Now, we consider a partial compactification of $N=\cup_{i=1}^k N_i$ in the following sense. Consider a local symplectic trivialization of the symplectic disk bundle induced by $p_1$, $B_1 \times \mathbb{D}^2$, where $B_1 \subset C_1$ is symplectomorphic to the standard symplectic closed disk with radius $\tau$. We assume that $4\epsilon_1 < \tau$. We recall that $\mathbb{D}^2$ is equipped with a standard symplectic form with radius $\epsilon_1$. Choose a symplectic embedding of $\mathbb{D}^2_{\epsilon_1}$ to $S^2_{\epsilon}$ with $\epsilon$ slightly large than $\epsilon_1$, where $S^2_{\epsilon}$ is a symplectic sphere of area $\pi \epsilon^2$. We glue $\cup_{i=1}^k N_i$ with $B_1 \times S^2_{\epsilon}$ along $B_1 \times \mathbb{D}^2_{\epsilon_1}$ with the identification above. This glued manifold is called $\bar{N}$ and the compatible almost complex structure constructed above can be extended to $\bar{N}$, which we denote as $J_{\bar{N}}$. We further require that $\{ q \} \times S^2_{\epsilon}$ is $J_{\bar{N}}$-holomorphic for every $q \in B_1$. We want to get an energy uniform lower bound for $J$-holomorphic curves representing certain fixed homology class, for those $J$ that are equal to $J_{\bar{N}}$ away from a neighborhood of the divisor $D$. Let $N^{\delta} = \cup_{i=1}^k \{ r_i \le \delta \} \subset \bar{N}$, where $r_i$ are the radial coordinates for the disk fibration $p_i$. \[energy lower bound\] Let $\delta_{\min} >0$ be small and $\delta_{\max}>0$ be slightly less than $\epsilon_1$. Let $q_{\infty} \in B_1 \times S^2_{\epsilon}$ be a point in $\bar{N}-N$ and the first coordinate of which is the center of $B_1$. Let $J$ be an $\omega$-compatible almost complex structure such that $J=J_{\bar{N}}$ on $\bar{N}-N^{\delta_{\min}}$. If $u:\mathbb{C}P^1 \to \bar{N}$ is a non-constant $J$ holomorphic curve passing through $q_{\infty}$, then either $u^*\omega([\mathbb{C}P^1]) > 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) $ or the image of $u$ stays inside $N^{\delta_{\max}} \cup B_1^{\frac{\tau}{2}} \times S^2_{\epsilon}$, where $B_1^{\frac{\tau}{2}}$ is a closed sub-disk of $B_1$ with the same center but radius $\frac{\tau}{2}$. Let us assume $u^*\omega([\mathbb{C}P^1]) \le 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) $. Otherwise, we have nothing to prove. Also, we can assume $u$ intersect $\partial N^{\delta_{\min}}$ and $\partial N^{\delta_{\max}}$ transversally, by slightly adjusting $\delta_{\min}$ and $\delta_{\max}$. Passing to the underlying curve if necessary, we can also assume $u$ is somewhere injective. Consider the portion of $u$ inside $Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}}$. Let $\bar{p_1}$ be the projection to the first factor. We have $\bar{p_1} \circ u|_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})}$ is a holomorphic map because $J=J_{\bar{N}}$ in $Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}}$ and $J_{\bar{N}}$ splits as a product. This map is also proper. Therefore, the map is either a surjection or a constant map. If it is a surjection, then $u^*\omega([\mathbb{C}P^1]) > \pi (\frac{\tau}{2})^2 >1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min})$. Contradiction. If it is a constant map, then the image of $u|_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})}$ is the fiber. Hence, we get $u^*\omega([\mathbb{C}P^1]) \ge \pi (\epsilon^2 - \delta_{\min}^2) > \pi (\delta_{\max}^2 - \delta_{\min}^2)$. If the image of $u$ does not stay inside $N^{\delta_{\max}} \cup B_1^{\frac{\tau}{2}} \times S^2_{\epsilon}$, then there is a point $q_*$ outside this region, lying inside the image of $u$ and $N^{\epsilon_1}-N^{\delta_{\max}}$. We can assume $q_*$ is an injectivity point of $u$. In particular, it also means that $u^{-1}(\bar{N}-N^{\delta_{\min}})$ is disconnected. Consider the connected component $\Sigma$ of $u^{-1}(\bar{N}-N^{\delta_{\min}})$, which contains the preimage of $q_*$ under $u$. Using one of the projections $p_i$, depending on the position of $q_*$, we can identify a neighborhood of $q_*$ as $Int(\mathbb{D}^2_{\delta_{\max}-\delta_{\min}}) \times (Int(\mathbb{D}^2_{\epsilon_1}) -\mathbb{D}^2_{\delta_{\min}})$, where $\mathbb{D}^2_{\delta_{\min}}$ has the same center as $\mathbb{D}^2_{\epsilon_1}$ and they are closed disks with radii $\delta_{\min}$ and $\epsilon_1$, respectively. We call this neighborhood $N_{q_*}$. Also, we still have $J=J_{\bar{N}}$ and $J_{\bar{N}}$ splits as a product in $N_{q_*}$. Similar as before, by projection to the factors, we see that $\int_{\Sigma \cap u^{-1}(N_{q_*})} u^*\omega \ge \min \{ \pi(\delta_{\max}-\delta_{\min})^2, \pi \epsilon_1^2-\pi \delta_{\min}^2 \}$. Therefore, we have $$u^*\omega([\mathbb{C}P^1]) \ge \int_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})} u^*\omega+\int_{\Sigma \cap u^{-1}(N_{q_*})} u^*\omega > 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min})$$ Contradiction. ### Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\] We recall the terminology [*GW triple*]{} used in [@McL14]. For a symplectic manifold $(W,\omega)$ (possibly non-compact), a homology class $[A] \in H_2(W;\mathbb{Z})$ and a family of compatible almost complex structures $\mathcal{J}$ such that \(1) $\mathcal{J}$ is non-empty and path connected. \(2) there is a relative compact open subset $U$ of $W$ such that for any $J \in \mathcal{J}$, any compact genus $0$ nodal $J$-holomorphic curve representing the class $[A]$ lies inside $U$. \(3) $c_1(TW)([A])+n-3=0$. $GW_0(W,[A],\mathcal{J})$ is called a GW triple. The key property of a GW triple is the following. \[GW\] Suppose $GW_0(W,[A],\mathcal{J})$ is a GW triple. Then, the GW invariants $GW_0(W,[A],J_0)$ and $GW_0(W,[A],J_1)$ are the same for any $J_0,J_1 \in \mathcal{J}$. In particular, if $GW_0(W,[A],J_0) \neq 0$, then for any $J \in \mathcal{J}$, there is a nodal closed genus $0$ $J$-holomorphic curve representing the class $[A]$. One more technique that we need to use is usually called neck-stretching (See [@BEHWZ03] and the references there-in). Given a contact hypersurface $Y \subset W$ separating $W$ with Liouville flow $X$ defined near $Y$. We call the two components of $W-Y$ as $W^-$ and $W^+$, where $W^-$ is the one containing $D$. Then, $Y$ has a tubular neighborhood of the form $(-\delta,\delta)\times Y$ induced by $X$, which can be identified as part of the symplectization of $Y$. By this identification, we can talk about what it means for an almost complex structure to be translation invariant and cylindrical in this neighborhood. If one choose a sequence of almost complex structures $J_i$ that “stretch the neck” along $Y$ and a sequence of closed $J_i$-holomorphic curve $u_i$ with the same domain such that there is a uniform energy bound, then $u_i$ will have a subseguence ’converge’ to a $J_{\infty}$-holomorphic building. The fact that we need to use is the following. \[SFT\] Suppose we have a sequence of $\omega$-compatible almost complex structure $J_i$ and a sequence of nodal closed genus $0$ $J_i$-holomorphic maps $u_i$ to $W$ representing the same homology class in $W$ such that the image of $u_i$ stays inside a fixed relative compact open subset of $W$. Assume $J_i$ stretch the neck along a separating contact hypersurface $Y \subset W$ with respect to a Liouville flow $X$ defined near $Y$. Assume that the image of $u_i$ has non-empty intersection with $W^-$ and $W^+$, respectively, for all $i$. Then, there are proper genus $0$ $J_{\infty}$-holomorphic maps (domains are not compact) $u_{\infty}^-:\Sigma^- \to W^-$ and $u_{\infty}^+:\Sigma^+ \to W^+$ such that $u_{\infty}^-$ and $u_{\infty}^+$ are asymptotic to Reeb orbits on $Y$ with respect to the contact form $\iota_X \omega$. In our notations, $u_{\infty}^-$ and $u_{\infty}^+$ are certain irreducible components of the top/bottom buildings but not necessary the whole top/bottom buildings. Also, $u_{\infty}^-$ does not necessarily refer to the bottom building because we do not declare the direction of the Liouville flow near $Y$. We are finally ready to prove Theorem \[obstruction-GS\]. The following which we are going to prove implies Theorem \[obstruction-GS\]. \[obstruction\] Let $D \subset (W,\omega)$ be an $\omega$-orthogonal symplectic divisor with area vector $a=(\omega[C_1], \dots, \omega[C_k])$. Let $z=(z_1,\dots,z_k)$ be a solution of $Q_Dz=a$. If one of the $z_i$ is non-positive (resp. positive), there is a small neighborhood $N^{\delta_{\min}} \subset W$ of $D$ such that there is no plumbing $(P(D),\omega|_{P(D)}) \subset (N^{\delta_{\min}},\omega|_{N^{\delta_{\min}}})$ of $D$ being a capping (resp. filling) of its boundary $(\partial P(D),\alpha)$ with $\alpha$ being the contact form, where $\alpha$ is any primitive of $\omega$ defined near $\partial P(D)$ with wrapping numbers $-z$. \[fig Liouville flow\] (0,2) node (yaxis) \[above\] [$y$]{} |- (2,0) node (xaxis) \[right\] [$x$]{}; (0,-2) |- (-2,0); (1,0) .. controls (2,1) and (-0.3,0.2) .. (-1.3,-0.1); (-1.3,-0.1) .. controls (-1.5,-0.3) and (-0.5,-2) .. (1,0); (-0.5,-1.2) node [$(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$]{}; (0.5,0.5) – (0.4,0.3); (-1,0.2) – (-0.8,-0.2); (0.5,-0.8) – (0.4,-0.5); (0.05,0.05) – (0.15,0.15); (0.05,-0.05)– (0.15,-0.15); (-0.05,-0.05) – (-0.15,-0.15); (-0.05,0.05) – (-0.15,0.15); Figure \[fig Liouville flow\] The direction of the arrows indicates the direction of the Liouville flow $X_{\Sigma^-}$ and $q_0$ is identified with the origin. This is a schematic picture and we do not claim that $(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$ is connected. We first prove the case that one of the $z_i$ is non-positive. Without loss of generality, assume $z_1 \le 0$. We use the notation in Lemma \[energy lower bound\]. In particular, we have symplectic disk fibration $p_i:N_i \to C_i$, the partial compactification $\bar{N}$ and its $\omega$-compatible almost complex structure $J_{\bar{N}}$. We also have $N^{\delta} = \cup_{i=1}^k \{ r_i \le \delta \} \subset \bar{N}$ and so on. We want to prove the statement with $N^{\delta_{\min}}$ being a small neighborhood of $D$, where $\delta_{\min}$ is so small such that $1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) > \pi \epsilon^2$. We recall that we have $\epsilon$ slightly larger than $\epsilon_1$ and $\epsilon_1$ is slightly larger than $\delta_{\max}$. We also recall that we have $B_1 \times S^2_{\epsilon} \subset \bar{N}$ and $\{q\} \times S^2_{\epsilon}$ has symplecitc area $\pi \epsilon^2$ for any $q \in B_1$. Suppose the contrary, assume $(P(D),\omega|_{P(D)}) \subset (N^{\delta_{\min}},\omega|_{N^{\delta_{\min}}})$ caps its boundary $(\partial P(D),\alpha)$. Let $X$ be the corresponding Liouville flow near $\partial P(D)$. We do a small symplectic blow-up centered at $q_{\infty}$ and this blow-up is so small that it is done in $\bar{N}-N$. We call this blown-up manifold $(\bar{N}',\omega_{\bar{N}'})$ and pick a $\omega_{\bar{N}'}$-compatible almost complex structure $J_{\bar{N}'}$ such that the blow-down map is $(J_{\bar{N}'},J_{\bar{N}})$-holomorphic, the exceptional divisor is $J_{\bar{N}'}$-holomorphic and $J_{\bar{N}'}=J_{\bar{N}}$ in $N$. Let the exceptional divisor be $E$ and the proper transform of the sphere fiber in $B_1 \times S^2_{\epsilon}$ containing $q_{\infty}$ be $A$. We have $GW_0(\bar{N}',[A],J_{\bar{N}'})=1$ by automatic transversality or argue as in the end of Step $4$ of the proof of Theorem 6.1 in [@McL14] for higher dimensions. Since blow-up decreases the area, $\omega_{\bar{N}'}([A])<\omega_{\bar{N}}(Bl_*[A])=\pi \epsilon^2$, where $Bl$ is the blow-down map. This gives us the energy upper bound. By the same argument as in Lemma \[energy lower bound\], we have that for any $\omega_{\bar{N}'}$-compatible almost complex structure $J$ such that $J=J_{\bar{N}'}$ on $\bar{N}'-N^{\delta_{\min}}$, any (nodal) $J$-holomorphic curve representing the class $[A]$ stays inside a fixed relative compact open subset of $\bar{N}'$. As a result, we have a GW triple $GW_0(\bar{N}',[A],\mathcal{J})$, where $\mathcal{J}$ is the family of compatible almost complex structure that equals $J_{\bar{N}'}$ on $\bar{N}'-N^{\delta_{\min}}$. By Proposition \[GW\], we have a nodal closed genus $0$ $J$-holomorphic curve for any $J \in \mathcal{J}$. Now, since $P(D) \subset N^{\delta_{\min}}$, we can choose a sequence $J_i$ in $\mathcal{J}$ such that it stretches the neck along $(\partial P(D),\alpha)$ and $J_i=J_{\bar{N}'}$ very close to $D$. We have a corresponding sequence of nodal closed genus $0$ $J_i$-holomorphic curve $u_i$ to $\bar{N}'$. By Proposition \[SFT\], we have a proper genus $0$ $J_{\infty}$-holomorphic maps $u_{\infty}^-:\Sigma^- \to Int(P(D))$ such that $u_{\infty}^-$ is asymptotic to Reeb orbits on $\partial P(D)$ with respect to the contact form $\alpha$. By the direction of the flow, $u_{\infty}^-$ corresponds to the top building. In general, the top building can be reducible. In our case, since $[A]\cdot [C_1]=1$ and $ [A] \cdot [C_i]=0$ for $i=2,\dots,k$, if the top building is reducible, there is some irreducible component lying inside $Int(P(D))-D$, by positivity of intersection and $D$ being $J_{\infty}$-holomorphic. Since $\omega_{\bar{N}'}$ is exact on $Int(P(D))-D$, any irreducible component lying inside $Int(P(D))-D$ most have non-compact domain and converge asymptotically to Reeb orbits on $Y$. By the direction of the Reeb flow, we get a contradiction by Stoke’s theorem. (cf. Proposition 8.1 of [@McL14] or Step 3 of proof of Theorem 6.1 in [@McL14] or Lemma 7.2 of [@AbSe10]) Therefore, we conclude that there is only one irreducible component which is exactly $u_{\infty}^-$ and the image of $u_{\infty}^-$ intersect $C_1$ transversally once. Let $q_0 \in \Sigma^-$ be the point that maps to the intersection. Let also $\mathbb{D}^2_{q_0}$ be a Darboux disk around $q_0$ and $i=u_{\infty}^-|_{\mathbb{D}^2_{q_0}}$. Now, we want to draw contradiction using the existence of $u_{\infty}^-$. Since $\omega$ is exact on $\partial P(D)$, it is exact in $N^{\delta_{\min}}-D$. Extend $\alpha$ to be a primitive of $\omega$ in $N^{\delta_{\min}}-D$ and we still denote it as $\alpha$. By assumption, $[\omega-d \alpha_c]$ is Lefschetz dual to $\sum\limits_{i=1}^k z_i[C_i]$. In particular, $i^* \alpha$ has wrapping number $-z_1$ around $q_0$ on $\mathbb{D}^2_{q_0}-q_0$. In other words, $[i^*\alpha-\frac{r^2}{2}d\theta]$ is cohomologous to $\frac{-z_1}{2\pi} d\theta$ in $H^1(\mathbb{D}^2_{q_0}-q_0,\mathbb{R})$, where $(r,\theta)$ are the polar coordinates. We have $$i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta+df$$ for some function $f$ on $\mathbb{D}^2_{q_0}-q_0$. When we choose the extension of $\alpha$ to $N^{\delta_{\min}}-D$, we can choose in a way that $i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta$ because the image of $i$ is away from $\partial P(D)$. Notice that the $(u_{\infty}^-)^*\omega$ dual of $(u_{\infty}^-)^*\alpha$ defines a Liouville vector field $X_{\Sigma^-}$ on $\Sigma^- -q_0$ away from critical points of $u_{\infty}^-$ (if any). This Liouville flow equals to the component of $X$ in $T(u_{\infty}^-(\Sigma^-))$ near $\partial P(D)$, when we write down the decomposition of $X$ into $T(u_{\infty}^-(\Sigma^-))$-component and its $\omega_{\bar{N}'}$-orthogonal complement component. Here, $T(u_{\infty}^-(\Sigma^-))$ denotes the tangent bundle of the image of $u_{\infty}^-$, which is well-defined near $\partial P(D)$. Since $u_{\infty}^-$ is $J_{\infty}$-holomorphic and it asymptotic converges to Reeb orbits of $(\partial P(D),\alpha)$, $X=-J_{\infty}R_{\alpha}$ has non-zero $T(u_{\infty}^-(\Sigma^-))$-component near $\partial P(D)$. Moreover, since $X$ points inward with respect to $P(D)$, so is $X_{\Sigma^-}$ on $\Sigma^-$ near infinity. In particular, if we take $\partial_{\eta}P(D)$ to be the flow of $\partial P(D)$ with respect to $X$ for a sufficiently small time, then $X_{\Sigma^-}$ is pointing inward along $(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$. On the other hand, $i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta$. Therefore, the Liouville vector field $X_{\Sigma^-}$ near $q_0$ equals $(\frac{r}{2}+\frac{-z_1}{2\pi r}) \partial_r$ and hence points outward with respect to $\mathbb{D}^2_{q_0}$ (This is where we use $z_1 \le 0$). As a result, we get a compact codimension $0$ submanifold $\Sigma^-_0$ with boundary in $\Sigma^--q_0$ that has Liouville flow pointing inward along the boundaries and $(u_{\infty}^-|_{\Sigma^-_0})^*\omega$ has a globally defined primitive $(u_{\infty}^-|_{\Sigma^-_0})^*\alpha$. It gives a contradiction by Stoks’s theorem and $\int_{\Sigma^-_0} (u_{\infty}^-)^*\omega \ge 0$. See Figure \[fig Liouville flow\]. For the other case, we assume $z_1$ is positive. In this case, the argument is basically the same but we need to use $u_{\infty}^+$ instead of $u_{\infty}^-$. We have $u_{\infty}^+$ intersecting the exceptional divisor $E$ transversally exactly once. By blowing down, we get a corresponding pseudo-holomorphic map $Bl \circ u_{\infty}^+ $. Let the point on $\Sigma^+$ that maps to the exceptional divisor be $q_0$ as before. Then, the $(u_{\infty}^+)^*\omega$ dual of $(u_{\infty}^+)^*\alpha$ defines a Liouville vector field on $\Sigma^+ -q_0$ away from critical points of $u_{\infty}^+$ (if any). Similar as before, we get a contradiction by Stoke’s theorem and the fact that $u_{\infty}^+$ restricted to any subdomian has non-negative energy. This completes the proof. We remark that the same proof can be generalized to higher dimensional $\omega$-orthogonal divisors. The only thing that need to be changed is to use Monotonicity lemma to get the energy lower bound in Lemma \[energy lower bound\] instead of using surjectivity. We leave it to interested readers. On the other hand, this is not obvious to the authors that how one can remove the $\omega$-orthogonal assumption in Theorem \[obstruction\]. Therefore, Theorem \[obstruction\] is not enough for our application. To deal with this issue, we have the following Theorem \[obstruction-closed case\]. \[obstruction-closed case\] Let $(D,\omega)$ be a symplectic divisor in a closed symplectic manifold $(W,\omega)$. There exists a neighborhood $N$ of $D$ such that there is no concave neighborhood $P(D)$ inside $N$ with the Liouville form $\alpha$ on $\partial P(D)$ having a non-negative wrapping number among its wrapping numbers. The proof is in the same vein of that of Theorem \[obstruction\]. However, we need $W$ to be closed to help us to run the argument this time. \[fig obstruction closed\] (-3.5,0) – (3.5,0) node (xaxis) \[right\] [$C_1$]{}; (-2,0) node \[below\] [$4\tau$]{} – (-2,2); (2,0) node \[below\] [$4\tau$]{} – (2,2); (-2,0.2) – (-2,2); (2,0.2) – (2,2); (-2,2) – (2,2); (-2,0.2) – (2,0.2); (-1.5,1.3) node \[left\] [$N_1$]{}; (-1.5,0.8) – (-1.5,1.7); (1.5,0.8) – (1.5,1.7); (-1.5,1.7) – (1.5,1.7); (-1.5,0.8) – (-0.25,0.8); (0.25,0.8) – (1.5,0.8); (-0.25,0.8) – (-0.25,0.8); (-1,1) – (-1,1.5); (1,1) – (1,1.5); (-1,1.5) – (1,1.5); (-1,1) – (1,1); (-1,1) – (-0.5,1); (0.5,1) – (1,1); (-1,1.3) node \[left\] [$N_2$]{}; (-0.5,1) – (-0.5,1.2); (0.5,1) – (0.5,1.2); (-0.5,1.2) – (0.5,1.2); (-0.25,0.8) – (-0.25,1.2); (0.25,0.8) – (0.25,1.2); (-0.5,1.2) – (-0.25,1.2); (0.25,1.2) – (0.5,1.2); (0,1.35) node [$q_{\infty}$]{}; (0,1.2) node [$\ast$]{}; (-1.5,0.8) –(-1.5,0) node \[below\] [$3\tau$]{}; (1.5,0.8) –(1.5,0) node \[below\] [$3\tau$]{}; (-1,1) –(-1,0) node \[below\] [$2\tau$]{}; (1,1) –(1,0) node \[below\] [$2\tau$]{}; (-0.5,1) –(-0.5,0) node \[below\] [$\tau$]{}; (0.5,1) –(0.5,0) node \[below\] [$\tau$]{}; (-0.25,0.8) –(-0.25,0) node \[below\] [$\frac{\tau}{2}$]{}; (0.25,0.8) –(0.25,0) node \[below\] [$\frac{\tau}{2}$]{}; (2,0.2) –(2.5,0.2) node \[right\] [$\epsilon_5$]{}; (1.5,0.8) –(2.5,0.8) node \[right\] [$\epsilon_4$]{}; (1,1) –(2.5,1) node \[right\] [$\epsilon_3$]{}; (0.5,1.2) –(2.5,1.2) node \[right\] [$\epsilon_2$]{}; (1,1.5) –(2.5,1.5) node \[right\] [$\epsilon_1$]{}; (1,1.7) –(2.5,1.7) node \[right\] [$\epsilon_0$]{}; (2,2) –(2.5,2) node \[right\] [$\epsilon$]{}; Figure \[fig obstruction closed\]: The bottom line represents $C_1$. The region enclosed by thickest solid lines represents $N_2$. The region enclosed by less thick solid lines represents $N_1$ For any $J=J_{\overline{W}}$ on $N_1$, every closed genus $0$ $J$-holomorphic curve $u$ passing through $q_{\infty}$ stays away from $N_2$ if energy of $u$ is sufficiently small. Same as before, we start with an energy estimate. Suppose $\alpha$ is a primitive of $\omega$ defined near $D$ but not defined on $D$. Let the wrapping numbers of $\alpha$ be $-z_i$. Suppose $z_1$ is non-positive. Identify a neighborhood $M_1$ of $C_1$ with a symplectic disk bundle over $C_1$ with symplectic connection rotating the fibers. Assume the fibers are symplecticomorphic to standard symplectic disk of radius $\epsilon$. Pick a point $p$ on $C_1$ such that there is a Darboux disk of radius $4\tau$ and $\tau > c\epsilon$ for some $c>0$ to be determined (we can achieve this by choosing a small $\epsilon$ in advance). Identify the neighborhood of $p$ as a product of closed disks $\mathbb{D}^2_{4\tau} \times \mathbb{D}^2_{\epsilon}$. Choose $\epsilon_i$ for $i=0,1,\dots,5$ to be determined such that $\epsilon > \epsilon_0 > \epsilon_1 > \dots > \epsilon_5 > 0$. Now, cut out the closed region $\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3}))$ from $W$ and call it $W_0$. We partially compactify $W_0$ to be $\overline{W}$ by gluing $W_0$ and $Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$ along $Int(\mathbb{D}^2_{\tau}) \times Int(\mathbb{D}^2_{\epsilon_3})$ by identifying $Int(\mathbb{D}^2_{\epsilon_3})$ with a choose of symplectic embedding to $S^2_{\epsilon_2}$, where $S^2_{\epsilon_2}$ is a symplectic sphere of symplectic area $\pi \epsilon_2^2$. We define $N_3 \subset N_2 \subset N_1 \subset N \subset \overline{W}$ to be the following subset of $\overline{W}$. Notice that $N$, $N_1$ and $N_2$ are closed (but not compact) and $N_3$ is open. $$N=(\mathbb{D}^2_{4\tau} \times \mathbb{D}^2_{\epsilon}-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3})))\cup Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$$ $$N_1=(\mathbb{D}^2_{4\tau} \times (\mathbb{D}^2_{\epsilon}-Int(\mathbb{D}^2_{\epsilon_5}))-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3})))\cup Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$$ $$\begin{aligned} N_2&=&((\mathbb{D}^2_{3\tau} \times (\mathbb{D}^2_{\epsilon_0}-Int(\mathbb{D}^2_{\epsilon_4}))-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3}))-Int(\mathbb{D}^2_{\frac{\tau}{2}}) \times Int(\mathbb{D}^2_{\epsilon_3})) \\ &&\cup (Int(\mathbb{D}^2_{\tau})-Int(\mathbb{D}^2_{\frac{\tau}{2}})) \times S^2_{\epsilon_2}\end{aligned}$$ $$N_3=Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}-Int(\mathbb{D}^2_{\tau}) \times \mathbb{D}^2_{\epsilon_3}$$ Then, $\overline{W}$ is our desired manifold to run the argument above (See Figure \[fig obstruction closed\]). Let $J_{\overline{W}}$ be an $\omega$-compatible almost complex structure on $\overline{W}-N_3$ such that $J_{\overline{W}}$ is split as a product in $N-N_3$. Then, extend $J_{\overline{W}}$ naturally over $N_3$ such that it is ’product-like’ making the $S^2_{\epsilon_2}$ sphere fibers $J_{\overline{W}}$-holomorphic. We still call this $J_{\overline{W}}$. Let $q_{\infty} \in N_3$ be a point in $N$ such that it lies in the $S^2_{\epsilon_2}$ sphere fiber at $p$. Similar as before, for any $\omega$-compatible almost complex structure $J$ such that $J=J_{\overline{W}}$ on $N_1$ and any closed genus $0$ $J$-holomorphic curve $u$ to $\overline{W}$ passing through $q_{\infty}$, we must have the image of $u$ stays inside $\overline{W}-N_2$ or the energy of $u$, $\int_{\mathbb{C}P^1}u^*\omega$, greater than a lower bound depending on $\epsilon_5$ (once $c$ and $\epsilon_i$ for $i=0,\dots,4$ are determined). It should be convincing that one can choose a choice of $c$ and $\epsilon_i$ such that any $J$-holomorphic curve representing the class $[S^2_{\epsilon_2}]$, the spherical fiber class at $p$, and passing through $q_{\infty}$ has to stay inside $\overline{W}-N_2$. Since $W$ is closed, $\overline{W}-N_2$ is a relative compact open subset which we can use to define the GW triple below. We claim that $D$ does not have a concave neighborhood $P(D)$ lying inside $W-N_1$ with the Liouville contact form $\alpha'$ defined near $\partial P(D)$ having the same wrapping numbers as that of $\alpha$. Suppose on the contrary, there were such a $P(D)$. Then, by a $C^0$ perturbation near the intersection points of $C_i$ in $D$, we can assume that $D$ is $\omega$-orthogonal and it still lies inside $P(D)$. We do a sufficiently small blow-up at $q_{\infty}$ as before and let the proper transform of the $S^2_{\epsilon_2}$ fiber containing $q_{\infty}$ be $A$. We have a GW triple $GW_0(\overline{W},[A],\mathcal{J})$, where $\mathcal{J}$ is the family of $\omega$-compatible almost complex structure $J$ such that $J=J_{\overline{W}}$ on $N_1$. Since $D$ is now $\omega$-orthogonal, we can find $J\in \mathcal{J}$ such that $D$ is $J$-holomorphic (notice that, there exists symplectic divisor with no almost complex structure making all irreducible components pseudo-holomorphic simultaneously). Then, we find a sequence $J_i \in \mathcal{J}$ making $D$ $J_i$-holomorphic for all $i$ and stretch the neck along $\partial P(D)$ as before to draw contradiction. Uniqueness ---------- In this subsection we show that any contact structure obtained from the GS construction is contactomorphic to one from the McLean’s construction. Then Theorem \[uniqueness-GS\] follows from the uniqueness of McLean’s construction. In fact, we are going to prove the following more precise version of Theorem \[uniqueness-GS\]. \[canonical contact structure\_proof\] Suppose $D=\cup_{i=1}^k C_i$ is a symplectic divisor with each intersection point being $\omega$-orthogonal such that the augmented graph $(\Gamma,a)$ satisfies the positive (resp. negative) GS criterion. Then, the contact structures induced by the positive (resp. negative) GS criterion are contactomorphic, independent of choices made in the construction and independent of $a$ as long as $(\Gamma,a)$ satisfies positive GS criterion. Moreover, if $D$ arises from resolving an isolated normal surface singularity, then the contact structure induced by the negative GS criterion is contactomorphic to the contact structure induced by the complex structure. On the other hand, if $D$ is the support of an effective ample line bundle, then the contact structure induced by the positive GS criterion is contactomorphic to that induced by a positive hermitian metric on the ample line bundle. ### Uniqueness of McLean’s construction We recall the uniqueness part of McLean’s construction, which can be regarded as a more complete version of Proposition \[McLean0\]. \[McLean\]\[cf. Corollary 4.3 and Lemma 4.12 of [@McL14]\] Suppose $f_0,f_1: W-D \to \mathbb{R}$ are compatible with $D$ and $D$ is a symplectic divisor with respect to both $\omega_0$ and $\omega_1$ having positive transversal intersections. Suppose $\theta_j \in \Omega^1(W-D)$ is a primitive of $\omega_j$ on $W-D$ such that it has positive (resp. negative) wrapping numbers for all $i=1, \dots, k$ and for both $j=0,1$. Suppose, for both $j=0,1$, there exist $g_j: W-D \to \mathbb{R}$ such that $df(X^j_{\theta_j+dg_j}) > 0$ (resp. $df(-X^j_{\theta_j+dg_j}) > 0$) near $D$, where $X^j_{\theta_j+dg_j}$ is the dual of $\theta_j+dg_j$ with respect to $\omega_j$. Then, for sufficiently negative $l$, we have that $(f_0^{-1}(l),\theta_0+dg_0|_{f_0^{-1}(l)})$ is contactomorphic to $(f_1^{-1}(l),\theta_1+dg_1|_{f_1^{-1}(l)})$. Moreover, when $(W,D,\omega)$ arises from resolving a normal isolated surface singularity, then the link with contact structure induced from complex line of tangency is contactomorphic to this canonical contact structure. The first thing to note is that the choice of $g_j$ for $j=0,1$ always exist (cf. Proposition \[McLean0\] above, Proposition 4.1 and Proposition 4.2 in [@McL14]). Moreover, by the definition of compatible function, it also always exist. In other words, Proposition \[McLean\] implies that in dimension four, for any symplectic form $\omega_0$ and $\omega_1$ making $D$ a divisor such that they have primitives $\theta_0$ and $\theta_1$ on $W-D$ with positive (resp. negative) wrapping numbers, the contact structures constructed by McLean’s construction with respect to $\theta_0$ and $\theta_1$ are contactomorphic. Proposition \[McLean\] is literally not exactly the same as Corollary 4.3 and Lemma 4.12 in [@McL14] so we want to make clear why it is still valid after we have made the changes. We remark that if $\theta_0 $ and $\theta_1$ have positive wrapping numbers, then $\theta_t=(1-t)\theta_0+t\theta_1$ has positive wrapping numbers for all $t$ and $f_t=(1-t)f_0+tf_1$ is compatible with $D$ for all $t$. As a symplectic divisor, we always assume $C_i$ have positive orientations with respect to the symplectic form for all $i$. In other words, both $\omega_0|_{C_i}$ and $\omega_1|_{C_i}$ are positive and hence $D$ is a symplectic divisor with respect to $d\theta_t$ for all $t$. Therefore, we get a deformation of $\omega_t$ and the first half of Proposition \[McLean\] with $\theta_0 $ and $\theta_1$ having positive wrapping numbers follows from Corollary 4.3 of [@McL14]. The analogous statement for the first half of Proposition \[McLean\] with $\theta_0 $ and $\theta_1$ having negative wrapping numbers follows similarly as in the case where $\theta_0 $ and $\theta_1$ have positive wrapping numbers. On the other hand, Lemma 4.12 of [@McL14] requires that the resolution is obtained from blowing up. Although there exist a resolution such that it is not obtained from blowing up in complex dimension three or higher, every resolution for an isolated normal surface singularity can be obtained by blowing up the unique minimal model, where the minimal model is obtained from blowing up the singularity. Therefore, the second half of Proposition \[McLean\] follows. ### Proof of Theorem \[uniqueness-GS\] To prove Proposition \[canonical contact structure\_proof\] using Proposition \[McLean\], the remaining task is to construct an appropriate disc fibration having a connection rotating fibers for the local models in the GS-construction. Then, the constructions of $\theta$, $f$ and $g$ will be automatic. We give the fibration in the following Lemma. \[canonical contact structure\_tech\] Let $z_1'$ and $z_2'$ be two positive numbers. Let $\mu: \mathbb{S}^2 \times \mathbb{S}^2 \to [z_1',z_1'+1] \times [z_2',z_2'+1]$ be the moment map of $\mathbb{S}^2 \times \mathbb{S}^2$ onto its image. Fix a small $\epsilon >0$ and let $D_1=\mu^{-1}(\{ z_1'\} \times [z_2', z_2' + 2\epsilon])$ be a symplectic disc. Fix a number $s \in \mathbb{R}$ first and then let $\delta>0$ be sufficiently small. Let $Q$ be the closed polygon with vertices $(z_1',z_2'), (z_1'+\delta,z_2'), (z_1'+\delta,z_2'+2\epsilon-s\delta), (z_1',z_2'+2\epsilon)$. Using the $(p_1,q_1,p_2,q_2)$ coordinates described in the GS-construction above, we define a map $\pi: \mu^{-1}(Q) \to D_1$ by sending $(p_1,q_1,p_2,q_2)$ to $(z_1',*,p_2+\frac{(2\epsilon-t(p_1,p_2)-\rho(t(p_1,p_2)))p_1}{\delta}, q_2)$, where $\rho: [0, 2\epsilon] \to [0,2\epsilon-s\delta]$ is a smooth strictly monotonic decreasing function with $\rho(0)=2\epsilon-s\delta$ and $\rho(2\epsilon)=0$ such that $\rho'(t)=-1$ for $t\in [0, \epsilon]$ and near $t=2\epsilon$. This can be done as $\delta$ is sufficiently small. Moreover, $t(p_1,p_2)$ is the unique $t$ solving $p_2-(z_2'+2\epsilon-t)=\frac{(\rho(t)-(2\epsilon-t))(p_1-z_1')}{\delta}$ and $*$ means that there is no $q_1$ coordinate above $(z_1',x)$ for any $x$ so $q_1$ coordinate is not relevant. Then, we have that $\pi$ gives a symplectic fibration with each fibre symplectomorphic to $(\mathbb{D}^2_{\sqrt{2\delta}},\omega_{std})$ and the symplectic connection of $\pi$ has structural group lies inside $U(1)$. Moroever, fibres are symplectic orthogonal to the base. \[fig disk fibration\] (0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{}; (1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.4,1.8) coordinate (d\_1)– (5.4,1.8) coordinate (d\_2); (2.4,0.9) coordinate (e\_1)– (2.4,3) coordinate (e\_2); (2.2,2.5) node \[left\] [$Q$]{}; (2.4,3) coordinate (j\_1)– (2.4,4.3) coordinate (j\_2); (1.5,3.9) coordinate (f\_1)– (2.4,4.3) coordinate (f\_2); (1.5,2.4) coordinate (g\_1)– (2.4,2.8) coordinate (g\_2); (1.5,3.6) – (2.4,4); (1.5,3.4) – (2.4,3.8); (1.5,3.2) – (2.4,3.6); (1.5,3.0) – (2.4,3.4); (1.5,2.8) – (2.4,3.2); (1.5,2.6) – (2.4,3); (1.5,1.1) – (2.4,1.1); (1.5,1.3) – (2.4,1.3); (1.5,1.5) – (2.4,1.6); (1.5,1.7) – (2.4,1.9); (1.5,1.9) – (2.4,2.2); (1.5,2.1) – (2.4,2.5); (3,0.9) coordinate (h\_1)– (3.9,1.8) coordinate (h\_2); (4.5,0.9) coordinate (i\_1)– (5.4,1.8) coordinate (i\_2); (c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{2}'$]{} -| (xaxis -| c) node\[below\] [$z_{1}'$]{}; (1.5,2.4) – (0,2.4) node\[left\] [$z_{2}'+\epsilon$]{}; (1.5,3.9) – (0,3.9) node\[left\] [$z_{2}'+2\epsilon$]{}; (3,0.9) – (3,0) node\[below\] [$z_{1}'+\epsilon$]{}; (4.5,0.9) – (4.5,0) node\[below\] [$z_{1}'+2\epsilon$]{}; Figure \[fig disk fibration\]. The arrows give the schematic picture for the projection $\pi$. First, we want to explain what $t(p_1,p_2)$ means geometrically. $\rho(2 \epsilon -t)$ is an oriented diffeomorphism from $[0, 2\epsilon] \to [0,2\epsilon-s\delta]$ so it can be viewed as a diffeomorphism from the left edge of $Q$ to the right edge of $Q$. $p_2-(z_2'+2\epsilon-t)=\frac{(\rho(t)-(2\epsilon-t))(p_1-z_1')}{\delta}$, which we call $L_t$, is the equation of line joining the point $(z_1',z_2'+2\epsilon-t)$ and $(z_1'+\delta,z_2'+\rho(t))$. Therefore, for a point $(p_1,p_2)$, $t(p_1,p_2)$ is such that $L_{t(p_1,p_2)}$ contains the point $(p_1,p_2)$. Moreover, $p_2+\frac{(2\epsilon-t(p_1,p_2)-\rho(t(p_1,p_2)))p_1}{\delta}$ is the $p_2$-coordinate of the intersection between line $L_{t(p_1,p_2)}$ and the left edge of $Q$, $\{ p_1=z_1' \}$. See Figure \[fig disk fibration\]. To prove the Lemma, we pick $\kappa$ close to $2 \epsilon$ from below such that $\rho'(t)=-1$ for all $t \in [\kappa, 2\epsilon]$. Let $\Delta$ be $\pi^{-1}(\mu^{-1}(\{ z_1' \} \times [z_2'+2\epsilon-\kappa,z_2'+2\epsilon]))$ We give a smooth trivialization of $\pi|_{\Delta}$ as follows. Let $\Phi: [0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}} \to \Delta$ be given by sending $(t,\vartheta_1,\tau,\vartheta_2)$ to $(p_1,q_1,p_2,q_2)=(z_1'+\tau,-s \vartheta_1+\vartheta_2,(z_2'+2\epsilon-t)+\frac{(\rho(t)-(2\epsilon-t))\tau}{\delta},-\vartheta_1)$, where $t,\vartheta_1$ are the coordinates of $[0,\kappa]$ and $\mathbb{R}/2\pi$, respectively, and $(\tau=\frac{r^2}{2},\vartheta_2)$ is such that $(r,\vartheta_2)$ is the standard polar coordinates of $\mathbb{D}^2_{\sqrt{2\delta}}$. In particular, $\tau \in [0,\delta]$. Note that, $\Phi$ is well-defined and it is a diffeomorphism. Let $\pi_{\Phi}:[0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}} \to [0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ be the projection to the first two factors. Then, we have $\pi \circ \Phi= \Phi \circ \pi_{\Phi}$. Notice that, when $\tau=0$, the $\vartheta_2$-coordinate degenerates and it corresponds to $p_1=z_1'$ and the $q_1$-coordinate degenerates. To investigate this fibration under the trivialization, we have $$\begin{aligned} \Phi^{*}\omega & = \Phi^{*}(dp_1 \wedge dq_1+dp_2 \wedge dq_2) \\ & = d\tau \wedge (-sd\vartheta_1+d\vartheta_2) \\ & \quad + (-dt+\frac{\tau}{\delta}dt+\frac{\rho'(t)\tau}{\delta}dt+ \frac{\rho(t)-(2\epsilon-t)}{\delta}d\tau) \wedge (-d\vartheta_1)\\ & = (1-\frac{\tau}{\delta}-\frac{\rho'(t)\tau}{\delta})dt \wedge d\vartheta_1 + d\tau \wedge d\vartheta_2 + (\frac{2\epsilon-t-\rho(t)}{\delta}-s) d\tau \wedge d\vartheta_1\end{aligned}$$ For a fibre, we have $t$ and $\vartheta_1$ being contant so the the symplectic form restricted on the fibre is $ d\tau \wedge d\vartheta_2$, which is the standard one. Hence, each fibre is symplectomorphic to $(\mathbb{D}^2_{\sqrt{2\delta}},\omega_{std})$. When $\tau=0$, the symplectic form equals $dt \wedge d\vartheta_1$ so the base is symplectic and fibres are symplectic orthogonal to the base. Moreover, the vector space that is symplectic orthogonal to the fibre at a point is spanned by $\partial_{t}$ and $\partial_{\vartheta_1}-(\frac{2\epsilon-t-\rho(t)}{\delta}-s)\partial_{\vartheta_2}$ so the symplectic connection has structural group inside $U(1)$. Finally, we remark that $\rho(0)=2\epsilon-s\delta$ and $\rho'(t)=-1$ when $t$ is close to $0$, hence $\Phi^{*}\omega=dt \wedge d\vartheta_1 + d\tau \wedge d\vartheta_2$. Therefore, when $t$ is close to $0$, the trivialization $\Phi$ actually coincide with the gluing symplectomorphism in the GS construction from preimage of $R_{e_{\alpha \beta},v_{\alpha}}$ to $[x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ , up to a translation in $t$-coordinate. On the other hand, $\pi|_{\mu^{-1}(Q)-\Delta}$ is clearly a symplectic fibration with all the desired properties described in the Lemma as it corresponds to the trivial projection by sending $(p_1,q_1,p_2,q_2)$ to $(z_1', *, p_2,q_2)$. This finishes the proof of this Lemma. We remark that the disc fibration above gives a fibration on the local models $N_{e_{\alpha \beta}}$ and it is compatible with the trivial fibration on the local models $N_{v_{\alpha}}$ so they give a well-defined fibration after gluing all the local models $N_{v_{\alpha}}$ and $N_{e_{\alpha \beta}}$. Now, we are ready to prove Proposition \[canonical contact structure\_proof\]. Let $(D=\cup_{i=1}^k C_i,W, \omega)$ be a symplectic plumbing. First, we assume the intersection form of $D$ is negative definite (or equivalently, the augmented graph satisfies the negative GS criterion). By [@GaSt09], $D$ satisfies the negative GS criterion. Therfore, by possibly shrinking $W$, we can assume $W$ is a symplectic plumbing constructed from the negative GS criterion. A byproduct of the construction is the existence of a primitive of $\omega$ on $W-D$, $\theta$, given by contracting $\omega$ by the Liouville vector field. From the construction, in the $N_{v_{\alpha}}$ local model, we have $\theta=\iota_{\bar{X}_{v_{\alpha}}+(\frac{r}{2}+\frac{z_{v_{\alpha}}'}{r})\partial_r} (\bar{\omega}_{v_{\alpha}}+rdr \wedge d\vartheta_2)=\iota_{\bar{X}_{v_{\alpha}}}\bar{\omega}_{v_{\alpha}}+(\frac{r^2}{2}+z_{v_{\alpha}}')d\vartheta_2$. When we restrict it to a fibre, we can see that the wrapping numbers of $\theta$ with respect to $C_{v_{\alpha}}$ is $2 \pi z_{v_{\alpha}}'$, which is positive. Here, $C_{v_{\alpha}}$ is the smooth symplectic submanifold corresponding to the vertex $v_{\alpha}$. Note that we have $\lambda=-z=2\pi z'$ in our convention above. By tracing back the negative GS construction, we see that Lemma \[canonical contact structure\_tech\] provides a desired symplectic fibrations we needed to apply Proposition \[McLean\]. In particular, this symplectic fibrations give us well-defined $r_i$-coordinates near the divisor. As a result, one can set $f=\sum\limits_{i=1}^k \log(\rho(r_i))$ and $g=0$ and get that $df(X_\theta) > 0$ near $D$ and $(f^{-1}(l),\theta|_{f^{-1}(l)})$ is precisely the contact manifold obtained from the negative GS criterion. In particular, $(f^{-1}(l),\theta|_{f^{-1}(l)})$ is the canonical one with respect to $(W,D)$. If we have made another set of choices in the construction, we get that $(\bar{f}^{-1}(l),\theta|_{\bar{f}^{-1}(l)})$ is the canonical one with respect to $(\bar{W},\bar{D})$. Then, since $(W,D)$ is diffeomorphic to $(\bar{W},\bar{D})$, we can pull back the compatible function and the $1$-form on $(\bar{W},\bar{D})$ to $(W,D)$. By Proposition \[McLean\], the two contact manifolds are contactomorphic. Same argument works to show that this contact structure is independent of symplectic area $a$ as long as $(\Gamma,a)$ still satisfies GS criteria. Also, when $D$ is arising from isolated normal surface singularity, contact structure of its link is contactomorphic that induced by GS-criterion, by Proposition \[McLean\], again. This finishes the case when $D$ is negative definite. Now, we assume that $(D,\omega)$ satisfies the positive GS criterion and $D$ is $\omega$-orthogonal. By the same reasoning as bove, the contact structure induced by the positive GS criterion is independent of choices. Suppose $D$ is also the support of an effective ample line bundle. Pick a hermitian metric $\| . \|$ and a section $s$ with zero being $\sum\limits_{i=1}^k z_i C_i$, where $z_i > 0$. Let $f=-\log \| s\|$, $\theta= -d^cf$ and $\omega= d\theta$, where $d^cf=df \circ J_{std}$. Then, $\theta$ induces a contact structure on the boundary of plumbing of $(D,\omega)$ with negative wrapping numbers (See Lemma 5.19 of [@McL12]). Moreover, $f$ is compatible with $D$ and $df(-X_{\theta}) > 0$ near $D$ (See Lemma 2.1 of [@McL12] or Lemma 4.12 of [@McL14]). Hence the contact structure induced by $\theta$ is contactomorphic to the canonical one by Proposition \[McLean\], which is contactomorphic to the one induced by the positive GS criterion. Examples of Concave Divisors {#Comments on the Induced Contact Structure} ---------------------------- In this subsection, we are going to see five illuminating examples. The first one is the simplest kind of symplectic divisor. The second one illustrates Theorem \[obstruction-GS\] is no longer valid if the plumbing chosen is not close to the divisor. In particular, there is a concave divisor which admits a convex neighborhood but it is not a convex divisor. The third one is a frequently used example when studying Lefschetz fibration. The forth one is a concave divisor with non-fillable contact structure on the boundary. The last one shows that the constructed contact structure on the boundary is not necessarily contactomorphic to the standard one that one might expect if the divisor is concave. \[single vertex\] A symplectic surface with self-intersection $n$ admits a concave (resp convex) boundary when $n>0$ (resp $n<0$). When $n=0$, a symplectic form cannot make both the surface symplectic and the restriction to boundary be exact so it has no convex or concave neighborhood. In fact, more is true, by a result of Eliasberg [@El90], $\mathbb{S}^1 \times \mathbb{S}^2$ cannot be a convex boundary of any symplecyic form on $\mathbb{D}^2 \times \mathbb{S}^2$. In contrast, althought a symplectic torus with self-intersection zero has no concave nor convex neighborhood, a Lagrangian torus has self-intersection zero and has a convex neigborhood. ([@Mc91]) In [@Mc91], McDuff constructed a symplectic form on $(S\Sigma_g \times [0,1],\omega)$ such that it has disconnected convex boundary, where $S\Sigma_g$ is a circle boundle of a genus $g$ surface and $g>1$. The contact structure near $S\Sigma_g \times \{0\}$ is contactomorphic to the concave boudary near a self-intersection $2g-2$ symplectic genus g surface. The contact structure near $S\Sigma_g \times \{1\}$ is contactomorphic to the convex boundary near a Lagrangian genus g surface. If one glues a symplectic closed disc bundle $P(D)$ over a symplectic genus g surface $D$ with $(S\Sigma_g \times [0,1],\omega)$ along $S\Sigma_g \times \{0\}$. One gets a plumbing of the surface with convex boundary. This suggests that a symplectic genus $g$ ($g > 1$) surface can have both concave and convex neighborhood, depending on the symplectic form and the neighborhood. Notice that $D$ is trivially $\omega$-orthogonal. It illustrates that the assumption on $P(D)$ being sufficiently close to $D$ in Theorem \[obstruction-GS\] cannot be dropped. Moreover, by Theorem \[obstruction-GS\], $D$ is a concave divisor but not a convex divisor although it admits convex neighborhood. \[section fiber\] Suppose there is a symplectic Lefschetz fibration $(X,\omega)$ over $\mathbb{CP}^1$ with generic fibre $F$ and a symplectic section $S$ of self-intersection $-n$ ($n \ge 0$). Let $D=F \cup S$, then the augmented graph of $D$ always satisfies the positive GS criterion regardless the area weights of the surfaces. Suppose also that $S$ is perturbed to be $\omega$-orthogonal to $F$. Then, Proposition \[MAIN2\] (or, Proposition \[McLean0\] if one does not want to perturb) shows that $D$ is a concave divisor. In other words, the complement of a concave neighborhood of $D$ is a convex filling of its boundary. This fits well to the well-known fact that the complement of a regular neighborhood of $D$ is a Stein domain. Moreover, this construction has been successfully used to find exotic Stein fillings [@AkEtMaSm08]. \[b\_2\^+\] Let $(\Gamma,a)$ be an augmented graph satisfying the positive GS criterion and $D$ be a realization. Suppose there are two genera zero vertices with self-intersection $s_1,s_2$ such that either \(i) they are adjacent to each other and $s_1 >s_2\ge 1$, or \(ii) they are not adjacent to each other with $s_1 \ge 1$ and $s_2 \ge 0$. Then, $D$ is a concave divisor but not a capping divisor. Suppose on the contrary, the boundary has a convex fillings $Y$. Then, we can glue $D$ with $Y$ to obtain a closed symplectic 4 manifold $W$. By McDuff’s theorem [@Mc90] (see also Theorem \[McDuff\] below), $W$ is rational or ruled and hence have $b_2^+ =1$. For (i), the two spheres generates a positive two dimensional subspace of $H_2(W)$ with respect to the intersection form. Thus, we get a contradiction. For (ii), it suffices to consider tha case $s_1=1$ and $s_2=0$. By the Theorem in [@Mc90], one can assume the sphere with self-intersection $1$ represent the hyperplane class $H$, with respect to an orthonormal basis $\{H,E_1,\dots,E_n \}$ for $H_2(W)$. The two spheres being disjoint implies the one with self-intersection $0$ has homology class being a linear combination of exceptional classes. Since the sphere is symplectic, the linear combination is non-trivial. Thus, we get a contradiction. Let $\Gamma$ be the graph in Example \[2-1\]. $$Q_{\Gamma}= \left( \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right),$$ Then the boundary fundamental group of $\Gamma$ is the free group generated by $e_1$ and $e_2$ modulo the relations $e_1e_2^1=e_2^1e_1$, $1=e_1^2e_2$ and $1=e_1e_2$ (See Lemma \[representation\] below). Therefore, the boundary of the plumbing according to $\Gamma$ has trivial fundamental group and hence diffeomorphic to a sphere. It is easily see that the corresponding augmented graph $(\Gamma,a)$ satisfies the positive GS criterion if and only if the area weights satisfy $a_1 < a_2 < 2a_1$, where $a_i$ is the area weight of $v_i$. In other words, if $a_1 < a_2 < 2a_1$, by Proposition \[MAIN2\] and Lemma \[b\_2\^+\], we get an overtwisted contact structure on $S^3$ ($S^3$ has only one tight contact structure which is fillable). \[non-standard example\] There is a capping divisor $D$ with graph as in the following Figure, by Theorem \[main classification theorem\]. However, $D$ is not conjugate to any other divisor (See Definition \[Conjugate Definition\_Divisor\]), by Lemma \[first chern class\]. $\partial P(D)$ is diffeomorphic to the boundary of the plumbing of the resolution of a tetrahedral singularity and the latter one which is equipped with a standard contact structure has a conjugate. Therefore, the fillable contact strucutre on $\partial P(D)$ is not the standard one. Applying the method in [@Li08],[@BhOn12] and [@St13], one can obtain a finiteness result on the number of minimal symplectic manifolds that can be compactified by $D$, up to diffeomorphism (See Proposition \[bounds\]). $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{1} \\ &\bullet^{-2} \\ }$$ Operations on Divisors {#Operation on Divisors} ====================== In this section we first apply the inflation operation to establish Theorem \[MAIN\]. Then, we explain in details the resulting flowchart and illustrate how to reduce the classification problem into a problem of graphs. Finally, we introduce blow up and the [**dual**]{} blow up operations, which are essential to the next section. Theorem \[MAIN\] ---------------- The proof of Theorem \[MAIN\] involes two inputs. The first one is a linear algebraic lemma, which is simple but important. The second one which is called inflation lemma allows us to deform the symplectic form to our desired one so as to apply Propositon \[MAIN2\]. ### A key lemma The following linear algebraic Lemma related to the positive GS criterion will be crucial. \[trichotomy\] Let $Q$ be a k by k symmetric matrix with off-diagonal entries being all non-negative. Assume that there exist $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. Suppose also that $Q$ is not negative definite. Then, there exists $z \in (0,\infty)^k$ such that $Qz \in (0,\infty)^k$. When $k=1$, it is trivial. Suppose the statement is true for (k-1) by (k-1) matrix and now we consider a k by k matrix $Q$. Let $q_{i,j}$ be the $(i,j)^{th}$-entry of $Q$. First observe that if $q_{i,i} \ge 0$, for all $i=1,\dots,k$, then the statement is true. The reason is that if each row has a positive entry, then $z=(1,\dots,1)$ works. If there exist a row with all $0$, then there is no $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. Therefore, we might assume $q_{k,k}<0$. Let $l_j=-\frac{q_{k,j}}{q_{k,k}} \ge 0$, for $j <k$, and let $B$ be the lower triangular matrix given by $$b_{i,j} = \left \{ \begin{array}{l l} \delta_{i,j} & \quad \text{if $i \neq k$ or $(i,j)=(k,k)$}\\ l_j & \quad \text{if $i = k$ and $(i,j) \neq (k,k)$} \end{array} \right.$$ Let $M=B^TQB$. Then, $$m_{i,j} = \left \{ \begin{array}{l l} q_{i,j}-\frac{q_{i,k}q_{k,j}}{q_{k,k}} & \quad \text{if $(i,j) \neq (k,k)$}\\ q_{k,k} & \quad \text{if $(i,j) = (k,k)$} \end{array} \right.$$ In particular, $m_{i,k}=m_{k,j}=0$, for all $i$ and $j$ less than $k$. We can write $M$ as a direct sum of a k-1 by k-1 matrix $M'$ with the 1 by 1 matrix $q_{k,k}$ in the obvious way. Notice that the off diagonal entries of $M'$ are all non-negative. Let $a=(a_1, \dots,a_k)^T$ and $z=(z_1,\dots,z_k)^T$ such that $Qz=a$. Let also $\overline{z}=(\overline{z}_1,\dots,\overline{z}_k)^T=B^{-1}z$ and $\overline{a}=(\overline{a}_1, \dots,\overline{a}_k)^T=B^Ta$. Then, $Qz=a$ is equivalent to $M\overline{z}=\overline{a}$. Here, $\overline{z}_i=z_i$, for $i <k$, and $\overline{z}_k=z_k-\sum\limits_{i=1}^{k-1}l_iz_i$. On the other hand, $\overline{a}_i=a_i+l_ia_k$, for all $i < k$, and $\overline{a}_k=a_k$. By assumption, there exist $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. So we have $(\overline{a}_1,\dots, \overline{a}_{k-1})^T \in (0,\infty)^k$ and $M' (z_1,\dots,z_{k-1})^T=(\overline{a}_1,\dots, \overline{a}_{k-1})^T$. Apply induction hypothesis, we can find $y \in (0,\infty)^{k-1}$ such that $M'y \in (0,\infty)^{k-1}$. Pick $y_k >0$ such that $q_{k,k}(y_k-\sum\limits_{i=1}^{k-1}l_iy_i) >0$ but sufficient close to zero. Then, let $\overline{z}=(y_1,\dots,y_{k-1},y_k-\sum\limits_{i=1}^{k-1}l_iy_i)^T$ and tracing it back. We have $Q(y_1,\dots,y_k)^T \in (0,\infty)^k$. Regarding the negative GS criterion, we remark that one can show the following. (It is mentioned in [@GaMa11] with additional assumption but the additional assumption can be removed.) Suppose $Q$ is a symmetric matrix with off-diagonal entries being non-negative. Then, the following statements are equivalent. \(a) For any $a \in (0,\infty)^n$, there exist $z \in (-\infty,0)^n$ satisfying $Qz=a$. (a2) For any $a \in (0,\infty)^n$, there exist $z \in (-\infty,0]^n$ satisfying $Qz=a$. \(b) There exist $a \in (0,\infty)^n$ such that there exist $z \in (-\infty,0)^n$ satisfying $Qz=a$. (b2) There exist $a \in (0,\infty)^n$ such that there exist $z \in (-\infty,0]^n$ satisfying $Qz=a$. \(c) $Q$ is negative definite. The implication from (a) to (b), (a2) to (b2), (a) to (a2), (b) to (b2) are trivial. (c) implying (a2) is Lemma 3.3 of [@GaSt09] and a moment thought will justify (c)+(a2) implying (a), which is hiddenly used in [@GaSt09]. (b) implying (c) is similar to the proof of Lemma \[trichotomy\]. To be more precise, one again use induction on the size of $Q$ and change the basis using $B$. Therefore, an augmented graph $(\Gamma,a)$ satisfies the negative GS criterion if and only if $Q_{\Gamma}$ is negative definite. In particular, when a graph $\Gamma$ is negative definite, the negative GS criterion is always satisfied, independent of the area weights. ### Inflation Now, it comes the second input. \[inflation\](Inflation, See [@LaMc96] and [@LiUs06]) Let $C$ be a smooth symplectic surface inside $(W,\omega)$. If $[C]^2 \ge 0$, then there exists a family of symplectic form $\omega_t$ on $W$ such that $[\omega_t]=[\omega]+tPD(C)$ for all $t \ge 0$. If $[C]^2 < 0$, then there exists a family of symplectic form $\omega_t$ on $W$ such that $[\omega_t]=[\omega]+tPD(C)$ for all $0 \le t < -\frac{\omega[C]}{[C]^2}$. Also, $C$ is symplectic with respect to $\omega_t$ for all $t$ in the range above. Moreover, if there is another smooth symplectic surface $C'$ intersect $C$ positively and $\omega$-orthogonally, then $C'$ is also symplectic with respect to $\omega_t$ for all $t$ in the range above. Here, $PD(C)$ denotes the Poincare dual of $C$. When $[C]^2 < 0$, one can see that $([\omega]+tPD(C))[C] > 0$ if and only if $t < -\frac{\omega[C]}{[C]^2}$. Therefore, the upper bound of $t$ in this case comes directly from $\omega_t[C] > 0$. We remarked that one can actually do inflation for a larger $t$ but one cannot hope for $C$ being symplectic anymore when $t$ goes beyond $-\frac{\omega[C]}{[C]^2}$. ### Proof \[Proof of Theorem \[MAIN\]\] First of all, we can isotope $D$ to $D'$ such that every intersection of $D'$ is $\omega_0$-orthogonal, using Theorem 2.3 of [@Go95]. Since every intersection of $D$ is transversal and no three of $C_i$ intersect at a common point, such an isotopy can be extended to an ambient isotopy. Now, instead of isotoping $D$, we can deform $\omega_0$ through the pull back of $\omega_0$ along the isotopy. As a result, we can assume $D$ is $\omega_0$-orthogonal. Now, we want to construct a family of realizations $D_t$ of $\Gamma$, by deforming the symplectic form, such that the augmented graph of $D_1$ satisfies the positive GS criterion. Let $D=D_0=C_1 \cup \dots \cup C_k$ and let also the area weights of $D_0$ with respect to $\omega_0$ be $a$. Since $\omega$ is exact on $\partial P(D)$, there exists $z$ such that $Q_{\Gamma}z=a$. Also, by assumption and Lemma \[trichotomy\], there exists $\overline{z} \in (0, \infty)^k$ such that $Q_{\Gamma}\overline{z}=\overline{a} \in (0, \infty)^k$. Let $z^t=z+t(\overline{z}-z)$ and $a^t=a+t(\overline{a}-a)=Q_{\Gamma}z^t \in (0, \infty)^k$. We want to construct a realization $D_1$ of $\Gamma$ with area weights $a^1$. If this can be done, then the augmented graph of $D_1$ will satisfy the positive GS criterion. Observe that, it suffices to find a family of symplectic forms $\omega_t$ such that $[\omega_t]=[\omega_0]+t \sum\limits_i (\overline{z_i}-z_i)PD([C_i])$ and a corresponding family of $\omega_t$-symplectic divisor $D_t=C_1 \cup \dots \cup C_k$. The reason is that $C_i$ has symplectic area equal the $i^{th}$ entry of $a^t$ under the symplectic form $[\omega_t]=[\omega_0]+t \sum\limits_i (\overline{z_i}-z_i)PD([C_i])$. However, we need to modify this natural choice of family a little bit. Without loss of generality, we can assume $\overline{z_i}>z_i$ for all $1 \le i \le k$. We can choose a piecewise linear path $p^t$ arbitrarily close to $z^t$ such that each piece is parallel to a coordinate axis and moving in the positive axis direction. Since satisfying the positive GS criterion is an open condition, we can choose $p^t$ such that $Q_{\Gamma}p^t \in (0, \infty)^k$. The fact that $p^t$ is chosen such that $Q_{\Gamma}p^t$ is entrywise greater then zero allows us to do inflation along $p^t$ to get out desired family of $\omega_t$ and $D_t$, by Lemma \[inflation\]. Therefore, we arrive at a symplectic form $\omega_1$ such that the augmented graph of $(D,\omega_1)$, denoted by $(\Gamma,a)$, satisfies the positive GS criterion. We finish the proof by applying Propositon \[MAIN2\]. \[symplecitc cone\] The proof of Theorem \[MAIN\] implies that for any $a \in (0,\infty)^k \cap Q_D (0,\infty)^k$, there is a symplectic deformation making the augmented graph of $(D,\omega_1)$ to be $(\Gamma,a)$. First suppose $D$ is not negative definite. By Theorem \[MAIN\], $\omega$ being exact on the boundary implies $D$ is a concave divisor after a symplectic deformation. If $D$ is negative definite, then $\omega$ is necessarily exact on the boundary with unique lift of $[\omega]$ to a relative second cohomology class. Moroever, the discussion after the proof of Lemma \[trichotomy\] implies that $D$ satisifes negative GS criterion and hence $D$ is a convex divisor. Flowchart and Reducing Classification Problem to Graph ------------------------------------------------------ We offer a detailed explanation of the flowchart. Given a divisor $(D,\omega)$ (not necessarily $\omega$-orthogonal, see Proposition \[McLean0\]), the first obstruction of whether $D$ admits a concave or convex neighborhood comes from $\omega$ being not exact on the boundary of $D$. In this case, $[\omega]$ cannot be lifted to a relative second cohomology class and $Q_{D}z=a$ has no solution for $z$. If $\omega$ is exact on the boundary, we look at the solutions $z$ for the equation $Q_{D}z=a$. When $Q_D$ is negative definite (in this case $\omega$ is necessarily exact on the boundary), there is a unique solution for $z$ and all the entries for this solution is negative. Therefore, $(D,\omega)$ satisfies the negative GS criterion and $D$ is convex (Proposition \[MAIN2\] or Proposition \[McLean0\]). If $\omega$ is exact on the boundary but $Q_D$ is not negative definite, the situation becomes a bit more complicated. There might be more than one solution for $z$ (when $Q_D$ is degenerate). If we are lucky that there is one solution $z$ with all entries being positive, then $D$ is concave (Proposition \[MAIN2\] or Proposition \[McLean0\]). However, it is possible that all the solutions $z$ have at least one entry being non-positive. In this case, if $D$ is $\omega$-orthogonal or $D$ lies inside a closed symplectic manifold, there is a small neigborhood $N$ of $D$ such that $D$ has no convex nor conave neighborhood inside $N$ (Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\]). However, we can choose an area vector $\bar{a}$ such that there is a solution $\bar{z}$ for $Q_D\bar{z}=\bar{a}$ with all entries of $\bar{z}$ being positive (Lemma \[trichotomy\]). Geometrically, we can do inflation (Lemma \[inflation\]) to deform the symplectic form such that $(D,\bar{\omega})$ has area vector $\bar{a}$. Then, $(D,\bar{\omega})$ is concave (Proposition \[MAIN2\] or Proposition \[McLean0\]). This is exactly the proof of Theorem \[MAIN\]. (exact) \[startstop\] [$\omega|_{\partial P(D)}$ exact?]{}; (not exact) \[startstop2, below of=exact\] [No concave nor convex neighborhood]{}; (definite) \[startstop, right of=exact\] [$Q_D$ negative definite?]{}; (convex) \[startstop2, below of=definite\] [Admits a convex neighborhood]{}; (GS criterion) \[startstop, right of=definite\] [$(D,\omega)$ satisfies positive GS criterion?]{}; (concave) \[startstop2, right of=GS criterion\] [Admits a concave neighborhood]{}; (deformation) \[startstop3, below of=GS criterion\] [No small concave neighborhood, but admits one after a deformation]{}; (exact) – node\[right\][no]{}(not exact); (exact) – node\[above\][yes]{}(definite); (definite) – node\[right\][yes]{}(convex); (definite) – node\[above\][no]{}(GS criterion); (GS criterion) – node\[above\][yes]{}(concave); (GS criterion) – node\[right\][no]{}(deformation); In Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\], we investigate capping (i.e. embeddable and concave) divisors with boudary fundamental group. Before doing this, we want to see how we use Theorem \[MAIN\], Proposition \[McLean0\], Theorem \[obstruction-closed case\] and Proposition \[McLean\] to reduce the problem to the realizability of its graph. \[reduce to graph\] Suppose $\Gamma$ is realizable. If the graph of $(D,\omega)$ is $\Gamma$, then $(D,\omega)$ is a capping divisor if and only if $(D,\omega)$ satisfies the positive GS criterion. Let $(\overline{D},\overline{\omega}) \subset \overline{W}$ be a realization of $\Gamma$ (See Definition \[realizable definition\]). In other words, $\overline{W}$ is a closed symplectic manifold and $\Gamma$ is the graph of $\overline{D}$. We first assume $(D,\omega)$ satisfies the positive GS criterion. By Theorem \[MAIN\] (See Remark \[symplecitc cone\]), we can find an $\overline{\omega}'$-orthogonal capping divisor $(\overline{D},\overline{\omega}')$ by doing symplectic deformation in $\overline{W}$ as long as its augmented graph $(\Gamma,\overline{a})$ satisfies the positive GS criterion (this is equivalent to $\overline{a} \in (0,\infty)^k \cap Q_{\Gamma} (0,\infty)^k$). Therefore, we can choose $\overline{a}$ such that $\overline{a}=a$, where $a$ is the area vector of $(D,\omega)$. Hence the augmented graphs of $(D,\omega)$ and $(\overline{D},\overline{\omega}')$ are the same and satisfy the positive GS criterion. By Proposition \[McLean0\], $(D,\omega)$ is a concave divisor. Moreover, Proposition \[McLean\] implies that the contact structures constructed on $P(D)$ and $P(\overline{D})$ are contactomorphic. Therefore, we can cut $Int(P(\overline{D}))$ from $\overline{W}$ and glue it with $P(D)$ along the boundary to get a closed symplectic manifold $W$. As a result, $(D,\omega)$ is a capping divisor. For the other direction, $(D,\omega)$ is a capping divisor. In particular, $(D,\omega)$ can be embedded into a closed symplectic manifold. By Theorem \[obstruction-closed case\], there is a neighborhood $N$ of $D$ such that there is no concave neighborhood $P(D)$ of $D$ lying inside $N$ if $(D,\omega)$ does not satisfies positive GS criterion. Therefore, $(D,\omega)$ is not a concave divisor. Contradiction. Having this, we are going to focus our study on graphs and Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\] is solely the classification of realizable graphs with finite boundary fundamental group. Blow Up and Dual Blow Up ------------------------ The symplectic blow up and blow down operations have obvious analogues in the category of graphs and augmented graphs. We will describe these operations for augmented graphs. Both of these operations will play an important role in the classification of capping divisors with finite boundary $\pi$ in the next section. Let $(\Gamma,a)$ be an augmented graph. In the following figure, $(\tilde{\Gamma},\tilde{a})$ is obtained by blowing up $(\Gamma,a)$ at $v_1$ and $(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}})$ is obtained by blowing up $(\tilde{\Gamma},\tilde{a})$ at an edge between $v_1$ and $v_0$. If the weight for the first blow up is $a_0$, then the area of $v_1$ and $v_0$ in $(\tilde{\Gamma},\tilde{a})$ is $a_1-a_0$ and $a_0$, respectively. If the weight for the second blow up is $a_{-1}$, then the area of $v_1$, $v_{-1}$ and $v_0$ in $(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}})$ is $a_1-a_0-a_{-1}$, $a_{-1}$ and $a_0-a_{-1}$, respectively. \[blow up figure\] $$\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] & \dots &\dots \ar@{-}[r] & \bullet_{v_1}^{-1-y} \ar@{-}[r] \ar@{-}[d] & \dots & \dots \ar@{-}[r] & \bullet_{v_1}^{-2-y} \ar@{-}[r] \ar@{-}[d] & \dots \\ & & & & \bullet_{v_0}^{-1} & & & \bullet_{v_{-1}}^{-1} \ar@{-}[d] & \\ (a)~(\Gamma, a) & & & (b)~(\tilde{\Gamma},\tilde{a}) & & & (c)~(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}}) & \bullet_{v_0}^{-2} & }$$ If one graph $\Gamma$ can be obtained from another graph $\Gamma'$ through blow ups and blow downs, then we call $\Gamma$ and $\Gamma'$ [*equivalent*]{}. A graph is called minimal if no blow down can be performed. \[0-0&gt;1\] $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $ is equivalent to $\xymatrix{ \bullet^{1}} $: $\xymatrix@R=1pc @C=2.5pc{ \bullet^{0} \ar@{-}[r] & \bullet^{0} & \ar[r] & & \bullet^{-1} \ar@{-}[r] & \bullet^{-1} \ar@{-}[r] & \bullet^{-1}\\ \ar[r] & & \bullet^{-1} \ar@{-}[r] & \bullet^{0} & \ar[r] & & \bullet^{1} \\} $ We remark that both realizablility and strongly realizability (See Definition \[realizable definition\]) are stable under blow ups and blow downs for graphs. However, there is no obvious reason for (strong) realizability to be stable under blow ups for augmented graphs (See Lemma \[stability of criterion\_blow up\] below). \[stability of criterion\_blow up\] Suppose $(\Gamma,a)$ is an augmented graph and $(\tilde{\Gamma},\tilde{a})$ be obtained from a single blow up of $(\Gamma,a)$ with weight $a_0$. If $(\Gamma,a)$ satisfies the negative GS criterion, then so does $(\tilde{\Gamma},\tilde{a})$. If $(\Gamma,a)$ satisfies the positive GS criterion, then so does $(\tilde{\Gamma},\tilde{a})$ for $a_0$ being sufficiently small. Suppose that $Q_{\Gamma}z=a$ for a vector $z=(z_1, \dots, z_k)^{T}$. We need to consider two cases. First, if $\tilde{\Gamma}$ is obtained from blowing up at the vertex of $v_1$ of $\Gamma$, then $\tilde{z}=(\tilde{z_0},\tilde{z_1},\dots,\tilde{z_k})^T=(z_1-a_0,z_1,z_2, \dots, z_k)^{T}$ satisfies $Q_{\tilde{\Gamma}}\tilde{z}=\tilde{a}$. Secondly, if $(\tilde{\Gamma},\tilde{a})$ is obtained from blowing up an edge between $v_1$ and $v_2$, then $\tilde{z}=(\tilde{z_0},\tilde{z_1},\dots,\tilde{z_k})^T=(z_1+z_2-a_0,z_1,z_2, \dots, z_k)^{T}$ satisfies $Q_{\tilde{\Gamma}}\tilde{z}=\tilde{a}$. In any of the above two cases, the Lemma follows. This illustrates the difference between convex and concave boundary. After blowing up a concave neighborhood 5-tuple that is obtained from the positive GS criterion, we might no longer be able to apply the GS criterion to get a concave neighborhood 5-tuple if the weight of blow up is too large. However, if we blow up a convex neighborhood 5-tuple that is obtained from the negative GS criterion, we can still get a convex neighborhood 5-tuple by the criterion again. For a graph $\Gamma$ and a vertex $v_1$ of $\Gamma$, we use $\Gamma^{(v_1)}$ to denote the graph that is obtained by adding two genera zero and self-intersection number zero vertices to the vertex $v_1$ of $\Gamma$ as illustrated in the following Figure. It is clear that $\Gamma^{(v_1)}$ is equivalent to attaching a single vertex of genus $0$ and self-intersection $1$ to $v_1$ and adding the self-intersection of $v_1$ by $1$ as in the following Figure (See Example \[0-0&gt;1\]). We denote it by $\overline{\Gamma^{(v_1)}}$ and call it the [**dual**]{} blow up of $\Gamma$ at $v_1$. \[dual blow up example\] $$\xymatrix @R=1pc @C=1pc { \dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] & \dots &\dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] \ar@{-}[d] & \dots & \dots \ar@{-}[r] & \bullet_{v_1}^{1-y} \ar@{-}[r] \ar@{-}[d] & \dots \\ & & & & \bullet_{v_0}^{0} \ar@{-}[d] & & & \bullet_{v_0}^{1} \\ (a)~\Gamma & & & (b)~\Gamma^{(v_1)} & \bullet_{v_{-1}}^{0} & & (c)~\overline{\Gamma^{(v_1)}} & }$$ By comparing $\overline{\Gamma^{(v_1)}}$ and the blown-up graph of $\Gamma$ at $v_1$ (See Figure \[blow up figure\](b)), we can regard the dual blow up as a dual operation of blow up. We remark that in [@Ne81] the dual blow up operation is also called blow up, and it is mentioned in [@Ne81] that the blow up and dual blow up operations do not change the oriented diffeomorphism type of the boundary of the plumbing. Capping Divisors with Finite Boundary $\pi_1$ {#Classification of Symplectic Divisors Having Finite Boundary Fundamental Group} ============================================= In this section, we classify capping divisors with finite boundary fundamental group. For completion and illustration, the classification of filling divisors with finite boundary fundamental group is given in subsection \[Filling classification\]. Different from the proof of filling divisors, the study for capping divisors requires essential symplectic input. Then, we illustrate the (strong) realizability of the graphs in type (P) and thus finish the proof of Theorem \[main classification theorem\]. Moreover, we sketch the proof of finiteness of fillings result (Proposition \[bounds\]) and study a conjugate phenomenon in subsection \[Fillings\]. Statement of Classification --------------------------- We use $<n,\lambda>$ to denote the following linear graph, where $\lambda$ and $n$ are both positive integers and $\lambda < n$, $$\xymatrix{ \bullet^{-d_1} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ }$$ where each vertex has genus zero and $d_i \ge 2$ are the minus of the self-intersection numbers such that $$\frac{n}{\lambda}=d_1-\frac{1}{d_2-\frac{1}{\dots-\frac{1}{d_k}}}.$$ In what follows, we use $[d_1,\dots,d_k]$ to denotes the continuous fraction so the condition above is just $\frac{n}{\lambda}=[d_1,\dots,d_k]$. Moreover, we use $<y;n_1,\lambda_1;n_2,\lambda_2;n_3,\lambda_3>$ to denote the following graph with exactly one branching point. $$\xymatrix{ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1} \ar@{-}[r]& \bullet^{-y} \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1} \ar@{-}[d] \\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ where all vertices have genera zero and we require the self intersection numbers satisfies $\frac{n_1}{\lambda_1}=[d_1,\dots,d_k] $, $\frac{n_2}{\lambda_2}=[b_1,\dots,b_l] $ and $\frac{n_3}{\lambda_3}=[c_1,\dots,c_m]$. We call the vertex with self-intersection $-y$ to be the central vertex. \[Five Types\] We define eight special types of graphs as follows. Type(N1): empty graph, Type(N2): linear graph $<n,\lambda>$, for $0 < \lambda < n$, $(n,\lambda)=1$, Type(N3): one branching point graph $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$, where $(n_2, n_3)$ is one of the pairs $(3, 3)$, $(3, 4)$, $(3, 5)$, or $(2, n)$, for some $n \ge 2$ and $0 < \lambda_i < n_i$, $(n_i,\lambda_i)=1$, and $y \ge 2$, Type(P1): linear graph $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $, Type(P2): (linear) dual blown up graph $\overline{\Gamma^{(v)}}$ where $\Gamma=<n,n-\lambda>$ is of type (N2) and $v$ is the left-end vertex, Type(P3): one branching point graph $<3-y;2,1;n_2,n_2-\lambda_2;n_3,n_3-\lambda_3>$, where $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is of type (N3), Type(P4): (one branch point) dual blown up graph $\overline{\Gamma^{(v)}}$ where $\Gamma$ is of type (N2) and $v$ is not an end vertex, Type(P5): (one or two branch points) dual blown up graphs $\overline{\Gamma^{(v)}}$ where $\Gamma$ is of type (N3) and $v$ is any vertex in $T$. Type (N) graphs have $b_2^+=0$ and type (P) graphs have $b_2^+=1$. These graphs are going to be our focus for the remaining of the paper. We remark that the set of type (N2) graphs is the same as the set of linear graphs with all self-intersection less than $-1$. Therefore, the set of type (P2) graphs is the same as the set of dual blow up at the right-end vertex of (N2) graphs, by symmetry. \[main classification theorem\] Let $\Gamma$ be a graph with finite boundary fundamental group. If $Q_{\Gamma}$ is not negative definite, then $\Gamma$ is realizable if and only if $\Gamma$ satisfies one of the following conditions. \(A) $\Gamma$ is equivalent to a graph in types (P1), (P2), (P3), (P4), or \(B) $\Gamma$ is equivalent to a graph $T^{(v)}$ in type (P5) such that (B)(i) $y \neq 2$, or (B)(ii) $y=2$ and $v$ is a vertex labeled by a subscript $Y$ in a graph from Figure \[Tetrahedral\] to Figure \[Dihedral\], where $-y$ is the self-intersection of the central vertex of $T$ for (B)(i) and (B)(ii). In particular, if $\Gamma$ is realizable, then we have $b_2^+(Q_{\Gamma})=1$ , $\delta_{\Gamma} \neq 0 $ and $\Gamma$ is strongly realizable. \[main classification theorem\_convex\] Suppose $\Gamma$ is a graph with finite boundary fundamental group. If $Q_{\Gamma}$ is negative definite, which means that $b_2^+(Q_{\Gamma})=0$ here, then $\Gamma$ is equivalent to a graph in type (N). Moreover, any type (N) graph can be realized as a resolution graph of an isolated quotient singularity. Notice that a graph with finite boundary fundamental group must have non-degenerate intersection form (See Lemma \[order\]) except the empty graph. Therefore, $b_2^+(Q_{\Gamma})=0$ is equivalent to $Q_{\Gamma}$ being negative definite. To be consistent, the empty graph is considered to be negative definite in this paper. Using Theorem \[main classification theorem\] and Proposition \[main classification theorem\_convex\], we give the classification of filling divisors and capping divisors with finite boundary fundamental group. \[complete classification\] Let $(D,\omega)$ be a divisor (not necessarily $\omega$-orthogonal) with finite boundary fundamental group. Then $(D,\omega)$ is a capping divisor if and only if it satisifies the positive GS criterion and its graph is equivalent to a realizable graph in type (P). On the other hand, $(D,\omega)$ is a filling divisor if and only if its graph is equivalent to a graph in type (N). The statement for capping divisor follows directly from Proposition \[reduce to graph\] and Theorem \[main classification theorem\]. On the other hand, if $(D,\omega)$ is of type $N$, it is negative definite and hence satisfies the negative GS criterion. By Proposition \[McLean\], $(D,\omega)$ is convex and we can close it up (by [@EtHo02]), thus is a filling divisor. If $(D,\omega)$ is a filling divisor but not in type (N), then $(D,\omega)$ is in type (P) and has $b_2^+(D)=1$, by Theorem \[main classification theorem\] and Proposition \[main classification theorem\_convex\]. We can close it up to a closed symplectic manifold $(W,\omega)$, which must be rational since it contains the divisor $D$ (for the reason why $W$ is rational if the graph of $D$ is equivalent to one in type (P3), see [@BhOn12], for the other, see Theorem \[McDuff\]). Therefore, $b_2^+(W)=1$. However, $\omega|_{W-P(D)}$ descends to a relative class in the cap, $W-P(D)$, and thus has positive square (i.e $[\omega|_{W-P(D)} -d\alpha_c]^2 >0$ for any choice of primitive $\alpha$ of $\omega$ defined near $P(D)$). Therefore, $b_2^+(W-P(D)) \ge 1$. However, $b_2^+(W)=b_2^+(P(D))+b_2^+(W-P(D))$ as $\partial P(D)$ is a rational homology sphere. Contradiction. Filling Divisors with Finite Boundary $\pi_1$ {#Filling classification} --------------------------------------------- We prove Proposition \[main classification theorem\_convex\] in this subsection. ### Topological Restrictions We first recall some topological constraints for a configuration to have finite boundary fundamental group. \[graph theoretic definition\] Suppose we have a graph $\Gamma$. The boundary fundamental group of $\Gamma$, denoted by $\pi_1(\Gamma)$, is the fundamental group of the boundary of the plumbing of the configuration represented by $\Gamma$. We call $\Gamma$ spherical, cyclic, finite cyclic if $\pi_1(\Gamma)$ is trivial, cyclic, finite cyclic. A branch point (or branch vertex) of a graph is a vertex with at least three branches. A branch at a vertex $v$ also refers to the sub-graph $\Gamma$ obtained by deleting $v$ and all other branches linking to $v$. A simple branch $\gamma$ is a branch that is linear. An extremal branch point is a branch point with only one non-simple branch. Finally, for a connected sub-graph $\gamma$, $\delta_{\gamma}$ denotes the determinant of the intersection form of $\gamma$. \[tree\] Let $T$ be a graph with finite $\pi_1(T)$. Then, all of its vertices have genera zero and $T$ is a finite tree. Therefore, from now on, all vertices are assumed to have genera zero and the number above a vertex is the self-intersection number of the vertex. Here, we give the concrete representation for boundary fundamental group. \[representation\]([@Hi66]) Let $T$ be a finite tree such that the genera of all vertices are zero. Label the vertices as $v_i$ for $i=1, \dots, n$ and let $q_{ij}=[v_i][v_j] \in \mathbb{Z}$ be the $(i,j)^{th}$-entry of the intersection form of $T$. Then, $\pi_1(T)$ is isomorphic to the free group generated by $e_1,\dots,e_n$ modulo the relations $$e_ie_j^{q_{ij}}=e_j^{q_{ij}}e_i,\quad \hbox{for any } i, j$$ and $$1=\prod_{1 \le j\le n}e_j^{q_{ij}}, \quad \hbox{for any } i.$$ \[order\] ([@Hi66]) Let $T$ be a finite tree such that the genera of all vertices are zero. Then, the order of the abelianization of $\pi_1(T)$ is finite if and only if $\delta_T \neq 0$. In this case, $\delta_T$ equals the order of the abelianization of $\pi_1(T)$. \[non-spherical\] For a type (N2) linear graph $T$, $\delta_T \neq 1$ and hence $\pi_1(T)$ is nontrivial by Lemma \[order\]. \[key\] Let $T$ be a minimal tree and $v$ a vertex in $T$. \(i) If $\pi_1(T)$ is cyclic, then there are at most two non-spherical branches at $v$. \(ii) If $\pi_1(T)$ is finite, then there are at most three non-spherical branches at $v$. Moreover, if there are three non-spherical branches, then they are all finite cyclic. The proof is based on the representation in Lemma \[representation\] and the group theoretical result in the following lemma. See Lemma 3.1 and 3.2 of [@Sh85] \[quotient\] Let $G_1, \dots, G_n$ be non-trivial groups and let $t_i \in G_i$ be an arbitrary element. Then, \(i) for $n \ge 4$, $G_1 * \dots *G_n/(\prod_{i=1}^{n}t_i=id)$ is infinite. \(ii) for $n \ge 3$, $G_1 * \dots *G_n/(\prod_{i=1}^{n}t_i=id)$ is non-trivial and non-cyclic. \(iii) $G_1*G_2*G_3/(\prod_{i=1}^{3}t_i=id)$ is finite if and only if $G_i$ are all cyclic groups generated by $t_i$ with $(G_1,G_2,G_3)$ isomorphic to one of the following unordered triples $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/k\mathbb{Z})$, ($k \ge 2$), $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/3\mathbb{Z}) $, $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/4\mathbb{Z})$, or $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/5\mathbb{Z})$. \[lens space analysis\] Suppose $T$ is of the form $\xymatrix@R=1pc @C=1pc{ T_1 \ar@{.}[r] & \bullet_{v_0} \ar@{-}[r] \ar@{.}[d]& \dots \ar@{-}[r] & \bullet_{v_r} \ar@{.}[d] \ar@{.}[r] & T_3 \\ & T_2 & & T_4 } $ with $r \ge 1$, where $T_i$ are non-spherical branches (not necessarily simple) such that $\delta_{T_1}\delta_{T_2} \neq 0$ and $\delta_{T_3}\delta_{T_4} \neq 0$. Suppose also that the boundary of the plumbing of $\xymatrix@R=1pc @C=1pc{ T_1 \ar@{.}[r] & \bullet_{v_0} \ar@{.}[r] & T_2 \\} $ and $\xymatrix@R=1pc @C=1pc{ T_3 \ar@{.}[r] & \bullet_{v_r} \ar@{.}[r] & T_4 \\} $ are diffeomorphic to a lens space or $\mathbb{S}^2 \times \mathbb{S}^1$. Then, $\pi_1(T)$ contains $\mathbb{Z} \oplus \mathbb{Z} $ as a subgroup. See Lemma 3.3 and 3.4 of [@Sh85] It is time to mention the following [**Fact**]{}: type (N) graphs have finite boundary fundamental group. To be more precise, (N1) graph is spherical, (N2) graphs are finite cyclic, and (N3) graphs are finite and non-cyclic. This is well known in algebraic geometry: graphs in type (N2) correspond to resolution graphs of cyclic quotient singularities and the graphs in type (N3) correspond to resolution graphs of dihedral, tetrahedral, octahedral and icosahedral singularities (cf. [@Br68] Satz $2.11$). One can also prove this fact directly by Lemma \[representation\] and Lemma \[quotient\]. It is easy for (N2) graphs. And for an (N3) graph $T$, $\pi_1(T)$ is finite as it can be realized as a finite extension of a finite group (basically by Lemma \[quotient\]), and it is non-cyclic because it has a non-cyclic quotient. ### Proof of Proposition \[main classification theorem\_convex\] {#classification} In this subsection we are going to make use of the constraints above to prove Proposition \[main classification theorem\_convex\]. \[simple branch\] Let $T$ be a negative definite, minimal tree with no branch point. Then $T$ is of type (N1) or (N2). In particular, $T$ is finite cyclic. A connected genus zero tree has no branch point, so it is linear. Linearity and minimality ensure no $-1$ vertices, while being negative definite ensures that each vertex has self-intersection less than 0. So $T$ is the empty graph (N1), or a linear graph with all vertices having self-intersection less than $-1$, which is a type (N2) graph. \[T shape\] Let $T$ be a minimal tree with exactly one branch point $v$. Suppose all the self-intersection of vertices in the branches are negative (satisfied if $T$ is negative definite). Then, $\pi_1(T)$ is finite if and only if $T$ is a (N3) or (P3) graph. In particular, $\pi_1(T)$ is not cyclic if $\pi_1(T)$ is finite. See Theorem 4.3 of [@Sh85]. The proof is purely topological. Now, we deal with the case that there are more than $1$ branch points. \[Tech\] Let $T$ be a negative definite, minimal tree with $k \ge 2$ branch points. Then, $\pi_1(T)$ is non-cyclic and infinite. The proof follows [@Sh85] closely. It is convenient to make two observations first. [**Observation 1**]{}: For any branch point, by the minimality assumption, all self-intersections of vertices in simple branches are less than $-1$. Therefore every simple branch of $T$ is not spherical from Example \[non-spherical\]. [**Observation 2**]{}: \[observation\] Let $T$ be a negative definite, minimal tree with $k \ge 2$ branch points. Suppose $\pi_1(T)$ is finite or cyclic. Let $v$ be a branch point and $\Gamma$ a branch at $v$. Suppose there are at least two non-spherical branches at $v$ other than $\Gamma$ (it is satisfied if $v$ is an extremal branch point and $\Gamma$ is the non-simple branch), then $\Gamma$ is finite cyclic and is either (a)a negative definite minimal tree with $k-1$ branch points, OR (b)a negative definite minimal tree with $k-2$ branch points, OR (c)not minimal. In case (c), there exists a branch point of $T$, $v_2$, which is linked to $v$, with exactly three branches and the self-intersection of $v_2$ is $-1$. Suppose first that $\pi_1(T)$ is cyclic. Since there are at least two non-spherical branches at $v$ other than $\Gamma$, $\Gamma$ is spherical by Lemma \[key\](i) applied to $v$. If $\pi_1(T)$ is finite, then $\Gamma$ is finite cyclic by Lemma \[key\](ii) applied to $v$. Therefore in either case $\pi_1(T)$ is finite cyclic. Moreover, if $\Gamma$ is minimal, it is either in (a) or (b). If $\Gamma$ is not minimal, it is in (c). In this case, the only possible $-1$ vertex that can be blown down is the vertex in $\Gamma$ linked to $v$, which we call it $v_2$. Since $T$ is minimal, $v_2$ has self-intersection $-1$ means that it is a branch point of $T$ but it can be blown down in $\Gamma$ means that it is not a branch point of $\Gamma$. Therefore, the result follows. An extremal branch point satisfies the assumption because there are at least two simple branches at $v$, which are non-spherical by Observation 1. We are going to first establish the claim of Lemma \[Tech\] for the cases $k=2$ and $k=3$, then prove by contradiction using induction on $k$. Label the vertices of $T$ as $v_1, \dots, v_m$ with the corresponding self-intersection $s_1, \dots, s_m$. First suppose $k=2$ and $v_1$, $v_2$ are the two branch points of $T$ with $\pi_1(T)$ cyclic or finite. If one of $v_1$, $v_2$ has three simple branches, say $v_2$, we denote the non-simple branch at $v_1$ by $\gamma$. Apply Lemma \[observation\] to $v_1$, $\gamma$ is in (a). However, negative definite minimal tree with exactly one branch point is not cyclic (See Lemma \[T shape\]). Contradiction. Thus, both $v_1$ and $v_2$ have only two simple branches. Let the two simple branches at $v_1$ be $T_1$ and $T_2$ and those at $v_2$ be $T_3$ and $T_4$. Then, the assumptions of Lemma \[lens space analysis\] for $v_0=v_1$ and $v_r=v_2$ are satisfied. Thus, $\pi_1(T)$ contains $\mathbb{Z} \oplus \mathbb{Z}$, contradiction. For $k=3$, let $v_2$, $v_1$ and $v_3$ be the three branch points of $T$ and suppose $T$ is finite or cyclic. We have two of the three branch points are extremal, say $v_2$ and $v_3$. Let $\Gamma$ be the non-simple branch at $v_2$ and $\Gamma'$ be the non-simple branch at $v_3$. Apply Lemma \[observation\] at $v_2$, we have $\Gamma$ is not minimal because we have shown that negative definite minimal trees with exactly one or two branch points are not finite cyclic. Thus, we must have $v_1$ is linked to $v_2$ and $v_1$ has only one simple branch in $T$, which we denote by $T_0$. Moreover, we have $s_1 = -1$ and by symmetry, $v_1$ is also linked to $v_3$. Observe that, we must have all vertices in $T_0$ having self-intersection $-2$, otherwise $\pi_1(\Gamma)$ is still not cyclic. Since $T$ is negative definite, both $s_2$ and $s_3$ are not $-1$. If $v_2$ or $v_3$ has three or more simple branches, then we can apply Lemma \[key\](ii) at $v_1$ and Lemma \[T shape\] to the branch at $v_1$ which is minimal and having exactly one branch point to get a contradiction. In other words, $v_2$ or $v_3$ has exactly two simple branches. Therefore, we have $T$ is of the following form with all $T_i$ being simple branches. $\xymatrix{ T_1 \ar@{.}[r] & \bullet_{v_2} \ar@{-}[r] \ar@{.}[d]& \bullet^{-1}_{v_1} \ar@{-}[d] \ar@{-}[r] & \bullet_{v_3} \ar@{.}[d] \ar@{.}[r] & T_3 \\ & T_2 & T_0 & T_4 } $ Notice that, both $\Gamma$ and $\Gamma'$ are equivalent to a linear graph because all vertices in $T_0$ have self-intersection $-2$. Let $\gamma=\xymatrix {T_1 \ar@{-}[r] & \bullet_{v_2} \ar@{-}[r] & T_2}$. Since $s_2 \neq -1$, we have $\delta_{\gamma} \neq 0$ and $\pi_1(\gamma)$ nontrivial. Hence, we can apply Lemma \[lens space analysis\] for $v_0=v_1$ and $v_r=v_3$. This is because the $T_i$ in Lemma \[lens space analysis\] are not assumed to be simple branches and $\Gamma'$ is equivalent to a linear graph. Hence, we get a contradiction. This finishes the study of $k=3$. In general, we assume the statement is true for $n <k$ and we deal with the case with $T$ having $k \ge 4$ branch points. Let $v_1$ be an extremal branch point and $\Gamma$ is the non-simple branch. The induction hypothesis and Lemma \[observation\] imply that $\Gamma$ is not minimal. In particular, there is a branch point of $T$, say $v_2$, is linked to $v_1$ with $s_2=-1$. If none of the three branches of $v_2$ in $T$ is simple, no matter how we blow down $\Gamma$, its minimal model has at least two branch points, contradicting to the above $k=2$ case or induction hypothesis. Hence, $v_2$ has a simple branch. Since $T$ is negative definite and $s_2=-1$, we must have $s_1 \le -2$. Thus, the branch at $v_2$ containing $v_1$ is not spherical. Let $\Gamma'$ be the non-simple branch at $v_2$ not containing $v_1$. Applying Lemma \[observation\] at $v_2$, we get that $\Gamma'$ is not minimal and hence there exist $v_3$, which is linked to $v_2$ with $s_3=-1$. The existence of two adjacent vertices, $v_2$ and $v_3$, having self-intersections $-1$ contradicts to $T$ being negative definite. Finally, we can complete the proof of Proposition \[main classification theorem\_convex\] by Lemma \[simple branch\], Lemma \[T shape\], Lemma \[Tech\] and the classification of isolated quotient surface singularities in [@Br68]. In [@Li08] and [@BhOz14], they study the filling of the lens spaces with the canonical contact structure. These correspond to the graphs in (N2). It is proved that the divisor filling is the maximal one among all the fillings and all other fillings can be obtained by rational blow downs of the divisor filling [@BhOz14]. Therefore, divisor filling is interesting to investigate. More Topological Restrictions ----------------------------- \[0-0\] Let $T$ be a tree and $v$ a vertex of $T$. Then, $\pi_1(T)=\pi_1(T^{(v)})$. It is a direct computation using Lemma \[representation\]. Label the vertices of $T$ as $v_1,\dots,v_n$ and let $v=v_1$. Label the two additional self-intersection $0$ vertices in $T^{(v)}$ as $v_{-1}$ and $v_0$, where $v_0$ is the one linked to $v_1$. We compare $\pi_1(T^{(v)})$ and $\pi_1(T)$. In terms of generators, $\pi_1(T^{(v)})$ has two additional generators, namely $e_{-1}$ and $e_0$. In terms of relations, there are two new relation and one of the relation in $\pi_1(T)$ is changed. The two new relations are given by $1=e_0$ and $1=e_{-1}e_1$. The relation $1=\prod_{1 \le j\le n}e_j^{q_{1j}}$ is changed to $1=e_0\prod_{1 \le j\le n}e_j^{q_{1j}}$. However, we have $1=e_0$, which means the changed relation is actually unchanged. Moreover, adding the generator $e_{-1}$ with the relation $1=e_{-1}e_1$ is doing nothing to the group so we arrive the conclusion. It is time to state the following fact. \[P graph\] Type P graphs have finite boundary fundamental groups. More precisely, (P1) graph is spherical, (P2) and (P4) graphs are finite cyclic, (P3) and (P5) graphs are finite and non-cyclic. Clear for the (P1) graph. Since (N2) graph is finite cyclic, so are (P2) and (P4) graphs by by Lemma \[0-0\]. (P3) and (P5) graphs and finite and non-cyclic by Lemmas \[T shape\] and \[0-0\]. In [@Ne81], it is mentioned that the dual blow up (which is called blow up there) does not change the oriented diffeomorphism type of the boundary of the plumbing. \[00\] Let $T$ be a minimal tree and $v$ a vertex in $T$. Suppose $\gamma$ is a simple branch at $v$ with $\gamma$ not equivalent to $\xymatrix{\bullet^0 \\} $ and some of the vertices having non-negative self-intersection. Then, $T$ is equivalent to a minimal tree $T'$ obtained by replacing the branch $\gamma$ by an equivalent branch $\xymatrix@R=1pc @C=1pc{ \bullet_{v'} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{0}_{x} \\ } $. where $v'$ is the vertex linked to $v$, $x$ is an end vertex and the self-intersection of $v$ may possibly be changed. We first make the following observations. If $T$ has a sub-tree of the form $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q} \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{b} \ar@{-}[r]& \dots \\ } $, then $T$ is equivalent to $T'$ where $T'$ is obtained by changing the sub-tree to $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q+1} \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{b-1} \ar@{-}[r]& \dots \\ } $. Also, if $T$ has a sub-tree of the form $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q} \ar@{-}[r] & \bullet^{0} \\ } $, then $T$ is equivalent to $T'$ where $T'$ is obtained by changing the sub-tree to $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q+1} \ar@{-}[r] & \bullet^{0} \\ } $. Let $u$ be the vertex in $\gamma$ having self-intersection non-negative, By possibly blowing up successively at edges linked to $u$, we assume that $u$ has self-intersection $0$. By the first observation, we can make the end vertex $w$ of $\gamma$ to have self-intersection $0$. Since $\gamma$ is not equivalent to $\xymatrix{ \bullet_{0} \\ } $, there is a vertex $w'$ which is different from $v$ and is linked to the end vertex $w$. Now, we blow down all $-1$ vertices in $\gamma-\{w,w'\}$. By the second observation and the fact that $w$ has self-intersection $0$, we obtain $T'$ as we want. For details, see Lemma 4.1 of [@Sh85]. \[negative definite spherical\] A spherical negative definite tree is not minimal. In other words, it is equivalent to an empty graph. See Section $3$ of [@Hi66] Symplectic Restrictions {#symplectic} ----------------------- We are going to provide symplectic input to give more constraints on the trees that we are interested in. In this subsection, $L_i$ appearing as a superscript of a vertex represents the homology class of the corresponding sphere. First, we recall a theorem of McDuff. \[McDuff\]([@Mc90]) Let $M$ be a closed symplectic 4-manifold. Suppose there exist an embedded symplectic sphere $C$ with positive (resp. zero) self-intersection. Then, $M$ is symplectic rational (resp. ruled). Moreover, if $(M,C)$ is relatively minimal and $[C]^2=0$, then there exists a symplectic deformation equivalent from $(M,C)$ to $(N,F)$, where $N$ is a symplectic sphere bundle over a closed symplectic surface and $F$ is a fibre. \[E\] Let $M$ be a symplectic 4-manifold. Suppose there exist an embedded symplectic sphere $C$ with \(i) positive self-intersection, or \(ii) zero self-intersection and there exists another embedded symplectic sphere $C'$ that intersect $C$ transversally once. Then $H_1(M)=0$, $M$ is rational and $b_2^+=1$. \[R\] Let $D$ be a symplectic divisor in a closed symplectic manifold $W$. Then, the graph of $D$ does not have a sub-tree of the form, $$\xymatrix{ \bullet^{L_1} \ar@{-}[r] & \bullet^{L_3} \ar@{-}[d] \ar@{-}[r] & \bullet^{L_4} \ar@{-}[r] & \bullet^{L_5}_x\\ & \bullet^{L_2} \\ }$$ with $[L_3]^2=-1$ and $[L_5]^2 \ge 0$, and all vertices having genera zero. By Corollary \[E\], we get $H_1(W)=0$ and $W$ is rational. Without loss of generality, we may assume $[L_5]^2=0$ (by possibly blowing up at regular points for the sphere representing $x$). Since $[L_3]^2=-1$, $W$ is not minimal and thus symplectomorphic deformation equivalent to blown-up of a Hizerburch surface, and $[L_5]$ is the fibre class of the Hizerburch surface by Theorem \[McDuff\]. Let $\{f,s,e_1,\dots,e_N\}$ be a basis for $H_2(W)$ such that $f$, $s$ and $e_i$ correspond to the fibre class, section class and the exceptional classes, respectively. Suppose $s^2=n$. Then, we recall the first chern class of the Hirzeburch surface is $(2-n)f+2s$, thus the first chern class of $W$ is $c_1(W)=(2-n)f+2s-e_1-\dots-e_N$. Moreover, $e_i$ can be a prior chosen so that $[L_3]=e_1$. Suppose there is an embedded symplectic sphere in $W$ with class $[S]=\alpha f+\beta s +a_1e_1+\dots+a_Ne_N$. Then, adjunction formula gives $$(2-n)\beta+2\alpha +2n\beta+a_1+\dots+a_N=2\alpha \beta+\beta^2n-a_1^2-\dots-a_N^2+2.$$ Suppose $[S]f=1$. Then $\beta=1$ and the formula reduces to $$a_1^2+\dots+a_N^2+a_1+\dots+a_N=0$$ and hence $a_i=0$ or $-1$ for all $i$. Suppose on the contrary $[S]f=0$. Then $\beta=0$ and the formula reduces to $$2\alpha+a_1^2+\dots+a_N^2+a_1+\dots+a_N=2.$$ Now, we want to study the homology of $L_i$ and draw contradiction. We recall that $[L_5]=f$ and $[L_3]=e_1$. Since $1=[L_4][L_5]=[L_4]f$, we apply the adjunction formula derived above and get $[L_4]=\alpha f+s+\epsilon_1 e_1+\dots+\epsilon_N e_N$ for some $\alpha$, where $\epsilon_i$ equals $0$ or $-1$ for all $i$. Since $1=[L_4][L_3]=[L_4]e_1$, $[L_4]$ is of the form $\alpha f+s-e_1+\epsilon_2 e_2+\dots+\epsilon_N e_N$. If we write $[L_1]=\overline{\alpha} f+\beta s +a_1e_1+\dots+a_Ne_N$, then $[L_1]f=0$, $[L_1]e_1=1$ and adjunction imply $[L_1]=\overline{\alpha} f-e_1+a_2e_2+\dots+a_Ne_N$ and $$\begin{aligned} \label{1} 2\overline{\alpha}+a_2^2+\dots+a_N^2+a_2+\dots+a_N=2\end{aligned}$$ Moreover, by $[L_1][L_4]=0$, we have $$\begin{aligned} \label{2} \overline{\alpha}-1-\epsilon_2a_2-\dots-\epsilon_N a_N=0\end{aligned}$$ By equations \[1\] and \[2\], we have $$\begin{aligned} &&a_2^2+\dots+a_N^2+(1+2\epsilon_2)a_2+\dots+(1+2\epsilon_N)a_N \\ &=& a_2(a_2+(-1)^{\epsilon_2})+\dots+a_N(a_N+(-1)^{\epsilon_N}) \\ &=&0\end{aligned}$$ Therefore, $a_i=0,-1$ if $\epsilon_i=0$ and $a_i=0,1$ if $\epsilon_i=-1$, for all $i$. Similarly, $[L_2]=\overline{\beta}f-e_1+b_2e_2+\dots+b_Ne_N$ with $b_i=0,-1$ if $\epsilon_i=0$ and $b_i=0,1$ if $\epsilon_i=-1$, for all $i$. By $[L_1][L_2]=0$, we have $-1-a_2b_2-\dots-a_Nb_N=0$. When $\epsilon_i=0$, we have both $a_i$ and $b_i$ equals $0$ or $-1$, thus $a_ib_i \ge 0$. Similarly, if $\epsilon_i=-1$, we still have $a_ib_i \ge 0$. Therefore, $-1-a_2b_2-\dots-a_Nb_N=0$ gives a contradiction. By the previous two Lemmas, we can now state one more basic consequence. \[0\] Let $T$ be a minimal graph of an embeddable divisor with finite $\pi_1$. Then, at any branch point $v$, no branch can be a single vertex $u$ with self-intersection zero. Suppose there is such a vertex $u$. Since $\pi_1(u)$ is infinite and $v$ has at least three branches, by Lemma \[key\](ii), there is a spherical branch at $v$, which we call $\gamma$. By Theorem \[McDuff\] again, the homology class $[u]$ represents a fiber class. Suppose $b_2^+(\gamma) \neq 0$, then there exists a class $[P]$ which is a linear combination of homology classes of vertices of $\gamma$ such that $[P]^2 >0$. Moreover, $[P][u]=0$. A basis for blown-up Hizerburch surface is given by $\{[u],s,e_1,\dots,e_n\}$, where $s$ is a section class and $e_i$ are the exceptional classes. Therefore, $[P][u]=0$ implies $[P]$ is a linear combination of $[u]$ and $e_i$. As a result, we have $[P]^2 \le 0$, which is a contradiction. Therefore, $b_2^+(\gamma)=0$. Since $\delta_{\gamma} \neq 0$, $b_2^+(\gamma)=0$ implies $\gamma$ is negative definite (Lemma \[order\]). Hence, by Lemma \[negative definite spherical\], $\gamma$ is not minimal. Therefore, $T$ has a sub-tree of the form, $$\xymatrix{ \bullet^{L_1} \ar@{-}[r] & \bullet^{-1}_w \ar@{-}[d] \ar@{-}[r] & \bullet^{L_4}_v \ar@{-}[r] & \bullet^{0}_u\\ & \bullet^{L_2} \\ }$$ Contradiction to Lemma \[R\]. Suppose $T$ is a realizable minimal tree with $\pi_1(T)$ being finite. Suppose also that there is a non-negative self-intersection vertex in a simple branch of $T$. Then, $T$ is equivalent to $T'^{(v)}$ for another minimal tree $T'$. We can apply Lemma \[0\] to ensure that the proof of Lemma \[00\] goes through and hence the result follows. By Corollary \[E\], $T$ has $b_2^+=1$ and hence $T'$ is negative definite. Moreover, by Lemma \[representation\], we can see that $\pi_1(T)=\pi_1(T')$ and hence finite. Therefore, what we need to do next is the classification minimal tree $T$ with $\pi_1(T)$ being finite and all self-intersection in simple branches being negative. By Lemma \[T shape\], we just need to consider the case that there are more than $1$ branch point. \[Tech2\] Let $T$ be a minimal realizable graph. Suppose $k \ge 2$ be the number of branch points of $T$. Suppose also all self-intersection of vertices in simple branches are less than $-1$. Then, $\pi_1(T)$ is non-cyclic and infinite. In particular, if $T$ is minimal, non-negative definite and $\pi_1(T)$ is finite, then $T$ is equivalent to a type (P) graph. The essence of the proof is the same as in Lemma \[Tech\] (See [@Sh85]). Since we do not assume our tree $T$ satisfies $b_2^+=1$, we cannot apply the result in [@Sh85] directly. Instead, we need to use Corollary \[E\] to guarantee $b_2^+=1$ whenever it is needed in the proof. This is required when we study the case that $T$ has $k=3$ branch points. We remark that in that case, there are two $-1$ vertices linked to each other so that blowing down one of them gives us an embedded symplectic sphere with self-intersection $0$. Therefore, we can apply Corollary \[E\] in that case to finish the proof of the first assertion. Moreover, by Lemma \[T shape\], the second assertion also follows. Proof of Theorem \[main classification theorem\] ------------------------------------------------ Having Lemma \[Tech2\], we can now focus on the study of type (P) graphs. Type (P1), (P2), (P3) are relatively easy to study and we are going to first go through it. Then, complete classification of realizability of type (P4) and (P5) graphs are given, which in turn completes the proof of Theorem \[main classification theorem\]. Finally, we are going to show that many graphs in type (P5) do not have their conjugate. ### Type (P1), (P2), (P3) {#Type (P1) to (P3)} We start with type (P1). By Example \[0-0&gt;1\], we have $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $ is equivalent to $\xymatrix{ \bullet^{1}} $. Then, by Example \[single vertex\], it is strongly realizable. Moreover, it corresponds to a capping divisor of the empty graph. Instead of answering the realizability of graphs in type (P2) and (P3) directly, we observe that graphs in type (2) and type (3) are all considered in [@BhOn12]. The (P2) graphs correspond to compactifying divisors for cyclic quotient singularities and the (P3) graphs correspond to compactifying divisors for the dihedral, tetrahedral, octahedral and icosahedral singularities. In particular, all (P2) and (P3) graphs are strongly realizable. The only less obvious correspondence between (P2), (P3) graphs and the graphs considered in [@BhOn12] are (P3) graphs of the form $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ with $y \le 1$ and $(n_2, n_3)=(2, n)$. We denote the following graph in [@BhOn12] as $(c,c_1, \dots, c_k)$, where $[c,c_1,\dots,c_k]=\frac{n}{n-q}>1$ and $c,c_i \ge 2$ for all $i$. These are the graphs of compactifying divisors of dihedral singularities used in [@BhOn12]. $$\xymatrix{ \bullet^{-2} \ar@{-}[r]& \bullet^{-1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-c+1} \ar@{-}[r] & \bullet^{-c_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-c_k}\\ &\bullet^{-2} \\ }$$ Observe that $(c,c_1, \dots, c_k)$ is the same as $<1;2,1;2,1;q,n-q>$ if one extends the definition to the case that $q < n-q$. Every graph $(c,c_1, \dots, c_k)$ in [@BhOn12] with $c,c_i \ge 2$ is equivalent to a (P3) graph $<y;2,1;2,1;n,\lambda>$ with $y \le 1$ and $0 < \lambda < n$ and vice versa. Observe that $<y;2,1;2,1;n,\lambda>$ is equivalent to $$\xymatrix@R=1pc @C=1pc{ \bullet^{-2} \ar@{-}[r]& \bullet^{-1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-1}_{v} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-d_1-1}_{w} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ &\bullet^{-2} \\ }$$ where there are $-y$ many self-intersection $-2$ spheres between the spheres named $v$ and $w$ as subscript and $\frac{n}{\lambda}=[d_1,\dots,d_k]$. Hence, it is of the form $(c,c_1, \dots, c_{-a+k-1})$ with $$[c,c_1,\dots,c_{-d},c_{-d+1},c_{-d+2}, \dots,c_{-d+k-1}] =[2,2,\dots,2,d_1+1,d_2,\dots,d_k].$$ This defines a map from the set of $<y;2,1;2,1;n,\lambda>$ with $y \le 1$ and $0 < \lambda < n$ to the set of $(c,c_1, \dots, c_k)$ and $c,c_i \ge 2$. Moreover, the inverse exists. Knowing that the graphs in type (P1) to type (P3) are realizable, as remarked before, we can determine whether a divisor with its graph being in type (P1) to type (P3) is a capping divisor or not, by Proposition \[reduce to graph\]. We remarked that for any graph $T$ in (P2), there is a unique (N2) graph $T'$ such that the dual blow up of $T'$ at the left-end vertex is $T$. In fact, by symmetry, there is also a unique (N2) graph $T"$ such that the dual blow up of the right-end vertex of $T"$ is $T$. To be more precise, $T"$ is obtained from $T'$ by rewriting the self-intersections from left to right to from right to left. ### Type (P4) and (P5) {#Type (P4) and (P5)} Suppose a graph $\overline{T^{(v)}}$ in type (P4) or (P5) admits a realization $D$ in a closed symplectic manifold $W$. By Theorem \[McDuff\] again, the existence of the self-intersection $1$ sphere implies that $W$ is rational. After preparation, now we are ready to study the realizability of type (P4) and (P5) graphs. We recall that for a type (P4) graph, $T^{(v)}$, $v$ is not an end vertex of $T$. We show that all (P4) graphs are realizable but we some graphs in type (P5) are not realizable. \[standard realization\] Suppose $T=<n,\lambda>= \xymatrix{ \bullet^{-d_1} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ }$ and $v_1$ is the vertex with self-intersection $-d_j$ and $j \neq 1,k$. Then, $\overline{T^{(v_1)}}$ and hence the (P4) graph $T^{(v_1)}$ is realizable by some symplectic divisor $D$. On the other hand, suppose $T_2=<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is a graph in type (N3). Then, $T_2^{(v)}$ is realizable if $y \neq 2$. We start with two algebraic lines in $\mathbb{CP}^2$ and call it $C_0$ and $C_1$. We blow up $\mathbb{CP}^2$ at $d_j$ distinct regular points at $C_1$ away from the intersection point of $C_0$ and $C_1$. Label the exceptional spheres as $E_1, E_{j+1}, E_1^{2}, \dots, E_1^{d_j-1}$. Call the proper transform of $C_0$ and $C_1$ as $C_0$ and $C_1$ again. Moreover, we call $E_1$ and $E_{j+1}$ to be $C_2$ and $C_{j+1}$, respectively. Then, we blow up $d_{j-1}-1$ many distinct regular points on $C_2$ that are away from the intersection points. Denote the exceptional spheres as $E_2, E_2^{2}, E_2^{3}, \dots, E_2^{d_{j-1}-1}$. Call the proper transform of $C_i$’s as $C_i$’s again and we call $E_2$ as $C_3$. We keep blowing up at regular points inductively and similarly on $C_3$ up to $C_{j-1}$ and denotes $E_{j-1}$ as $C_j$. Now,we blow up $C_j$ at $d_1-1$ many distinct regular points and call the exceptional spheres as $E_j, E_j^{2}, E_j^{3}, \dots, E_j^{d_{1}-1}$. This time, we do not let $C_{j+1}$ to be $E_j$ (we actually defined $C_{j+1}=E_{j+1}$). We get the second branch $C_2 \cup \dots \cup C_j$ of $C_1$ ($C_0$ is the first branch of $C_1$). Now, we blow up similarly for $E_{j+1}=C_{j+1}$ and we can get the last branch $C_{j+1} \cup \dots \cup C_k$ of $C_1$. This gives an embeddable symplectic divisor $D=C_0 \cup \dots \cup C_k$ that realize $\overline{T^{(v_1)}}$. For the type (P5) graph, we also consider $\overline{T_2^{(v)}}$ instead of $T_2^{(v)}$. We assume that $v$ is a vertex with two branches. The case when $v$ is the vertex with three branches is similar. We start with $\mathbb{CP}^2$ with $D$ being union of two distinct $\mathbb{CP}^1$, denoted by $C_1$ and $C_2$. Without loss of generality, we can assume $\overline{T_2^{(v)}}$ is of the form $$\xymatrix@R=1pc @C=1pc{ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1} \ar@{-}[r]& \bullet^{-y}_c \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{1-b_j}_v \ar@{-}[r] \ar@{-}[d]& \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1} \ar@{-}[d] & & & \bullet^{1}_p &\\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ Let we denote the sphere with self-intersection $-d_s$, $-b_s$ ($s \neq j$) and $-c_s$ as $C^\alpha_s$, $C^\beta_s$ ($s \neq j$) and $C^\gamma_s$, respectively. Also, denote the sphere with self-intersection $-y$, $1-b_j$ and $1$ as $C^c$ $C^v$ and $C^p$, respectively. It suffices to consider the case that $d_s=b_s=c_s=2$ for all $s$ because we can obtain the other cases by extra blow-ups. It is possible to obtain $\overline{T_2^{(v)}}$ with homology of the spheres indicated below by iterative blow-ups starting from $D$ similar to that in Lemma \[standard realization\]. Here $h$ is the hyperplane class and $e_i$’s are the exceptional classes resulting from single blow-ups. $[C^\alpha_s]=e_{i^{\alpha_{s-1}}_2}-e_{i^{\alpha_{s}}_2}$ for $2 \le s \le k$; $[C^\alpha_1]=e_{i^{\alpha_1}_1}-e_{i^{\alpha_1}_2}$; $[C^\gamma_s]=e_{i^{\gamma_{s-1}}_2}-e_{i^{\gamma_{s}}_2}$ for $2 \le s \le m$; $[C^\gamma_1]=e_{i^{\gamma_1}_1}-e_{i^{\gamma_1}_2}$; $[C^c]=e_{i^{\beta_1}_2}-e_{i^{\alpha_1}_1}-e_{i^{\gamma_1}_1}$; $[C^\beta_s]=e_{i^{\beta_{s+1}}_2}-e_{i^{\beta_{s}}_2}$ for $1 \le s \le j-2$; $[C^\beta_{j-1}]=e_{i^{\beta_{j-1}}_1}-e_{i^{\beta_{j-1}}_2}$; $[C^v]=h-e_{i^{\beta_{j-1}}_1}-e_{i^{\beta_{j+1}}_1}$; $[C^\beta_{j+1}]=e_{i^{\beta_{j+1}}_1}-e_{i^{\beta_{j+1}}_2}$; $[C^\beta_s]=e_{i^{\beta_{s-1}}_2}-e_{i^{\beta_{s}}_2}$ for $j+2 \le s \le l$, and $[C^p]=h$. This shows the existence of a realization for the type (P5) graph. To aid the non-realizablility study of some type (P5) graphs, we recall a combinatorical argument given by Lisca (See Proposition 4.4 of [@Li08]). \[combinaotric Lisca\] Suppose $W=\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}$ equipped with a symplectic form $\omega$ coming from blown-up of the Fubini-Study form $\omega_{FS}$. Let $D=C_0 \cup C_1 \cup \dots \cup C_k$ be a symplectic divisor with linear graph and $C_0$ corresponds to one of the two end vertices. Suppose the self-intersection of $C_i$ is $-b_i$ for $2 \le i \le k$, $[C_0]^2=1$ and $[C_1]^2=1-b_1$, where $b_i \ge 1$ for all $i$. Suppose $\{h,e_1,\dots,e_N\} \subset H_2(\mathbb{CP}^2\#N\overline{\mathbb{CP}^2};\mathbb{Z})$ forms an orthogonal basis with $h$ being the line class and $e_i^2=-1$. Assume also $[C_0]=h$. Then, $[C_1]=h-e_{i^1_1}-e_{i^1_2}-\dots-e_{i^1_{b_1}}$ and $[C_j]=e_{i^j_1}-e_{i^j_2}-\dots-e_{i^j_{b_j}}$ for $2 \le j\le k$, where, for any $\alpha$, $e_{i^\alpha_m} \neq e_{i^\alpha_n}$ for $m \neq n$. From now on, when we write the homology of a sphere, say $C$, we might simply write $h-e.-e.-\dots-e.$ and $e.-e.-\dots-e.$ to represent the homology class of $C$. In this case, the different $e.$’s in $[C]$ are understood to be distinct exceptional classes as in the conclusion of the Proposition \[combinaotric Lisca\]. \[realizable\] Suppose $T=<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is a graph in type (N3). If $v$ is the vertex with three branches, then $T^{(v)}$ is realizable if and only if $y \neq 2$. The realizability part is already covered by Lemma \[standard realization\] so we are going to show the other direction. Suppose $y=2$ and, on the contrary, there were a realization of $T^{(v)}$ in a closed symplectic manifold. Then, we have $\overline{T^{(v)}}$ is also realizable and we have the following graph. $$\xymatrix{ & & & \bullet^{1}_p \ar@{-}[d] \\ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1}_{v_1} \ar@{-}[r]& \bullet^{-1}_v \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1}_{v_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1}_{v_3} \ar@{-}[d] \\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ By Theorem \[McDuff\] we can assume the positive sphere (the one with subscript $p$) has homology class $h$. Then, the only vertex with 4 branches ($v$) has to have homology class of the form $h-e_1-e_2$, by Proposition \[combinaotric Lisca\]. Here, as usual, $e_1$ and $e_2$ are exceptional classes formed by blowups. We recall that Proposition \[combinaotric Lisca\] ensure that the vertices $v_i$ has homology of the form $e_{j_1}-e_{j_2}-\dots-e_{j_t}$ for some distinct $e_{j_s}$ $1 \le s \le t$. To give the positive one contribution of the intersection of vertex $v_i$ with $v$, modulo symmetry, two of three vertices, $v_1$, $v_2$ and $v_3$ has homology class of the form $e_1-e.-\dots-e.$, where $e.$ are distinct exceptional classes not equal to $e_1$. However, it contradict to the zero intersection of any pair of $v_i$, $i=1,2,3$. Using the same line of reasoning, one can determine completely which graph is realizable and which is not and we put the results in the following. Therefore, the proof of Theorem \[main classification theorem\] is completed. A vertex with subscribe $Y$ indicates that if it is $v$, then the corresponding $T^{(v)}$ is realizable. Otherwise, the subscribe is $X$. \[Tetrahedral\] Tetrahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Octahedral\] Octahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & & \bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-4}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-4}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Icosahedral\] Icosahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-3}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \ar@{-}[r] & \bullet^{-2}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-3}_Y \\ & \bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-5}_Y\\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \ar@{-}[r] & \bullet^{-2}_Y \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-5}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Dihedral\] Dihedral $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-d_1}_N \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}_N \\ & \bullet^{-2}_Y \\ }$$ where $N=Y$ if $d_1 \ge 3$ and $N=X$ if $d_1=2$. This completes the classification of realizable graph with finite boundary fundamental group. Remarks on Fillings {#Fillings} ------------------- In this subsection, we study the fillings for a given capping divisor $D$. First, we sketch the proof of the finiteness of fillings. Then, we study the conjugate phenomena. Finally, Liouville domain as a filling is considered. ### Finiteness {#Finiteness} \[bounds\] Suppose $D$ is a capping divisor with finite boundary fundamental group. Then, up to diffeomorphism, only finitely many minimal symplectic manifolds can be compactified by $D$. \[Stetch of proof\] We follow the strategy in [@Li08],[@BhOn12] and [@St13]. We remark that this question is answered in [@BhOn12] for graphs in type (P1), (P2), (P3). Therefore, it suffices to consider the case that the graph of $D$ is a graph in type (P4) or (P5), which are all dual blown up graphs. Suppose $D$ is a capping divisor for a symplectic manifold $Y$, and let $W$ be the resulting closed manifold. By Theorem \[McDuff\], $W$ is rational since there is a positive sphere $Q$ in $D$ due to the dual blow up. We can pick an orthonormal basis for $\{h,e_1, \dots, e_n \} $ for $H_2(W)$ such that $h^2=1$, $e_i^2=-1$ and $\omega(e_i) > 0$ for all $1 \le i \le N$. Moreover, we can assume the positive sphere $Q$ is of class $h$ by Theorem \[McDuff\]. Let $C_j$ be the sphere corresponding to the vertex $v$ and suppose that its self-intersection is $1-d_j$. By Proposition \[combinaotric Lisca\], the homology of $C_j$ is $[C_j]=h-e_{i^j_1}-\dots-e_{i^j_{d_j}}$ for some $i^j_1, \dots, i^j_{d_j}$ distinct. Moreover, we know that for the other spheres, the homology is of the form $e.-e.-\dots-e.$. If we do iterative symplectic blow-downs away from the positive sphere $Q$, we will end up with $(\mathbb{CP}^2,\omega_0)$, and the image of $D$ under the blow-down maps can be made to be union of exactly $2$ $J$-holomorphic spheres for some $\omega_0$-tamed $J$ if the blow-down maps are carefully chosen. By keeping track of the homological effects of the blow-downs, one can classify all possible $Y$ using the same reasoning as in [@Li08], [@OhOn05] and [@BhOn12]. In particular, one can obtain finiteness. Fixing a capping divisor $D$, we would like to investigate whether there are bounds for topological complexity among all minimal symplectic manifold that can be compactified by $D$. The answer is no in general. By Donaldson’s celebrated construction of Lefschetz pencil, any closed symplectic 4 manifold can be decomposed into a disc bundle over a closed symplectic surface $\Sigma$ glued with a Stein domain. In this case, we can view the symplectic surface $\Sigma$ as a symplectic capping divisor for the Stein domain. One of the interesting problems in this case is to bound the topological complexity for a given genus $g$. Some finiteness results of the topological complexity are obtained in [@Sm01] when the Lefschetz pencil has small genus. However, it is proved in [@BaMo12] that there is no bound of the Euler number of the filling when the genus is greater than $10$. In our setting, we allow ’reducible’ symplectic capping divisor so one should hope for obtaining some finiteness results when the symplectic capping divisor has small geometric genus. By Proposition \[bounds\], bounds for diffeomorphic invariants are obtained when $D$ has finite boundary fundamental group, which is a special case of geometric genus being zero. ### Non-Conjugate Phenomena {#Non-Conjugate Phenomena} Graphs in type (N) are resolution graphs of distinct quotient singularities. In particular, if $T_1$ and $T_2$ are graphs in (N), then they have different boundary fundamental groups except both $T_1$ and $T_2$ are resolution graphs of cyclic singularities (See e.g. [@Br68] Satz $2.11$, fourth column of the table). When two graphs $T$ and $T^c$ admit strong realizations $D$ and $D^c$ respectively such that $D$ and $D^c$ are conjugate to each other, we say that $T$ is conjugate to $T^c$. In Section \[Type (P1) to (P3)\], we mentioned that each graph $T^N$ in type (N) has a conjugate graph $T^P$ in type (P1), (P2) or (P3), and vice versa. We are going to show that many type (P5) graphs do not share this phenomena. There should be many ways to do it and we would like to use the first Chern class. When $T$ admits a realization $D=C_1 \cup \dots \cup C_k$ inside a closed symplectic manifold $W$, the first Chern class of $W$ descends to the first Chern class $c_1^D$ for $P(D)$, where $P(D)$ is a plumbing of $D$. Since $\partial P(D)$ is a rational homology sphere, $c_1^D$ lifts to a class uniquely in $H^2(P(D),\partial P(D), \mathbb{Q})$ by the Mayer-Vietoris sequence, which we still denote as $c_1^D$. Then, by the Lefschetz-Poincare duality, we can identify it with a class in $H_2(P(D), \mathbb{Q})$, which is generated by $[C_1],\dots,[C_k]$. Keeping the notations as in the previous paragraph, we call $(c_1^T)^2+k$ the characterizing number of $T$ and denote it by $n^T$. For $Y=W-P(D)$, we define the characterizing number of $Y$ to be $n^Y=(c_1^Y)^2+b_2(Y)$, where $c_1^Y$ and $b_2(Y)$ are the first Chern class and second Betti number of $Y$, respectively. \[first chern class\] Suppose $T$ is a graph in special types that admits a realization $D=C_1 \cup \dots \cup C_k$ in $W$. Let $b=(s_1+2,\dots,s_k+2)^T$ and write $c_1^T=\sum\limits_{i=1}^k w_i[C_k]$. Then, \(i) $w=(w_1,\dots,w_k)^T$ satisfies $Q_Tw=b$, \(ii) if $T$ is of type (P) and $Y=W-P(D)$, then we have $n^T+n^Y=10$. \(iii) if $T^{(v)}$ is of type (P5) and $n^{T}+n^{T^{(v)}} \neq 10$, then there is no graph conjugate to $T^{(v)}$. \[non-standard contact structure\] Suppose $T$ is a type (N3) graph and $T^{v}$ is a dual blow up of $T$. If $n^{T}+n^{T^{(v)}} \neq 10$ and $T^{(v)}$ is realizable, then it follows from (iii) that, on the diffeomorphic boundaries of plumbings, the contact structure $\xi^{T^{v}}$ induced by the positive GS criterion on $T^{(v)}$ is not contactomorphic to the canonical contact structure $\xi^T$, which is induced by the negative GS criterion on $T$. In this case, we can actually use the capping divisor $T^{(v)}$ to classify the symplectic fillings of the non-standard contact structure $\xi^{T^{v}}$ on the boundary of plumbing of $T^{(v)}$ (See also subsection \[Finiteness\]). Since the first Chern class is induced by a symplectic form, adjunction formula works in $P(D)$. Therefore, we have $2+s_i=c_1^T[C_i]$, where $s_i$ is the self-intersection of $C_i$. Hence, (i) follows immediately. For (ii), since $W$ is rational, we have $(c_1^W)^2+b_2(W)=10$. By the Mayer-Vietoris sequence, we have $H_2(P(D),\mathbb{Q}) \bigoplus H_2(Y,\mathbb{Q})= H_2(W, \mathbb{Q})$. Thus, $b_2(W)=b_2(P(D))+b_2(Y)$ and $(c_1^W)^2=(c_1^T)^2+(c_1^Y)^2$, which proves (ii). Finally, if the complement of plumbing of $T^{(v)}$ is a plumbing of a symplectic divisor, say $D'$, then $D'$ must be negative definite. By Lemma \[0-0\], we know that $\pi_1(D')=\pi_1(T^{(v)})=\pi_1(T)$. Among the type (N3) graphs, the boundary fundamental group uniquely characterize the graph (See [@Br68] Satz $2.11$, fourth column of the table). Therefore, by the classification Theorem \[main classification theorem\], the graph of $D'$ must be $T$ and hence, (ii) implies (iii). Consider the resolution graph of $E_8$ singularities, which is given by $$\xymatrix{ \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] &\bullet^{-2} \ar@{-}[r] &\bullet^{-2}\\ & &\bullet^{-2} \\ }$$ By Lemma \[first chern class\](i), the first Chern class is $c_1^{E_8}=0$ The following graph is a symplectic capping divisor of a plumbing of $E_8$, which we call $E_8^c$. $$\xymatrix{ \bullet^{-2}_{v_2} \ar@{-}[r]& \bullet^{-1}_{v_1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_{v_3} \\ &\bullet^{-5}_{v_4} \\ }$$ Then, by Lemma \[first chern class\](i), we have $c_1^{E_8^c}=2[C^{v_1}]+[C^{v_2}]+[C^{v_3}]+[C^{v_4}]$, where $C^{v_i}$ is the sphere corresponding to $v_i$. Direct calculation gives $(c_1^{E_8})^2=-2$ which is also predicted by Lemma \[first chern class\](ii). By a direct computation using mathematica, if $T$ in (N3) does not correspond to dihedral singularity, then there are only seven different (P5) graphs $T^{(v)}$ that satisfy $n^{T}+n^{T^{(v)}}=10$. Moreover, by Theorem \[main classification theorem\], only four of them are realizable and they are given by the followings. Therefore, these four graphs are the only exception that Lemma \[first chern class\](iii) cannot conclude anything among all graphs in (P5) not arising from dihedral resolution graph. For the following four type (N3) graphs $T$, the corresponding four type (P5) graphs $T^{(v)}$ satisfy $n^{T}+n^{T^{(v)}}=10$. $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-7} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ & & \bullet^{-2}_v \\ }$$ $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-4} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-8} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-3} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_v \ar@{-}[r] & \bullet^{-2} \\ &\bullet^{-2} \\ }$$ ### Liouville Domain It is also interesting to know when a symplectic manifold, in particular, Liouville domain, can be compactified by a symplectic capping divisor. Affine varieties are this kind of Liouville domains. Thus, in some sense we can regard such Liouville domains as symplectic analogues of affine varieties. For an affine surface $X$, log Kodaira dimension can be defined for the pair $(V,D)$, where $V$ is the completion of $X$ and $V-D=X$. Moreover, this holomorphic invariant is independent of the compactification. When the affine surface $X$ is a homology plane (also called affine acyclic), McLean actually showed in [@McL12-2] that the log Kodaira dimension is also a symplectic invariant. Therefore, among all the Louville domains, (rational) homology planes are particularly interesting. It is a classical question in algebraic geometry to classify all (rational) homology planes (See the last Section of [@Mi00] and [@Za98]). A common feature for such an affine variety is that its completion is a rational surface. As we have seen, symplectic $4-$manifolds that can be compactified by a capping divisor with finite boundary fundamental group also share this phenomena. In particular, it would be interesting to know what symplectic capping divisors can compactify a Liouville domain that is a rational homology disk but the completion is not a rational surface. Another classical question is to determine all singularities that admits a rational homology disks smoothing. If the resolution graph for one such singularity is $\Gamma$, then in particular, the plumbing of $\Gamma$ can be symplectically filled by the rational homology disk. We remark that this question is completely answered using techniques ranging from smooth topology, symplectic topology and algebraic geometry (See [@StSzWa08], [@BhSt11] and [@PaShSt14]). Using the same reasoning as in Lemma \[first chern class\], we have the following Lemma. \[rational homology disks filling\] Suppose $T$ is a realizable type (P) graph. If $T$ can symplectic divisorial compactify a rational homology disk, then we have $n^T=10$. By mathematica, we find that there are only four type (P5) graphs $T^{(v)}$ that are not arising from dihedral resolution graph and satisfy $n^{T^{(v)}}=10$. Among these four, only three of them are realizable and they are listed in the following. In particular, it means that apart from these three, all other strongly realizable graphs in type (P5) not arising from dihedral resolution graph cannot sympelctic divisorial compactify a rational homology disk. For the following three (N3) graphs $T$, the corresponding three (P5) graphs $T^{(v)}$ satisfy $n^{T^{(v)}}=10$. $$\xymatrix{ \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3} \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_v \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-6} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2}_v \\ & &\bullet^{-2} \\ }$$ Note that since most of the graphs in type (P5) do not have conjugate, the contact structure on the boundary is non-standard. Therefore, the consideration above is not covered by [@StSzWa08], [@BhSt11] and [@PaShSt14] (See Remark \[non-standard contact structure\]). [99]{} M. Abouzaid and P. Seidel. 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ArXiv

[Ground States of Quantum Electrodynamics with Cutoffs ]{}\ $\;$\ [Toshimitsu Takaesu ]{}\ $\;$\ *Faculty of Science and Technology, Gunma University,\ Gunma, 371-8510, Japan* > **Abstract** In this paper, we investigate a system of quantum electrodynamics with cutoffs. The total Hamiltonian is defined on a tensor product of a fermion Fock space and a boson Fock. It is shown that, under spatially localized conditions and momentum regularity conditions, the total Hamiltonian has a ground state for all values of coupling constants. In particular, its multiplicity is finite.\ > $\; $\ > [MSC 2010 : 47A10, 81Q10. $\; $\ > key words : Fock spaces, Spectral analysis, Quantum Electrodynamics]{}. Introduction ============= $\;$ This articles is concerned with a system of quantum electrodynamics with cutoffs. In quantum field theory, the interactions of charged particles and photons are described by quantum electrodynamics. We consider the system of a massive Dirac field coupled to a radiation field. The radiation filed is quantized in the Coulomb gauge. In this system, the process of electron-positron pair production and annihilation occurs. We mathematically investigate the spectrum of the total Hamiltonian for the system. The Hilbert space for the system is defined by a tensor product of a fermion Fock space and boson Fock space, which is called a boson-fermion Fock space. The total Hamiltonian is given by $$\begin{aligned} {H_{\textrm{QED}}}= H_{{ \textrm{D} }}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}& + {\kappa_{\textrm{I}}}\sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}A_j ({\ensuremath{\mathbf{x}}} ) ) d{\ensuremath{\mathbf{x}}} \notag \\ & \; \; + {\kappa_{\textrm{II}}}\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}}) }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}} \notag \end{aligned}$$ on the Hilbert space. Here $H_{{ \textrm{D} }}$ and $H_{{\textrm{rad}}} $ denote the energy Hamiltonians of the Dirac field and radiation field, respectively, $\psi ({\ensuremath{\mathbf{x}}})$ the Dirac field operator, ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})=(A_j({\ensuremath{\mathbf{x}}}))_{j=1}^3$ the radiation field operator, ${\ensuremath{\mathbf{\alpha}}}=(\alpha_j)_{j=1}^3$ $4\times 4$ Dirac matrices, and ${\chi_{\textrm{I}} (\mathbf{x}) }$ and ${\chi_{\textrm{II}} (\mathbf{x}) }$ the spatial cutoffs. The constants ${\kappa_{\textrm{I}}}\in {\ensuremath{\mathbf{R}}}$ and ${\kappa_{\textrm{II}}}\in {\ensuremath{\mathbf{R}}}$ are called coupling constants. Ultraviolet cutoffs are imposed on ${ \psi ({\ensuremath{\mathbf{x}}}) }$ and ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})$, respectively.\ By making use of the spacial cutoffs and ultraviolet cutoffs, ${H_{\textrm{QED}}}$ is self-adjoint operator on the Hilbert space, and the spectrum of ${H_{\textrm{QED}}}$ is bounded from below. The main interest in this paper is the lower bound of the spectrum of ${H_{\textrm{QED}}}$. If the infimum of the spectrum of a self-adjoint operator is eigenvalue, the eigenvector is called ground state. The infimum of the spectrum of $H_{0}= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}}}$ is eigenvalue, but it is embedded in continuous spectrum. This is because the radiation field is a massless field. It is not clear that ${H_{\textrm{QED}}}$ has a ground state since the embedded eigenvalue is not stable when interactions are turned on.\ The ground state of ${H_{\textrm{QED}}}$ for sufficiently small values of coupling constants was proven in [@Ta09]. The aim of this paper is to prove that ${H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite. For the ground states of other QED models, Dimassi-Guillot [@DiGu03] and Barbaroux-Dimassi-Guillot [@BDG04] investigated the system of the Dirac field in external potential coupled to the radiation field. They proved the existence of the ground state of the total Hamiltonian with generalized interactions for sufficiently small values of coupling constants. As far as we know, the existence of the ground states for the systems of a fermionic field coupled to a massless bosonic field, which include QED models, has not been proven for all values of coupling constants until now.\ To prove the existence of the ground state of ${H_{\textrm{QED}}}$ for all values of coupling constants, we apply the methods for systems of particles coupled bosonic fields. The spectral analysis and scattering theory for these systems, which include the non-relativistic QED models, have been progressed since the middle of ’90s. The existence of the ground states was established by Arai-Hirokawa [@AH97], Bach-Fr[ö]{}hlich-Sigal [@BFS98; @BFS99], G[é]{}rard [@Ge00], Griesemer-Lieb-Loss [@GLL01], Lieb-Loss [@LiLo03], Spohn [@Sp98] and many researchers. The strategy is as follows. $\; $\ **\[1st Step\]** We introduce approximating Hamiltonians $H_{m}$, $m>0$. Physically, $m>0$ denotes the artificial mass of photon, and we call $H_{m}$ a massive Hamiltonian. To prove the existence of ground states of $H_{m}$, we use partition of unity on Fock space, which was developed by Dereziński-Gérard [@DeGe99]. We especially need the partitions of unity for both Dirac field and radiation field. By the partitions of unity and the Weyl sequence method, we prove that a positive spectral gap above the infimum of the spectrum exists for all values of coupling constants. From this, the existence of the ground states of $H_m$ for all values of coupling constants follows. $\; $\ **\[2nd Step\]** Let $\Psi_m $ be the ground state of $H_{m}$, $m>0$. Without loss of generality, we may assume that the $\Psi_m$ is normalized. Then, there exists a subsequence of $\{ \Psi_{m_j} \}_{j=1}^\infty$ with $m_{j+1} < m_{j}$, $j\in {\ensuremath{\mathbf{N}}}$, such that the weak limit of $ \{ \Psi_{m_j}\}_{j=1}^{\infty}$ exists. The key point is to show that the the weak limit is non-zero vector. To prove this, we consider a combined method of Gerard [@Ge00] and Griesemer-Lieb-Loss in [@GLL01]. We use the electron positron derivative bounds and photon derivative bounds. To derive these bounds, the argument of the spatially localization is needed. For the spatially localized conditions, we suppose $$\int_{{\mathbf{R}^{3} }} |{\ensuremath{\mathbf{x}}}| \, | \chi_{{\textrm{I}}}({\ensuremath{\mathbf{x}}}) | dx\, < \, \infty , \qquad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{| \chi_{{\textrm{II}}}({\ensuremath{\mathbf{x}}}) \chi_{{\textrm{II}}}({\ensuremath{\mathbf{y}}}) |}{|{\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{y}}}|} |{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }< \infty.$$ I addition, We imposed momentum regularity conditions on the Dirac field and radiation field, which include the infrared regularity condition $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{| {\ensuremath{\mathbf{k}}} |^{5} } {d \mathbf{k} }< \infty .$$ $\;$\ $\; $\ We briefly review the results for the systems of fermionic fields coupled bosonic fields. For QED models, the Gell-Mann - Low formula of ${H_{\textrm{QED}}}$ was obtained by Futakuchi-Usui [@FuUs14]. For the Yukawa model, which is the system for a massive Dirac field interacting with a massive Klein-Gordon field, the existence of the ground state was proven in [@Ta11]. The spectaral analysis for the the weak interaction models has been analyzed, and refer to Barbaroux-Faupin-Guillot [@BFG14], Guillot [@Gu15] and the reference therein. $\;$\ This paper is organized as follows. In section 2, full Fock spaces, fermion Fock spaces and boson Fock spaces are introduced, and Dirac field operators and radiation field operators are defined on a Fermion Fock space and boson Fock space, respectively. The total Hamiltonian is defined on a boson-fermion Fock space and the main theorem is stated. In Section 3, partitions of unity for the Dirac field and radiation field are investigated. Then the existence of the ground state of $H_{m}$ is proven. In section 4, the derivative bounds for electrons-positrons and photons are derived. In Section 5, we give the proof of the main theorem.\ Notations and Main Results ========================== Fock Spaces ----------- **(i) Full Fock Space**\ The full Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Z}}}$ is defined by $ {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})= \oplus_{n=0}^\infty ( {\otimes^{n}}{\ensuremath{\mathscr{Z}}})$ where ${\otimes^{n}}{\ensuremath{\mathscr{Z}}}$ is the $n$ fold tensor product of $Z$. The Fock vacuum is defined by $\Omega = \{1, 0,0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Z}}})$. Let $ {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}})$ be the set which consists of all linear operators on ${\ensuremath{\mathscr{Z}}}$. The functor of $Q\in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is defined by $\Gamma (Q) = \oplus_{n=0}^{\infty} \left( {\otimes^{n}}Q \right)$ and the second quantization of $T\in{\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$ is given by $ {\ensuremath{d\Gamma({T}) }} = \oplus_{n=0}^{\infty} \tilde{T}^{(n)} $ with $ \tilde{T}^{(n)} =\sum\limits_{j=1}^{n} ( ({\otimes}^{j-1} {{\small \text{1}}\hspace{-0.32em}1}) {\otimes}T {\otimes}( {\otimes}^{n-j} {{\small \text{1}}\hspace{-0.32em}1}) )$. The number operator is defined by $N= {\ensuremath{d\Gamma({{{\small \text{1}}\hspace{-0.32em}1}}) }} $.\ $\;$\ **(ii) Fermion Fock Space**\ The fermion Fock space over a complex Hilbert space ${\ensuremath{\mathscr{X}}}$ is defined by $ {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}})$ where $ {\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}}$ denotes the $n$-fold anti-symmetric tensor product of ${\ensuremath{\mathscr{X}}}$. The Fock vacuum is defined by $\Omega_{{\textrm{f}}} = \{1, 0, 0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{X}}})$. Let $T_{{\textrm{f}}}$ and $Q_{{\textrm{f}}}$ be linear operators on ${\ensuremath{\mathscr{X}}}$. We set ${\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{f}}}}) }}_{\upharpoonright {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})}$ and $\Gamma_{{\textrm{f}}}( Q_{{\textrm{f}}})= \Gamma (Q_{{\textrm{f}}})_{\upharpoonright {\mathscr{F}_{\textrm{f}} }( {\ensuremath{\mathscr{X}}})}$ where $ X_{\upharpoonright {\ensuremath{\mathscr{M}}}} $ is the restriction of the operator $X$ to the subspace ${\ensuremath{\mathscr{M}}} $. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $C^{\, \dagger}(f)$, $f \in {\ensuremath{\mathscr{X}}}$, is defined by $(C^{\, \dagger}(f) \Psi )^{(n)} = \sqrt{n} \, U_{\textrm{a}}^{ n}(f {\otimes}\Psi^{(n-1)})$, $n \geq 1$, and $(C^{\, \dagger }(f) \Psi )^{(0)} = 0 $ where $ U_{\textrm{a}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{X}}} $ to ${\otimes^{n}_{\textrm{a}}}{\ensuremath{\mathscr{X}}} $. The annihilation operator $C(f)$ is defined by $C(f)=(C^{\, \dagger}(f))^{\ast}$ where $X^{\ast} $ denotes the adjoint of the operator $X$. For each subspace ${\ensuremath{\mathscr{M}}} \subset {\ensuremath{\mathscr{X}}}$, the finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{f}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{M}}}) $ is defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $C^{\, \dagger}(f_{1}) , \cdots C^{\, \dagger}(f_{n}) \Omega_{{\textrm{f}}} $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical anti-commutation relations on ${\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})$ : $$\qquad \{ C(f) , \, C^{ \, \dagger}(f') \} = (f , f' ) , \quad \{ C^{ \, \dagger }(f) , C^{ \, \dagger }(f') \} = \{ C(f) , C (f') \} =0 , \quad f, f' \in {\ensuremath{\mathscr{X}}},$$ where $\{ X ,Y \} = XY +YX $. $\;$\ **(ii) Boson Fock Space**\ The boson Fock space over a complex Hilbert space ${\ensuremath{\mathscr{Y}}}$ is defined by $ {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})= \oplus_{n=0}^\infty ( {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}})$ where $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}}$ denotes the $ n$-fold symmetric tensor product of ${\ensuremath{\mathscr{Y}}}$. The Fock vacuum is given by $\Omega_{{\textrm{b}}} = \{1, 0, 0, \cdots \} \in {\ensuremath{\mathscr{F}}}( {\ensuremath{\mathscr{Y}}})$ Let $T_{{\textrm{b}}}$ and $Q_{{\textrm{b}}}$ be linear operators on ${\ensuremath{\mathscr{Y}}}$. Then we define ${\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}}}) }}= {\ensuremath{d\Gamma({T_{{\textrm{b}}} }) }}_{\upharpoonright {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})}$ and $\Gamma_{{\textrm{b}}}( Q)= \Gamma (Q_{{\textrm{b}}})_{\upharpoonright {\mathscr{F}_{\textrm{b}} }( {\ensuremath{\mathscr{Y}}})}$. The number operator is defined by $N_{{\textrm{f}}}= {\ensuremath{d\Gamma_{\textrm{f}}({{{\small \text{1}}\hspace{-0.32em}1}}) }} $. The creation operator $A^{\dagger}(g)$, $g \in {\ensuremath{\mathscr{Y}}}$, is defined by $(A^{\dagger}(g) \Phi )^{(n)} = \sqrt{n} \, U_{\textrm{s}}^{ n}(f {\otimes}\Phi^{(n-1)})$, $n \geq 1$, and $(A^{\dagger}(g) \Phi )^{(0)} = 0 $ where $ U_{\textrm{s}}^{ n} $ is the projection from $ {\otimes^{n}}{\ensuremath{\mathscr{Y}}} $ to $ {\otimes^{n}_{\textrm{s}}}{\ensuremath{\mathscr{Y}}} $. The annihilation operator $A(f)$ is defined by $A(g)= ( A^{\dagger}(g) )^{\ast} $. The finite particle space $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $ on the subspace ${\ensuremath{\mathscr{N}}} \subset {\ensuremath{\mathscr{Y}}}$ defined by the linear hull of $ \Omega_{{\textrm{f}}} $ and $A^{\, \dagger}(g_{1}) , \cdots A^{\, \dagger}(g_{n}) \Omega_{{\textrm{f}}} $, $j=1 , \cdots , n$, $n \in {\ensuremath{\mathbf{N}}}$. The creation and annihilation operators satisfy the canonical commutation relations on $ {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{N}}}) $: $$\qquad [ A(g),A^{ \dagger }(g ') ] = (g , g' ) , \quad [ A^{ \dagger }(g) , A^{ \dagger }(g') ] = [ A(g) , A (g') ] =0 , \quad g, g' \in {\ensuremath{\mathscr{Y}}} ,$$ where $[X ,Y]=XY-YX $.\ Dirac field ----------- Let ${\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} = {\mathscr{F}_{\textrm{f}} }(L^2( {\mathbf{R}^{3} };{\ensuremath{\mathbf{C}}}^4))$. The energy Hamiltonian of the Dirac field is defined by $$H_{{ \textrm{D} }} = {\ensuremath{d\Gamma_{\textrm{f}}({\omega_{\,M} }) }}$$ where $\omega_M ({\ensuremath{\mathbf{p}}}) = \sqrt{| {\ensuremath{\mathbf{p}}}|^2 + M^2 }$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, and $M>0$. Let $C^{\, \dagger } (^{t}(f_{1} , \cdots , f_{4}))$, $f_{l} \in L^2( {\mathbf{R}^{3} })$, $l=1, \cdots ,4$, be the creation operator on ${\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}$. For each $f \in L^2( {\mathbf{R}^{3} })$, we set $$\begin{aligned} &b_{1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t} (f,0,0,0) ), \quad \quad b_{-1/2}^{\dagger}(f) = C^{\, \dagger}({}^{t}(0,f,0,0) ), \\ &d_{1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,f,0) ), \quad \quad d_{-1/2}^{\, \dagger}(f) = C^{\, \dagger}({}^{t}(0,0,0,f) ) . \end{aligned}$$ We define $b_s ( f)$ and $d_{s} ( g)$ by the conjugate of $b_s^{\dagger} ( f)$ and $d_{s}^{ \, \dagger } ( g)$, respectively. Then the canonical anti-commutation relations $$\begin{aligned} & \{ b_s (f) ,b_{s'}^{\dagger} (g ) \} = \{ d_s (f) ,d_{s'}^{\, \dagger } (g ) \}= \delta_{s, s'} (f,g ), \quad \label{ACR1} \\ & \{ b_s (f) ,b_{s'} (g) \} =\{ d_s (f) ,d_{s'} (g) \}= \{ b_s (g) ,d_{s'}^{\, \dagger} (g) \} = 0 , \quad \label{ACR2}\end{aligned}$$ are satisfied and it holds that $$\|b_s (f) \| = \|b_s^{\dagger }(f) \| = \| f \| , \qquad \|d_s(g) \| = \|d_s^{\, \dagger }(g) \|= \| g \| . \label{bdNorm}$$ Let $h_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}})={\ensuremath{\mathbf{\alpha}}} \cdot {\ensuremath{\mathbf{p}}}+ M\beta $ be the Fourier transformed Dirac operator with $4 \times 4$ Dirac matrices $\boldsymbol{\alpha} = (\alpha^j)_{j=1}^3$ and $\beta $. Let ${\ensuremath{\mathbf{S}}}({\ensuremath{\mathbf{p}}})= {\ensuremath{\mathbf{S}}} \cdot {\ensuremath{\mathbf{p}}}$, ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, where ${\ensuremath{\mathbf{S}}}= -\frac{i}{4} \boldsymbol{\alpha} \wedge \boldsymbol{\alpha} $ is the spin angular momentum. The spinors $ u_{s}({\ensuremath{\mathbf{p}}} ) = (u_{s}^{l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4} \; $ and $v_{s}({\ensuremath{\mathbf{p}}} ) = (v_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) )_{l=1}^{4} \; $ are function which satisfy the following : $$\begin{aligned} \textbf{(D.1)}\; \; \; &h_{D} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}) = E_{M} ({\ensuremath{\mathbf{p}}}) u_{s} ({\ensuremath{\mathbf{p}}}), \quad h_{D} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}) = -E_{M} ({\ensuremath{\mathbf{p}}}) v_{s} ({\ensuremath{\mathbf{p}}}), \\ \textbf{(D.2)} \; \; \; & S({\ensuremath{\mathbf{p}}} ) u_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | u_{s} ({\ensuremath{\mathbf{p}}}), \quad S({\ensuremath{\mathbf{p}}} ) v_{s} ({\ensuremath{\mathbf{p}}}) = s | {\ensuremath{\mathbf{p}}} | v_{s} ({\ensuremath{\mathbf{p}}}) , \\ \textbf{(D.3)} \; \; \; & \sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} u_{s'}^{\,l} ({\ensuremath{\mathbf{p}}}' ) = \sum_{l=1}^4 v_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' ) =\delta_{s,s'} ,\quad \sum_{l=1}^4 u_{s}^{\, l} ({\ensuremath{\mathbf{p}}} )^{\ast} v_{s'}^{\, l} ({\ensuremath{\mathbf{p}}}' ) = 0 . \end{aligned}$$ \[exa1\] We review the example of spinors in the standard representation (see [@Tha] ; Section 1). The Pauli matrices are defined by $ \sigma_{1} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) $, $ \sigma_{2} = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right) $ and $ \sigma_{3} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right) $. Then, the Dirac marices are $ \alpha = \left( \begin{array}{cc} O & \sigma \\ \sigma & O \end{array} \right) $, $ \beta = \left( \begin{array}{cc} {{\small \text{1}}\hspace{-0.32em}1}& O \\ O & -{{\small \text{1}}\hspace{-0.32em}1}\end{array} \right) $, and the spin angular momentum is ${\ensuremath{\mathbf{S}}} = \frac{1}{2} \left( \begin{array}{cc} \ \sigma & O \\ O & \sigma \end{array} \right) $. Let $ O_{\textrm{SR}} = \{ {\ensuremath{\mathbf{p}}} = (p^{1} , p^{2}, p^{3} ) \in {\mathbf{R}^{3} }\; \left| \frac{}{} \right. |{\ensuremath{\mathbf{p}}}| -p^3 = 0 \} $. We see that the Lebesgue measure of $O_{\textrm{SR} }$ is zero. We set $$\eta_{+} ({\ensuremath{\mathbf{p}}} ) = \left\{ \begin{array}{c} \frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | ( | {\ensuremath{\mathbf{p}}} | -p^{3} ) } } \begin{pmatrix} p^{1} - i p^{2} \\ | {\ensuremath{\mathbf{p}}} | - p^{3} \end{pmatrix} , \; {\ensuremath{\mathbf{p}}} \notin O_{\textrm{SR} } , \\ \qquad \qquad \begin{pmatrix} 1 \\ 0 \end{pmatrix} , \quad \quad \qquad {\ensuremath{\mathbf{p}}} \in O_{\textrm{SR} } , \end{array} \right. \; \; \eta_{-} ({\ensuremath{\mathbf{p}}} ) = \left\{ \begin{array}{c} \frac{1}{\sqrt{ 2 | {\ensuremath{\mathbf{p}}} | ( | {\ensuremath{\mathbf{p}}} | - p^{3} ) } } \begin{pmatrix} p^{3} - | {\ensuremath{\mathbf{p}}} | \\ p^{1} + i p^{2} \end{pmatrix} , \, {\ensuremath{\mathbf{p}}} \notin O_{\textrm{SR} } , \\ \qquad \quad \begin{pmatrix} 0 \\ 1 \end{pmatrix} , \quad \qquad \qquad {\ensuremath{\mathbf{p}}} \in O_{\textrm{SR} } . \end{array} \right.$$ Let $$u_{\pm1/2} ({\ensuremath{\mathbf{p}}}) = \begin{pmatrix} \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \\ \pm \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \end{pmatrix} , \qquad v_{\pm 1/2} ({\ensuremath{\mathbf{p}}}) = \begin{pmatrix} \mp \lambda_{-} ({\ensuremath{\mathbf{p}}} ) \phi_{\pm} ({\ensuremath{\mathbf{p}}}) \\ \pm \lambda_{+} ({\ensuremath{\mathbf{p}}} ) \eta_{\pm} ({\ensuremath{\mathbf{p}}}) \end{pmatrix} ,$$ with $ \lambda_{\pm} ({\ensuremath{\mathbf{p}}}) = \frac{1}{\sqrt{2}} \sqrt{ 1 \pm M \, E_{M} ({\ensuremath{\mathbf{p}}})^{-1 }}$. Here note that $ u_{s} $ and $ v_{s} $ satisfy $ u_{s} , v_s \in \oplus^4 ( C^{1} ( {\mathbf{R}^{3} }\backslash O_{\textrm{SR}} ) ) $.\ $\;$ $\;$\ The Dirac field operator $ \psi({\ensuremath{\mathbf{x}}}) = {}^{t} (\psi_{1}({\ensuremath{\mathbf{x}}}) , \cdots , \psi_{4}({\ensuremath{\mathbf{x}}}))$ is defined by $$\qquad \qquad \quad \psi_{l}({\ensuremath{\mathbf{x}}}) = \sum_{s=\pm 1/2} \left( b_s ( f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ) + d^{\, \dagger}_s ( g_{s , {\ensuremath{\mathbf{x}}}}^l) \frac{}{} \right) , \quad \qquad l=1, \cdots , 4 ,$$ where $f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}})= f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $f_{s }^{\, l} ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) u_s^{l}({\ensuremath{\mathbf{p}}})$ and $g_{s , {\ensuremath{\mathbf{x}}}}^l ({\ensuremath{\mathbf{p}}})= g_{s }^l ({\ensuremath{\mathbf{p}}})e^{-{\ensuremath{\mathbf{p}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $g_{s }^l ({\ensuremath{\mathbf{p}}})= \frac{1}{\sqrt{ (2 \pi )^3 }}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \tilde{v}_s^{l}({\ensuremath{\mathbf{p}}})$ and $\tilde{v}_s^{\, l}({\ensuremath{\mathbf{p}}})= v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}})$. Here $\chi_{{ \textrm{D} }} $ satisfy the following condition.\ > **(A.1 ; Ultraviolet Cutoff for Dirac Field)** $$\int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty .$$ Then it holds that $$\qquad \|\psi_l ({\ensuremath{\mathbf{x}}}) \| \leq c_{\, { \textrm{D} }}^{ \, l } , \label{psiBound}$$ where $ c_{\, { \textrm{D} }}^{\, l } = \frac{1}{\sqrt{(2\pi)^3}} \sum\limits_{s= \pm 1/2} \left( \| \chi_{{ \textrm{D} }} \, u_{s}^{\,l} \| + \| \chi_{{ \textrm{D} }} \, \tilde{v}_{s}^{\, l} \| \right)$, $l=1 , \cdots , 4.$\ Radiation Field in the Coulomb Gauge ------------------------------------ Let ${\mathscr{F}_{\textrm{rad}}}= {\mathscr{F}_{\textrm{b}} }( \oplus_{r= 1,2} L^2( {\mathbf{R}^{3} }))$. The free Hamiltonian is defined by $${H_{\textrm{rad}}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega }) }}$$ where $\omega ({\ensuremath{\mathbf{k}}}) = |{\ensuremath{\mathbf{k}}}|$, ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. Let $A^{\ast }(h_1 , h_2 )$, $h_{r} \in L^2( {\mathbf{R}^{3} })$, $r=1,2$, be the creation operators on ${\mathscr{F}_{\textrm{rad}}}$. Let $$\qquad a^{\dagger }_{1} (h ) = A (( h, 0)) , \quad a^{\dagger }_{2} (h )= A (( 0, h )) , \qquad h \in L^2( {\mathbf{R}^{3} }) ,$$ and $a_{r}(h')=(a^{\dagger} (h'))^{\ast}$, $h' \in L^2( {\mathbf{R}^{3} }) $, $r=1,2$. The creation operators and annihilation operators satisfy the canonical commutation relations $$\begin{aligned} & [a_r (h) ,a_{r'}^{\dagger} (h') ] = \delta_{r, r'} (h, \, h' ), \; \; \label{radCCR1} \\ & [a_r (h) ,a_{r'} ( h' ) ] = [a_r^{\dagger } (h' ) ,a_{r'}^{\dagger} (h') ] = 0 , \label{radCCR2}\end{aligned}$$ on ${\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}}({\ensuremath{\mathscr{M}}})$ where ${\ensuremath{\mathscr{M}}} $ is a subspace of $\oplus_{r= 1,2} L^2( {\mathbf{R}^{3} }) $. For all $h \in {\ensuremath{\mathscr{D}}}(\omega^{-1/2}) $, it follows that $$\quad \| a_r (h ) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq \| \frac{h}{\sqrt{\omega }} \| , \quad \| a^{\dagger }_r (h ) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq \| \frac{h}{\sqrt{\omega }} \| + \| h \| . \label{radfiedBound}$$ The polarization vectors ${\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) = (e_r^{j} ({\ensuremath{\mathbf{k}}}))$, $r=1,2$, satisfy the following relations. $$\textbf{(R.1)} \; \qquad {\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}}) \cdot {\ensuremath{\mathbf{e}}}_{r'} ({\ensuremath{\mathbf{k}}}) =0 , \quad {\ensuremath{\mathbf{e}}}_{r} ({\ensuremath{\mathbf{k}}})\cdot {\ensuremath{\mathbf{k}}} = 0 , \quad \quad {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} . \notag$$ $\;$ \[exa2\] We check the example of the polarization vectors. For all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} $, we set $$\qquad {\ensuremath{\mathbf{e}}}_1({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{(k^1)^2 + (k^2) ^2}} \left( \begin{array}{c} - k^{2} \\ k^{1} \\ 0 \end{array} \right) , \; \; \; {\ensuremath{\mathbf{e}}}_2({\ensuremath{\mathbf{k}}}) =\frac{1}{|{\ensuremath{\mathbf{k}}}|\sqrt{k_1^2 +k_2^2}} \left( \begin{array}{c} k^{1} k^3 \\ k^{2} k^3 \\ - (k^1)^2 - (k^2)^2 \end{array} \right) . \quad$$ Then **(R.1)** is satisfied. Here it is noted that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash \{ {\ensuremath{\mathbf{0}}} \} ))$, $r=1,2$. $\;$\ The radiation field operator ${\ensuremath{\mathbf{A}}}({\ensuremath{\mathbf{x}}})= ( A_j ({\ensuremath{\mathbf{x}}}))_{j=1}^3$ is defined by $$A_j({\ensuremath{\mathbf{x}}}) = \sum_{r=1,2} \left( a_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) + a^{\dagger}_r (h_{r , {\ensuremath{\mathbf{x}}}}^j) \frac{}{} \right)$$ where $h_{r , {\ensuremath{\mathbf{x}}}}^j ({\ensuremath{\mathbf{k}}})= h_{r }^j ({\ensuremath{\mathbf{k}}})e^{-{\ensuremath{\mathbf{k}}}\cdot {\ensuremath{\mathbf{x}}} }$ with $h_{r }^j ({\ensuremath{\mathbf{k}}})= \frac{1}{\sqrt{(2 \pi )^3}} \frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) e_r^{j}({\ensuremath{\mathbf{k}}})}{\sqrt{2 \omega({\ensuremath{\mathbf{k}}})}}$, and $\chi_{{\textrm{rad}}}$ satisfy the following condition.\ > **(A.2 : Ultraviolet Cutoff for Radiation Field)** $$\qquad > \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ \omega({\ensuremath{\mathbf{k}}})^{l}} {d \mathbf{k} }< \infty , \quad l= 1,2 .$$ Then $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}}}+1 )^{-1/2} \| \leq c^{\, j}_{{\textrm{rad}}}$$ where $c^{\,j}_{{\textrm{rad}}}= \frac{1}{\sqrt{(2\pi )^3}}\sum\limits_{r=1,2}\left( \sqrt{2} \| \frac{\chi_{{\textrm{rad}}} e_r^j }{ \omega} \| + \| \frac{\chi_{{\textrm{rad}}} e_r^j }{\sqrt{2 \omega}} \| \right) $.\ Total Hamiltonian and Main Theorem ---------------------------------- We define the system of the Dirac field interacting with the radiation field. The Hilbert space for the system is defined by ${\mathscr{F}_{\textrm{QED}}}= {\mathscr{F}_{\textrm{Dirac}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$. The free Hamiltonian is defined by $$H_0= H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}}\notag$$ on the domain ${\ensuremath{\mathscr{D}}}(H_{0}) = {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}}})$. To define the interactions, we introduce spatial cutoff $\chi_{{\textrm{I}}}$ and $\chi_{{\textrm{II}}}$, which satisfy the condition below.\ > **(A.3 : Spatial Cutoff )** $$\int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty , \qquad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} > \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} {d \mathbf{x} }{d \mathbf{y} }< \infty .$$ First we define a functional on $ {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} $ by $$\qquad \qquad \ell_{{\textrm{I}}}(\Phi , \Psi ) = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , ({\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}A_j ({\ensuremath{\mathbf{x}}}) ) \Psi \right) , \; \; \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} , \, \Psi \in {\ensuremath{\mathscr{D}}}(H_0 ) ,$$ where $ {\psi^{\dagger} (\mathbf{x}) }= ( \psi_{1}({\ensuremath{\mathbf{x}}})^{\ast} , \cdots , \psi_{4}({\ensuremath{\mathbf{x}}})^{\ast} ) $. We see that $$| \ell_{{\textrm{I}}}(\Phi , \Psi ) | \leq \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }\right) \, \sum\limits_{j=1}^3 \, \sum_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\, j } \, \| \Phi \| \, \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \| . \notag$$ By the Riesz representation theorem, we can define the operator ${H_{\textrm{I}}}$ which satisfy $ (\Phi , {H_{\textrm{I}}}\Psi ) = \ell_{{\textrm{I}}}(\Phi , \Psi ) $ and $$\| {H_{\textrm{I}}}\Psi \| \leq c_{\, {\textrm{I}}} \, \| {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}({H_{\textrm{rad}}}+1)^{1/2} \Psi \| , \label{HIbound}$$ where $ c_{\, {\textrm{I}}} = \| \chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \, \sum\limits_{l,l'=1}^4 |\alpha_{l,l'}^j | c_{\, { \textrm{D} }}^{\, l } c_{\, { \textrm{D} }}^{ \, l' } c_{{\textrm{rad}}}^{\, j }$. By the spectral decomposition theorem, it is proven that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0} \Psi \| +c_{\textrm{I}}\left( \frac{1 }{2 \epsilon} +1 \right) \| \Psi \| . \label{HIbound'}$$ Next we define a functional on $ {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} $ by $$\ell_{{\textrm{II}}}(\Phi , \Psi ) = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( \Phi , \left( {\psi^{\dagger} (\mathbf{x}) }{ \psi ({\ensuremath{\mathbf{x}}}) }{\psi^{\dagger} (\mathbf{y}) }{ \psi ({\ensuremath{\mathbf{y}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) \Psi \right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}} , \; \; \Phi , \, \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \\$$ We see that $$|\ell_{{\textrm{II}}}(\Phi , \Psi ) | \leq \left( \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \left| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right| {d \mathbf{x} }{d \mathbf{y} }\right) \, \sum_{l , l' =1}^4 (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2 \| \Phi \| \, \| \Psi \| . \notag$$ Then, by the Riesz representation theorem, we can define an operator ${H_{\textrm{II}}}$ satisfying $ (\Phi , {H_{\textrm{II}}}\Psi ) = \ell_{{\textrm{II}}}(\Phi , \Psi )$ and $$\|{H_{\textrm{II}}}\| \leq c_{{\textrm{II}}} , \label{HIIbound}$$ where $ c_{{\textrm{II}}} = \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{ |{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \right\|_{L^1} \, \sum\limits_{l , l' =1}^4 (c_{\, { \textrm{D} }}^{ \, l} c_{\, { \textrm{D} }}^{\, l'})^2$. By (\[HIbound’\]) and (\[HIIbound\]), it holds that $$\| ({\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}) \Psi \| \leq c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\epsilon \| H_0 \Psi \| + \left( c_{{\textrm{I}}} {\kappa_{\textrm{I}}}\left( \frac{1 }{2 \epsilon} +1 \right) + c_{{\textrm{II}}} {\kappa_{\textrm{II}}}\right) \| \Psi \| . \notag$$ Then the Kato-Rellich theorem yields that that ${H_{\textrm{QED}}}$ is self-adjoint on ${\ensuremath{\mathscr{D}}}(H_{0})$ and essentially self-djoint on any core of $H_{0}$. Hence, in particular, ${H_{\textrm{QED}}}$ is essentially self-adjoin on $${\ensuremath{\mathscr{D}}}_{0} = {\mathscr{F}_{\textrm{Dir}}}^{\, {\textrm{fin}}}{{\ensuremath{\mathscr{D}}}(\omega_{\, M})} \hat{{\otimes}}{\mathscr{F}_{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}}(\omega ))$$ where $\hat{{\otimes}}$ denotes the algebraic tensor product. $\; $\ To prove the existence of the ground state of ${H_{\textrm{QED}}}$, we suppose additional conditions below. > **(A.4 : Spatial Localization)** $$\int_{{\mathbf{R}^{3} }} |{\ensuremath{\mathbf{x}}}|| {\chi_{\textrm{I}} (\mathbf{x}) }| {d \mathbf{x} }< \infty , \quad \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} > \frac{|{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|}|{\ensuremath{\mathbf{x}}}| {d \mathbf{x} }{d \mathbf{y} }< \infty .$$ > **(A.5 : Momentum Regularity Condition for Dirac Field)**\ > There exists a subset $O_{{ \textrm{D} }} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that $u_{s}, v_{s} \in \oplus^4 \, ( C^1 ({\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }})) $, $s= \pm 1/2$. $\chi_{{ \textrm{D} }} \in C^{1}({\mathbf{R}^{3} }) $, and it satisfies that $$\int_{{\mathbf{R}^{3} }} | \partial_{p^{\nu}}\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty , > \; \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} u_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty , \; \int_{{\mathbf{R}^{3} }} | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p }}}) \partial_{p^{\nu }} v_{s}^{\, l} (-{\ensuremath{\mathbf{p}}}) |^2 d {\ensuremath{\mathbf{p}}} > < \infty ,$$ for all $\nu =1, 2,3$, $ l = 1 , \cdots ,4$, $ s= \pm 1/2 $. > **(A.6 : Momentum Regularity Condition for Radiation Field )**\ > There exists a subset $O_{{\textrm{rad}}} \subset {\mathbf{R}^{3} }$ with Lebesgue measure zero such that ${\ensuremath{\mathbf{e}}}_{r} \in \oplus^3 \, ( C^{1}({\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}} ))$, $r=1,2$, where $O_{{\textrm{rad}}} $. $\chi_{{\textrm{rad}}} \in C^{1}({\mathbf{R}^{3} })$ and it satisfies that $$\int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}}|^5} {d \mathbf{k} }< \infty , \quad > \int_{{\mathbf{R}^{3} }} \frac{ | \partial_{k^\nu }\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |^2}{ | {\ensuremath{\mathbf{k}}} |^3} {d \mathbf{k} }< \infty , \quad \int_{{\mathbf{R}^{3} }} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu } e_{r}^{j} ({\ensuremath{\mathbf{k}}}) |^2}{ |{\ensuremath{\mathbf{k}}}|^3} {d \mathbf{k} }< \infty ,$$ for all $\nu =1, 2, 3$, $j=1, 2, 3$, $r=1,2$. \[exa3\] Examples of $O_{{ \textrm{D} }}$ and $O_{{\textrm{rad}}}$ in **(A.5)** and **(A.6)** are as follows. In the case of the standard representation, $O_{{ \textrm{D} }} = O_{\textrm{SR}}$ where $ O_{\textrm{SR}} $ is defined in Remark \[exa1\]. For the polarization vectors considered in Remark \[exa2\], $O_{{\textrm{rad}}} = \{ {\ensuremath{\mathbf{0}}} \}$. $\; $\ The main theorem in this paper is as follows. \[Main-Theorem\] **(Existence of a Ground State)**\ Suppose **(A.1)** - **(A.6)**. Then $ {H_{\textrm{QED}}}$ has a ground state for all values of coupling constants. In particular, its multiplicity is finite. Ground States of Massive case ============================= In this section, we consider a massive Hamiltonian defined by $$H_m = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}, \notag$$ where ${H_{\textrm{rad}, \, m}}= {\ensuremath{d\Gamma_{\textrm{b}}({\omega_{m}}) }}$ with $\omega_m ({\ensuremath{\mathbf{k}}})=\sqrt{{\ensuremath{\mathbf{k}}}^2 + m^2} $, $m>0$.\ Fock Spaces on Direct Sum of Hilbert Spaces ------------------------------------------- $\, $ We review basic properties of Fock spaces on direct sum of Hilbert spaces. These are useful for constructing partitions of unity on Fock spaces (see, Dereziński-Gérard [@DeGe99]). $\, $\ **(i) Full Fock Space on** ${\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$\ Let $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $, $ Z_0 , Z_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $, where ${\ensuremath{\mathscr{Z}}}$ is a complex Hilbert space. We consider $ Z = \left[ \begin{array}{c} Z_0 \\ Z_\infty \end{array} \right] $ is an operator $ {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ which acts for $$\qquad \qquad Z h = \left[ \begin{array}{c} Z_0 \, h \\ Z_\infty \, h \end{array} \right] , \quad \; h \in {\ensuremath{\mathscr{D}}} (Z_{0}) \cap {\ensuremath{\mathscr{D}}} (Z_{\infty}) .$$ Let $J = \left[ \begin{array}{c} J_0 \, \\ J_\infty \, \end{array} \right]$, $ J_0 , J_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $ and $ B = \left[ \begin{array}{c} B_0 \, \\ B_\infty \, \end{array} \right] $, $ B_0 , B_\infty \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Z}}}) $. We define $d \Gamma ( J , B) : {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}}) \to {\ensuremath{\mathscr{F}}} ({\ensuremath{\mathscr{Z}}}) \oplus {\ensuremath{\mathscr{F}}}({\ensuremath{\mathscr{Z}}})$ by $$d \Gamma (J , B ) = \oplus_{n=0}^{\infty} \left( \sum\limits_{j=1}^n ( {\otimes}^{j-1} J ) {\otimes}B {\otimes}( {\otimes}^{n-j} J ) \right) .$$ If $B_{0}$ and $B_{\infty}$ are bounded, and $J_{0}^{\ast}J_0 + J_{\infty}^{\ast}J_{\infty} \leq 1 $, it holds that $$\| d \Gamma ( J, B ) (N +1)^{-1} \| \leq \sqrt{ \|B_{0} \|^2 + \| B_{\infty} \|^2 } . \label{3.1.1}$$ Let $T \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Z}}})$. Then it holds that $$\Gamma (J ) {\ensuremath{d\Gamma({T}) }} = d \Gamma \left( \left[ \begin{array}{cc} T & 0 \\ 0 & T \end{array} \right] \right) \Gamma ( J ) + d \Gamma ( J ,\tilde{\text{ad}}_{T}( J ) ) , \label{3.1.2}$$ where $ \tilde{\text{ad}}_{T}(J) : {\ensuremath{\mathscr{Z}}} \to {\ensuremath{\mathscr{Z}}} \oplus {\ensuremath{\mathscr{Z}}}$ is defined by $$\qquad \qquad \quad \tilde{\text{ad}}_{T}(J) h=\left[ \begin{array}{c} [T ,J_0 ] h \\ {[} T , J_\infty {]} h \end{array} \right] , \quad h \in {\ensuremath{\mathscr{D}}}([T ,J_0 ]) \cap {\ensuremath{\mathscr{D}}}([T ,J_\infty ] ) .$$ $\;$\ **(ii) Fermion Fock Space on** ${\ensuremath{\mathscr{X}}} \oplus {\ensuremath{\mathscr{X}}}$\ Let ${\ensuremath{\mathscr{X}}}$ be a complex Hilbert space. Let $ J_{{\textrm{f}}} = \left[ \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } \, \\ J_{{\textrm{f}}}^{\, \infty} \, \end{array} \right] $, $ J_{{\textrm{f}}}^{\, 0 } , J_{{\textrm{f}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $ and $ B_{{\textrm{f}}} = \left[ \begin{array}{c} B_{{\textrm{f}}}^{\, 0 } \, \\ B_{{\textrm{f}}}^{\, \infty}\, \end{array} \right] $, $ B_{{\textrm{f}}}^{\, 0 }, B^{\, \infty}_{{\textrm{f}}} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{X}}}) $. We set ${\ensuremath{d\Gamma_{\textrm{f}}({ J_{{\textrm{f}}} , B_{{\textrm{f}}} }) }} = {\ensuremath{d\Gamma({ J_{{\textrm{f}}} , B_{{\textrm{f}}} }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})} $. Suppose that $B_{{\textrm{f}}}^{\, 0 } $ and $B^{\, \infty}_{{\textrm{f}}} $ are bounded, and $ (J_{{\textrm{f}}}^{\, 0 })^\ast J_{{\textrm{f}}}^{\, 0 } + (J_{{\textrm{f}}}^{\, \infty})^{\ast} J_{{\textrm{f}}}^{\, \infty} \leq 1 $. By (\[3.1.1\]), it holds that $$\| d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} , B_{{\textrm{f}}} ) (N_{{\textrm{f}}} +1)^{-1} \| \leq \sqrt{ \| B_{{\textrm{f}}}^{\, 0 } \|^2 + \| B_{{\textrm{f}}}^{\, \infty} \|^2 } . \label{3.1.3}$$ Let $T_{{\textrm{f}}} \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{X}}})$. From (\[3.1.2\]), it holds that $${\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) {\ensuremath{d\Gamma_{\textrm{f}}({T_{{\textrm{f}}} }) }} = d {\Gamma_{\textrm{f}}}\left( \left[ \begin{array}{cc} T_{{\textrm{f}}} & 0 \\ 0 & T_{{\textrm{f}}} \end{array} \right] \right) {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{f}}}( J_{{\textrm{f}}} ,\tilde{\text{ad}}_{T_{{\textrm{f}}} }( J_{{\textrm{f}}} ) ). \label{3.1.4}$$ Let $ C(f)$ and $C^{ \, \dagger} (f)$, $f \in {\ensuremath{\mathscr{X}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{f}} }({\ensuremath{\mathscr{X}}})$, respectively. Then it follows that $$\begin{aligned} & \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) C (f) = C \left( \left[ \begin{array}{c} f \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) + \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) \, C \left( (1-( J_{{\textrm{f}}}^{\, 0 })^{\ast})f \frac{}{}\right) , \label{3.1.5} \\ & \Gamma ( J_{{\textrm{f}}} ) C^{\, \dagger} (f) = C^{\, \dagger} \left( \left[ \begin{array}{c} f \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) + C^{\, \dagger} \left( \left[ \begin{array}{c} J_{{\textrm{f}}}^{\, 0 } -1 \\ J_{{\textrm{f}}}^{\, \infty} \end{array} \right] f \right) \Gamma_{{\textrm{f}}} ( J_{{\textrm{f}}} ) . \label{3.1.6}\end{aligned}$$ $\;$\ $\; $\ **(iii) Boson Fock Space on** ${\ensuremath{\mathscr{Y}}} \oplus {\ensuremath{\mathscr{Y}}}$\ Let ${\ensuremath{\mathscr{Y}}}$ be a complex Hilbert space. Let $ J_{{\textrm{b}}} = \left[ \begin{array}{c} J_{{\textrm{b}}}^{\, 0 } \, \\ J_{{\textrm{b}}}^{\, \infty} \, \end{array} \right] $, $ J_{{\textrm{b}}}^{\, 0 } , J_{{\textrm{b}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $ and $ B_{{\textrm{b}}} = \left[ \begin{array}{c} B_{{\textrm{b}}}^{\, 0 } \, \\ B_{{\textrm{b}}}^{\, \infty} \, \end{array} \right] $, $ B_{{\textrm{b}}}^{\, 0 }, B_{{\textrm{b}}}^{\, \infty} \in {\ensuremath{\mathscr{L}}} ({\ensuremath{\mathscr{Y}}}) $. We define ${\ensuremath{d\Gamma_{\textrm{b}}({ J_{{\textrm{b}}} , B_{{\textrm{b}}} }) }} = {\ensuremath{d\Gamma({ J_{{\textrm{b}}} , B_{{\textrm{b}}} }) }}_{{\upharpoonright}{\mathscr{F}_{\textrm{b}} }({\ensuremath{\mathscr{Y}}})} $. Assume that $B_{{\textrm{b}}}^{\, 0 } $ and $ B_{{\textrm{b}}}^{\, \infty} $ are bounded, and $ (J_{{\textrm{b}}}^{\, 0 })^\ast J_{{\textrm{b}}}^{\, 0 } + (J_{{\textrm{b}}}^{\, \infty})^{\ast} J_{{\textrm{b}}}^{\, \infty} \leq 1 $. By (\[3.1.1\]), it follows that $$\| d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} , B_{{\textrm{b}}} ) (N_{{\textrm{b}}} +1)^{-1} \| \leq \sqrt{ \| B_{{\textrm{b}}}^{\, 0 } \|^2 + \| B_{{\textrm{b}}}^{\, \infty} \|^2 } . \label{3.1.7}$$ Let $ T_{{\textrm{b}}} \in {\ensuremath{\mathscr{L}}}({\ensuremath{\mathscr{Y}}})$. Then (\[3.1.2\]) yields that $${\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) {\ensuremath{d\Gamma_{\textrm{b}}({T_{{\textrm{b}}} }) }} = d {\Gamma_{\textrm{b}}}\left( \left[ \begin{array}{cc} T_{{\textrm{b}}} & 0 \\ 0 & T_{{\textrm{b}}} \end{array} \right] \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{f}}} ) + d {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ,\tilde{\text{ad}}_{T_{{\textrm{b}}} }( J_{{\textrm{b}}} ) ). \label{3.1.8}$$ Let $ A(g)$ and $A^{\dagger} (g)$, $g \in {\ensuremath{\mathscr{Y}}}$, be the annihilation and creation operators on ${\mathscr{F}_{\textrm{b}} }({\ensuremath{\mathscr{Y}}})$, respectively. Then it follows that $$\begin{aligned} & \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) A (g) = A \left( \left[ \begin{array}{c} g \\ 0 \end{array} \right] \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) + {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) \, A \left( (1-( J_{{\textrm{b}}}^{\, 0 } )^{\ast}) g \frac{}{}\right) , \label{3.1.9} \\ & {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) A^{\dagger } (g) = A^{\dagger} \left( \left[ \begin{array}{c} g \\ 0 \end{array} \right] \right) \Gamma_{{\textrm{b}}} ( J_{{\textrm{b}}} ) + A^{\dagger } \left( \left[ \begin{array}{c} J_{{\textrm{b}}}^{\, 0 } -1 \\ J_{{\textrm{b}}}^{\, \infty } \end{array} \right] g \right) {\Gamma_{\textrm{b}}}( J_{{\textrm{b}}} ) . \label{3.1.10}\end{aligned}$$ Partition of Unity for the Dirac Field -------------------------------------- We construct a partition of unity for the Dirac field. For general properties of partition of unity for fermionic fields, refer to Ammari [@Am04]. $\;$\ Let $$\quad c_{ \tau , s} (f) = \left\{ \begin{array}{c} b_{s}(f), \; \; \tau = + , \\ d_{s}(f) , \; \;\tau = - . \end{array} \right.$$ Let $U_{{\textrm{f}}}: {\mathscr{F}_{\textrm{f}} }\left( L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4 ) \oplus L^2( {\mathbf{R}^{3} }_{\bf{p}} ; {\ensuremath{\mathbf{C}}}^4 ) \right) \to {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} $ be an isometric operator which satisfy $U_{{\textrm{f}}} \, \Omega_{{ \textrm{D} }} =\Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} $ and $$\begin{aligned} &U_{{\textrm{f}}} \; c^{\dagger }_{\tau_1, s_1} \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots c^{\dagger }_{\tau_1 , s_1} \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) \Omega_{{ \textrm{D} }} \notag \\ & = \left( c^{\dagger}_{\tau_1 , s_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger}_{\tau_1 , s_1}(g_1) \frac{}{}\right) \cdots \left( c^{\dagger }_{\tau_n, s_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger}_{\tau_n , r_n}(g_n) \frac{}{} \right) \Omega_{{ \textrm{D} }} {\otimes}\Omega_{{ \textrm{D} }} . \notag\end{aligned}$$ Here note that $(-1)^{N_{{ \textrm{D} }}}\Psi = (-1)^n \Psi $ for the vector of the form $ \Psi = c^\dagger_{\tau_1 , s_1}(f_1) \cdots c^\dagger_{\tau_n, s_n} (f_n) \Omega_{{ \textrm{D} }} $, $f_{j} \in L^2 ({\mathbf{R}^{3} }) $, $j= 1, \cdots ,n$, $n \in {\ensuremath{\mathbf{N}}}$. Let $j_{0} , j_{\infty} \in C^{\, \infty} ({\ensuremath{\mathbf{R}}})$. We assume that $j_0 \geq 0 $, $j_{\infty} \geq 0$, $j_{0} ({\ensuremath{\mathbf{x}}})^2 + j_{\infty} ({\ensuremath{\mathbf{x}}})^2 =1$, $j_{0}({\ensuremath{\mathbf{x}}})=1 $ for $|{\ensuremath{\mathbf{x}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{x}}})=0 $ for $|{\ensuremath{\mathbf{x}}}| \geq 2 $. Let $j_{{\textrm{f}}, R } = \left[ \begin{array}{c} j_{{\textrm{f}}, R }^{\, 0} \\ j_{{\textrm{f}}, R }^{\, \infty} \end{array} \right]$ where $ j_{{\textrm{f}}, R }^{\, 0}= j_{0} (\frac{-i \, {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R})$ and $ j_{{\textrm{f}}, R }^{\, \infty}= j_{\infty} (\frac{-i \, {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{p}}}}}}}}{R}) $ with $\nabla_{{\ensuremath{\mathbf{p}}}}= (\partial_{p^1}, \partial_{p^2},\partial_{p^3} ) $.\ $\;$\ Let $X_{{\textrm{f}}, R} : {\mathscr{F}_{\textrm{Dir}}}\to {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} $ defined by $$\quad X_{{\textrm{f}}, R} = \, U_{{\textrm{f}}} \, \Gamma_{{\textrm{f}}}(j_{{\textrm{f}}, R}) . \notag$$ From (\[3.1.4\])-(\[3.1.6\]), it holds that $$\begin{aligned} & X_{{\textrm{f}}, R} \, H_{{ \textrm{D} }} = \left( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} \frac{}{} \right) X_{{\textrm{f}}, R} \, + U_{{\textrm{f}}} \, d \Gamma_{{\textrm{f}}} (j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{M} }(j_{{\textrm{f}},R})) , \label{3.2.1} \\ & X_{{\textrm{f}}, R} \, c_{\tau , s}(f) = (c_{\tau, s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \, + X_{{\textrm{f}}, R} \, c_{\tau , s}((1-j_{{\textrm{f}}, R}^{\, 0} ) f) , \label{3.2.2} \\ & X_{{\textrm{f}}, R} \, c^{\dagger}_{\tau , s }(f) = ( c^\ast_{\tau , s } (f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} + \left( c^{\dagger}_{\tau , s } ((j_{{\textrm{f}}, R}^{\, 0} -1 )f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}c^{\dagger }_{\tau , s }(j_{{\textrm{f}}, R}^{\, \infty} f ) \right) X_{{\textrm{f}}, R} . \label{3.2.3}\end{aligned}$$ \[lemma32a\] Assume **(A.1)**. Then, $$\begin{aligned} & \textbf{(i)} \left\| \left( X_{{\textrm{f}}, R} \, H_{{ \textrm{D} }} - ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} ) \right) X_{{\textrm{f}}, R} ( N_{{ \textrm{D} }} + 1 )^{-1} \right\| \leq \frac{c_{ \, {\textrm{f}}}}{R} , \\ & \textbf{(ii)} \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{1,l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) ,\quad l= 1, \cdots 4, \\ &\textbf{(iii)} \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{2 , l}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , \quad l= 1, \cdots 4 . \end{aligned}$$ Here $c_{ \, {\textrm{f}}} \geq 0$ is a constant, and $\delta^{i, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq0 $, $l=1 ,\cdots ,4$, $i= 1, 2$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ i , l }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{i,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$. $\;$\ **(Proof)** **(i)** By (\[3.2.1\]), we have $$\left\| \left( X_{{\textrm{f}}, R} H_{{ \textrm{D} }} - ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}H_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right) (N_{{ \textrm{D} }} +1)^{-1} \right\| \leq \| d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1} \| , \notag$$ and (\[3.1.3\]) yields that $$\| d \Gamma_{{\textrm{f}}} ( j_{{\textrm{f}}, R} , \tilde{\text{ad}}_{\omega_{\, M}}(j_{{\textrm{f}},R})) (N_{{ \textrm{D} }} +1)^{-1} \| \leq \sqrt{ \| [\omega_{\, M},j^{\, 0}_{{\textrm{f}}, R} ] \|^2_{B(L^2({\mathbf{R}^{3} }))} + \| [\omega_{\, M},j^{\, \infty}_{{\textrm{f}}, R} ] \|^2_{B(L^2({\mathbf{R}^{3} }) )} } \notag$$ By pseudo-differential calculus (e.g., [@FGS02] ; Appendix A, [@Hida11] ; Section IV), it follows that $\;$ $\| [\omega_{M},j^{\, \sharp}_{{\textrm{f}}, R} ] \|_{B(L^2({\mathbf{R}^{3} }) )} \leq$ $ \frac{ c_{\sharp }}{R}$, $\sharp = 0, \infty $, where $c_{\sharp} \geq 0$ are constants. Thus **(i)** is proven.\ **(ii)** By the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})= \sum\limits_{s= \pm 1/2}(b_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d^{\, \dagger }_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, we have from (\[3.2.2\]) and (\[3.2.3\]) that $$\begin{aligned} & X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \notag \\ & =\sum_{s= \pm 1/2} \left( X_{{\textrm{f}}, R} \, b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + \left( d^{\, \dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^{l\, } )\right) X_{{\textrm{f}}, R} \right) . \end{aligned}$$ Then we have $$\begin{aligned} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \notag \\ & \leq \sum_{s= \pm 1/2 } \left( \| b_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ( d^{\, \dagger}_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s ,{\ensuremath{\mathbf{x}}}}^l){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \| + \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}d_{s}^{\, \dagger } (j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l ) X_{{\textrm{f}}, R} \| \right) \notag \\ & \leq \sum_{s= \pm 1/2} \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s ,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ((j_{{\textrm{f}}, R}^{\, 0}-1) g_{s,{\ensuremath{\mathbf{x}}}}^l \| + \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l \| \frac{}{} \right) .\end{aligned}$$ Let $\delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( \| ((1-j_{{\textrm{f}}, R}^{\, 0}) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l}) \| + \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^l) \| + \| j_{{\textrm{f}}, R}^{\, \infty} g_{s,{\ensuremath{\mathbf{x}}}}^l \| \frac{}{} \right) $. We see that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{1, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq \sum\limits_{s= \pm 1/2 } \left( \|f_{s}^{\,l} \| + 2 \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty} \delta^{1,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Hence **(ii)** follows.\ $\; $\ **(iii)** From the definition of $\psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast}= \sum\limits_{s= \pm 1/2}(b^{\dagger }_{s} (f_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) + d_{s} (g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) )$, (\[3.2.2\]) and (\[3.2.3\]) yield that $$\begin{aligned} & X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \notag \\ & = \sum_{s= \pm 1/2} \left( \left( b^{\dagger }_{s} ((j_{{\textrm{f}}, R}^{\, 0}-1) f_{s,{\ensuremath{\mathbf{x}}}}^{\, l}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ (-1)^{N_{{ \textrm{D} }}} {\otimes}b_{s}^{\dagger } (j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^{\, l } ) \right) X_{{\textrm{f}}, R} + X_{{\textrm{f}}, R} \, d_{s} ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s, {\ensuremath{\mathbf{x}}}}^{\,l}) \right) . \end{aligned}$$ Then it follows that $$\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \| \leq \delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) , \notag$$ where $\delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{s= \pm 1/2 } \left( \| ((j_{{\textrm{f}}, R}^{\, 0} -1) f_{s,{\ensuremath{\mathbf{x}}}}^l) \| + \| j_{{\textrm{f}}, R}^{\, \infty} f_{s,{\ensuremath{\mathbf{x}}}}^l \| + \| ((1-j_{{\textrm{f}}, R}^{\, 0}) g_{s,{\ensuremath{\mathbf{x}}}}^{\, l}) \| \, \right) $. It is seen that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{2, l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}})| \leq \sum\limits_{s= \pm 1/2 } \left( 2 \|f_{s}^{\,l} \| + \|g_{s}^l \|\right) $ and $\lim\limits_{R \to \infty} \delta^{2,l}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain **(iii)**. $\blacksquare $\ \[coro32a\] Assume **(A.1)**. Then, for all $l ,l' = 1, \cdots 4$, $$\begin{aligned} \textbf{(i)} \, & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{3, l,l'}_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}) , \\ \textbf{(ii)} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}}) \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l}({\ensuremath{\mathbf{x}}}) \psi_{l'}({\ensuremath{\mathbf{y}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \leq \delta^{ \, 4, l, l' }_{ \, {\textrm{f}},R } ({\ensuremath{\mathbf{x}}}, {\ensuremath{\mathbf{y}}}) .\notag\end{aligned}$$ Here $\delta^{ 3 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) \geq 0 $ satisfies $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^{ 3, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ 3, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all ${\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$, and $\delta^{ 4 , l, l' }_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}} ) \geq0 $ satisfies $ \sup\limits_{ ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) \in {\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} | \delta^{ \, 4, l,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} ,{\ensuremath{\mathbf{y}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^{ \, 4, l ,l'}_{\, {\textrm{f}}, R} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) =0 $ for all $ {\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}} \in {\mathbf{R}^{3} }$. **(Proof)** **(i)** By Lemma \[lemma32a\] **(ii)** and **(iii)**, it is seen that $$\begin{aligned} & \left\| X_{{\textrm{f}}, R} \, \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right\| \notag \\ &\leq \left\| \left( X_{{\textrm{f}}, R} \, \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} - \left( ( \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right) \right) \psi_{l'}({\ensuremath{\mathbf{x}}}) \right\| \notag \\ & \qquad \qquad \qquad + \left\| (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \left( X_{{\textrm{f}}, R} \, \psi_{l'}({\ensuremath{\mathbf{x}}}) - ( \psi_{l'}({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}) X_{{\textrm{f}}, R} \right)\right\| \notag \\ & \leq \delta_{{\textrm{f}}, R}^{2,l} \|\psi_{l'}({\ensuremath{\mathbf{x}}}) \| + \delta_{{\textrm{f}}, R}^{1,l'} \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \| . \notag \end{aligned}$$ Note that $ \| \psi_{\, l'}({\ensuremath{\mathbf{x}}}) \| \leq c_{\, { \textrm{D} }}^{\, l'}$ and $ \| \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \| \leq c_{\, { \textrm{D} }}^{l} $. Hence **(i)** is obtained. Similarly, we can prove **(ii)** by using **(i)**. $\blacksquare$\ Partition of Unity for Radiation Field --------------------------------------- Let $U_{{\textrm{b}}}: {\ensuremath{\mathscr{F}}}_{{\textrm{b}}}( L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2 \} ) \oplus L^2( {\mathbf{R}^{3} }_{{\ensuremath{\mathbf{k}}}} \times \{ 1,2 \} ) ) \to {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ an isometric operator satisfying $U_{{\textrm{b}}} \, {\Omega_{\textrm{rad}}}= {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}$ and $$\begin{aligned} &U_{{\textrm{b}}} \, a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_1 \end{array} \right]\right) \cdots a_{r_1}^{\dagger } \left(\left[ \begin{array}{c} f_1 \\ g_n \end{array} \right]\right) {\Omega_{\textrm{rad}}}\\ = & \, \left( a^{\dagger }_{r_1} (f_1) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger }_{r_1}(g_1) \frac{}{}\right) \cdots \left( a^{\dagger}_{r_n} (f_n) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a^{\dagger}_{r_n}(g_n) \frac{}{} \right) {\Omega_{\textrm{rad}}}{\otimes}{\Omega_{\textrm{rad}}}. \end{aligned}$$ Let $j_{0} , j_{\, \infty} \in C^{\infty} ({\ensuremath{\mathbf{R}}})$. We suppose that $j_0 \geq 0$, $j_{\infty} \geq0 $, $j_{0}^2 + j_{\infty}^2 =1$, $j_{0}({\ensuremath{\mathbf{y}}})=1 $ if $|{\ensuremath{\mathbf{y}}}| \leq 1 $ and $j_{0}({\ensuremath{\mathbf{y}}})=0 $ if $|{\ensuremath{\mathbf{y}}}| \geq 2 $. We set $j_{{\textrm{b}}, R } =\left[ \begin{array}{c} j_{{\textrm{b}}, R }^{\, 0} \\ j_{{\textrm{b}}, R }^{\, \infty} \end{array} \right] $ where $ j_{{\textrm{b}}, R }^{\, 0}= j_{0} (\frac{-i {\ensuremath{\mathbf{\nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R})$ and $ j_{{\textrm{b}}, R }^{\, \infty}= j_{\infty} (\frac{-i {\ensuremath{\mathbf{ \nabla_{{\ensuremath{\mathbf{k}}}}}}}}{R}) $ with $\nabla_{{\ensuremath{\mathbf{k}}}}= (\partial_{k^1}, \partial_{k^2},\partial_{k^3} ) $.\ $\;$\ Let $Y_{{\textrm{b}}, R} : {\mathscr{F}_{\textrm{rad}}}\to {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} $ defined by $$Y_{{\textrm{b}}, R} = U_{{\textrm{b}}} \, \Gamma_{{\textrm{b}}}(j_{{\textrm{b}}, R}) . \notag$$ From (\[3.1.8\]) - (\[3.1.10\]), it follows that $$\begin{aligned} & Y_{{\textrm{b}}, R} \, {H_{\textrm{rad}, \, m}}= \left( {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}\frac{}{} \right) Y_{{\textrm{b}}, R} - U_{{\textrm{b}}} \, d \Gamma_{{\textrm{b}}} ( j_{{\textrm{b}}, R} , \tilde{\text{ad}}_{\omega_{m} }(j_{{\textrm{b}},R})) , \label{3.3.1} \\ & Y_{{\textrm{b}}, R} \, a_{r}(h) = (a_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} + Y_{{\textrm{b}}, R} \, a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h) , \label{3.3.2} \\ &Y_{{\textrm{b}}, R} \, a^{\dagger}_{r}(h) = (a^\ast_r (h) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} + \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h) \right) Y_{{\textrm{b}}, R} . \label{3.3.3}\end{aligned}$$ \[lemma33a\] Assume **(A.2)**. Then $$\begin{aligned} & \textbf{(i)} \left\| \left( Y_{{\textrm{b}}, R} \, {H_{\textrm{rad}, \, m}}- ( {H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}) \, Y_{{\textrm{b}}, R} \frac{}{} \right) (N_{{\textrm{rad}}} +1)^{-1} \right\| \leq \frac{c_{ \, {\textrm{b}}}}{R} , \\ & \textbf{(ii)} \left\| \left( Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \frac{}{} \right) (N_{{\textrm{rad}}}+1)^{-1/2} \right\| \leq \delta^j_{ \, {\textrm{b}},R } ({\ensuremath{\mathbf{x}}}) .\end{aligned}$$ Here $c_{ \, {\textrm{b}}} \geq 0$ is a constant and $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) \geq0 $, $j=1, 2,3$, are error terms which satisfy $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} | \delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) | < \infty$ and $\lim\limits_{R \to \infty }\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0 $ for all $ {\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }$. **(Proof)** **(i)** It is proven in a similar way to Lemma \[lemma32a\] **(i)**.\ **(ii)** By the definition of $ A_{j}({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2} \left( a_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) + a^{\dagger}_{r}(h^j_{r , {\ensuremath{\mathbf{x}}}}) \right)$, it follows from (\[3.3.2\]) and (\[3.3.3\]) that $$\begin{aligned} & Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \notag \\ &= \sum_{r=1,2} \left( Y_{{\textrm{b}}, R} \, a_{r}((1-j_{{\textrm{b}}, R}^{\, 0} )h_{r, {\ensuremath{\mathbf{x}}}}^j) + \left(a^{\dagger}_r ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r, {\ensuremath{\mathbf{x}}}}^j ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}+ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger }(j_{{\textrm{b}}, R}^{\, \infty} h_{r, {\ensuremath{\mathbf{x}}}}^j ) \right) Y_{{\textrm{b}}, R} \right) . \notag \end{aligned}$$ Since $\|a_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2} \| \leq \| h \|$ and $\|a^{\dagger }_{r}(h) (N_{{\textrm{rad}}} + 1)^{-1/2} \| \leq 2 \| h \|$, we have $$\begin{aligned} & \left\| \left( Y_{{\textrm{b}}, R} \, A_{j}({\ensuremath{\mathbf{x}}}) - ( A_{j}({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) Y_{{\textrm{b}}, R} \right) (N_{{\textrm{rad}}} +1)^{-1/2} \right\| \notag \\ & \leq \sum_{r=1,2} \left( \| ( a_{r} (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j) (N_{{\textrm{rad}}} +1)^{-1/2} \| \right. \notag \\ & \qquad \qquad \left. + \| (a^{\dagger}_{r} ((j_{{\textrm{b}}, R}^{\, 0}-1) h_{r,{\ensuremath{\mathbf{x}}}}^j)(N_{{\textrm{rad}}} +1)^{-1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) ( (N_{{\textrm{rad}}} +1)^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})Y_{{\textrm{b}}, R} (N_{{\textrm{rad}}} +1)^{-1/2} \| \right. \notag \\ &\qquad \qquad \qquad \left. + \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}^{\dagger} (j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j ) (N_{{\textrm{rad}}} +1)^{-1/2} ) ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}(N_{{\textrm{rad}}} +1)^{1/2} ) Y_{{\textrm{b}}, R} (N_{{\textrm{rad}}} +1)^{-1/2} \| \frac{}{} \right) \notag \\ & \leq \sum\limits_{r=1,2} \left( \| (1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j \| +2 \| (j_{{\textrm{b}}, R}^{\, 0}-1)h_{r,{\ensuremath{\mathbf{x}}}}^j \| + 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j \| \frac{}{} \right) . \notag \end{aligned}$$ Let $\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) = \sum\limits_{r=1,2} \left( 3 \| ((1-j_{{\textrm{b}}, R}^{\, 0}) h_{r,{\ensuremath{\mathbf{x}}}}^j) \| + 2 \| j_{{\textrm{b}}, R}^{\, \infty} h_{r,{\ensuremath{\mathbf{x}}}}^j \| \frac{}{} \right) $, $j=1,2 ,3$. We see that $ \sup\limits_{{\ensuremath{\mathbf{x}}} \in {\mathbf{R}^{3} }} |\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}})| \leq 5 \left( \|h_{1}^j \| + \|h_{2}^j \|\right) $ and $\lim\limits_{R \to \infty}\delta^j_{\, {\textrm{b}}, R} ({\ensuremath{\mathbf{x}}}) =0$ for all ${\ensuremath{\mathbf{x}}} \in {\ensuremath{\mathbf{R}}}$. Thus we obtain the proof. $\blacksquare$\ Existence of Ground State of $H_{m}$ ------------------------------------ We recall that the massive Hamiltonian is defined by $$H_{m} = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}+ {\kappa_{\textrm{I}}}{H_{\textrm{I}}}+ {\kappa_{\textrm{II}}}{H_{\textrm{II}}}. \notag$$ Throughout this subsection, we do not omit the subscripts of the identities ${{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and ${{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$.\ $\;$\ Since $\frac{1}{\omega_m({\ensuremath{\mathbf{k}}})^{\lambda}} \leq \frac{1}{\omega ({\ensuremath{\mathbf{k}}})^{\lambda}} $, $\lambda>0$, it holds that $$\|A_j ({\ensuremath{\mathbf{x}}}) ( {H_{\textrm{rad}, \, m}}+1 )^{-1/2} \| \leq \sum_{r=1,2} \left( 2 \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \omega_{m}} \| + \| \frac{ \chi_{{\textrm{rad}}} e_{r}^{\, j}}{ \sqrt{\omega_m}} \| \right) \leq c_{{\textrm{rad}}}^{\, j} . \label{9/1.1}$$ Then, we have $$\| {H_{\textrm{I}}}\Psi \| \leq c_{\,{\textrm{I}}} \, \| {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} \Psi \| , \quad \label{HImbound}$$ and it holds that for all $\epsilon > 0$, $$\|{H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \|H_{0 , m} \Psi \| + c_{\textrm{I}}\left( \frac{1 }{2 \epsilon} +1 \right) \| \Psi \| . . \label{HImbound'}$$ From (\[HImbound’\]) and $\| H_{{\textrm{II}}}\| < \infty $, it is proven that $H_{m}$ is self-adjoint and essentially self adjoint on any core of $H_{0,m}$.\ \[Massive-Case\] **(Existence of a Ground State of $H_m $)**\ Suppose **(A.1)** - **(A.3)**. Let $m <M$. Then $H_m $ has purely discrete spectrum in $[ E_{0} (H_m ) , E_{0} (H_m ) + m )$. In particular, $ {H_{m}}$ has a ground state. $\; $\ To prove Theorem \[Massive-Case\], we need some preparations. We define $ \tilde{X}_{{\textrm{f}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{X}_{{\textrm{f}}, R } = X_{{\textrm{f}}, R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} .$$ We introduce Hamiltonian $\tilde{H}_{m} : {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}\to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ defined by $$\tilde{H}_{m} = \tilde{H}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\tilde{H}_{{\textrm{rad}}} + {\kappa_{\textrm{I}}}\tilde{H}_{{\textrm{I}}} + {\kappa_{\textrm{II}}}\tilde{H}_{{\textrm{II}}} , \notag$$ where $\tilde{H}_{{ \textrm{D} }} = H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} $, $ \tilde{H}_{{\textrm{rad}}} = {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}$ and $$\begin{aligned} &\tilde{H}_{{\textrm{I}}} = \sum_{j=1}^{3} {\int_{\mathbf{R}^{3}}}\chi_{{\textrm{I}}} ({\ensuremath{\mathbf{x}}}) (\tilde{\psi}^{\dagger}({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi}({\ensuremath{\mathbf{x}}} ) {\otimes}A_j ({\ensuremath{\mathbf{x}}} )) d{\ensuremath{\mathbf{x}}} , \\ & \tilde{H}_{{\textrm{II}}} = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}} -{\ensuremath{\mathbf{y}}}|} \left( \tilde{\psi}^{\dagger } ({\ensuremath{\mathbf{x}}}) \tilde{\psi} ({\ensuremath{\mathbf{x}}}) \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{y}}}) \tilde{\psi} ({\ensuremath{\mathbf{y}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) d {\ensuremath{\mathbf{x}}} \, d {\ensuremath{\mathbf{y}}}. \end{aligned}$$ with $\tilde{\psi}({\ensuremath{\mathbf{x}}})=\psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$ and $\tilde{\alpha}^{j} = \alpha^j {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}$, $j=1, \cdots 3$. $\; $\ \[9/9.a\] Assume **(A.1)** - **(A.3)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_{m})$. Then, it holds that $$\begin{aligned} & \textbf{(i)} \; \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } ( H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) - ( \tilde{H}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \notag \\ & \quad \; \; \; \; \leq \frac{ c_{ \, {\textrm{f}}}}{R} \, \left( \left\| ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \right\| + \left\| \Psi \right\| \right) , \\ & \textbf{(ii)} \; \, \, \left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}- \tilde{{H_{\textrm{I}}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{I}}} (R ) \, \left( \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \| + \left\| \Psi \right\| \right), \\ & \textbf{(iii)} \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}- \tilde{{H_{\textrm{II}}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \, \| \Psi \| .\end{aligned}$$ Here $c_{{\textrm{f}}} \geq 0$ is the constant in Lemma \[lemma32a\]**(i)**, and $\delta_{\, {\textrm{f}}, {\textrm{I}}}(R ) \geq 0 $ and $\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \geq 0 $ are error terms satisfying that $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{I}}} (R) =0 $ and $\lim\limits_{R \to \infty }\delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0 $, respectively. **(Proof)**\ **(i)** It directly follows from Lemma \[lemma32a\] (**i**).\ **(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) $ and $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $. Then, $$\begin{aligned} & \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \notag \\ & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , \left( \left( X_{{\textrm{f}}, R} {\psi^{\dagger} (\mathbf{x}) }({\ensuremath{\mathbf{x}}}) \alpha^j \psi ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right) {d \mathbf{x} }. \notag \end{aligned}$$ Then we have $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, \left\| X_{{\textrm{f}}, R} \psi^{\dagger} ({\ensuremath{\mathbf{x}}}) \alpha^j \psi ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}^{\dagger} ({\ensuremath{\mathbf{x}}}) \tilde{\alpha}^j \tilde{\psi} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right\| \, \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right\| {d \mathbf{x} }\notag \\ & \leq \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha_{l , l'}^j | \int_{{\mathbf{R}^{3} }} \left| {\chi_{\textrm{I}} (\mathbf{x}) }\right| \, \| \left( X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'} ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) \| \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right\| {d \mathbf{x} }. \notag\end{aligned}$$ By Corollary \[coro32a\] **(i)**, we have $ \| \left( X_{{\textrm{f}}, R} \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l'} ({\ensuremath{\mathbf{x}}}) - \tilde{\psi}_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \tilde{\psi}_{l'} ({\ensuremath{\mathbf{x}}}) X_{{\textrm{f}}, R} \right) \| \leq \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) $. We also see that $ \| A_{j} ({\ensuremath{\mathbf{x}}}) (N_{{\textrm{rad}}} +1)^{-1/2}\| \leq 3 \sum\limits_{r=1,2}\| h_{r}^j\| $. Then it follows that $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \quad \qquad \leq \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha_{l , l'}^j | \| h_{r}^j \| \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \delta_{\, {\textrm{f}}, R}^{3, l, l'} ({\ensuremath{\mathbf{x}}}) {d \mathbf{x} }\frac{}{} \right) \, \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \| . \label{9/9.2}\end{aligned}$$ Since (\[9/9.2\]) holds for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, it follows that $$\left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{I}}}\, - \tilde{H}_{{\textrm{I}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{I}}} (R) \, \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}} +1 )^{1/2}) \Psi \| ,$$ where $\delta_{\, {\textrm{f}}, {\textrm{I}}} (R) = 3 \sum\limits_{r=1,2} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 \, |\alpha_{l , l'}^j | \, \| h_{r}^j \| \, \| \chi_{{\textrm{I}}} \, \delta_{\, {\textrm{f}}, R}^{3, l, l'} \|_{L^1} $. We see that $ \lim\limits_{R \to \infty} \delta_{\, {\textrm{f}},{\textrm{I}}} (R) =0 $, and hence **(ii)** follows.\ **(iii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. We set $Q_{ l }({\ensuremath{\mathbf{x}}})= \psi_{l} ({\ensuremath{\mathbf{x}}})^{\ast} \psi_{l} ({\ensuremath{\mathbf{x}}}) $. Then for all $\tilde{\Phi} \in {\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\ensuremath{\mathscr{F}}}_{{ \textrm{Dir} }} {\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Phi} \| =1 $, $$\begin{aligned} & \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \notag \\ & = \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \left( \tilde{\Phi} , \left( ( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - (( {Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l'} ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} \right) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }. \notag\end{aligned}$$ Then we have $$\begin{aligned} & \left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \left\| \left( X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - ( {Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right) \Psi \right\| {d \mathbf{x} }{d \mathbf{y} }. \notag \end{aligned}$$ From Corollary \[coro32a\] **(ii)**, it holds that $\left\| X_{{\textrm{f}}, R} Q_{ l }({\ensuremath{\mathbf{x}}}) Q_{l' } ({\ensuremath{\mathbf{y}}} ) - ({Q}_{ l }({\ensuremath{\mathbf{x}}}) {Q}_{l' } ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} ) X_{{\textrm{f}}, R} \right\| \leq \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}})$. Then we have $$\left| \left( \tilde{\Phi} , \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right) \right| \leq \left( \sum_{l, l'=1}^4\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }| }{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \delta_{{\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) {d \mathbf{x} }{d \mathbf{y} }\right) \, \left\| \Psi \right\| . \notag$$ This implies that $$\left\| \left( \tilde{X}_{{\textrm{f}}, R } {H_{\textrm{II}}}\, - \tilde{H}_{{\textrm{II}}} \tilde{X}_{{\textrm{f}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) \, \left\| \Psi \right\| , \notag$$ where $ \delta_{ \, {\textrm{f}}, {\textrm{II}}} (R) = \sum\limits_{l, l'=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{ | {\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }|}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} \delta_{\, {\textrm{f}}, R}^{4, l, l'} ({\ensuremath{\mathbf{x}}} , {\ensuremath{\mathbf{y}}}) {d \mathbf{x} }{d \mathbf{y} }$. We see that $\lim\limits_{R \to \infty } \delta_{\, {\textrm{f}}, {\textrm{II}}} (R) =0$, and thus the proof is obtained. $\blacksquare$\ $\;$\ We define $\tilde{Y}_{{\textrm{b}}, R } : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ by $$\tilde{Y}_{{\textrm{b}}, R } = {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}Y_{{\textrm{b}}, R} .$$ \[9/9.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned} & \textbf{(i)} \left\| \left( \tilde{Y}_{{\textrm{b}}, R } ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}) - ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{H_{\textrm{rad}, \, m}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{QED} }} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \notag \\ & \qquad \qquad \leq \frac{ c_{ \, {\textrm{b}}}}{R} \, \left( \left\| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi \right\| + \left\| \Psi \right\| \right) , \\ & \textbf{(ii)} \left\| \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \leq \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R ) \left( \| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \| + \| \Psi \| \right) ,\end{aligned}$$ where $c_{ \, {\textrm{b}}} \geq 0 $ and $\delta_{ \, {\textrm{b}},{\textrm{I}}} (R) \geq 0 $ satisfying $\lim\limits_{R \to \infty } \delta_{ \, {\textrm{b}}, {\textrm{I}}} (R) =0 $. **(Proof)** **(i)** It immediately follows from Lemma \[lemma33a\] **(i)**.\ **(ii)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$ and $\tilde{\Xi} \in {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$. We see that $$\begin{aligned} & \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{rad}}, R } \right) \Psi \right) \notag \\ & = \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \tilde{\Xi} , \left( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) \right) {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} ({\ensuremath{\mathbf{x}}}) \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right) {d \mathbf{x} }, \notag .\end{aligned}$$ Then, $$\begin{aligned} & \left| \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right) \right| \notag \\ & \leq \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \, \left\| ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right\| {d \mathbf{x} }\notag \\ & \leq \left( \sum_{j=1}^3 \sum_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \right) \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left\| \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \right) \Psi \right\| {d \mathbf{x} }. \notag \end{aligned}$$ From Lemma \[lemma33a\] **(ii)**, it holds that $$\left\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\left( Y_{{\textrm{b}}, R } A_{j} \, - ( A_{j} ({\ensuremath{\mathbf{x}}}){\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) Y_{{\textrm{b}}, R } ) \right) \Psi \right\| \leq \delta_{\, {\textrm{b}},R}^j ({\ensuremath{\mathbf{x}}}) \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| , \notag$$ where $ \delta_{{\textrm{b}}, R}^j ({\ensuremath{\mathbf{x}}}) \geq 0 $ is the error term, and hence, $$\left| \left( \tilde{\Xi} , \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right) \right| \leq \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| , \label{9/9.3}$$ where $\delta_{\, {\textrm{b}},{\textrm{I}}} (R) = \sum\limits_{l,l'=1 }^4 |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l} c_{\, { \textrm{D} }}^{\, l'} \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \delta^j_{\, {\textrm{b}}} ({\ensuremath{\mathbf{x}}}) dx $. Since (\[9/9.3\]) holds for all $\tilde{\Xi} \in {\mathscr{F}_{\textrm{Dir}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}{\otimes}{\mathscr{F}_{\textrm{rad}}}$ with $\| \tilde{\Xi} \| =1$, we have $$\left\| \left( \tilde{Y}_{{\textrm{b}}, R } \, {H_{\textrm{I}}}- ( {H_{\textrm{I}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{{\textrm{rad}}, R } \right) \Psi \right\| \leq \delta_{\, {\textrm{b}}, {\textrm{I}}} (R) \, \|({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}(N_{{\textrm{rad}}}+1 )^{1/2}) \Psi \| . \notag$$ Since $\lim\limits_{R \to \infty }\delta_{\, {\textrm{b}}, {\textrm{I}}}(R)=0$, the proof is obtained. $\blacksquare$\ $\; $\ Here we introduce a new norm defined by $$\qquad \| \Psi \|_{ \lambda , \, \lambda ' } \; = \; \|( N_{{ \textrm{D} }}^{\, \lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{ \, \lambda ' /2} )\Psi \| + \| \Psi \| , \quad \Psi \in {\ensuremath{\mathscr{D}}} (N_{{ \textrm{D} }}^{\lambda /2 } {\otimes}N_{{\textrm{rad}}}^{ \lambda ' /2} ) .$$ $\;$\ From Proposition \[9/9.a\] and Proposition \[9/9.b\], the next corollary follows. $\;$\ \[9/9.c\] Assume **(A.1)** - **(A.3)**. Then for all $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$, $$\begin{aligned} &\textbf{(i)} \; \left\| \left( \tilde{X}_{{\textrm{f}}, R } H_{m} - ( \tilde{H}_{m} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{X}_{{\textrm{f}}, R} \right) \Psi \right\| \leq \delta_{ \, {\textrm{f}}} (R) \| \Psi \|_{2,1 } , \\ &\textbf{(ii)} \; \left\| \left( \tilde{Y}_{{\textrm{b}}, R } H_{m} - ( H_{m} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{{\textrm{b}}, R } \right) \Psi \right\| \leq \delta_{ \,{\textrm{b}}} (R) \| \Psi \|_{0, 2} \, .\end{aligned}$$ Here $ \delta_{ \, {\textrm{f}}} (R) \geq 0 $ and $ \delta_{ \, {\textrm{b}}} (R) \geq 0 $ are error terms which satisfy that $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{f}}} (R)=0 $ and $ \lim\limits_{R \to \infty} \delta_{ \, {\textrm{b}}} (R) =0 $, respectively.\ \[9/9.d\] \[LformboundHm\] Assume **(A.1)** - **(A.3)**. Let $q_{\, {\textrm{f}}, R} = (j^{\, 0}_{\, {\textrm{f}}, R})^2$ and $q_{\, {\textrm{b}}, R} = (j^{\, 0}_{\, {\textrm{b}}, R})^2$. Then, for all $\Psi \in {\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$, $$\begin{aligned} (\Psi, H_m \Psi ) \geq & E_{0} (H_m) + \, m \, + \, (M-m) \left( \Psi, ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) ) \Psi \right) \\ & - M \left( \Psi , \left( \Gamma_{{\textrm{f}}} (q_{\, {\textrm{f}}, R}) {\otimes}\Gamma_{{\textrm{b}}} (q_{\, {\textrm{b}}, R}) \right) \Psi \right) + \left( \delta_{\, {\textrm{f}}} (R ) \| \Psi \|_{2,1 } + \delta_{\, {\textrm{b}}} (R ) \| \Psi \|_{0,2 } \right) . \end{aligned}$$ **(Proof)** Let $\Psi \in {\ensuremath{\mathscr{ D}}} (H_m ) $ with $\| \Psi \| =1$. By Lemma Corollary \[9/9.c\] **(ii)**, $$\begin{aligned} (\Psi , H_m \Psi ) & = \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{Y}_{\, {\textrm{b}},R} H_m \Psi \right) \notag \\ & \geq ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) + ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{H_{\textrm{rad}, \, m}}) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} . \notag \end{aligned}$$ We see that $ {H_{\textrm{rad}, \, m}}\geq m ( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {P_{\textrm{rad}}}) $ with $ {P_{\textrm{rad}}}= E_{N_{{\textrm{rad}}}}(\{ 0\})$ where $E_{X}(J)$ denotes the spectral projection on a Borel set $J \in {\ensuremath{\mathscr{B}}}({\ensuremath{\mathbf{R}}})$ for a self-adjoint operator $X$. Then $$\begin{aligned} (\Psi , H_m \Psi ) & \geq ( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{rad}},R} \Psi ) + m \notag \\ & \qquad \qquad \quad - m ( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^\ast ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{P_{\textrm{rad}}}) \tilde{Y}_{\, {\textrm{rad}},R} \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2 } \notag \\ & \geq ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) + m-m (\Psi , ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}( q_{\, {\textrm{b}}, R}) \Psi ) - \delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2 } . \label{9/10.1}\end{aligned}$$ Here we used $ {Y}_{\, {\textrm{b}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} {\otimes}{P_{\textrm{rad}}}) {Y}_{\, {\textrm{b}},R} = {\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) $ in the last line. We evaluate the first term in the right hand side of (\[9/10.1\]). Let $\tilde{\tilde{X}}_{{\textrm{f}},R}= \tilde{X}_{\, {\textrm{f}},R} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}$. By Corollary \[9/9.c\] **(i)**, $$\begin{aligned} & ( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( H_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) \notag \\ & = \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} ( ( \tilde{X}_{\, {\textrm{f}},R}^{\ast} \tilde{X}_{\, {\textrm{f}},R} H_m) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{\, {\textrm{b}},R} \Psi ) \right) \notag \\ &\geq \left( \Psi , \tilde{Y}_{\, {\textrm{rad}},R}^{\ast} \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} ( \tilde{H}_m {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \tilde{\tilde{X}}_{\, {\textrm{f}},R} \tilde{Y}_{\, {\textrm{b}},R} \Psi \right) \notag \\ & \qquad \qquad \qquad \quad + \left( \Psi_n , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{\tilde{X}}^{\ast}_{\, {\textrm{f}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}H_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{\tilde{X}}_{\, {\textrm{b}},R} \tilde{Y}_{ {\textrm{b}},R} \Psi \right) - \delta_{\, {\textrm{f}}} (R) \| \tilde{Y}_{ {\textrm{b}},R} \Psi \|_{2,1}^{\sim }, \label{9/9.3}\end{aligned}$$ where we set $$\| \tilde{ \Phi} \|_{ \lambda , \lambda ' }^{\sim} \; = \; \|( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{ \Phi} \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{ \Phi} \| + \| \tilde{ \Phi} \| ,$$ for $ \tilde{ \Phi} \in {\ensuremath{\mathscr{D}}} ( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}) \cap {\ensuremath{\mathscr{D}}}({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) $. We see that $$\begin{aligned} \| \tilde{Y}_{ {\textrm{b}},R} \Psi \|_{ 2,1 }^{\sim} & = \| \tilde{Y}_{ {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{Y}_{ {\textrm{b}},R} \Psi \| + \| \tilde{Y}_{ {\textrm{b}},R} \Psi \| \\ & \leq \| \tilde{Y}_{ {\textrm{b}},R} ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| + \|\tilde{Y}_{ {\textrm{b}},R} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi \|+ \| \tilde{Y}_{ {\textrm{b}},R} \Psi \| = \| \Psi \|_{2,1} ,\end{aligned}$$ and $ H_{{ \textrm{D} }} \geq M ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }}- P_{{ \textrm{D} }}) $ with $ P_{{ \textrm{D} }} = E_{N_{{ \textrm{D} }}}(\{ 0 \})$. Then we have $$\begin{aligned} (\ref{9/9.3})& \geq E_{0} (\tilde{H}_m ) + M - M \left( \Psi , \tilde{Y}_{\, {\textrm{b}},R}^{\ast} \tilde{\tilde{X}}_{\, {\textrm{f}},R}^{\ast} ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{P_{\Omega_{\textrm{D}}}}{\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \tilde{\tilde{X}}_{\, {\textrm{f}},R} \tilde{Y}_{\, {\textrm{b}},R} \Psi \right) - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{ 2,1} \notag \\ & \geq E_{0} (H_m ) + M - M ( \Psi , ( {\Gamma_{\textrm{f}}}( q_{ \, {\textrm{f}}, R} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi ) - \delta_{\, {\textrm{f}}, m } (R) \| \Psi \|_{2,1} . \notag\end{aligned}$$ Here we used $ E_{0} (\tilde{H}_m)= E_{0} (H_m)$ and $ {X}_{\, {\textrm{f}},R}^{\ast} ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}P_{{ \textrm{D} }} ) {X}_{\, {\textrm{f}},R} = {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}}, R} ) $ in the last line. Thus we have $$\begin{aligned} (\Psi, H_m \Psi ) & \geq E_{0} (H_m ) + m + M - M \left( \Psi , ( {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \right) \notag \\ & \qquad \qquad -m \left( \Psi , \left( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) \right) \Psi \right) -\delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{2,1} .\end{aligned}$$ Note that $$\begin{aligned} {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} & \geq {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+ ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} - {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) ) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) \notag \\ & = {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}}+ {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) - {\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) . \notag\end{aligned}$$ Then we have $$\begin{aligned} (\Psi, H_m \Psi ) & \geq E_{0} (H_m) + \, m \, + \, (M-m) \left( \Psi, ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) ) \Psi \right) \\ & \quad- M \left( \Psi , ({\Gamma_{\textrm{f}}}\left( q_{ \, {\textrm{f}}, R} \right) {\otimes}{\Gamma_{\textrm{b}}}\left( q_{ \, {\textrm{b}}, R} \right) ) \Psi \right) -\delta_{\, {\textrm{b}}} (R) \| \Psi \|_{0,2} - \delta_{\, {\textrm{f}}} (R) \| \Psi \|_{2,1} .\end{aligned}$$ Thus the proof is obtained. $\blacksquare$\ \[9/9.e\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\qquad \qquad \| H_{0, m} \Psi \| \leq L_{\epsilon} \| H_{m} \Psi \| + R_{\epsilon} \| \Psi \| , \qquad \Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \notag$$ where $L_{\epsilon} = \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon} $ and $R_{\epsilon}= \frac{1}{1- c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|\epsilon}\left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, ( \frac{1 }{2 \epsilon} +1 ) + |{\kappa_{\textrm{II}}}| \, \| {H_{\textrm{II}}}\| \right)$. **(Proof)** Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m)$. Since $H_{0,m} = H_m - {\kappa_{\textrm{I}}}{H_{\textrm{I}}}- {\kappa_{\textrm{II}}}{H_{\textrm{II}}}$, we see that $$\|H_{0,m} \Psi \| \leq \| H_m \Psi \| + \left| {\kappa_{\textrm{I}}}\right| \, \left| {H_{\textrm{I}}}\Psi \right\| + \left| {\kappa_{\textrm{II}}}\right| \, \, \| {H_{\textrm{II}}}\| \, \| \Psi \| . \notag$$ From (\[HImbound’\]), it holds that $ \| {H_{\textrm{I}}}\Psi \| \leq c_{{\textrm{I}}} \epsilon \| H_{0 , m } \Psi \| + c_{{\textrm{I}}} ( \frac{1 }{2 \epsilon} +1 ) \| \Psi \| $ for all $\epsilon > 0$. Hence $$(1- c_{\, {\textrm{I}}} |{\kappa_{\textrm{I}}}| \epsilon )\| H_{0,m} \Psi \| \leq \| H_m \Psi \| + \left( c_{\textrm{I}}|{\kappa_{\textrm{I}}}| \, \left( \frac{1 }{2 \epsilon} +1 \right) + | {\kappa_{\textrm{II}}}| \, \| {H_{\textrm{II}}}\| \, \right) \| \Psi \| .$$ Taking $\epsilon >0$ such that $ \epsilon < \frac{1}{c_{\, {\textrm{I}}}|{\kappa_{\textrm{I}}}|}$, we obtain the proof. $\blacksquare $ $\;$\ Since $\| N_{{ \textrm{D} }} \Psi \| \leq \frac{1}{M} \| H_{{ \textrm{D} }} \Psi \|$, $\Psi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, and $\| N_{{\textrm{rad}}} \Phi \| \leq \frac{1}{m} \| H_{{\textrm{rad}}} \Phi \|$, $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}})$, the next corollary follows from Lemma \[9/9.e\].\ \[9/9.f\] Assume **(A.1)** - **(A.3)**. Then for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$\begin{aligned} & \textbf{(i)} \; \| (N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi \| \leq \frac{L_{\epsilon}}{M} \|H_{m} \Psi \| + \frac{R_\epsilon}{M} \| \Psi \| , \\ & \textbf{(ii)} \;\| ({{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi \| \leq \frac{L_{\epsilon }}{m} \|H_{m} \Psi \| + \frac{R_\epsilon}{m} \| \Psi \| . \end{aligned}$$ $\;$\ [**(Proof of Theorem \[Massive-Case\] )**]{}\ It is enough to show that $ \sigma_{{ \textrm{ess} }} (H_m ) \subset [E_{0} (H_m ) + m , \infty )$. Let $\lambda \in \sigma_{{ \textrm{ess} }} (H_m )$. Then by the Weyl’s theorem, there exists a sequence $\{ \Psi_n \}_{n=1}^{\infty}$ of ${\ensuremath{\mathscr{D}}}(H_m )$ such that (i) $\| \Psi_n \| =1$, $n \in {\ensuremath{\mathbf{N}}}$, (ii) s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, and (iii) w-$\lim\limits_{n \to \infty} \Psi_n = 0$. Since $ | \lambda- (\Psi_n , H_m \Psi_n ) | \leq | ( \Psi_n , ( H_m- \lambda )\Psi_n | \leq \| ( H_m- \lambda )\Psi_n \| $, it holds that $ \lambda = \lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) $. Here we show that $$\lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) \geq E_{0} ({H_{m}}) + m , \notag$$ and then, the proof is obtained. Let $m \leq M$. From Lemma \[LformboundHm\], $$\begin{aligned} ( \Psi_{n} ,H_m \Psi_{n} ) \geq & E_{0} (H_m) + m - M (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n ) \notag \\ & \qquad \qquad \qquad \quad \;\;- \delta_{\, {\textrm{f}}} (R) \| \Psi_n \|_{2,1} -\delta_{\, {\textrm{b}}, m } (R) \| \Psi_n \|_{0,2 }. \notag \end{aligned}$$ Since s-$\lim\limits_{ n \to \infty} (H_m -\lambda ) \Psi_n =0 $, we can set $$E_{m} = \sup_{n \in {\ensuremath{\mathbf{N}}}} \| H_m \Psi_n \| < \infty .$$ Let $0 \leq \lambda \leq 2 $ and $0 \leq \lambda' \leq 2$. From Corollary \[9/9.f\], it is seen that for all $ 0< \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned} \| \Psi_{n} \|_{\lambda ,\lambda ' } & = \|( N_{{ \textrm{D} }}^{\lambda /2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{\lambda ' } ) \Psi_n \| + \| \Psi_n \| \notag \\ & \leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_n \| + \|( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}} ) \Psi_n \| + 3 \| \Psi_n \| \notag \\ & \leq (\frac{1}{M} + \frac{1}{m} ) \left( L_{\epsilon } \|H_{m} \Psi_n \| + 2R_{\epsilon} \right) +3\| \Psi_n \| \notag \\ & \leq E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) + 2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3 . \notag\end{aligned}$$ Then we have $$( \Psi_{n} ,H_m \Psi_{n} ) \geq E_{0} (H_m) + m - M (\Psi_n , ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) ,\Psi_n ) - \delta_{m, \epsilon} (R) , \label{9/9.6}$$ where $\delta_{m , \epsilon} (R) = c_{\, m , \epsilon} ( \delta_{\, {\textrm{b}}} (R) + \delta_{ \, {\textrm{f}}} (R) ) $ with $c_{\, m , \epsilon } = E_{m} L_{\epsilon} (\frac{1}{M} + \frac{1}{m} ) + 2 (\frac{1}{M} + \frac{1}{m} )R_{\epsilon} +3$. It is seen that $$\begin{aligned} &\left| ( \Psi_n , ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n ) \right| \notag \\ & \qquad \qquad \leq \| (H_{0,m} +1)^{1/2} \Psi_n \| \, \| (H_{0,m} +1)^{-1/2} ({\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n \| . \label{9/16.1}\end{aligned}$$ From Lemma \[9/9.e\], we see that $$\| (H_{0,m} +1)^{1/2} \Psi_n \| \leq \| H_{0,m} \Psi_n \| + \|\Psi_n \| \leq L_\epsilon \| H_{m} \Psi_n \| + ( R_\epsilon +1) \|\Psi_n \| = E_{0} (H_m) L_{\epsilon} + R_\epsilon +1 ,$$ and hence, $$\sup\limits_{n \in {\ensuremath{\mathbf{N}}}} \| (H_{0,m} +1)^{1/2} \Psi_n \| \leq E_{m} L_\epsilon + R_\epsilon +1 . \label{9/16.2}$$ It holds that $$\begin{aligned} (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) = & (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} ) \notag \\ & \; \times ( (H_{{ \textrm{D} }} + 1)^{-1/2} {\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R} ) ) {\otimes}( ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R} ) ) ) , \notag\end{aligned}$$ and hence, $(H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) )$ is compact, since $\| (H_{0,m} +1)^{-1/2} ( (H_{{ \textrm{D} }} + 1)^{1/2} {\otimes}({H_{\textrm{rad}, \, m}}+1)^{1/2} )\| \leq 1$ and $ \left( (H_{{ \textrm{D} }} + 1)^{-1/2}{\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R}) \right) {\otimes}\left( ({H_{\textrm{rad}, \, m}}+1)^{-1/2} {\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}}, R}) \right)$ is compact. Therefore it holds that $$\lim_{n \to \infty} \left\| (H_{0,m} +1)^{-1/2} ( {\Gamma_{\textrm{f}}}(q_{\, {\textrm{f}},R} ) {\otimes}{\Gamma_{\textrm{b}}}(q_{\, {\textrm{b}}, R}) ) \Psi_n \right\|=0. \label{9/16.3}$$ From (\[9/16.1\]) - (\[9/16.3\]) we have $ \lim\limits_{n \to \infty}\left| \left( \Psi_n , ({\Gamma_{\textrm{f}}}(q_{ \, {\textrm{f}},R}) {\otimes}{\Gamma_{\textrm{b}}}(q_{ \, {\textrm{b}},R}) ) ,\Psi_n \right) \right| =0 $. Then by taking the limit of (\[9/9.6\]) as $R \to \infty$, we have $ \lim\limits_{n \to \infty} (\Psi_n , H_m \Psi_n ) \geq E_{0}(H_m ) + m $. $\blacksquare$\ Derivative Bounds ================= From Theorem \[Massive-Case\], $H_{m}$ has the ground state. Let $\Psi_{m}$ be the normalized ground state of $H_{m}$, i.e. $$\qquad \qquad H_{m} \Psi_m = E_{0}(H_m ) \Psi_m , \quad \| \Psi_m \| = 1. \notag$$ Electron-Positron Derivative Bounds ----------------------------------- We introduce the distribution kernel of the annihilation operator for the Dirac field. For all $ \Psi = \left\{ \Psi^{(n)} = {}^{t}\left( \Psi^{(n)}_1 , \cdots , \Psi^{(n)}_4 \right) \right\}_{n=0}^{\infty} \in {\ensuremath{\mathscr{D}}} ( H_{{ \textrm{D} }} )$, we set $$\qquad C_{l}({\ensuremath{\mathbf{p}}})\Psi^{(n , \nu )}( {\ensuremath{\mathbf{p}}}_{1} , \cdots , {\ensuremath{\mathbf{p}}}_{n} ) = \delta_{\, l , \nu } \sqrt{n+1} \Psi^{(n+1 , \nu )}( {\ensuremath{\mathbf{p}}} , {\ensuremath{\mathbf{p}}}_{1} , \cdots , {\ensuremath{\mathbf{p}}}_{n} ) . \quad l=1 ,\cdots 4 .$$ Let $$b_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{1} ({\ensuremath{\mathbf{p}}}), \; \; b_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{2} ({\ensuremath{\mathbf{p}}}) , \; \; d_{1/2} ({\ensuremath{\mathbf{p}}}) = C_{3} ({\ensuremath{\mathbf{p}}}), \; \; d_{-1/2} ({\ensuremath{\mathbf{p}}}) = C_{4} ({\ensuremath{\mathbf{p}}}) .$$ $\;$\ Then it follows that for all $\Phi \in {\mathscr{F}_{\textrm{Dirac}}}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned} \qquad \quad & ( \Phi , b_{s} (f) \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi , b_{s} ({\ensuremath{\mathbf{p}}}) \Psi ) d {\ensuremath{\mathbf{p}}} , \quad \quad \; f \in {\ensuremath{\mathscr{D}}}(\omega_{\, M}) , \\ & ( \Phi , d_{s} (g) \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} ( \Phi , d_{s} ({\ensuremath{\mathbf{p}}}) \Psi ) d {\ensuremath{\mathbf{p}}} , \quad \quad \; g \in {\ensuremath{\mathscr{D}}}(\omega_{\, M}) . \end{aligned}$$ The number operator for electrons and positrons are defined by $$N_{{ \textrm{D} }}^{+} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} {{\small \text{1}}\hspace{-0.32em}1}& O \\ O& O \end{array} \right) \right) , \qquad N_{{ \textrm{D} }}^{-} = d {\Gamma_{\textrm{f}}}\left( \left( \begin{array}{cc} O & O \\ O& {{\small \text{1}}\hspace{-0.32em}1}\end{array} \right) \right) ,$$ respectively. It holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}( H_{{ \textrm{D} }} )$, $$\begin{aligned} &(\Phi , N_{{ \textrm{D} }}^{+} \Psi ) = \sum_{\pm1/2}\int_{{\mathbf{R}^{3} }}( b_{s}({\ensuremath{\mathbf{p}}}) \Phi , b_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}} , \notag \\ &(\Phi , N_{{ \textrm{D} }}^{-} \Psi ) = \sum_{\pm1/2} \int_{{\mathbf{R}^{3} }}( d_{s}({\ensuremath{\mathbf{p}}}) \Phi , d_{s}({\ensuremath{\mathbf{p}}}) \Psi) d {\ensuremath{\mathbf{p}}} \notag .\end{aligned}$$ $\; $\ By the canonical anti-commutation relation, it is proven in ([@Ta09] ; Section III) that $$\begin{aligned} &[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }, b_{s}(f ) ] = - \sum_{l,l'=1}^4 \alpha_{l,l'}^{j} ( f , f_{s,{\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l'}({\ensuremath{\mathbf{x}}}) , \label{ori1} \\ &[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }, d_{s}(g ) ] = \sum_{l,l'=1}^4 \alpha_{l,l'}^{j} ( g ,g_{s ,{\ensuremath{\mathbf{x}}}}^{l'} ) \; \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} , \label{ori2} \end{aligned}$$ and for $\rho({\ensuremath{\mathbf{x}}})={\psi^{\dagger} (\mathbf{x}) }\psi ({\ensuremath{\mathbf{x}}})$, $$\begin{aligned} &[ \rho({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; b_{s}(f ) ] = -\sum_{l=1}^4 \left( ( f , f_{s, {\ensuremath{\mathbf{y}}}}^{l} ) \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) + ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) \right) , \label{ori3} \\ &[ \rho ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}), \; d_{s}( g ) ] = \sum_{l=1}^4 \left( ( g , g_{s, {\ensuremath{\mathbf{y}}}}^{l} ) \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) ^{\ast} + (g , g_{s, {\ensuremath{\mathbf{x}}}}^{l} ) \; \psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \rho ({\ensuremath{\mathbf{y}}}) \right) . \label{ori4}\end{aligned}$$ $\; $\ Let $X$ and $Y$ be operators on a Hilbert space. The weak commutator is defined by $$[X, \,Y ]^0 (\Phi , \Psi ) = (X^{\ast}\Phi , Y \Psi ) - (Y^{\ast}\Phi , X \Psi ) ,$$ where $ \Psi \in {\ensuremath{\mathscr{D}}}(X) \cap {\ensuremath{\mathscr{D}}} ( Y) $ and $ \Phi \in {\ensuremath{\mathscr{D}}}(X^{\ast}) \cap {\ensuremath{\mathscr{D}}} ( Y^{\ast}) $.\ \[9/12.a\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $f \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned} \textbf{(i)} & \; \; [{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} , \quad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\; \Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \\ \textbf{(ii)} & \; \; [{H_{\textrm{II}}}, b_{s}(f){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, +} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}} , \quad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \end{aligned}$$ Here $ K_s^{+}({\ensuremath{\mathbf{p}}})$, $S_s^{\, +} ({\ensuremath{\mathbf{p}}}) $ and $T_s^{+} ({\ensuremath{\mathbf{p}}}) $ are operators which satisfy $$\begin{aligned} & (\Phi , K_s^{+}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }, \notag \\ & (\Phi , S^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} f_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) \left( \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag \\ &(\Phi , T^{+}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }. \notag\end{aligned}$$ **(Proof)**\ **(i)** Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ and $ \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. By (\[ori1\]), we have $$\begin{aligned} [{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) & =\sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) , b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 ( \Phi , \Psi ) {d \mathbf{x} }\notag \\ & = \sum_{j=1}^3\int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \Phi , \left( [ {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}} ), b_{s}(f) ] {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) \right) \Psi \right) {d \mathbf{x} }\notag \\ & = - \sum_{ j=1}^3 \sum_{ l, l'=1}^4 \alpha_{l , l'}^j \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\, (f, f_{s, {\ensuremath{\mathbf{x}}}}^{\, l}) \left( \Phi , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }. \notag \end{aligned}$$ Let $\ell_{s, {\ensuremath{\mathbf{p}}}} : {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \to {\ensuremath{\mathbf{C}}} $ be a functional defined by $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ') = - \sum_{j=1}^3 \sum_{l, l'=1}^4 \alpha_{l , l'}^j \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\, f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi ' , ( \psi_{l'} ({\ensuremath{\mathbf{x}}} ) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}})) \Psi '\right) {d \mathbf{x} },$$ for $ \Phi ' \in {\mathscr{F}_{\textrm{QED}}}, \Psi ' \in {\ensuremath{\mathscr{D}}}(H_{0, m} )$. We see that $$\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) \leq c_{ \, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi ' \| \, \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {H_{\textrm{rad}, \, m}}+1)^{1/2}) \Psi ' \| ,$$ where $c_{\, {\textrm{I}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{j=1}^3 \sum\limits_{l, l'=1}^4 | \alpha_{l , l'}^j| \, \| \chi_{\, {\textrm{I}}} \|_{L^1} \, |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} $. Then from the Riesz Representation theorem, we can define an operator $ K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})$ which satisfy $\ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi ', \Psi ' ) = (\Phi ' , K_{s}^{\, +} ({\ensuremath{\mathbf{p}}}) \Psi' ) $. Then it holds that $$[{H_{\textrm{I}}}, b_{s}(f) ]^0 (\Phi , \Psi) = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \ell_{s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) d {\ensuremath{\mathbf{p}}} = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} .$$ **(ii)** From (\[ori3\]), we see that for all $\Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned} [{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) & = \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } [ \rho ({\ensuremath{\mathbf{x}}} ) \rho ({\ensuremath{\mathbf{y}}} ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi ,\Psi ) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & =\int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } \left( \Phi , \left( [ \rho ({\ensuremath{\mathbf{x}}} ) \rho ({\ensuremath{\mathbf{y}}} ) , b_{s}(f) ] {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } \left\{ ( f , f_{s, {\ensuremath{\mathbf{y}}}}^{\, l} ) ( \Phi , ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) \right. \notag \\ & \qquad \qquad \qquad \qquad \left. + ( f , f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ) (\Phi , \, ( \psi_{l} ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi ) \right\} {d \mathbf{x} }{d \mathbf{y} }. \notag \end{aligned}$$ We set functionals $q_{s, {\ensuremath{\mathbf{p}}}} $ and $r_{s,{\ensuremath{\mathbf{p}}}} $ on ${\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} \times {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ by $$\begin{aligned} &\quad q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{y}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) ( \Phi ' , ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l} ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ' ) {d \mathbf{x} }{d \mathbf{y} }, \quad \Phi ', \, \Psi ' \in {\mathscr{F}_{\textrm{QED}}}, \\ &\quad r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '', \Psi '') = -\sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{| {\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}} | } f_{s, {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) ( \Phi '' , (\psi_{l} ({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi '' ) {d \mathbf{x} }{d \mathbf{y} }, \quad \Phi '' , \, \Psi '' \in {\mathscr{F}_{\textrm{QED}}}.\end{aligned}$$ We see that $$\begin{aligned} & q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) \leq c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi ' \| \, \| \Psi '\| , \notag \\ & r_{s, {\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' ) \leq c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} \| \Phi '' \| \, \| \Psi '' \| ,\end{aligned}$$ where $c_{ \, {\textrm{II}}, \, s, {\ensuremath{\mathbf{p}}}} = \sum\limits_{l, l'=1}^4 \left\| \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}- {\ensuremath{\mathbf{y}}}| } \right\|_{L^1} \, |f_{s}^{\, l}({\ensuremath{\mathbf{p}}})|(c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l} $. Then from Riesz Representation theorem, we can define operators $ S_s^{+} ({\ensuremath{\mathbf{p}}})$ and $ T_s^{+} ({\ensuremath{\mathbf{p}}})$ such that $q_{s, {\ensuremath{\mathbf{p}}}} (\Phi ' , \Psi ' ) = (\Phi ', S_{s}^{+} ({\ensuremath{\mathbf{p}}}) \Psi ' ) $ and $r_{{\ensuremath{\mathbf{p}}}} (\Phi '' , \Psi '' ) = (\Phi '' , T_s^{+} ({\ensuremath{\mathbf{p}}}) \Psi '' ) $, respectively. Then it holds that $$\begin{aligned} [{H_{\textrm{II}}}, b_{s}(f) ]^0 (\Phi , \Psi) & = \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( q_{s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) + r_{ s, {\ensuremath{\mathbf{p}}}} (\Phi , \Psi ) \right) d {\ensuremath{\mathbf{p}}} \\ &= \int_{{\mathbf{R}^{3} }} \overline{f({\ensuremath{\mathbf{p}}})} \left( \Phi , (S_s^{+} ({\ensuremath{\mathbf{p}}}) + T_s^{+} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}}.\end{aligned}$$ Thus proof is obtained. $\blacksquare $ $\; $\ In a similar way to Lemma \[9/12.a\], the following lemma is also proven.\ \[9/12.b\] Assume **(A.1)** - **(A.3)**. Then it holds that for all $g \in L^2 ({\mathbf{R}^{3} })$, $$\begin{aligned} \textbf{(i)} \; & \; \; [{H_{\textrm{I}}}, d_{s}(g) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi ) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , K_{s}^{\,-} ({\ensuremath{\mathbf{p}}}) \Psi \right) d {\ensuremath{\mathbf{p}}} , \qquad \Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} ,\Psi \in {\ensuremath{\mathscr{D}}}(H_m ) , \\ \textbf{(ii)} & \; \; [{H_{\textrm{II}}}, d_{s}(g){\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} g({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , (S_s^{\, -} ({\ensuremath{\mathbf{p}}}) + T_s^{-} ({\ensuremath{\mathbf{p}}}) ) \Psi \right) d {\ensuremath{\mathbf{p}}} , \qquad \Phi , \Psi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }} . \end{aligned}$$ Here $ K_s^{\, -}({\ensuremath{\mathbf{p}}})$, $S_s^{\, -} ({\ensuremath{\mathbf{p}}}) $ and $T_s^{-} ({\ensuremath{\mathbf{p}}}) $ are operators which satisfy $$\begin{aligned} & (\Phi , K_s^{\, -}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }g_{s , {\ensuremath{\mathbf{x}}}}^{\, l'} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} }, \notag \\ & (\Phi , S^{-}_{s}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} g_{s , {\ensuremath{\mathbf{y}}} }^{\, l}({\ensuremath{\mathbf{p}}}) \left( \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}})^{\ast} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag \\ &(\Phi , T^{-}_{-}({\ensuremath{\mathbf{p}}}) \Psi ) = \sum_{l=1}^4 \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} g_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}})^{\ast} \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi \right) {d \mathbf{x} }{d \mathbf{y} }, \notag\end{aligned}$$ $\; $ \[9/12.c\] Assume **(A.1)** - **(A.5)**. Let $\Psi \in {\ensuremath{\mathscr{D}}}(H_m )$. Then, $K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $S_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$ and $T_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi$, $s= \pm 1/2$, are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$. **(Proof)** We show that $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is strongly differentiable. Let $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$. From **(A.4)**, $K_{s}^{+}({\ensuremath{\mathbf{p}}}) \Psi$ is weakly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$, and we have $$\partial_{p^{\nu}} (\Phi , K_{s}^{+ }({\ensuremath{\mathbf{p}}}) \Psi ) = -\sum_{j=1}^3 \sum_{l,l'=1}^4 \alpha^{j}_{l,l'}\, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) \left( \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi \right) {d \mathbf{x} },$$ and $ |\partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )| \leq \sum\limits_{j=1}^3 \sum\limits_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} \left( \int_{{\mathbf{R}^{3} }}| \partial_{p^\nu}f_{s , {\ensuremath{\mathbf{x}}}}^{\, l} ({\ensuremath{\mathbf{p}}}) | {d \mathbf{x} }\, \right) \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}{H_{\textrm{rad}, \, m}}^{1/2} ) \Psi \| $. Then the Riesz representation theorem shows that there exists a vector $\Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ such that $( \Phi , \Xi_{\Psi} ({\ensuremath{\mathbf{p}}}) )= \partial_{p^{\nu}} (\Phi , K_{s}^{\pm }({\ensuremath{\mathbf{p}}}) \Psi )$. Let ${\ensuremath{\mathbf{e}}}_{\nu} = (\delta_{\nu , j})_{j=1}^3 $. It is seen that $$\begin{aligned} &(\Phi , \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi) - ( \Phi , \Xi ({\ensuremath{\mathbf{p}}}) ) \notag \\ & = -\sum_{j=1}^3 \sum_{l , l'=1}^4 \alpha^{j}_{l, l'} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }\left( \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\,l } ({\ensuremath{\mathbf{p}}})}{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{l } ({\ensuremath{\mathbf{p}}})\right) (\Phi , ( \psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_{j} ({\ensuremath{\mathbf{x}}}) ) \Psi ) {d \mathbf{x} }, \notag \end{aligned}$$ and hence, $$\begin{aligned} &| (\Phi , ( \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi - \Xi ({\ensuremath{\mathbf{p}}})) | \notag \\ & \leq \sum_{j=1}^3 \sum_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}})\right| {d \mathbf{x} }\right) \| \Psi \| . \label{9/13.1}\end{aligned}$$ Since (\[9/13.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $ \| \Phi \| =1$, we have $$\begin{aligned} & \left\| \frac{K_{s}^{+ }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - K_{s}^{+ }({\ensuremath{\mathbf{p}}} )}{\epsilon} \Psi - \Xi ({\ensuremath{\mathbf{p}}}) \right\| \notag \\ & \leq \sum_{j=1}^3 \sum_{l , l'=1}^4 | \alpha^{j}_{l, l'} | c_{{ \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j'} \left( \int_{{\mathbf{R}^{3} }} | {\chi_{\textrm{I}} (\mathbf{x}) }| \left| \frac{f_{s , {\ensuremath{\mathbf{x}}}}^{\, l }({\ensuremath{\mathbf{p}}} + \epsilon {\ensuremath{\mathbf{e}}}_{\nu}) - f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}}) }{\epsilon} - \partial_{p^{\nu}} f_{s , {\ensuremath{\mathbf{x}}}}^{\, l } ({\ensuremath{\mathbf{p}}})\right| {d \mathbf{x} }\right) \| \Psi \| \to 0 , \notag\end{aligned}$$ as $\epsilon \to 0$. Thus $K_{s}^{+ }({\ensuremath{\mathbf{p}}} ) \Psi $ is strongly differentiable. Similarly, it is proven that $ K_{s}^{\, -}({\ensuremath{\mathbf{p}}} ) \Psi$, $ S_{s}^{\pm }({\ensuremath{\mathbf{p}}} ) \Psi$ and $ T_{s}^{\pm }({\ensuremath{\mathbf{p}}} ) \Psi$ are strongly differentiable for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }\backslash O_{{ \textrm{D} }}$. $\blacksquare $\ \[9/12.d\] For all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$, it holds that $$\begin{aligned} &\textbf{(i)} \; \; [H_{{ \textrm{D} }} , b_{s}(f) ]^0 (\Phi , \Psi )=- \left( \Phi , b_{s}( \omega_{\, M} f )\Psi \right) , \\ & \textbf{(ii)} \; \; [H_{{ \textrm{D} }} , d_{s}(f) ]^0 (\Phi , \Psi )=- \left( \Phi , d_{s}( \omega_{\, M} f )\Psi \right) .\end{aligned}$$ **(Proof)** It holds that for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} )$, $$[H_{{ \textrm{D} }}, b^{\dagger}_{s}(f) ]\Phi = b^{\dagger}_{s}( \omega_{\, M} f) \Phi .$$ Let $\Psi \in {\ensuremath{\mathscr{D}}} (H_m )$. Then $$(H_{{ \textrm{D} }} \Phi, b_{s}(f) \Psi ) - ( b_{s}(f) \Phi , H_{{ \textrm{D} }} \Psi ) = ( [ b^{\dagger}_{s}(f) , H_{{ \textrm{D} }} ] \Phi , \Psi ) = (- b^{\dagger}_{s}( \omega_{\, M} f) \Phi , \Psi) ,$$ and hence, $$(H_{{ \textrm{D} }} \Phi, b_{s}(f) \Psi ) - ( b_{s}(f) \Phi , H_{{ \textrm{D} }} \Psi ) = -( \Phi , b_{s}( \omega_{\, M} f) \Psi) . \label{9/11.1}$$ Since $ {\ensuremath{\mathscr{F}}}_{{\textrm{Dirac}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, M} ))$ is a core of $H_{{ \textrm{D} }}$ and $ b_{s}(f)$ is bounded, (\[9/11.1\]) holds for all $\Phi \in {\ensuremath{\mathscr{D}}}(H_{{ \textrm{D} }})$. Hence **(i)** follows. Similarly, we can also prove **(ii)**. $\blacksquare $\ \[EP-Pullth\] **(Electron-Positron Pull-Through Formula)**\ Assume **(A.1)** - **(A.3)**. Then that $$\begin{aligned} & \textbf{(i)} \; \; \; (b_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m , \\ & \textbf{(ii)} \; \; (d_s ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = (H_m- E_0 (H_m) + \omega_{M}({\ensuremath{\mathbf{p}}}))^{-1} \left( {\kappa_{\textrm{I}}}K^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,-}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m , \end{aligned}$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$. **(Proof)** Let $\Phi, \in {\ensuremath{\mathscr{D}}} (H_m )$. By Lemma \[9/12.d\] **(i)**, we have $$\begin{aligned} & [{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) \notag \\ &= - \left( \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) + {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) + {\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) . \notag\end{aligned}$$ On the other hand, $H_m \Psi_m=E_{0}(H_m) \Psi_m$ yields that $$[{H_{m}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) = \left( ( {H_{m}}- E_{0}({H_{m}}) ) \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) . \notag$$ Then, we have $$\begin{aligned} & ( ({H_{m}}- E_{0}({H_{m}})) \Phi , ( b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) + ( \Phi , ( b_{s}( \omega_M f ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) \notag \\ &\qquad \qquad \qquad = {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) + {\kappa_{\textrm{II}}}[{H_{\textrm{II}}}, b_{s}(f) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}]^0 (\Phi , \Psi_m ) . \notag\end{aligned}$$ By Lemma \[9/12.a\], it follows that $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) \Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \frac{}{} \right) d {\ensuremath{\mathbf{p}}} \notag \\ &\quad \qquad \quad = \int_{{\mathbf{R}^{3} }} f({\ensuremath{\mathbf{p}}})^{\ast} \left( \Phi , \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m \right) d {\ensuremath{\mathbf{p}}}. \label{9/12.3}\end{aligned}$$ Since (\[9/12.3\]) holds for all $f \in L^2 ({{\mathbf{R}^{3} }} )$, it follows that $$( ({H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) \Phi , ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) =( \Phi , \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m ) , \notag$$ for almost everywhere ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$. This implies that $ ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \in {\ensuremath{\mathscr{D}}} ({H_{m}}) $ and $$( {H_{m}}- E_{0}({H_{m}}) + \omega_M ({\ensuremath{\mathbf{p}}})) ( b_{s}({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m = \left( {\kappa_{\textrm{I}}}K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + {\kappa_{\textrm{II}}}( S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) + T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \right) \Psi_m . \label{9/12.7}$$ From (\[9/12.7\]), we obtain **(i)**. Similarly, **(ii)** is also proven. $\blacksquare $\ \[EP-DB\] (**Electron-Positron Derivative Bounds**)\ Assume **(A.1)** - **(A.5)**. Then, it holds that for all ${\ensuremath{\mathbf{p}}} \in {\ensuremath{\mathbf{R}}}^3 \backslash O_{{ \textrm{D} }}$ and $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\begin{aligned} &\textbf{(i)} \; \; \left\| \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \right\| \leq \left( ( L_{\epsilon} E_{0}(H_{m} ) + R_\epsilon + 1 \,) |{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{} \right) F_{s , +}^{\nu}({\ensuremath{\mathbf{p}}}) , \\ &\textbf{(ii)} \; \; \left\| \partial_{p^{\nu}} (d_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \right\| \leq \left( ( L_{\epsilon} E_{0}(H_{m} ) + R_\epsilon + 1 \,) |{\kappa_{\textrm{I}}}|+ 2|{\kappa_{\textrm{II}}}| \frac{}{} \right) F_{s , -}^{\nu}({\ensuremath{\mathbf{p}}}) . \end{aligned}$$ Here $ F_{s ,\pm }^{\nu} $ are functions satisfying $ F_{s , \pm }^{\nu} \in L^{2} ({\mathbf{R}^{3} }) $, $s= \pm 1/2$, $\nu=1 , \cdots , 3$. **(Proof)** Let $R_{m,M}({\ensuremath{\mathbf{p}}}) =(H_m- E_0 (H_m) + \omega_{\, M}({\ensuremath{\mathbf{p}}}))^{-1} $. From Proposition \[EP-Pullth\] it holds that for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$ with $ \| \Phi \|=1$, $$\begin{aligned} ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) & = {\kappa_{\textrm{I}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) + {\kappa_{\textrm{II}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & \qquad \qquad + {\kappa_{\textrm{II}}}\left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) . \label{9/13.3} \end{aligned}$$ Here we evaluate the three terms in the right-hand side of (\[9/13.3\]) as follows.\ (First term) We see that $$\begin{aligned} & \left( \Phi,\partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) K^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =- \sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \partial_{p^{\nu}} \left( f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right) , \notag \\ & = -\sum_{j=1}^{3} \sum_{l,l'=1}^{4} \alpha^{j}_{l,l'}\, \left\{ ( \partial_{p^{\nu}} f_{s}^{\, l} ({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right. , \notag \\ & \qquad \qquad \qquad -i f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\notag \\ & \qquad \qquad \qquad \left. - \frac{ f_{s}^{\, l} ({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right\} \notag .\end{aligned}$$ Since $ \| R_{m,M}({\ensuremath{\mathbf{p}}})\| \leq \frac{1}{\omega_{\, M}({\ensuremath{\mathbf{p}}})} \leq \frac{1}{M}$ and $\|\Phi \|=1$, we have [$$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{ c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M} \| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|, \\ & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{ c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M} \, \| |{\ensuremath{\mathbf{x}}}| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \|, \\ & \left| \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l'}({\ensuremath{\mathbf{x}}}) {\otimes}A_j ({\ensuremath{\mathbf{x}}})) \Psi_m \right) {d \mathbf{x} }\right| \leq \frac{c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j}}{ M^2} \, \| \chi_{{\textrm{I}}} \|_{L^1} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}+1)^{1 /2} ) \Psi_{m} \| .\end{aligned}$$ ]{} It is seen that $ \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}({H_{\textrm{rad}, \, m}}^{1/2} +1)^{1/2}) \Psi_m \| \leq \| H_{0,m} \Psi_{m} \| + \| \Psi_m\| = \| H_{0,m} \Psi_{m} \| +1 $, and hence, $$\begin{aligned} & \left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) K_{s}^{\, +} ({\ensuremath{\mathbf{p}}})\Psi_m \right) \right| \notag \\ & \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{j=1}^{3} \sum_{l,l'=1}^{4} c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) \, \left( \| H_{0,m} \Psi_m \| +1 \frac{}{} \right) . \label{9/13.I} \end{aligned}$$ (Second term) It is seen that $$\begin{aligned} & \left( \Phi, \partial_{p^{\nu}}R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =- \sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) \notag \\ & =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}} f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\ &\qquad \quad - i f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, ( \rho ({\ensuremath{\mathbf{x}}}) \psi_{l}({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.9}\end{aligned}$$ By evaluating the right-hand side of (\[9/13.9\]), we have $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) S^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \right| \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4} ( c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l}\left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) . \label{9/13.II}$$ (Third term) We see that $$\begin{aligned} & \left( \Phi, \partial_{p^{\nu}} R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \notag \\ & =-\sum_{l=1}^4 \partial_{p^{\nu}} \left( f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, ( \psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) \notag \\ & =- \sum_{l=1}^4 \left\{ ( \partial_{p^{\nu}} f_{s }^{\, l}({\ensuremath{\mathbf{p}}})) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right. \notag \\ &\qquad \quad - i f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} y^{\nu} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}}) \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\notag \\ & \qquad \quad \left. - \frac{f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) p^{\nu} }{ \omega_{\, M}({\ensuremath{\mathbf{p}}})} \int_{{\mathbf{R}^{3} }\times {\mathbf{R}^{3} }} \frac{{\chi_{\textrm{II}} (\mathbf{x}) }{\chi_{\textrm{II}} (\mathbf{y}) }}{|{\ensuremath{\mathbf{x}}}-{\ensuremath{\mathbf{y}}}|} e^{-i {\ensuremath{\mathbf{p}}} \cdot {\ensuremath{\mathbf{y}}}} \left( R_{m,M}({\ensuremath{\mathbf{p}}})^2 \Phi , \, (\psi_{l}({\ensuremath{\mathbf{x}}}) \rho ({\ensuremath{\mathbf{y}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})) \Psi_m \right) {d \mathbf{x} }{d \mathbf{y} }\right) . \label{9/13.10}\end{aligned}$$ We estimate the right-hand side of the absolute value of (\[9/13.10\]), and then, $$\left| \partial_{p^{\nu}} \left( \Phi, R_{m,M}({\ensuremath{\mathbf{p}}}) T^{\,+}_{s} ({\ensuremath{\mathbf{p}}} ) \Psi_m \right) \right| \leq \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \sum_{l,l'=1}^{4}c_{\, { \textrm{D} }}^{\, l} ( c_{\, { \textrm{D} }}^{\, l'})^2 \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M^2} \right) . \label{9/13.III}$$ From (\[9/13.I\]), (\[9/13.II\]) and (\[9/13.III\]), we have $$\begin{aligned} & \left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| \notag \\ & \leq \sum_{l=1}^4 c_{+}^{\,l} \left( \frac{| \partial_{p^{\nu}}f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) |}{ M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ | f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) | \, }{ M^2} \right) \left( |\kappa_{\, {\textrm{I}}}| \, \| H_{0,m} \Psi_m \| + |{\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) , \notag \end{aligned}$$ where $c_{+}^{\,l} = \| (1+ |{\ensuremath{\mathbf{x}}}|) \chi_{{\textrm{I}}} \|_{L^1} \, \times \max \left\{ \sum\limits_{j=1}^{3}\sum\limits_{l'=1}^{4} |\alpha^j_{l,l'}| c_{\, { \textrm{D} }}^{\, l'} c_{{\textrm{rad}}}^{\, j} , \; \sum\limits_{l'=1}^{4} ( c_{\, { \textrm{D} }}^{\, l'})^2 c_{\, { \textrm{D} }}^{\, l} \right\}$. By the definition of $f_{s }^{\, l}({\ensuremath{\mathbf{p}}}) = \frac{\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \, u_{s}^{\,l}({\ensuremath{\mathbf{p}}})}{ \sqrt{(2 \pi )^3 } }$, we have $$\left| ( \Phi , \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right| \leq F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) \, \left( |\kappa_{\, {\textrm{I}}}| \, \| H_{0,m} \Psi_m \| + |{\kappa_{\textrm{I}}}|+ 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) , \label{9/13.11}$$ where $$F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) = \frac{1}{\sqrt{(2 \pi )^3}} \sum_{l=1}^4 c_{+}^{\, l} \left( \frac{| \partial_{p^{\nu}} \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \, |}{ M} + \frac{ | \chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) \partial_{p^{\nu}} u_{s}^{\,l}({\ensuremath{\mathbf{p}}}) | \, }{M} + \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) | \, }{ M} + \frac{ |\chi_{{ \textrm{D} }}({\ensuremath{\mathbf{p}}}) | \, }{ M^2} \right) .$$ We see that (\[9/13.11\]) holds for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$ with $\| \Phi \| =1$, and this implies that $$\left\| \partial_{p^{\nu}} ( b_{s} ({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} ) \right\| \leq F_{s ,+}^{\nu}({\ensuremath{\mathbf{p}}}) \, \left( |\kappa_{\, {\textrm{I}}}| \| H_{0,m} \Psi_m \| +| {\kappa_{\textrm{I}}}| + 2 |\kappa_{\, {\textrm{II}}}| \frac{}{} \right) . \notag$$ From Lemma \[9/9.e\], it holds that for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $$\| H_{0,m} \Psi_{m} \| \leq \, L_{\epsilon} \| H_{m} \Psi_m \| + R_{\epsilon} \| \Psi_m \| = \, L_{\epsilon} E_{0}(H_m) + R_{\epsilon} .$$ Thus **(i)** is obtained. Similarly, **(ii)** is also proven in a same way as **(i)**. $\blacksquare $\ Photon Derivative Bound ----------------------- In a similar to the Dirac field, we introduce the distribution kernel of the annihilation operator for the radiation field. For all $ \Psi = \left\{ \Psi^{(n)} = \left( \Psi^{(n)}_1 , \Psi^{(n)}_2 \right) \right\}_{n=0}^{\infty} \in {\ensuremath{\mathscr{D}}} ( H_{{\textrm{rad}}, m } )$, we define $a_{r}({\ensuremath{\mathbf{k}}})$, by $$a_{r}({\ensuremath{\mathbf{k}}})\Psi^{(n)}_{\varrho} ( {\ensuremath{\mathbf{k}}}_{1} , \cdots , {\ensuremath{\mathbf{k}}}_{n} ) = \delta_{\, r , \varrho } \sqrt{n+1} \Psi^{(n+1 )}_{\varrho }( {\ensuremath{\mathbf{k}}} , {\ensuremath{\mathbf{k}}}_{1} , \cdots , {\ensuremath{\mathbf{k}}}_{n} ) , \qquad \varrho = 1,2.$$ It holds that $$\qquad \qquad (\Phi , a_{r}(h) \Psi )= \int_{{\mathbf{R}^{3} }}h({\ensuremath{\mathbf{k}}})^{\ast}(\Phi , a_{r}({\ensuremath{\mathbf{k}}}) \Psi) d {\ensuremath{\mathbf{k}}}, \quad \Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}} , \; \Psi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}}, m}) .$$ \[9/13.c\]Assume (**A.2**). Then for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_{{\textrm{rad}},m })$, $$\begin{aligned} & \textbf{(i)} \; \; [H_{{\textrm{rad}}, m } \, , a_{r} (h)]^0 (\Phi , \Psi )= \left( \Phi , a_{r}( \omega_m h )\Psi \right) , \\ &\textbf{(ii)} \; \; [A_{j}({\ensuremath{\mathbf{x}}} ) , a_{r}(h) ]^0 (\Phi , \Psi )= - (h, h_{r , {\ensuremath{\mathbf{x}}}}^j )\left( \Phi , \Psi \right) . \end{aligned}$$ **(Proof)** It holds that for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{\textrm{rad}}}^{\, {\textrm{fin}}} ({\ensuremath{\mathscr{D}}} ( \omega_{\, m} ))$, $$\begin{aligned} & [ H_{{\textrm{rad}}, m}, a^{\dagger}_{r}(h) ]\Phi = - a^{\dagger}_{r}(\omega_{\,m }h)\Phi , \label{9/14.1} \\ &[ A_{j} ({\ensuremath{\mathbf{x}}}), a^{\dagger}_{r}(h) ]\Phi = ( h_{r , {\ensuremath{\mathbf{x}}}}^j ,h ) \Phi .\label{9/14.2}\end{aligned}$$ In a similar way to Lemma \[9/12.d\], we can prove **(i)** by (\[9/14.1\]) and **(ii)** by (\[9/14.2\]). $\blacksquare $\ \[9/13.d\] Assume **(A.1)** - **(A.3)**. Then\ **(i)** it holds that for all $\Phi , \Psi \in {\ensuremath{\mathscr{D}}}(H_m )$, $$[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( h )]^0 (\Phi , \Psi) \, = \, \int_{{\mathbf{R}^{3} }} h({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi , Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi \right) d {\ensuremath{\mathbf{k}}} . \notag$$ Here $ Q_r ({\ensuremath{\mathbf{k}}})$ is an operator which satisfy $$(\Phi , Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi ) = - \sum_{j=1}^3 \, \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }h_{r, {\ensuremath{\mathbf{x}}}}^{\, j} ({\ensuremath{\mathbf{k}}}) \left( \Phi , \, ({\psi^{\dagger} (\mathbf{x}) }\alpha^j { \psi ({\ensuremath{\mathbf{x}}}) }{\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi \right) {d \mathbf{x} }$$ with $ \| Q_{r} ({\ensuremath{\mathbf{k}}}) \| \leq \|\chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|$.\ **(ii)** Additionally assume **(A.4)** and **(A.6)**. Then, $Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi$ is strongly differential for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$. **(Proof)** **(i)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m)$ From Lemma \[9/13.c\], $$\begin{aligned} [{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi ) &= \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}A_{j}({\ensuremath{\mathbf{x}}}), {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)]^0( \Phi , \Psi ) {d \mathbf{x} }\notag \\ &= \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }[ {{\small \text{1}}\hspace{-0.32em}1}{\otimes}A_{j}({\ensuremath{\mathbf{x}}}) , {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 ( \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) {d \mathbf{x} }\notag \\ & =- \sum_{j=1}^3 \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }( h, h_{r ,{\ensuremath{\mathbf{x}}}}^j )( \Phi ,( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi ) {d \mathbf{x} }.\end{aligned}$$ We define $\ell_{r, {\ensuremath{\mathbf{k}}}}:{\mathscr{F}_{\textrm{QED}}}{\otimes}{\mathscr{F}_{\textrm{QED}}}\to {\ensuremath{\mathbf{C}}}$ by $$\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ') = - \sum_{j=1}^3 h_{r}^j ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \, ( ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi ({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Phi ', \Psi ' ) {d \mathbf{x} }$$ We see that $|\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ' ) | \leq \|\chi_{{\textrm{I}}} \|_{L^1} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |h_{r}^j ({\ensuremath{\mathbf{k}}})| \, |\alpha^{j}_{l,l'}| \, |c_{\, { \textrm{D} }}^{\, l}| \, |c_{\, { \textrm{D} }}^{\, l'}|\, \| \Phi ' \| \, \| \Psi' \| $. By Riesz representation theorem, we can define an operator $Q_{r} ({\ensuremath{\mathbf{k}}})$ such that $\ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi ' , \Psi ') = (\Phi ', Q_{r}({\ensuremath{\mathbf{k}}}) \Psi ')$. Then we have $$[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(f) ]^0 (\Phi , \Psi ) = \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \ell_{r, {\ensuremath{\mathbf{k}}}} (\Phi , \Psi ) {d \mathbf{k} }= \int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi \right) {d \mathbf{k} }. \notag$$ Then **(i)** is obtained.\ **(ii)** The strong differentiability of $ Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi $ is proven by **(A.4)** and **(A.6)** in a similar way to Lemma \[9/12.c\], and the proof is omitted. $\blacksquare $.\ \[PF-P\] ()\ Assume **(A.1)** - **(A.3)**. Then it holds that for almost everywhere ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$, $$({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m ={\kappa_{\textrm{I}}}(H_m- E_0 (H_m) + \omega_{m}({\ensuremath{\mathbf{k}}}))^{-1} Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m .$$ **(Proof)** Let $\Phi \in {\ensuremath{\mathscr{D}}}(H_m) $. By Lemma \[9/13.c\] **(i)**, $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h ) ]^0 (\Phi , \Psi_m )= - \left( \Phi , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ( \omega_m h)) \Psi_m \right) + {\kappa_{\textrm{I}}}\, [{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m ) . \notag$$ It also holds that $$[{H_{m}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h) ]^0 (\Phi , \Psi_m ) = \left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) . \notag$$ Then we have $$\left( (H_{m}-E_{0}({H_{m}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h)) \Psi_m \right) + \left( \Phi, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(\omega_m h)) \Psi_m \right) = {\kappa_{\textrm{I}}}[{H_{\textrm{I}}}, {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}(h )]( \Phi, \Psi_{m} ). \notag$$ By Lemma \[9/13.d\], $$\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right) {d \mathbf{k} }= {\kappa_{\textrm{I}}}\int_{{\mathbf{R}^{3} }} h ({\ensuremath{\mathbf{k}}})^{\ast} \left( \Phi ,Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m \right) {d \mathbf{k} }. \label{9/13.14}$$ Note that (\[9/13.14\]) holds for all $h \in L^2 ({\mathbf{R}^{3} })$. Then we have $$\left( (H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) \Phi , ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right) = \left( \Phi ,\, {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m \right) , \label{9/13.15}$$ for almost everywhere $ {\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }$. In addition, (\[9/13.15\]) yields that $ ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \in {\ensuremath{\mathscr{D}}} (H_m )$ and $$(H_{m}-E_{0}({H_{m}})+\omega_{m}({\ensuremath{\mathbf{k}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m = {\kappa_{\textrm{I}}}Q_r ({\ensuremath{\mathbf{k}}}) \Psi_m . \notag$$ Thus the proof is obtained. $\blacksquare $ \[P-DB\] (**Photon Derivative Bounds**)\ Assume **(A.1)**-**(A.4)** and **(A.6)**. Then it holds that for all ${\ensuremath{\mathbf{k}}} \in {\mathbf{R}^{3} }\backslash O_{{\textrm{rad}}}$, $$\left\| \partial_{k^\nu} ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m \right\| \leq |{\kappa_{\textrm{I}}}| F_{r}^{\, \nu} ({\ensuremath{\mathbf{k}}}) \notag$$ where $ F_{r}^{\, \nu} $ is a function which satisfy $ F_{r}^{\, \nu} \in L^2 ({\mathbf{R}^{3} }) $. **(Proof)** $\; $\ Let $ R_{m}({\ensuremath{\mathbf{k}}}) = (H_m- E_0 (H_m) + \omega_{\, m}({\ensuremath{\mathbf{k}}}))^{-1} $. From Proposition \[PF-P\], it holds that ${{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m = R_{m}({\ensuremath{\mathbf{k}}}) Q_{r} ({\ensuremath{\mathbf{k}}}) \Psi_m $. Then for all $\Phi \in {\ensuremath{\mathscr{F}}}_{{ \textrm{QED} }}$, $$\begin{aligned} & = ( \Phi , \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_m ) \notag \\ & =- {\kappa_{\textrm{I}}}\sum_{j=1}^3 \, \partial_{k^{\nu}} \left( h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right) \notag \\ & =- {\kappa_{\textrm{I}}}\sum_{j=1}^3 \, \left\{ ( \partial_{k^{\nu}} h_{r}^{\, j} ({\ensuremath{\mathbf{k}}})) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right. \notag \\ & \qquad \qquad \qquad -i h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }x^{\nu} \, e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}}) \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\notag \\ & \qquad \qquad \qquad \left. - \frac{ h_{r}^{\, j} ({\ensuremath{\mathbf{k}}}) k^{\nu} }{\omega_{\, m}({\ensuremath{\mathbf{k}}})} \int_{{\mathbf{R}^{3} }} {\chi_{\textrm{I}} (\mathbf{x}) }e^{-i {\ensuremath{\mathbf{k}}} \cdot {\ensuremath{\mathbf{x}}}} \left( R_{m}({\ensuremath{\mathbf{k}}})^2 \Phi , \, ( {\psi^{\dagger} (\mathbf{x}) }\alpha^j \psi({\ensuremath{\mathbf{x}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) {d \mathbf{x} }\right\} . \label{9/13.16}\end{aligned}$$ By estimating the absolute value of the right-hand side of (\[9/13.16\]), we have $$\begin{aligned} & \left| \partial_{k^{\nu}} \left( \Phi, R_{m}({\ensuremath{\mathbf{k}}})Q ({\ensuremath{\mathbf{k}}})\Psi_m \right) \right| \notag \\ & \leq \| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}|{\kappa_{\textrm{I}}}| \sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, | c_{\, { \textrm{D} }}^{\,l'}| \, \left( \frac{| \partial_{k^{\nu}} h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) |}{\omega_{\, m}({\ensuremath{\mathbf{k}}})} + \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \, }{\omega_{\, m}({\ensuremath{\mathbf{k}}})} + \frac{ | h_{r }^{\, j}({\ensuremath{\mathbf{k}}}) | \, }{\omega_{\, m}({\ensuremath{\mathbf{k}}})^2} \right) . \notag \end{aligned}$$ From the definition of $h_{r}^{\, j}({\ensuremath{\mathbf{k}}})=\frac{\chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) e_{r}^{\,j}({\ensuremath{\mathbf{k}}}))}{\sqrt{2 (2 \pi )^3 \omega ({\ensuremath{\mathbf{k}}}) }}$, we have $$\partial_{k^\nu} h_{r}^{\, j}({\ensuremath{\mathbf{k}}}) = \frac{1}{\sqrt{2 (2 \pi )^3 }} \left( \frac{ ( \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) ) e_{r}^{\,j}({\ensuremath{\mathbf{k}}}) }{\omega({\ensuremath{\mathbf{k}}})^{1/2}} + \frac{ \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})}{\omega ({\ensuremath{\mathbf{k}}})^{1/2}} - \frac{1}{2} \frac{ \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) k^{\nu}}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}} \right) . \notag$$ Hence, it holds that $$\left| ( \Phi , \partial_{k^\nu } ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r} ({\ensuremath{\mathbf{k}}})) \Psi_{m} ) \right| \leq | {\kappa_{\textrm{I}}}| \, F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) , \label{9/13.17}$$ where $$\begin{aligned} F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) =\frac{\| (1+ |{\ensuremath{\mathbf{x}}}| ) \chi_{{\textrm{I}}} \|_{L^1}}{\sqrt{2 (2 \pi )^3 }} & \sum_{j=1}^3 \sum_{l,l'=1}^4 \, | \left\{ \alpha^j_{l,l'} | \, | c_{\, { \textrm{D} }}^{\,l}| \, | c_{\, { \textrm{D} }}^{\,l'}| \, \frac{}{} \right.\notag \\ & \times \left. \left( \frac{| \partial_{k^\nu} \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |+| \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) \partial_{k^\nu} e_{r}^{\,j}({\ensuremath{\mathbf{k}}})| + | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) | }{ \omega ({\ensuremath{\mathbf{k}}})^{3/2}} + \frac{3}{2} \frac{ | \chi_{{\textrm{rad}}}({\ensuremath{\mathbf{k}}}) |}{ \omega ({\ensuremath{\mathbf{k}}})^{5/2}} \right) \right\} . \notag \end{aligned}$$ Since (\[9/13.17\]) holds for all $\Phi \in {\mathscr{F}_{\textrm{QED}}}$, we have $$\| \partial_{k^{\nu}} ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_{m} \| \leq | {\kappa_{\textrm{I}}}| F_{r}^{\nu}({\ensuremath{\mathbf{k}}}) .$$ The condition **(A.6)** yields that $ F_{r}^{\nu} \in L^2 ({\mathbf{R}^{3} }) $, and hence the proof is obtained. $\blacksquare $ Proof of Theorem \[Main-Theorem\] ================================== Let $\{ \Psi_m\}_{m>0}$ be the sequence of the normalized ground state of $H_m$, $m>0$. Then there exists a subsequence of $\{ \Psi_{m_{j}} \}_{j=1}^{\infty}$ with $m_{j+1} < m_{j} $, $j \in {\ensuremath{\mathbf{N}}}$, such that the weak limit $\Psi_{0} := $w-$\lim\limits_{j \to \infty } \Psi_{m_{j}} $ exists. \[9/16.a\] Suppose **(A.1)** - **(A.3)**. Then,\ $\qquad $ **(i)** ${\ensuremath{\mathscr{D}}}_{0}$ is a common core of $H_{{ \textrm{QED} }}$ and $H_{m}$, $m>0$, and $H_{m}$ strongly converges to $H_{{ \textrm{QED} }}$ on ${\ensuremath{\mathscr{D}}}_{0}$\ $\qquad $ **(ii)** $ \lim\limits_{m \to \infty} E_{0}(H_m) = E_{0}(H_{{ \textrm{QED} }} ) $. **(i)** Since ${\ensuremath{\mathscr{D}}}_{0}$ is a core of $H_{0,m}$, ${\ensuremath{\mathscr{D}}}_{0}$ is also a core of $H_{m}$. It is directly proven that $ \lim\limits_{m \to 0} H_{m} \Psi = H_{{ \textrm{QED} }} \Psi $ for all $\Psi \in {\ensuremath{\mathscr{D}}}_0$.\ **(ii)** We see that $(\Psi , H_{m} \Psi ) \geq ( \Psi, H_{{ \textrm{QED} }} \Psi ) \geq {E_{0}(H_{{ \textrm{QED} }})}$, for all $\Psi \in {\ensuremath{\mathscr{D}}}_{0}$. Hence $\inf\limits_{m>0}E_{0}(H_{m}) \geq E_{0}(H_m )$. From **(i)**, it follows that $H_{m}$ converges to $H_{{ \textrm{QED} }}$ as $m \to 0$ in the strong resolvent sense, and this yields that $\limsup\limits_{m \to 0}E_{0}(H_{m}) \leq E_{0}({H_{\textrm{QED}}})$. Hence **(ii)** follows. $\blacksquare $\ $\;$\ From Lemma \[9/16.a\] **(ii)**, we can set $$E_{\infty} = \sup\limits_{j\in {\ensuremath{\mathbf{N}}}} |E_{0}(H_{m_j })| \; < \infty . \\$$ \[9/14.a\] **(Number Operator Bounds)**\ Suppose **(A.1)** - **(A.6)**. Then, for all $ 0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}|} ,$ $$\begin{aligned} \quad {\ensuremath{\mathbf{(i)}}} \; \; & \sup_{j\in {\ensuremath{\mathbf{N}}} } \| (N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m_j } \| \leq \left( \frac{L_{\epsilon}}{M}E_{\infty} + \frac{R_{\epsilon} }{M}\right)^{1/2} , \notag \\ {\ensuremath{\mathbf{(ii)}}} \; \;& \sup_{j\in {\ensuremath{\mathbf{N}}} } \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j } \| \leq c_0 |{\kappa_{\textrm{I}}}| \, \left\| \frac{ \chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| , \notag \end{aligned}$$ where $c_{0} = \sqrt{\frac{11}{2(2\pi)^3}} \sum\limits_{j=1}^3 \sum\limits_{l,l'=1}^4 |\alpha^{j}_{l,l'}| c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} $. **(Proof)** **(i)** We see that $ \|( N_{{ \textrm{D} }}^{1/2} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 = (\Psi_m , ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} )\leq \|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|$, and Corollary \[9/9.f\] yields that for all $0 < \epsilon < \frac{1}{c_{\textrm{I}}| {\kappa_{\textrm{I}}}| }$, $$\|( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \| \leq \frac{L_{\epsilon}}{M} \| H_m \Psi_m \| + \frac{ R_{\epsilon} }{M}\|\Psi_m \| = \frac{L_{\epsilon}}{M}E_{0}(H_m ) + \frac{R_{\epsilon}}{M} . \notag$$ Hence **(i)** follows.\ **(ii)** From the photon pull-through formula in Proposition \[PF-P\], it follows that $$\begin{aligned} (\Psi_m , ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}) \Psi_m ) & = \sum_{r=1,2} \int_{{\mathbf{R}^{3} }} \| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & = |{\kappa_{\textrm{I}}}|^2 \sum_{r=1,2} \int_{{\mathbf{R}^{3} }}\| (H_m -E_{0}(H_m ) + \omega_{m}({\ensuremath{\mathbf{k}}})) Q_{r}({\ensuremath{\mathbf{k}}}) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq |{\kappa_{\textrm{I}}}|^2 \frac{11}{{2(2\pi)^3}} \sum_{r=1,2} \sum_{j=1}^3 \sum_{l,l'=1}^4 |\alpha^{j}_{l,l'}|^2 ( c_{{ \textrm{D} }}^{\,l} c_{{ \textrm{D} }}^{\,l'} )^2 \left( \int_{{\mathbf{R}^{3} }} \frac{| \chi_{{\textrm{rad}}}|^2}{|{\ensuremath{\mathbf{k}}}|^3} d {\ensuremath{\mathbf{k}}} \right) . \label{9/15.1}\end{aligned}$$ From (\[9/15.1\]), we obtain **(ii)**. $\blacksquare $\ \[9/15.a\] Assume **(A.1)**-**(A.6)**. Let $ F \in C_0^{\, \infty} ({\ensuremath{\mathbf{R}}}^3)$ which satisfy $0 \leq F \leq 1$ and $F({\ensuremath{\mathbf{x}}})=1$ for $|{\ensuremath{\mathbf{x}}}| \leq 1$, and set $ F_{ R}({\ensuremath{\mathbf{x}}}) = F (\frac{ {\ensuremath{\mathbf{x}}} }{R} ) $. Let $\hat{{\ensuremath{\mathbf{p}}}} = - i \nabla_{{\ensuremath{\mathbf{p}}}} $ and $\hat{{\ensuremath{\mathbf{k}}}} = - i \nabla_{{\ensuremath{\mathbf{k}}}} $. Then for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} {\kappa_{\textrm{I}}}}$, $R \geq 1$ and $R' \geq 1$, $$\begin{aligned} &\textbf{(i)} \; \; \, \sup_{j\in {\ensuremath{\mathbf{N}}} } \| ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}( F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m_j } \|) \leq \frac{c_{1,\epsilon }}{\sqrt{R}} , \\ &\textbf{(ii)} \; \; \, \sup_{ j\in {\ensuremath{\mathbf{N}}}} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )) \Psi_{m_j } \| ) \leq \frac{c_2}{\sqrt{R'}} , \notag \end{aligned}$$ where $$c_{1, \epsilon } = \left( 4 \frac{L_{\epsilon} E_{\infty} + R_{\epsilon}}{M}\right)^{1/4} \left( \left( \frac{L_{\epsilon} E_{\infty} + R_{\epsilon}}{M}\right)^{1/2} + ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 2|{\kappa_{\textrm{II}}}| \sum_{s=\pm 1/2} \sum_{\nu =1}^3 \sum_{\tau= \pm } \|F_{s ,\tau}^{\nu} \| \right)^{1/2} \notag$$ and $$c_2 = |{\kappa_{\textrm{I}}}|^{1/2} \left( c_0 \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| \,\right)^{1/2} \, \left( c_0 \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} \right\| + \, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right)^{1/2} . \notag$$ **(Proof)** It follows that $ ({{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )^2 \leq {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \leq {\ensuremath{d\Gamma_{\textrm{f}}({1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}})}) }} $, and then, $$\begin{aligned} \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 &\leq ( \Psi_m ,\left( {d \Gamma_{\textrm{f}}}( 1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}\right) \Psi_{m} ) \notag \\ & = \sum_{s= \pm 1/2} \left( \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) d {\ensuremath{\mathbf{p}}} \right. \notag \\ & \qquad \quad \left. + \int_{{\mathbf{R}^{3} }} \left( (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m } \right) d {\ensuremath{\mathbf{p}}} \right) . \label{9/15.2}\end{aligned}$$ We evaluate the two terms in the right-hand side of (\[9/15.2\]). The first term is estimated as $$\begin{aligned} &\left| \int_{{\mathbf{R}^{3} }} \left( (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_{m} , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \left( \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\ & = \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2}. \notag\end{aligned}$$ It is seen that $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \leq 4 \int_{{\mathbf{R}^{3} }} \| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2} (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \qquad \qquad \qquad \qquad + 4 \sum_{\nu=1}^3 \int_{{\mathbf{R}^{3} }} \| (1-F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) ) \, \frac{(\hat{p}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{p}}}}^2} \, (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \|^2 d {\ensuremath{\mathbf{p}}} . \notag\end{aligned}$$ Note that for all ${\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }$, $$\sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left| \left( 1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{1}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq \frac{1}{R^2} \; , \quad \; \; \; \; \sup_{{\ensuremath{\mathbf{p}}} \in {\mathbf{R}^{3} }} \left| \left( 1- F_{ R} ({\ensuremath{\mathbf{p}}})\right) \, \frac{p^{\nu}}{ 1+ {\ensuremath{\mathbf{p}}}^2} \right| \leq \frac{1}{R} .$$ Then by the electron derivative bounds in Theorem \[EP-DB\] **(i)** and the spectral decomposition theorem, we have $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \left\| (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}))(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \ \notag \\ & \leq \frac{4}{R^4} \int_{{\mathbf{R}^{3} }} \left\| (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} + \frac{4}{R^2} \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }} \left\| \partial_{p^{\nu}}(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \notag \\ & \leq \frac{4}{R^4} \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \|^2 + \frac{ c_m( \epsilon )^2}{R^2} \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }}|F_{s, +}^{\nu}({\ensuremath{\mathbf{p}}})|^2 d {\ensuremath{\mathbf{p}}} , \notag \end{aligned}$$ where $c_m ( \epsilon ) = 2 ( L_{\epsilon} E_{0} (H_m ) + R_{\epsilon} +1) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|$. Therefore, $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} (b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(b_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \| ( N_{{ \textrm{D} }}^{+} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \frac{2}{R^2} \| ( N_{{ \textrm{D} }}^+ {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3 \|F_{s ,+}^{\nu} \|_{L^2} \right) . \label{9/15.3}\end{aligned}$$ In a same way as the first term, we can estimate the second term in the right-hand side of (\[9/15.2\]) by the positron derivative bounds in Theorem \[EP-DB\] **(ii)**, and then, $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} (d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1})\Psi_m , (1- F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )(d_s({\ensuremath{\mathbf{p}}}) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_m ) d {\ensuremath{\mathbf{p}}} \right| \notag \\ & \leq \| ( N_{{ \textrm{D} }}^{-} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \times \left( \frac{2}{R^2} \| ( N_{{ \textrm{D} }}^- {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + \frac{c_m ( \epsilon )}{R} \sum_{\nu =1}^3 \|F_{s , -}^{\nu} \|_{L^2} \right) . \label{9/15.4}\end{aligned}$$ From (\[9/15.3\]) and (\[9/15.4\]), we have for all $R>1$,\ $$\begin{aligned} & \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{f}}}(F_{ R} (\hat{{\ensuremath{\mathbf{p}}}}) )) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}) \Psi_{m} \|^2 \notag \\ &\leq \frac{1}{R}\sum_{\tau = \pm} \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| \left( 2 \| ( N_{{ \textrm{D} }}^{\tau} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_m \| + c_m ( \epsilon ) \sum_{s=\pm 1/2} \sum_{\nu =1}^3 \|F_{s ,\tau}^{\nu} \|_{L^2} \right) . \label{9/15.7}\end{aligned}$$ From Lemma \[9/14.a\] **(i)**, $$\sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }}^{\pm} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \| \leq \sup_{j \in {\ensuremath{\mathbf{N}}}} \| ( N_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1})^{1/2} \Psi_{m_j} \| \leq \left( \frac{L_{\epsilon}}{M} E_{\infty} +\frac{R_{\epsilon}}{M} \right)^{1/2} ,$$ and we see that $$\sup_{j \in {\ensuremath{\mathbf{N}}}} c_{m_{j}} ( \epsilon ) = \sup_{j \in {\ensuremath{\mathbf{N}}}} \left(2 ( L_{\epsilon} E_{0}(H_{m_j}) + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}| \right) \; \leq \; 2 ( L_{\epsilon} E_{\infty} + R_{\epsilon} +1 ) |{\kappa_{\textrm{I}}}| + 4|{\kappa_{\textrm{II}}}|.$$ Hence **(i)** follows.\ **(ii)** In a similar way to the proof of **(i)**, it follows that $ ({{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})))^2 \leq {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}(F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})) \leq {\ensuremath{d\Gamma_{\textrm{b}}({1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) }) }} $, and hence, $$\begin{aligned} \| ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})) ) \Psi_{m} \|^2 &\leq ( \Psi_m ,\left( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}{d \Gamma_{\textrm{b}}}( 1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \right) \Psi_{m} ) \notag \\ & = \sum_{r= 1,2} \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} . \label{9/15.5}\end{aligned}$$ We see that $$\begin{aligned} &\left| \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} \right| \notag \\ & \leq \left( \int_{{\mathbf{R}^{3} }} \left\| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \right\|^2 d {\ensuremath{\mathbf{p}}} \right)^{1/2} \notag \\ & = \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \int_{{\mathbf{R}^{3} }} \left\| (1-F_{{\textrm{b}}, R'} )( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} \right)^{1/2} . \notag\end{aligned}$$ By the photon derivative bounds in Theorem \[P-DB\] and the spectral decomposition theorem, $$\begin{aligned} & \int_{{\mathbf{R}^{3} }} \| (1-F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) )( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq 4 \int_{{\mathbf{R}^{3} }} \| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{1}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}})) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \qquad \qquad + 4 \sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }} \| (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) \, \frac{(\hat{k}^\nu )^2}{1+ \hat{{\ensuremath{\mathbf{k}}}}^2} \, ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq \frac{4}{R'^4} \int_{{\mathbf{R}^{3} }} \left\| ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} + \frac{4}{R'^2} \sum_{\nu=1}^3\int_{{\mathbf{R}^{3} }} \left\| \partial_{k^{\nu}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right\|^2 d {\ensuremath{\mathbf{k}}} \notag \\ & \leq \frac{4}{R'^4} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2 + \frac{ 4 |{\kappa_{\textrm{I}}}|^2}{R'^2}\sum_{\nu =1}^3 \int_{{\mathbf{R}^{3} }}|F_{r}^{\nu}({\ensuremath{\mathbf{k}}})|^2 d {\ensuremath{\mathbf{k}}} . \notag \end{aligned}$$ Then we have $$\begin{aligned} & \left| \int_{{\mathbf{R}^{3} }} \left( ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) )\Psi_{m} , (1- F_{R'}(\hat{{\ensuremath{\mathbf{k}}}}) ) ({{\small \text{1}}\hspace{-0.32em}1}{\otimes}a_{r}({\ensuremath{\mathbf{k}}}) ) \Psi_m \right) d {\ensuremath{\mathbf{k}}} \right| \notag \\ & \leq \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \frac{4}{{R'}^4} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \|^2 + \frac{ 4|{\kappa_{\textrm{I}}}|^2 }{{R'}^2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}^2 \right)^{1/2} \notag , \end{aligned}$$ and hence, for all $R'>1$, $$\begin{aligned} & \| ( ( {{\small \text{1}}\hspace{-0.32em}1}- {\Gamma_{\textrm{b}}}( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) ) \Psi_{m} \|^2 \notag \\ &\leq \frac{2}{R'} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| \times \left( \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_m \| +|{\kappa_{\textrm{I}}}|\, \sum_{r=1,2} \sum_{\nu =1}^3 \| F_{r}^{\nu} \|_{L^2}\right) .\end{aligned}$$ From Lemma \[9/14.a\] (**ii**), it holds that $ \sup\limits_{j \in {\ensuremath{\mathbf{N}}}} \| ( {{\small \text{1}}\hspace{-0.32em}1}{\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j} \| < c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}} , \right\| $. Therefore the proof is obtained. $\blacksquare$\ $\;$\ [**(Proof of Theorem \[Main-Theorem\])**]{}\ From Proposition \[9/16.a\] and a general theorem ([@AH97] ; Lemmma 4.9), it is enough to show that w-$ \lim\limits_{j \to \infty} \Psi_{m_j} \neq 0$. We see that $$\begin{aligned} {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} & = ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) )) \notag \\ & \quad + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n] )) + {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([ n+1, \infty) )) . \notag \end{aligned}$$ Then by Proposition \[9/15.a\], we have for all $0 < \epsilon < \frac{1}{c_{{\textrm{I}}} |{\kappa_{\textrm{I}}}|}$, $R>1$ and $R'> 1 $, $$\begin{aligned} & \left\| \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) E_{N_{{\textrm{rad}}}}([0, n ] ) )\right) \Psi_{m_j} \right\| \notag \\ & \geq 1- \left( \frac{}{} \| (( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} -{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ))) {\otimes}{{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} ) \Psi_{m_j} \| \right. \notag \\ & \qquad \qquad \qquad \left. + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}( {{\small \text{1}}\hspace{-0.32em}1}_{{\textrm{rad}}} - {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) ) ) \Psi_{m_j} \| + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \frac{}{} \right) . \notag \\ & \geq 1- \left( \frac{c_{1, \epsilon }}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \right) , \notag\end{aligned}$$ It is seen that $$\sqrt{n+1 }\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \leq \| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}N_{{\textrm{rad}}}^{1/2} ) \Psi_{m_j} \| \leq c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| .$$ Then from Lemma \[9/14.a\] **(ii)**, we have $$\sup\limits_{j \geq 1}\| ( {{\small \text{1}}\hspace{-0.32em}1}_{{ \textrm{D} }} {\otimes}E_{N_{{\textrm{rad}}}}([n+1, \infty) )) \Psi_{m_j} \| \leq \frac{c_3 }{(n+1)^{1/2}}. \notag$$ where $c_{3}= c_{0} |{\kappa_{\textrm{I}}}| \left\| \frac{\chi_{{\textrm{rad}}}}{\omega^{3/2}}\right\| $. Then it follows that $$\| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \geq 1 - \left( \frac{c_{1 , \epsilon}}{R} + \frac{c_2}{R'} + \frac{c_3}{(n+1)^{1/2}} \right) . \label{9/15.9}$$ We also see that $$\begin{aligned} & \| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \|^2 \notag \\ & = ((H_0 +1) E_{N_{{\textrm{rad}}}}([0, n ] ) \Psi_{m_j} ,(H_0 +1)^{-1} ( {\Gamma_{\textrm{f}}}(F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] )) ) \Psi_{m_j} ) \notag \\ & \leq \| (H_0 +1)\Psi_{m_j} \| \, \| (H_0 +1)^{-1} \, ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))) \, \Psi_{m_j} \| . \notag\end{aligned}$$ We see that $\| (H_0 +1)\Psi_{m_j} \| \leq \| H_0 \Psi_{m_j} \| +1 \leq \| H_{0,m_{j}} \Psi_{m_j} \| +1 $ and Lemma \[9/9.e\] yields that $$\| H_{0,m_{j}} \Psi_{m_j} \| \leq \frac{L_{\epsilon } }{M} \| H_{m_j} \Psi_{m_j} \| + \frac{R_{\epsilon }}{M} \leq \frac{L_{\epsilon}}{M} E_{\infty} + \frac{R_{\epsilon}}{M} .$$ Then we have $$\begin{aligned} &\| {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )) {\otimes}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \notag \\ & \qquad \qquad \qquad \leq c_{4 ,\epsilon}\| (H_0 +1)^{-1} \, \left( ( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} ) ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}} )^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) \, \Psi_{m_j} \|^{1/2}, \label{9/15.10}\end{aligned}$$ where $c_{4 , \epsilon }= (\frac{L_{\epsilon}E_{\infty} + R_{\epsilon} }{M} +1)^{1/2}$. From (\[9/15.9\]) and (\[9/15.10\]) $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{m_j} \| \geq \frac{1}{c_{4, \epsilon }}^2 \left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 \notag$$ Since $(H_0 +1)^{-1} \left( {\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}} )^2) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2) E_{N_{{\textrm{rad}}}}([0, n ] ))) \right) $ is compact, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Psi_{0} \| \geq \frac{1}{c_{4, \epsilon }^2}\left( 1 - \left( \frac{c_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 , \label{9/15.20}$$ where we set $\Psi_{0} =$ w-$ \lim\limits_{j \to \infty} \Psi_{m_j} $. Then for sufficiently large $R>0$, $R'>0 $ and $n >0$, the right-hand side of (\[9/15.20\]) is greater than zero, and hence $ \Psi_{0} \ne 0 $.\ $\;$\ **(Multiplicity)** Assume dim ker $(H -E_{0}(H_{{ \textrm{QED} }})) = \infty $. Let $ \Psi_{l} $, $l \in {\ensuremath{\mathbf{N}}}$, be the ground states. Let ${\ensuremath{\mathscr{M}}} $ be the closure of the linear hull of $\{ \Psi_l \}_{l=0}^{\infty}$. Then ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ is decomposed as ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }} = {\ensuremath{\mathscr{M}}} \oplus {\ensuremath{\mathscr{M}}}^{\bot} $. Let $\{ \Phi_l \}_{l=0}^{\infty}$ be a complete orthogonal system of ${\ensuremath{\mathscr{M}}}^{\bot}$. We can set a complete orthonormal system $\{ \Xi_l \}_{l=0}^{\infty}$ of ${\ensuremath{\mathscr{H}}}_{{ \textrm{QED} }}$ by $ \Xi_{2 l-1 } = \Psi_{l} $ and $ \Xi_{2 l} = \Phi_{l} $ for all $l \in {\ensuremath{\mathbf{N}}}$. Since $\{ \Xi_l \}_{l=0}^{\infty}$ is a complete orthonormal system , w-$\lim\limits_{l \to \infty}\Xi_l =0$. On the other hand, $\Xi_{2 l-1 }$ is ground state for all $l \in {\ensuremath{\mathbf{N}}}$, and hence $H_{{ \textrm{QED} }}\Xi_{2 l-1 }=E_{0}({H_{\textrm{QED}}}) \Xi_{2 l-1 } $. In a same argument of the proof of the existence of the ground state, we have $$\|(H_0 +1)^{-1}{\Gamma_{\textrm{f}}}( F_{R}(\hat{{\ensuremath{\mathbf{p}}}})^2 ) {\otimes}( {\Gamma_{\textrm{b}}}( F_{R'}(\hat{{\ensuremath{\mathbf{k}}}})^2 ) E_{N_{{\textrm{rad}}}}([0, n ] ))\Xi_{2l-1} \| \geq \frac{1}{\tilde{c}_{4, \epsilon }^2}\left( 1 - \left( \frac{ \tilde{c}_{1,\epsilon}}{\sqrt{R}} + \frac{c_2}{\sqrt{R'}} + \frac{c_3}{(n+1)^{1/2}} \right) \right)^2 , \label{9/15.15}$$ where $\tilde{c}_{1, \epsilon} $ and $\tilde{c}_{4, \epsilon} $ are the constants $ c_{1 , \epsilon}$ and $ c_{1 , \epsilon}$ replacing $E_{\infty}$ with $E_{0}({H_{\textrm{QED}}})$. Then by taking sufficiently large $R>0$, $R'>0 $ and $n >0$, we have w-$\lim\limits_{l \to \infty}\Xi_{2l-1} \ne 0$, but this is contradict to w-$\lim\limits_{l \to \infty}\Xi_l = 0$. Hence dim ker $(H_{{ \textrm{QED} }} -E_{0}(H_{{ \textrm{QED} }})) < \infty $. $\blacksquare $\ $\;$\ $\;$\ [**\[Concluding remarks\]**]{}\ **(1) The case of Massless Dirac field**\ It is not realistic model, but we can consider the system of a massless Dirac field coupled to the radiation field. In such a case, by replacing (**A.5**) with similar conditions to (**A.6**), we can also prove the existence of the ground state in a same ways as ${H_{\textrm{QED}}}$. $\; $\ **(2) Infrared divergent problem**\ For some systems of particles coupled to massless Bose fields, the existence of the ground states without infrared regularity conditions was obtained (refer to e.g., Bach-Fröhlich-Sigal [@BFS99], Griesemer-Lieb-Loss [@GLL01] and Hasler-Herbst [@HaHe11]), and non-existence of the ground states for other other systems was also investigated (see e.g., Arai-Hirokawa-Hiroshima [@AHH99]). To prove the existence or non-existence of the ground state of ${H_{\textrm{QED}}}$ without infrared regularity conditions is left for future study. [99]{}

ArXiv

--- abstract: 'With a self-similar magnetohydrodynamic (MHD) model of an exploding progenitor star and an outgoing rebound shock and with the thermal bremsstrahlung as the major radiation mechanism in X-ray bands, we reproduce the early X-ray light curve observed for the recent event of XRO 080109/SN 2008D association. The X-ray light curve consists of a fast rise, as the shock travels into the “visible layer" in the stellar envelope, and a subsequent power-law decay, as the plasma cools in a self-similar evolution. The observed spectral softening is naturally expected in our rebound MHD shock scenario. We propose to attribute the “non-thermal spectrum" observed to be a superposition of different thermal spectra produced at different layers of the stellar envelope.' author: - 'Ren-Yu Hu' - 'Yu-Qing Lou' title: | Rebound Shock Breakouts of Exploding\ Massive Stars: A MHD Void Model --- [ address=[Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing 100084, China]{}, email=[[email protected]]{} ]{} [ address=[Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua University, Beijing 100084, China]{}, email=[[email protected]]{} ]{} Introduction ============ SN 2008D, the best type Ibc supernova detected so far, is preceded by a X-Ray Outburst (XRO) captured by SWIFT satellite on 2008 January 9, and this XRO is interpreted as a shock breakout of a Wolf-Rayet (WR) progenitor with a radius of $\sim 10^{11}$ cm [Soderberg2008]{}. The isotropic X-ray energy is estimated to be $\sim 2\times10^{46}$ erg, and there seems no collimation detected so the event is not regarded as a GRB. This XRO showed a rapid rise, peaked at $\sim 63$ s, and a decay modelled to be exponential with an e-folding time of $\sim 129$ s [@Soderberg2008]. The follow-up optical and ultraviolet observations indicate a total supernova kinetic energy of $\sim 2-4\times10^{51}$ erg and a mass of SN ejecta to be $\sim 3-5$ M$_{\odot}$ [@Soderberg2008]. Some authors estimate from a detailed spectral analysis that SN 2008D, originally a $\sim 30$ M$_{\odot}$ star, has a spherical symmetric explosion energy of $\sim 6\times10^{51}$ erg and an ejected mass $\sim 7$ M$_{\odot}$ [@Mazzali]. The evolution of optical spectra of XRO-SN 2008D resembles that of XRO-SN 2006aj, whose progenitor is also believed to be a WR star [@Campana2006]. The production of $\gamma$-rays and X-rays by shock breakouts has been proposed earlier [@Colgate; @Chevalier]. This XRO and the associated SN present an unprecedented case to be investigated in details, especially on interpretations for the rise and decay times of the X-ray light curve. The claim of an exponential decay may be premature given a fairly large scatter, and it may have concealed valuable physical clues offered by this XRO. During the XRO, the observed spectroscopic softening still lacks a convincing explanation. Here, we advance a self-similar MHD rebound shock model in an attempt to reproduce the observed X-ray light curve. The next section contains an overall description of the self-similar MHD model and the procedure of analysis; in the third section, we compare our model results with data; and conclusions are summed up in the last section. A Self-Similar MHD Void Shock Model =================================== For a polytropic magnetofluid in quasi-spherical symmetry under the self-gravity, the governing magnetohydrodynamic (MHD) equations include mass conservation, momentum conservation (Euler equation), magnetic induction equation, and an equation of specific entropy conservations along streamlines to approximate energetic processes. For this more general polytropic equation of state, we regard the polytropic index $\gamma$ as a parameter [@WangLou08]. These coupled nonlinear MHD partial differential equations (PDEs) can be reduced to nonlinear ordinary differential equations (ODEs) by introducing a self-similar transformation $r=k^{1/2}xt^n$, where $r$ is the radius, $t$ is the time and $k$ is a scale parameter relevant to the local sound speed, rendering the independent self-similar variable $x$ dimensionless. The corresponding transformation of the dependent MHD variables can be found in refs. [@WangLou08; @LouHu]. The exponent $n$ is a key parameter that determines the dynamic behaviour of a polytropic fluid. For $n+\gamma=2$, the formulation reduces to that of a conventional polytropic gas in which the specific entropy remains constant everywhere [@SutoSilk1988; @Yahil1983; @LouWang06; @LouWang07; @WangLou07]. The special case of $n=1$ and $\gamma=1$ corresponds to the isothermal case [@BianLou2005; @YuLouBianWu06]. Such self-similar evolutions represent an important subclass of all possible evolutions. We also introduce a dimensionless magnetic parameter to represent the strength of a magnetic field $h\equiv<B_t^2>/(16\pi^2G\rho^2r^2)$, where $<B_t^2>$ is the ensemble average of a random transverse magnetic field squared, $G$ is the gravity constant and $\rho$ is the mass density. Meanwhile, MHD shocks are necessary to connect different branches of self-similar solutions. The conservation laws impose constraints on physical variables across a MHD shock front. We can then derive downstream physical quantities (density, velocity, pressure and temperature) from the upstream physical quantities or vice versa. Self-similar solutions produce radial profiles of density, radial velocity, pressure and temperature at any time of evolution, and the detailed procedure of analysis can be found in the reference of Wang & Lou [@WangLou08]. It is also sensible to invoke the plasma cooling function and obtain radiation diagnostics from a magnetofluid of high temperatures $\sim 10^7-10^8$K [@Sutherland1993]. Recently, we obtained a new class of self-similar “void" solutions within a certain radius $r^*$ referred to as the void boundary. In general, such a void solution describes an expanding fluid envelope with a central cavity and possibly associated with an outgoing shock [@LouCao]. The self-similar evolution implies that the central void expands as a power-law in time $r^*\propto t^n$. We study detailed behaviours of void solutions under different parameters in a general polytropic MHD framework [HL, LouHu]{}. Here, we propose to utilize such void shock solutions to model the explosion of a massive progenitor star in the process of a rebound MHD shock breakout. The Bondi-Parker radius of a remnant compact object if any left in the center is defined as $$r_{\rm BP}=\frac{GM_*}{2a^2}\ ,\label{BP}$$ where $M_*$ is the mass of the central object and $a$ is the sound speed at the inner void edge of the surrounding gas. Far beyond this radius $r_{\rm BP}$, the gravity of the central object becomes negligible compared to the thermal pressure. For supernovae, $M_*$ would be of the order of M$_{\odot}$ [@Mazzali]. At $\sim 1$ s after the core bounce, the temperature of the stellar envelope is of the order of $10^8$ K, and the sound speed $a^2\sim10^{17}$ cm$^2$ s$^{-2}$, and then $r_{\rm BP}\sim 10^{8}$ cm. Meanwhile, the void radius $r^*$ expands to larger than $10^8$ cm [@Janka]. Furthermore, the Bondi-Parker radius expands slower than the void boundary does [@LouHu]. Therefore, the cavity assumption may be justifiable. MHD Model and X-ray Light Curve =============================== The self-similar MHD void shock model of a WR stellar envelope in explosion associated with a shock breakout and the corresponding X-ray light curve are shown in Figure \[Figure1\]. ![Our MHD void shock model for a shock breakout in a progenitor of SN 2008D (left) and the resulting X-ray light curve (right). On the left from top to bottom, the panels show the radial profiles of density, radial velocity, pressure, enclosed mass and temperature of the stellar envelope within radial range $10^8$ cm (void boundary) and $\sim10^{11}$ cm (outer boundary) at 1 s after the core collapse and rebounce. The model is obtained with the self-similar parameters as $n=0.8$, $\gamma=1.2$ (conventional polytropic) and $h=0$ (non-magnetized fluid). On the right, we compare the X-ray light curve calculated from our MHD void shock model (red curve) and data from the X-Ray Telescope (XRT) on board the SWIFT satellite [@Soderberg2008] (solid circles with error bars suppressed). X-ray fluxes are normalized to the peak flux. The X-ray light curve is shown as a function of time since the XRT trigger, noted as $t_{\rm obs}$. The core collapse happened $\sim 552$ s before the trigger. The X-ray light curve is calculated with the radiation layer thickness to be $9\times10^{9}$ cm. The calculated X-ray light curve is in a shape of fast-rise-and-decay. The rise time is $\sim 62$ s, and the decay obeys a power law to the time since core collapse with the index to be $-4.3$. The equivalent e-folding is 128 s (i.e., the timescale when the emission intensity drops to $\sim 37\%$ of the peak value).[]{data-label="Figure1"}](Figure1.eps){height=".3\textheight"} Following observational inferences [@Soderberg2008] for the progenitor radius, we cut off our model at this “outer boundary" ($r_{\rm out}\sim 10^{11}$ cm). This approximation is reasonable, as the radial density profile of the star drops rapidly at the stellar surface. We do not consider dynamical effects and the X-ray contributions of the gas outside $r_{\rm out}$. Our MHD model gives an enclosed mass at $\sim 10^{11}$ cm to be $3.8$ M$_{\odot}$, comparable to the estimated mass of ejecta $(\sim 3-5$ M$_{\odot})$. The gas kinetic energy is $\sim 3\times10^{51}$ erg, also in the observed range. The gravitational binding energy given by our model is $\sim 10^{50}$ erg, much less than the kinetic energy corresponding to an exploding stellar envelope during a MHD rebound shock breakout. In Figure \[Figure1\], we see features of a rebound MHD shock surrounding a central void in self-similar expansion. From the upstream to downstream sides across the shock, the density, pressure and temperature increases suddenly. The radial velocity also increases, but the velocity in the shock comoving reference framework decreases as expected from the upstream to downstream sides. Note that the temperature is $\sim 10^8$ K and it drops to $\sim 10^7$ K in the process of the evolution under consideration, corresponding to the energy range of X-ray photons detected. Typically the part of gas near the downstream shock front, where the density and temperature are the highest, is most efficient in producing X-ray emissions. We suggest that X-ray emissions observed are from the thermal bremsstrahlung radiation mainly produced around the downstream side of a rebound shock. We compute the X-ray light curve using the plasma cooling rate result of reference [@Sutherland1993 see also Lou & Zhai 2008 in preparation for X-ray diagnostics of isothermal voids in self-similar expansion], in which both the free-free and free-bound emissions are taken into account. The optical depth in the stellar envelope is unknown, so we treat it as another parameter to search for the best fit of X-ray light curve. Here we introduce an “inner boundary" $r_{\rm in}$, and presume that only X-ray emissions in the layer between $r_{\rm in}$ and $r_{\rm out}$ can be observed and should be integrated. The thickness of such “radiative layer" noted as $s$ is another parameter to adjust. Our scenario is as follows. The MHD rebound shock front expands outward obeying a power law in time $t$ since the core collapse, and the shock strength weakens with increasing $t$. Before the shock front reaches $r_{\rm in}$, the density and temperature are low in the radiative layer and cannot produce detectable X-ray emissions. Once the shock front reaches $r_{\rm in}$ and runs into the radiative layer, more and more downstream part enters the radiative layer, and X-ray emissions increase rapidly. The X-ray emission reaches its maximum when the shock front reaches $r_{\rm out}$ (shock breakout) and the entire radiative layer is occupied by the downstream part. Thereafter, the density and temperature inside the radiative layer decrease self-similarly, and the X-ray emission decreases obeying a power law. The radiative layer thickness $s$ and the shock speed determine the rise time and the power law index of the subsequent decay. The average temperature of the radiative layer decreases self-similarly in the shock breakout process, naturally leading to the spectral softening as observed. Summary and Conclusions ======================= With a self-similar MHD void shock model and the thermal bremsstrahlung as the main radiation loss, we obtain a fairly good fit to the X-ray light curve observed and confirm that XRO 080109 is most likely a shock breakout event. We identify that the decay in the X-ray light curve follows a power law (instead of an exponential law) in time since the core collapse and rebounce, which occurred $\sim 552$ s before the observation of X-ray emissions. Meanwhile, the spectral softening is expected qualitatively. In this work, we use the most simplified radiation transfer presumption that the radiation produced in the ‘visible layer’ can be totally observed. Actually, the optical depth varies with radius and should be treated in a more elaborate manner. Additionally, we presume that the boundaries of the “visible layer" $r_{in}$ and $r_{out}$ do not vary with time. Despite all these idealizations, our self-similar dynamic approach appears to be suitable to couple with the radiation process and to model X-ray outbursts in supernova as observed. Regarding the X-ray spectra observed, they cannot be fitted with a simple blackbody profile, and a nonthermal power law profile was suggested [@Soderberg2008]. Several mechanisms have been proposed to explain such X-ray spectra, for example the bulk comptonization by scatterings of the photons between the ejecta and a dense circumstellar medium [@Soderberg2008], or diluted thermal spectra which require the thermalization occurs at a considerable depth in the supernova [@Chevalier2008]. We propose that the power-law profile might be a natural result of multi-colour superposition of blackbody spectra. Based on our scenario outlined here, X-ray emissions come from different layers within the radiative layer around the stellar surface, and the radiative layer has different temperatures at different depth. As a result, the observed X-ray spectra are the superposition of thermal blackbody components with different temperatures. We suggest that this might resolve issues of spectral profile and evolution. During breakouts of rebound shocks and in the presence of MHD shock accelerated relativistic electrons usually presumed with a power-law energy spectrum within a certain electron energy range, we could also compute synchrotron emissions associated with such kind of SN shock breakouts. Among others, it is then possible to follow the evolution of magnetic field strength associated with SN explosions [@Lou94; @LouWang07] and estimate the effectiveness of accelerating relativistic particles (i.e., high-energy cosmic rays [@Science06]). There is the freedom of choosing a few parameters to fit the data at a certain epoch. It is then possible to test the hypothesis of a self-similar shock evolution by further observations. In a more general perspective and on the basis of our dynamic models for rebound MHD shocks, we hope to further develop radiative diagnostics for shock breakouts of supernovae and thus for SN related GRBs. This research has been partially supported by Tsinghua Center for Astrophysics (THCA), by the National Natural Science Foundation of China (NSFC) grants 10373009 and 10533020 and by the National Basic Science Talent Training Foundation (NSFC J0630317) at the Tsinghua University, and by the SRFDP 20050003088 and the Yangtze Endowment from the Ministry of Education at Tsinghua University. [9]{} A. D. Soderberg, E. Berger, K. L. Page, P. Schady, J. Parrent, et al., *Nature* **453**, 469–474 (2008). P. A. Mazzali, S. Valenti, M. Della Valle, G. Chincarini, et al., *Science* **223**, L109–L112 (2008). S. Campana, V. Mangano, A. J. Blustin, P. Brown, D. N. Burrows, et al., *Nature* **442**, 1008–1010 (2006). S. A. Colgate, *Astrophys. J.* **187**, 333–335 (1974). R. I. Klein and R. A. Chevalier, *Astrophys. J.* **223**, L109–L112 (1978). Y.-Q. Lou and W.-G. Wang, *Mon. Not. R. Astron. Soc.*, **372**, 885–900 (2006). Y.-Q. Lou and W.-G. Wang, *Mon. Not. R. Astron. Soc.*, **378**, L54–L58 (2007). Y.-Q. Lou, *Astrophys. J. Lett.*, **428**, L21–L24 (1994). M. Amenomori et al., *Science*, **314**, 439-443 (2006). W.-G. Wang and Y.-Q. Lou, *Astrophys. Space Sci.*, **311**, 363–400 (2007). W.-G. Wang and Y.-Q. Lou, *Astrophys. Space Sci.*, **315**, 135–156 (2008). Y.-Q. Lou and Y. Cao, *Mon. Not. R. Astron. Soc.*, **384**, 611–629 (2008). F. Y. Bian and Y.-Q. Lou, *Mon. Not. R. Astron. Soc.*, **363**, 1315–1328 (2005). C. Yu, Y.-Q. Lou, F. Y. Bian, and Y. Wu, *Mon. Not. R. Astron. Soc.*, **370**, 121–140 (2006). R.-Y. Hu and Y.-Q. Lou, *Mon. Not. R. Astron. Soc.*, (2008arXiv0808.2090H) in press (2008). Y.-Q. Lou and R.-Y. Hu, *Mon. Not. R. Astron. Soc.*, submitted (2008). H. T. Janka and E. Müller, *Astron. Astrophys.* **306**, 167–198 (1996). 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ArXiv

--- abstract: | We generalise and improve a result of Stoll, Walsh and Yuan by showing that there are at most two solutions in coprime positive integers of the equation in the title when $b=p^{m}$ where $m$ is a non-negative integer, $p$ is prime, $(a,p)=1$, $a^{2}+p^{2m}$ not a perfect square and $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has an integer solution. This result is best possible. We also obtain best possible results for all positive integer solutions when $m=1$ and $2$. When $b$ is an arbitrary square with $(a,b)=1$ and $a^{2}+b^{2}$ not a perfect square, we are able to prove there are at most three solutions in coprime positive integers provided $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution and $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-b^{2}$ has only one family of solutions. Our proof is based on a novel use of the hypergeometric method that may also be useful for other problems. address: 'London, UK' author: - Paul M Voutier title: 'Sharp bounds on the number of solutions of $X^{2}-\left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$' --- Introduction ============ Diophantine equations of the form $aX^{2}-bY^{4}=c$ are linked to several important areas in number theory. They are a quartic model of elliptic curves, for example. They are also associated with squares in binary recurrence sequences too. Ljunggren (see [@L1; @L2; @L3; @L4] for some of his many results) made significant contributions to the study of the integer solutions of such equations, especially when $a$, $b$ are positive integers and $c=\pm 1, \pm 2, \pm 4$. They have been the subject of much attention since then too (see, for example, Akhtari’s result [@Akh] and the references there). For other values of $c$, the study of such equations appears to be much more difficult. In 2009, Stoll, Walsh and Yuan [@SWY] showed that for any non-negative integer $m$, there are at most three solutions in odd positive integers to $$X^{2} - \left( 1+2^{2m} \right) Y^{4} = -2^{2m}.$$ Here we generalise, and improve, their result to the equation $$\label{eq:2} X^{2} - \left( a^{2}+b^{2} \right) Y^{4} = -b^{2},$$ under the conditions stated in our theorems below. \[thm:1.1\] Let $a$, $m$ and $p$ be non-negative integers with $a \geq 1$, $p$ a prime, $\gcd \left( a,p^{m} \right)=1$ and $a+p^{2m}$ not a perfect square. Suppose $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. Then $\eqref{eq:2}$ has at most two coprime positive integer solutions. Note that the conditions in Theorem \[thm:1.1\] are always satisfied for $a=1$ and $p=2$, so the results here to include, and improve, the results in [@SWY]. \[rem:1\] Theorem \[thm:1.1\] is best possible. One can find infinitely many examples of $a,m$ and $p$ such that there are two solutions in coprime positive integers. Example 1: let $b$ be any odd positive integer not divisible by $5$ and $a=\left( b^{2}-5 \right)/4$. Then we have the obvious solution, $(a,1)$, of . The fundamental solution of the negative Pell equation here is $\left( a+2, 1 \right)$, so $\left( a+\sqrt{a^{2}+b^{2}} \right) \left( (a+2) + \sqrt{a^{2}+b^{2}} \right)^{2}$ gives rise, after simplifying, to another solution, $\left( \left( b^{6}+5b^{4}+15b^{2}-5 \right)/16, \left( b^{2}+1 \right) /2 \right)$, of . Example 2: let $b$ be any odd positive integer and $a=\left( 5b^{2}-1 \right)/4$. also has two solutions. In addition to the obvious solution, $(a,1)$, of , we also have the following solution, $\left( \left( 3125b^{6}+625b^{4}+75b^{2}-1 \right)/16, \left( 25b^{2}+1 \right)/2 \right)$. Of course, it would be satisfying to remove the condition that the coordinates of the integer solutions be coprime. We have not been able to do that in the same generality as in Theorems \[thm:1.1\], but we have been able to prove the following. \[cor:1.1\] Let $a$, $m$ and $p$ be positive integers with $a \geq 1$, $m=1,2$, $p$ a prime, $\gcd \left( a,p \right)=1$ and $a^{2}+p^{2m}$ not a perfect square. Suppose $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. Then $\eqref{eq:2}$ has at most three positive integer solutions. (of Corollary \[cor:1.1\]) From Theorem \[thm:1.1\], we know there are at most two coprime solutions. If there is a solution with $\gcd(x,y) \neq 1$, then for both $m=1$ and $m=2$, we can remove the common factors to get $-1$ on the right-hand side. We can now appeal to Theorem D of [@Chen1] to show there is at most one such solution. Corollary \[cor:1.1\] is also best possible. We can use Example 1 in Remark \[rem:1\] to see this. Suppose $b$ there is a perfect square, $b=b_{1}^{2}$. In addition to the two solutions given in Remark \[rem:1\], we also have the solution $\left( \left( b^{3}+3b \right)/4, b_{1} \right)$. We only found one example with $b$ prime and three solutions, namely $a=31$, $b=5$ with the solutions $(31, 1)$, $(785, 5)$, $(3076289, 313)$. It is natural to wonder what happens when $p^{m}$ is replaced by any positive integer $b$. Our technique here can be used to show that Theorem \[thm:1.1\] and Corollary \[cor:1.1\] are both true if we replace $p^{m}$ with $2p^{m}$. The proof is nearly identical to what follows, so we have not pursued this here. We are also able to prove the following result. \[thm:1.2\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has a solution and that all coprime integer solutions $(x,y)$ to the quadratic equation $$\label{eq:quad-eqnc} x^{2} - \left( a^{2}+b^{2} \right)y^{2}=-b^{2}$$ are given by $$\label{eq:14c} x+y \sqrt{a^{2}+b^{2}} = \pm \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \alpha^{2k}, \hspace{1.0mm} k \in {\mathbb{Z}},$$ where $\alpha = \left( T_{1}+U_{1} \sqrt{a^{2}+b^{2}} \right)/2$ and $\left( T_{1},U_{1} \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=-4$ in positive integers. Then $\eqref{eq:2}$ has at most three coprime positive integer solutions. We have not been able to find any equations satisfying these conditions that have three solutions, so we believe that there are at most two coprime solutions of such equations too. It would also be of interest to eliminate the condition that $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. However, we have not been able to do so. The obstacle is that Lemma \[lem:3.2aa\] is no longer true without that condition. An example where this fails is provided in the remark after Lemma \[lem:3.2aa\]. Diophantine Approximation via Hypergeometric Functions ====================================================== Recall that by an [*effective irrational measure*]{} for an irrational number, $\alpha$, we mean an inequality of the form $$\left| \alpha - \frac{p}{q} \right|>\frac{c}{|q|^{\mu}},$$ for all $p/q \in {\mathbb{Q}}$ with $\gcd(p,q)=1$ and $|q|>Q$, where $c$, $Q$ and $\mu$ are all effectively computable. By Liouville’s famous result [@Liou], where he constructed the first examples of numbers proven to be transcendental, we have such effective irrational measures for algebraic numbers of degree $n$, with $\mu=n$. But for most applications we require $\mu<n$. We can use the hypergeometric method to obtain effective irrationality measures that improve on Liouville’s result for the algebraic numbers that arise here. However, that does not suffice for us to prove our theorem. The problem here arises not because of the exponent, $\mu$, in the effective irrationality measure, but because the constant, $c(\alpha)$, is too large. Upon investigating this further, we found that we can complete the proof of Theorem \[thm:1.1\] if we use not the effective irrationality measures from the hypergeometric method, but rather consider more carefully the actual results that we obtain from the use of hypergeometric functions. The means of doing so is the following lemma. \[lem:2.1\] Let $\theta \in {\mathbb{C}}$ and let ${\mathbb{K}}$ be an imaginary quadratic field. Suppose that there exist $k_{0},\ell_{0} > 0$ and $E,Q > 1$ such that for all non-negative integers $r$, there are algebraic integers $p_{r}$ and $q_{r}$ in ${\mathbb{K}}$ with $\left| q_{r} \right| < k_{0}Q^{r}$ and $\left| q_{r} \theta - p_{r} \right| \leq \ell_{0}E^{-r}$ satisfying $p_{r}q_{r+1} \neq p_{r+1}q_{r}$. For any algebraic integers $p$ and $q$ in ${\mathbb{K}}$, let $r_{0}$ be the smallest positive integer such that $|q|<E^{r_{0}}/ \left( 2 \ell_{0} \right)$. [(a)]{} We have $$\left| q\theta - p \right| > \frac{1}{2k_{0}Q^{r_{0}+1}}.$$ [(b)]{} When $p/q \neq p_{r}/q_{r}$, we have $$\left| q\theta - p \right| > \frac{1}{2k_{0}Q^{r_{0}}}.$$ We can improve the constants here somewhat, replacing $1/\left( 2k_{0} \right)$ in both parts by $\left( 1-1/E \right)/k_{0}$ and defining $r_{0}$ by $\left( Q-1/E \right)/\left( Q-1 \right)\ell_{0}|q|<E^{r_{0}}$. This would be helpful when reducing the size of $c$ here is important. This would have reduced the size of the bound on $a^{2}+b^{2}$ in Case 3 of Lemma \[lem:thm\] below. But as the remaining calculation to finish the proof of Lemma \[lem:thm\] is so quick, we have not pursued this here. The proof is identical to that of Lemma 6.1 of [@V2] except at the end of the proof we do not convert the lower bounds into ones involving $|q|^{-(\kappa+1)}$. Construction of Approximations ------------------------------ \ Let $t, u_{1}$ and $u_{2}$ be rational integers with $t<0$. We let $u=\left( u_{1}+u_{2}\sqrt{t} \right)/2$ be an algebraic integer in ${\mathbb{K}}={\mathbb{Q}}\left( \sqrt{t} \right)$ with $\sigma(u)=\left( u_{1}-u_{2}\sqrt{t} \right)/2$ as its algebraic (and complex) conjugate. Put $\omega = u/\sigma(u)$ and write $\omega=e^{i\varphi}$, where $-\pi<\varphi \leq \pi$. For any real number $\nu$, we shall put $\omega^{\nu}= e^{i\nu\varphi}$. Suppose that $\alpha, \beta$ and $\gamma$ are complex numbers and $\gamma$ is not a non-positive integer, ${}_{2}F_{1}(\alpha, \beta, \gamma, z)$ shall denote the classical (or Gauss) hypergeometric function of the complex variable $z$. For positive integers $m$ and $n$ with $0 < m < n$, $(m,n) = 1$ and $r$ a non-negative integer, we put $\nu=m/n$ and $$X_{m,n,r}(z)={}_{2}F_{1}(-r-\nu, -r, 1-\nu, z), \quad Y_{m,n,r}=z^{r}X_{m,n,r} \left(z^{-1} \right)$$ and $$\begin{aligned} R_{m,n,r}(z) &= \frac{\Gamma(r+1+\nu)}{r!\Gamma(\nu)} \int_{1}^{z} (1-t)^{r}(t-z)^{r}t^{-r-1+\nu}dt \\ &= (z-1)^{2r+1} \frac{\nu \cdots (r+\nu)}{(r+1) \cdots (2r+1)} {} _{2}F_{1} \left( r+1-\nu, r+1; 2r+2; 1-z \right),\end{aligned}$$ where $0$ is not on the path of integration from $1$ to $z$. We collect here some facts about these functions that we will require. \[lem:2.2\] [(a)]{} Suppose that $|\omega-1|<1$. We have $$\omega^{\nu}Y_{m,n,r}(\omega)-X_{m,n,r}(\omega)=R_{m,n,r}(\omega).$$ [(b)]{} We have $$X_{m,n,r}(\omega)Y_{m,n,r+1}(\omega) \neq X_{m,n,r+1}(\omega)Y_{m,n,r}(\omega).$$ [(c)]{} If $|\omega|=1$ and $|\omega-1|<1$, then $$\left| R_{m,n,r}(\omega) \right| \leq \frac{\Gamma(r+1+\nu)}{r!\Gamma(\nu)} |\varphi| \left| 1-\sqrt{\omega} \right|^{2r}.$$ [(d)]{} If $|\omega|=1$ and $|\omega-1|<1$, then $$\left| X_{m,n,r}(\omega) \right| = \left| Y_{m,n,r}(\omega) \right| < 1.072\frac{r!\Gamma(1-\nu)}{\Gamma(r+1-\nu)} \left| 1 + \sqrt{\omega} \right|^{2r}.$$ [(e)]{} For $|\omega|=1$ and ${\operatorname{Re}}(\omega) \geq 0$, we have $$\left| {} _{2}F_{1} \left( r+1-\nu, r+1; 2r+2; 1-\omega \right) \right| \geq 1,$$ with the minimum value occurring at $\omega=1$. Part (a) is established in the proof of Lemma 2.3 of [@Chen1]. Part (b) is Lemma 4 of [@Baker]. Part (c) is Lemma 2.5 of [@Chen1]. Part (d) is a slight refinement of Lemma 7.3(a) of [@V2]. In the proof of that lemma, we showed that in our notation here $$\left| X_{m,n,r}(\omega) \right| \leq \frac{4}{\left| 1 + \sqrt{w} \right|^{2}} \frac{\Gamma(1-m/n) \, r!}{\Gamma(r+1-m/n)} \left| 1+\sqrt{\omega} \right|^{2r}.$$ Since $\omega$ is on the unit circle, we can write $1+\sqrt{\omega}=1+w_{1} \pm \sqrt{1-w_{1}^{2}}i$, where $0 \leq w_{1} \leq 1$. Here we have $|\theta|<\pi/3$ in order that $|\omega-1|<1$ holds. Hence $w_{1}=\cos(\theta/2)>\cos(\pi/6)$, and so $$\frac{4}{\left| 1 + \sqrt{w} \right|^{2}}<1.072.$$ For part (e), we use Pochammer’s integral (see equation (1.6.6) of [@Sl]), along with the transformation $t=1/s$, to write $$\begin{aligned} {}_{2} F_{1} \left( a,b; c; z \right) &=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{1}^{\infty} (s-1)^{c-b-1}s^{a-c}(s-z)^{-a}ds \\ &=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{0}^{\infty} s^{c-b-1}(s+1)^{a-c}(s+1-z)^{-a}ds.\end{aligned}$$ Thus $${}_{2} F_{1} \left( a,b; c; 1-z \right) =\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{0}^{\infty} s^{c-b}(s+1)^{a-c}(s+z)^{-a}ds/s$$ and our problem becomes one of showing that the absolute value of the function $$\int_{0}^{\infty} t^{\alpha} (t+1)^{-\beta}(t+z)^{-\gamma}\frac{dt}{t}$$ with $\alpha, \beta, \gamma>0$ and $\beta+\gamma>\alpha$ attains its minimum in $\Omega=\{z:|z|=1, z\notin[-1,0]\}$ at $z=1$. Note that here we have $\alpha=c-b$, $\beta=c-a$ and $\gamma=a$. We can change the integration path to any path that avoids the singularities of the integrand, i.e., any path that stays in the open angle bounded by the rays $\{-\tau z:\tau>0\}$ and $\{-\tau:\tau>0\}$ containing the positive semi-axis. So we will change it to the ray $\{\tau\sqrt{z}: \tau>0\}$. Thus out integral becomes $$\int_{0}^{\infty} \left( \sqrt{z} t \right)^{\alpha} \left( \sqrt{z}t+1 \right)^{-\beta} \left( \sqrt{z}t+z \right)^{-\gamma} \frac{dt}{t} = z^{(\alpha-\beta-\gamma)/2} \int_{0}^{\infty} t^{\alpha} \left( t+1/\sqrt{z} \right)^{-\beta} \left( t+z/\sqrt{z} \right)^{-\gamma} \frac{dt}{t}.$$ Putting $w=1/\sqrt{z}$ and recalling that $|z|=1$, we have $$\left| z^{(\alpha-\beta-\gamma)/2} \int_{0}^{\infty} t^{\alpha} \left( t+1/\sqrt{z} \right)^{-\beta} \left( t+z/\sqrt{z} \right)^{-\gamma} \frac{dt}{t} \right| = \left| \int_{0}^{\infty} t^{\alpha} \left( t+w \right)^{-\beta} \left( t+wz \right)^{-\gamma} \frac{dt}{t} \right|,$$ so the problem is reduced to establishing the following:\ let $w,z' \in {\mathbb{C}}$, ${\operatorname{Re}}(w), {\operatorname{Re}}(z')>0$ and $z'w \in {\mathbb{R}}_{+}$. Then $$\left|\int_0^\infty t^\alpha(t+w)^{-\beta}(t+z')^{-\gamma}\frac {dt}t\right|\ge \left|\int_0^\infty t^\alpha(t+|w|)^{-\beta}(t+|z'|)^{-\gamma}\frac {dt}t\right|\,.$$ Since $c=2a$, $|z|=1$ and our definition of $w$, here we have $\beta=\gamma$ and $z'=z/\sqrt{z}=\sqrt{\bar{z}}$, this is immediate because the integrand on the left is then positive and obviously greater than the one on the right. Since $|w|=|z'|=1$, we have $$\left|\int_0^\infty t^\alpha(t+|w|)^{-\beta}(t+|z'|)^{-\gamma}\frac {dt}t\right| =\left|\int_0^\infty t^\alpha(t+1)^{-\beta}(t+1)^{-\gamma}\frac {dt}t\right|,$$ which shows the integral attains its minimum at $z=1$, as stated. We let $D_{n,r}$ denote the smallest positive integer such that $D_{n,r} X_{m,n,r}(x) \in {\mathbb{Z}}[x]$ for all $m$ as above. For $d \in {\mathbb{Z}}$, we define $N_{d,n,r}$ to be the largest integer such that $\left( D_{n,r}/ N_{d,n,r} \right)X_{m,n,r}\left( 1-\sqrt{d}\,x \right) \in {\mathbb{Z}}\left[ \sqrt{d} \right] [x]$, again for all $m$ as above. We will use $v_{p}(x)$ to denote the largest power of a prime $p$ which divides into the rational number $x$. We put $$\label{eq:ndn-defn} {\mathcal{N}}_{d,n} =\prod_{p|n} p^{\min(v_{p}(d)/2, v_{p}(n)+1/(p-1))}.$$ In what follows, we shall restrict our attention to $m=1$, $n=4$ (so $\nu=1/4$) and $t=-1$. \[lem:2.4\] We have $$\label{eq:cndn-defn} \frac{\Gamma(3/4) \, r!}{\Gamma(r+3/4)} \frac{D_{4,r}}{N_{d,4,r}} <{\mathcal{C}}_{4,1} \left( \frac{e^{1.68}}{{\mathcal{N}}_{d,4}} \right)^{r} \text{ and } \hspace*{1.0mm} \frac{\Gamma(r+5/4)}{\Gamma(1/4)r!} \frac{D_{4,r}}{N_{d,4,r}} < {\mathcal{C}}_{4,2} \left( \frac{e^{1.68}}{{\mathcal{N}}_{d,4}} \right)^{r}$$ for all non-negative integers $r$, where ${\mathcal{C}}_{4,1}=0.83$, ${\mathcal{C}}_{4,2}=0.2$ and ${\mathcal{D}}_{4}=e^{1.68}$. From Lemma 7.4(c) of [@V2], we have $$\max \left( 1, \frac{\Gamma(3/4) \, r!}{\Gamma(r+3/4)}, 4\frac{\Gamma(r+5/4)}{\Gamma(1/4)r!} \right) \frac{D_{4,r}}{N_{d,4,r}} < 100 \left( \frac{e^{1.64}}{{\mathcal{N}}_{d,4}} \right)^{r}$$ However, the value $100$ results in us requiring a lot of computation to complete the proof of our theorem here. Therefore, we seek a smaller value at the expense of replacing $1.64$ by a larger value, whose value has less of an impact on our proof. For $r \geq 156$, $100\exp(1.64r)<0.2\exp(1.68r)$, so we compute directly the left-hand sides of for $r \leq 155$. We find that the maximum values of the left-hand sides of divided by $\exp(1.68r)$ both occur for $r=3$.The lemma follows. Put $$p_{r}'=\frac{D_{4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right) ^{r}, \quad q_{r}'=\frac{D_{4,r}}{N_{d,4,r}} Y_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r},$$ and $$R_{r}'=\frac{D_{4,r}}{N_{d,4,r}} R_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r},$$ where $r$ is any non-negative integer and $d$ will be determined below. By the definitions of $D_{4,r}$ and $N_{d,4,r}$, we easily observe that $p_{r}'$ and $q_{r}'$ are algebraic integers of ${\mathbb{Q}}(i)$. We can see that as follows. $X_{1,4,r}(z)$ is a polynomial of degree $r$ and $$p_{r}' = \frac{D_{1,4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r} = \frac{D_{1,4,r}}{N_{d,4,r}} X_{r} \left( 1-u_{2}i \frac{2}{u_{1}-u_{2}i} \right) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r}.$$ So with $d=u_{2}^{2}$, we have $p_{r}'$ is an algebraic integer of ${\mathbb{Q}}(i)$. The analogous expression for $q_{r}'$ shows that it is also an algebraic integer. In fact, there may be some further common factors. As in [@V3], put $$\begin{aligned} g_{1} & = \gcd \left( u_{1}, u_{2} \right), \\ g_{2} & = \gcd \left( u_{1}/g_{1}, t \right)=1,\\ g_{3} & = \left\{ \begin{array}{ll} 1 & \mbox{if $t \equiv 1 \bmod 4$ and $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$}, \\ 2 & \mbox{if $t \equiv 3 \bmod 4$ and $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$},\\ 4 & \mbox{otherwise,} \end{array} \right. \\ & = \left\{ \begin{array}{ll} 2 & \mbox{if $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$},\\ 4 & \mbox{otherwise,} \end{array} \right. \\ g & = g_{1}\sqrt{g_{2}/g_{3}}.\end{aligned}$$ Then we can put $$\label{eq:7} p_{r}=\frac{D_{4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right) ^{r}, \quad q_{r}=\frac{D_{4,r}}{N_{d,4,r}} Y_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)^{r},$$ and $$R_{r}=\frac{D_{4,r}}{N_{d,4,r}} R_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)^{r},$$ where $$d=\left( u-\sigma(u) \right)^{2}/g^{2} = u_{2}^{2}t/g^{2} = -u_{2}^{2}/g^{2}.$$ Since $\left( u_{1}-u_{2}i \right)/g$ is an algebraic integer, the argument above still applies to show that $p_{r}$ and $q_{r}$ are algebraic integers. So in Lemma \[lem:2.4\], we can take $$Q = \frac{{\mathcal{D}}_{4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{|g|{\mathcal{N}}_{d,4}}$$ and $$\label{eq:k-UB} k_{0}<1.072{\mathcal{C}}_{4,1}<0.89.$$ We also have $$E = \frac{|g|{\mathcal{N}}_{d,4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{{\mathcal{D}}_{4}u_{2}^{2}}$$ and $$\ell_{0}={\mathcal{C}}_{4,2}|\varphi|=0.2|\varphi|.$$ Other Preliminary Lemmas ======================== \[lem:3.1\] Let $a$, $m$ and $p$ be positive integers with $a \geq 1$ and $p$ a prime. Put $b=p^{m}$ or $b=2p^{m}$ and suppose that $\gcd \left( a,b \right)=1$ and $a^{2}+b^{2}$ not a perfect square. Furthermore, suppose that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution. All coprime integer solutions $(x,y)$ to the quadratic equation $$\label{eq:quad-eqnc-dup} x^{2} - \left( a^{2}+b^{2} \right)y^{2}=-b^{2}$$ are given by $$\label{eq:14c-dup} x+y \sqrt{a^{2}+b^{2}} = \pm \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \alpha^{2k}, \hspace{1.0mm} k \in {\mathbb{Z}},$$ where $\alpha = \left( T_{1}+U_{1} \sqrt{a^{2}+b^{2}} \right)/2$ and $\left( T_{1},U_{1} \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=-4$ in positive integers. The condition that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution is not required here, but will be needed required for Lemma \[lem:3.2aa\]. Here we can replace $\alpha^{2}$ by $T+U \sqrt{a^{2}+b^{2}}$ where $\left( T,U \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=1$ in positive integers. The proof uses the fact that for $\beta, \gamma \in {\mathcal{O}}_{{\mathbb{K}}}$ for some number field, ${\mathbb{K}}$, we have $(\beta)=(\gamma)$ if and only if $\beta=\gamma\epsilon$ where $\epsilon$ is a unit in ${\mathcal{O}}_{{\mathbb{K}}}$. In what follows, we let ${\mathbb{K}}= {\mathbb{Q}}\left( \sqrt{a^{2}+b^{2}} \right)$. First suppose that $b=p^{m}$ with $p \neq 2$. Since the Legendre symbol $\left( a^{2}+b^{2} / p \right)=\left( a^{2} / p \right)=1$ if $p \neq 2$ and $a^{2}+b^{2} \equiv 1 \bmod 8$ if $p=2$ (since $b^{2} \geq 8$ by our assumption), we know there is a prime ideal, ${\mathfrak{p}}$ in ${\mathcal{O}}_{{\mathbb{K}}}$, such that $(p)={\mathfrak{p}}\bar{{\mathfrak{p}}}$, where $\bar{{\mathfrak{p}}}=\left\{ a_{1}-b_{1}\sqrt{a^{2}+b^{2}}: a_{1}+b_{1}\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\right\}$. Let $(x,y)$ be any relatively prime solution of and consider $x+y \sqrt{a^{2}+b^{2}}$. It has norm $-b^{2}=-p^{2m}$, so it is a member of $(p)^{2m}$. Since $x$ and $y$ are relatively prime, we must have either $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$ or $x+y \sqrt{a^{2}+b^{2}} \in \bar{{\mathfrak{p}}}^{2m}$. Note that it cannot be a member of ${\mathfrak{p}}^{m_{1}} \bar{{\mathfrak{p}}}^{2m-m_{1}}$ for $1 \leq m_{1}<2m$, as such an ideal would have a power of $(p)$ as a factor and hence $x$ and $y$ would no longer be relatively prime – it is here where we need the assumption that $p \neq 2$. Without loss of generality, let us suppose that $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$ and also that $a+\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$. The proofs for the other cases follow by the fact that the other factor of $(p)$ is $\bar{{\mathfrak{p}}}$, the conjugate of ${\mathfrak{p}}$. Since $\left( a+\sqrt{a^{2}+b^{2}} \right)$, $\left( x+y\sqrt{a^{2}+b^{2}} \right)$ and ${\mathfrak{p}}^{2m}$ all have norm $b^{2}$ and $a+\sqrt{a^{2}+b^{2}}, x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$, it follows that ${\mathfrak{p}}^{2m} = \left( a+\sqrt{a^{2}+b^{2}} \right) = \left( a+\sqrt{a^{2}+b^{2}} \right)$. Therefore, $x+y \sqrt{a^{2}+b^{2}}$ must be a unit of norm $1$ times $a+\sqrt{a^{2}+b^{2}}$ and the result follows. If $p=2$, we also need to consider the possibility that it is a member of $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$ or $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m-1} \bar{{\mathfrak{p}}}$. As above, we may suppose that $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$ and also that $a+\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$. The same argument as above now holds to show that the lemma holds in this case too. We now consider $b=2p^{m}$. We may assume that $p \neq 2$, since the case of $p=2$ is covered above. Here $b^{2} \equiv 4 \bmod 8$ and hence $a^{2}+b^{2} \equiv 5 \bmod 8$. Here $(2)$ is a prime ideal in ${\mathcal{O}}_{{\mathbb{K}}}$ and, as shown above, $(p)$ splits into the product of two prime ideals, ${\mathfrak{p}}$ and $\bar{{\mathfrak{p}}}$, which are conjugates of each other. Let $(x,y)$ be any relatively prime solution of and consider $x+y \sqrt{a^{2}+b^{2}}$. It has norm $-b^{2}=-4p^{2m}$. Since $x$ and $y$ are relatively prime, we must have either $x+y \sqrt{a^{2}+b^{2}} \in (2){\mathfrak{p}}^{m}$ or $x+y \sqrt{a^{2}+b^{2}} \in (2) \bar{{\mathfrak{p}}}^{m}$. As above, it cannot be a member of $(2){\mathfrak{p}}^{m_{1}} \bar{{\mathfrak{p}}}^{2m-m_{1}}$ for $1 \leq m_{1}<2m$. Arguing as above, the result now follows. \[lem:3.2aa\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose that all coprime positive integer solutions of $\eqref{eq:quad-eqnc}$ are given by $\eqref{eq:14c}$ and that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution. If $(X,Y) \neq (a,1)$ is a coprime positive integer solution to $$X^{2} - \left( a^{2}+b^{2} \right)Y^{4}=-b^{2},$$ then $$\pm X \pm bi = \left( a+bi \right) \left( r \pm si \right)^{4}, \quad Y=r^{2}+s^{2},$$ where $r,s \in {\mathbb{Z}}$ with $\gcd(r,s)=1$ and $s>r>0$. Note that we can also express the solution $(X,Y)=(a,1)$ in this form, but with $r=1$ and $s=0$ (i.e., we remove the condition that $s>0$). The condition that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution is required here. It arises in the proof as it implies that any solution $(X,Y)$ of $X^{2} - \left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$ comes from an even power of $\alpha$. This provides us with the representations in that play a key role in obtaining the desired representation. For $k \geq 0$, we define $T_{k}$ and $U_{k}$ by $$\alpha^{k} = \frac{T_{k} + U_{k} \sqrt{a^{2}+b^{2}}}{2}.$$ Note that $T_{k}, U_{k} \in {\mathbb{Z}}$ and $T_{k} \equiv U_{k} \bmod 2$, with $T_{k} \equiv U_{k} \equiv 1 \bmod 2$ only possible if $a^{2}+b^{2} \equiv 1 \bmod 4$. Therefore, by expanding , a solution in coprime positive integers $(X,Y) \neq (a,1)$ to $X^{2}-\left( a^{2}+b^{2} \right)Y^{4}=-b^{2}$ arises from $$X+Y^{2}\sqrt{a^{2}+b^{2}} = \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( \frac{T_{2k}+U_{2k}\sqrt{a^{2}+b^{2}}}{2} \right)$$ and so such a solution is equivalent to $$\label{eq:15} 2X= \left( a^{2}+b^{2} \right) U_{2k} \pm aT_{2k}, \quad 2Y^{2}=T_{2k} \pm aU_{2k}$$ for some $k \geq 1$. Note we need $k \neq 0$ since $(X,Y) \neq (a,1)$. We now show that the expressions for $X$ and $Y^{2}$ in are actually positive. Since $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$, we have $T_{2k}>\sqrt{a^{2}+b^{2}}U_{2k}>aU_{2k}$. Hence $Y^{2}>0$ is satisfied. From $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$, we have $\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}^{2}/U_{2k}=-1/U_{2k}$ and $T_{2k}>\sqrt{a^{2}+b^{2}}U_{2k}$. So we have $$\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} >\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}\sqrt{a^{2}+b^{2}} >\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}^{2}/U_{2k}=-1/U_{2k}.$$ Since $U_{2k} \geq 1$ and $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} \in {\mathbb{Z}}$, it follows that $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} \geq 0$. If $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k}=0$ held, then since $\gcd(a,b)=1$, any prime divisor of $a^{2}+b^{2}$ must divide $T_{2k}$. But then $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$ would be impossible. Therefore $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k}>0$, and as a result $\left( a^{2}+b^{2} \right) U_{2k} \pm aT_{2k}>0$ holds. So $X$ is also positive, as required. Notice that this tells us that $$2X+2Y^{2}\sqrt{a^{2}+b^{2}} = -\left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( T_{2k}+U_{2k}\sqrt{a^{2}+b^{2}} \right)$$ is not possible. Also, corresponding to $k<0$, $$2X+2Y^{2}\sqrt{a^{2}+b^{2}} = \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( T_{2k}-U_{2k}\sqrt{a^{2}+b^{2}} \right)$$ gives us $2X=\pm aT_{2k}-\left( a^{2}+b^{2} \right) U_{2k}$ and $2Y^{2}=T_{2k} \mp aU_{2k}$. But from our argument above, we see that this value of $X$ can never be positive. Hence all the solutions must come from . Now we use the expressions arising from to prove the lemma. Note that $$\begin{aligned} \alpha^{2k} & = & \frac{T_{2k} + U_{2k} \sqrt{a^{2}+b^{2}}}{2} = \left( \frac{T_{k} + U_{k} \sqrt{a^{2}+b^{2}}}{2} \right)^{2} \\ & = & \frac{T_{k}^{2}+\left(a^{2}+b^{2} \right) U_{k}^{2} +2T_{k}U_{k} \sqrt{a^{2}+b^{2}}}{4}.\end{aligned}$$ Thus $$\label{eq:my-2} T_{2k}=\frac{T_{k}^{2}+\left(a^{2}+b^{2} \right) U_{k}^{2}}{2} \quad \text{ and } \quad U_{2k}=T_{k}U_{k}.$$ Since $T_{k} \equiv U_{k} \bmod 2$ and $T_{k} \equiv U_{k} \equiv 1 \bmod 2$ only if $a^{2}+b^{2} \equiv 1 \bmod 4$, we see that $T_{2k}, U_{2k} \in {\mathbb{Z}}$, $T_{2k} \equiv U_{2k} \bmod 2$ and $T_{2k} \equiv U_{2k} \equiv 1 \bmod 2$ only if $a^{2}+b^{2} \equiv 1 \bmod 4$. Also notice from the expression for $2X$ in that if $a^{2}+b^{2} \equiv 1 \bmod 4$ and $T_{2k} \equiv U_{2k} \equiv 1 \bmod 2$, then we must have $a$ odd and hence $b$ even. Otherwise, the right-hand side of the expression for $2X$ is odd. By the expressions in for $T_{2k}$ and $U_{2k}$, implies that $$(2Y)^{2}=T_{k}^{2}+\left( a^{2}+b^{2} \right)U_{k}^{2} \pm 2aT_{k}U_{k} = \left( T_{k} \pm aU_{k} \right)^{2} + \left( bU_{k} \right)^{2}.$$ Our statements above about the parity of $a$, $b$, $T_{k}$ and $U_{k}$ imply that both $T_{k} \pm aU_{k}$ and $bU_{k}$ are always even. Therefore, $$\label{eq:my-3aa} Y^{2}= \left( \frac{T_{k} \pm aU_{k}}{2} \right)^{2} + \left( \frac{bU_{k}}{2} \right)^{2}.$$ Observe that $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)$ divides $\left( T_{k} \pm aU_{k} \right)$ and $bU_{k}/2$. Since $\left( T_{k}+aU_{k} \right) \left( T_{k}-aU_{k} \right)/4= \left( bU_{k}/2 \right)^{2} \pm 1$, and $T_{k}+aU_{k}$ and $T_{k}-aU_{k}$ have the same parity, it follows that $\gcd \left( T_{k} \pm aU_{k}, bU_{k}/2 \right)$ divides $2$. Thus $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)$ divides $2$. We consider each of the possibilities for this gcd now. If $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)=1$, then there are positive integers $b_{1}, b_{2}$, $r_{1}$ and $s_{1}$ satisfying $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$ such that $b_{1}b_{2}=b$, $r_{1}s_{1}=U_{k}$ and $$\label{eq:16a} Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2}, \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2} \quad \text{ and } \quad b_{1}s_{1}>b_{2}r_{1}.$$ If $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)=2$, then both $Y+\left( T_{k} \pm aU_{k} \right)/2$ and $Y-\left( T_{k} \pm aU_{k} \right)/2$ are twice a square. So we have $$\label{eq:16b} Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2}, \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2} \quad \text{ and } \quad b_{1}s_{1}>b_{2}r_{1},$$ where $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$, and either $2b_{1}b_{2}=b$, $r_{1}s_{1}=U_{k}$; or $b_{1}b_{2}=b$ and $2r_{1}s_{1}=U_{k}$. In the first case, subtracting the two expressions in and substituting for $U_{k}$, we obtain $$U_{k}=r_{1}s_{1} \quad \text{ and } \quad T_{k}=b_{1}^{2}s_{1}^{2}-b_{2}^{2}r_{1}^{2} \mp ar_{1}s_{1}.$$ We have $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=\pm 4$. However, the proof is the same for both cases, so we consider only $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=4$ here. Substituting the above expressions for $T_{k}$ and $U_{k}$ into $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=4$ and then simplifying leads to the equation $$b_{2}^{4}r_{1}^{4} \pm 2ab_{2}^{2}r_{1}^{3}s_{1}-3b^{2}r_{1}^{2}s_{1}^{2} \mp 2ab_{1}^{2}r_{1}s_{1}^{3}+b_{1}^{4}s_{1}^{4}=4.$$ Multiplying both sides by $2bi$, we obtain $$\label{eq:my-3} (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4} -(a-bi) \left( b_{2}r_{1} \mp b_{1}s_{1}i \right)^{4}=32bi.$$ We can write $$\begin{aligned} & & (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4} +(a-bi) \left( b_{2}r_{1} \mp b_{1}s_{1}i \right)^{4} \\ & = & 8a \left( T_{k}^{2}+\left( a^{2}+b^{2} \right)U_{k}^{2} \right) -16 \left( a^{2}+b^{2} \right)T_{k}U_{k} = 32X,\end{aligned}$$ the last equality follows from applying to our expression for $X$ from with the signs there all positive. Combining this with yields $$16X+16bi= (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4}.$$ Had we considered $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=-4$ above, we would have found that $$16X-16bi= (a-bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4}.$$ We noted above that $b_{2}r_{1}$ and $b_{1}s_{1}$ have the same parity. Therefore, $b_{2}r_{1} \pm b_{1}s_{1}i$ is divisible by $1 \pm i$, say $b_{2}r_{1} \pm b_{1}s_{1}i=(1 \pm i)(r+si)$ for some integers $r$ and $s$ with $\gcd(r,s)=1$ and $s>r>0$ (since $b_{1}s_{1}>b_{2}r_{1}$ from ). The expression in the lemma for both $4X+4bi$ and $4X-4bi$ follow. From the expression for $4X+4bi$, we obtain $$16 \left( X^{2}+b^{2} \right) = 16\left( a^{2}+b^{2} \right) \left( r^{2}+s^{2} \right)^{4},$$ but we also have $16 \left( X^{2}+b^{2} \right)=16\left( a^{2}+b^{2} \right)Y^{4}$. The expression in the lemma for $2Y$ follows. When the signs appearing in are negative, a nearly identical argument to the above leads to $$-4X\pm 4bi= (a+bi) \left( b_{2}r_{1}-b_{1}s_{1}i \right)^{4}, \quad b_{1}b_{2}=b, \quad \gcd \left( r_{1},s_{1} \right)=1.$$ As above, this completes the proof of this lemma. Next, in Lemma \[lem:3.3\] below, we establish a gap principle separating possible solutions of . We need a few additional lemmas to help us first. Lemma \[lem:3.4\](b) will also play a key role in the proof of Theorem \[thm:1.1\] too. \[lem:3.4\] Suppose that $a$ and $b$ are relatively prime positive integers and $(X,Y)$ is a coprime positive integer solution of $x^{2}- \left( a^{2}+b^{2} \right)y^{4}=-b^{2}$ with $Y>1$. [(a)]{} $Y$ is only divisible by primes, $p \equiv 1 \bmod 4$. As a consequence, if $Y>1$, then $Y \geq 5$ and if $Y>5$, then $Y \geq 13$. [(b)]{} Let $a$ and $b$ be as in the statement of Theorem $\ref{thm:1.2}$. Then $Y>b/2$. [(c)]{} Let $a$, $m$ and $p$ be as in the statement of Theorem $\ref{thm:1.1}$ and put $b=p^{m}$, then $Y>b^{2}/4$. If $p \neq 2$, then $Y>b^{2}/2$. The additional conditions in parts (b) and (c) are required. Without them, the lower bound for $Y$ in part (b) does not hold in general. E.g., if $b$ is an element of the recurrence sequence, $b_{0}=-3$, $b_{1}=4$, $b_{n+2}=50b_{n+1}-b_{n}$ for $n \geq 0$, then $\left( 1+b^{2} \right) 5^{4}-b^{2}=624b^{2}+625$ is a perfect square and $(X,Y)=\left( \sqrt{624b^{2}+625}, 5 \right)$ is a solution of the diophantine equation for $a=1$ and such values of $b$. Hence $Y$ stays fixed as $b$ grows. Also note that the bound in part (b) is nearly best-possible. For any odd $b' \equiv 0,2,8 \bmod 10$ with $b'>2$, put $b=(b')^{2}-1$, $a=(b')^{3}/4-3b'/2$, then $(x,y)= \left( b' \left( b'^6+4b'^{4}+5b'^{2}+10 \right)/4, b+2 \right)$ is a solution of . So it appears that the correct bound is $Y>b$. The bound in part (c) is best-possible for $b$ odd, as can be seen by considering Example 1 in Remark \[rem:1\]. \(a) If $Y=2k$, then we are seeking solutions of $X^{2}=16k^{4}a^{2} + \left( 16k^{4}-1 \right)b^{2}$. Since $\gcd(a,b)=1$, if $a$ is even, then $b$ is odd and $16k^{4}a^{2} + \left( 16k^{4}-1 \right)b^{2} \equiv 3 \bmod 4$. This implies that $X^{2} \equiv 3 \bmod 4$, which is not possible, so there are no solutions with $Y$ even in this case. If $b$ is even, then $X$ is even, but that violates our assumption that $\gcd(X,Y)=1$. Suppose $Y=pk$, where $p$ is an odd prime. Then we have $X^{2}=a^{2}p^{4}k^{4}+ \left( p^{4}k^{4}-1 \right) b^{2}$. Thus $X^{2} \equiv -b^{2} \bmod p$. If $b \equiv 0 \bmod p$, then $X \equiv 0 \bmod p$, which is not allowed by our assumption that $X$ and $Y$ are relatively prime. For such $b$, $X^{2} \equiv -b^{2} \bmod p$ is not solvable if $p \equiv 3 \bmod 4$, as required. \(b) and (c) Recall from the proof of Lemma \[lem:3.2aa\] that we have either $$Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$, $\gcd \left( r_{1}, s_{1} \right)=1$ and $b_{1}b_{2}=b$; or $$Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2}$$ where $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$ and $2b_{1}b_{2}=b$. These relations are and . If $k=1$, then adding these expressions we have three possibilities. First, if $$Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$ and $b_{1}b_{2}=b$, then $$2Y=b_{1}^{2}s_{1}^{2}+b_{2}^{2}r_{1}^{2} = b_{1}^{2}s_{1}^{2}+\left( b/b_{1} \right)^{2}r_{1}^{2}.$$ Differentiating this expression with respect to $b_{1}$, we find that the derivative is only zero for $b_{1}$ positive if $b_{1}=\sqrt{r_{1}b/s_{1}}$. Here we have $2Y=2r_{1}s_{1}b \geq 2b$. If $b_{1}=1$, then we have $2Y=s_{1}^{2}+b^{2}r_{1}^{2}$. So $2Y \geq b^{2}+1 \geq 2b$. Similarly, if $b_{1}=b$, then we also have $2Y \geq 2b$. Furthermore, if $b=p^{m}$ is a prime power, then either $b_{1}=b$ or $b_{2}=b$. Here we obtain $2Y>b^{2}$. Second, if $$Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$ and $2b_{1}b_{2}=b$, then $$2Y=2b_{1}^{2}s_{1}^{2}+2b_{2}^{2}r_{1}^{2}=2b_{1}^{2}s_{1}^{2}+2 \left( b/ \left( 2b_{1} \right) \right)^{2}r_{1}^{2}.$$ Differentiating this expression with respect to $b_{1}$, we find that the derivative is only zero for $b_{1}$ positive if $b_{1}=\sqrt{r_{1}b/\left( 2s_{1} \right)}$. Here we have $2Y=2r_{1}s_{1}b \geq 2b$. If $b_{1}=1$, then we have $2Y=2s_{1}^{2}+b^{2}r_{1}^{2}/2$. Since $b$ is even, we have $b \geq 2$ and so $2Y \geq b^{2}/2+2 \geq 2b$. Similarly, if $b_{1}=b/2$, then we also have $2Y \geq 2b$. Furthermore, if $b=p^{m}$ is a prime power, then either $b_{1}=b/2$ or $b_{2}=b/2$. Here we obtain $2Y>2b^{2}/4$. This completes the proof for $k=1$. Now we consider $k>1$. If $a=U_{1}=1$, then we have $T_{1}^{2}-\left( 1+b^{2} \right) =-4$. This is only possible if $T_{1}=1$ and $b=2$. Using Magma, we find that there are no integer solutions of $X^{2}-5Y^{4}=-4$ with $Y>1$. Hence we can ignore this case and assume that $aU_{1} \geq 2$. Here we have $T_{1}^{2} \geq b^{2}$. Combining this with , we have $$Y^{2}= \left( T_{k} \pm aU_{k} \right)^{2}4 + \left( bU_{k}/2 \right)^{2} \geq \left( bU_{2}/2 \right)^{2}=b^{2}T_{1}^{2}U_{1}^{2} \geq b^{4}.$$ This argument holds since $\left\{ U_{k} \right\}$ satisfies the recurrence sequence $U_{k+2}=2T_{1}U_{k+1}+U_{k}$ and $T_{1} \geq 1$, from the minimal polynomial for $\alpha$, so $U_{k} \geq U_{2}$ for $k \geq 2$. Thus $Y \geq b^{2}$. \[lem:3.2bb\] Let $\omega=e^{i\theta}$ with $-\pi < \theta \leq \pi$ and put $\omega^{1/4}=e^{i\theta/4}$. If $$0 < \left| \omega^{1/4} - z \right| <c_{1},$$ for some $z \in {\mathbb{C}}$ with $|z|=1$ and $0 < c_{1} < 1$, then $$\left| \omega - z^{4} \right| > c_{2} \left| \omega^{1/4} - z \right|,$$ where $c_{2}=\left( 2-c_{1}^{2} \right) \sqrt{4-c_{1}^{2}}$. We can write $$\left| \omega - z^{4} \right| = \left| \omega^{1/4} - z \right| \times \prod_{k=1}^{3} \left| \omega^{1/4} - e^{2\pi ik/4}z \right|.$$ Multiplying by $\omega^{-1/4}$ and expanding the resulting expression, the product above equals $$\prod_{k=1}^{3} \left| e^{2\pi ik/4-i\theta/4}z-1 \right| = \left| e^{3i\varphi}+e^{2i\varphi}+e^{i\varphi}+1 \right|,$$ for some $-\pi < \varphi \leq \pi$. Squaring this quantity and simplifying, we obtain $$8\cos^{2}(\varphi) \left( \cos(\varphi)+1 \right).$$ If $\left| \omega^{1/4} - z \right|=c_{1}$, we have $2-2\cos(\varphi)=c_{1}^{2}$ and the lemma follows under this assumption by a routine substitution. Since $c_{2}$ is a decreasing function of $c_{1}$, the lemma also holds in general. \[lem:3.3\] Under the same assumptions as in Theorem $\ref{thm:1.1}$, suppose that $\left( X_{1},Y_{1} \right)$ and $\left( X_{2},Y_{2} \right)$ are two solutions in coprime positive integers to with $Y_{2}>Y_{1}>1$. Then $$Y_{2}>7.98\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}.$$ By Lemma \[lem:3.2aa\], there are integers $r_{1}, s_{1}, r_{2}, s_{2}$ such that $$\pm 4X_{j} \pm 4bi = \left( a+bi \right) \left( r_{j} \pm s_{j} i \right)^{4}, \quad 2Y_{j}=r_{j}^{2}+s_{j}^{2}, \hspace{1.0mm} \text{$j=1,2$}.$$ We will assume that $$\label{eq:3.2} 4X_{1} \pm 4bi = \left( a+bi \right) \left( r_{1} + s_{1} i \right)^{4}, \quad 4X_{2} \pm 4bi = \left( a+bi \right) \left( r_{2} + s_{2} i \right)^{4},$$ as the argument for the other cases is exactly the same. It follows that $$\label{eq:3.3} \left( a+bi \right) \left( r_{j}+s_{j}i \right)^{4} - \left( a-bi \right) \left( r_{j}-s_{j}i \right)^{4} = \pm 8bi, \quad \text{($j=1,2$)}.$$ Putting $\omega = \left( a-bi \right) / \left( a+bi \right)$, by Lemma \[lem:3.4\](a) and , we have $$\label{eq:3.4} \left| \omega - \left( \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right)^{4} \right| =\left| \frac{\pm 8bi}{\left( a+bi \right)\left( r_{j}-s_{j}i \right)^{4}} \right| = \frac{8b}{\sqrt{a^{2}+b^{2}} 4Y_{j}^{2}} < 2/25.$$ For $j=1,2$, let $\zeta_{4}^{(j)}$ be the $4$-th root of unity such that $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| = \min_{0 \leq k \leq 3} \left| \omega^{1/4} - \zeta_{4}^{k} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right|.$$ From , we immediately have $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| < (2/25)^{1/4},$$ for $j=1,2$. In fact, we will show that this quantity is even smaller. Applying Lemma \[lem:3.2bb\] with $c_{1}=(2/25)^{1/4}$, we can take $c_{2}=3.31$. Combining this with , we obtain $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| < (2/25)/3.31<0.025.$$ Applying Lemma \[lem:3.2bb\] again, now with $c_{1}=0.025$, we obtain $$\label{eq:20} \left| \omega - \left( \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right)^{4} \right| > 3.998 \left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right|$$ and $$\label{eq:3.6} \left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| <0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{j}^{2}}.$$ Hence $$\begin{aligned} \label{eq:3.7} \left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| & \leq \left| \omega^{1/4} - \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} \right| + \left| \omega^{1/4} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| \nonumber \\ & < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{1}^{2}} + 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}}.\end{aligned}$$ Next we obtain a lower bound for this same quantity. If $$\zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} =\zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i},$$ then from our expression for $Y_{j}$ in Lemma \[lem:3.2aa\] $$\zeta_{4}^{(1)}\frac{\left( r_{1}+s_{1}i \right)^{2}}{2Y_{1}} = \zeta_{4}^{(2)} \frac{\left( r_{2}+s_{2}i \right)^{2}}{2Y_{2}},$$ so $$\frac{\left( r_{1}+s_{1}i \right)^{4}}{4Y_{1}^{2}} = \pm \frac{\left( r_{2}+s_{2}i \right)^{4}}{4Y_{2}^{2}}.$$ From , it follows that $$\left( X_{1} \pm bi \right) Y_{2}^{2} = \pm \left( X_{2} \pm bi \right) Y_{1}^{2}.$$ Comparing the imaginary parts of both sides of this equation, we find that $Y_{1}=Y_{2}$, but this contradicts our assumption that $Y_{2}>Y_{1}$. Let $x+yi= \left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right)$. We can write $$\left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| = \frac{2\zeta_{4}^{(1)}x-\left( \zeta_{4}^{(1)}+\zeta_{4}^{(2)} \right)( x+yi)} {\left( r_{1}-s_{1}i \right) \left( r_{2}-s_{2}i \right)}.$$ Regardless of the values of $\zeta_{4}^{(1)}$ and $\zeta_{4}^{(2)}$, we always have $(1+i)| \left( \zeta_{4}^{(1)} + \zeta_{4}^{(2)} \right)$. Hence $1+i$ always divides the numerator of the above expression. Also notice that since $2Y_{j}=r_{j}^{2}+s_{j}^{2}$ is even, we have $r_{j} \equiv s_{j} \bmod 2$, so $x$ and $y$ must both be even and $$\left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| \geq \frac{2|1+i|}{\left| {\left( r_{1}-s_{1}i \right) \left( r_{2}-s_{2}i \right)} \right|} =\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}}.$$ Combining this with , we have $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}} \left( \frac{1}{Y_{1}^{2}} + \frac{1}{Y_{2}^{2}} \right).$$ From $Y_{2}>Y_{1}$, this immediately gives us $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}}\frac{2}{Y_{1}^{2}},$$ so $$Y_{2}>1.997\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}>1.997Y_{1}^{3}.$$ We can use this gap principle to improve its constant term. Applying Lemma \[lem:3.4\](a), $$Y_{2}^{2}>3.988Y_{1}^{6} \geq 2490Y_{1}^{2}$$ yielding $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}} \frac{1.0005}{Y_{1}^{2}}$$ and finally $$Y_{2}>7.98\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}.$$ completing our proof. Proof of Theorem \[thm:1.1\] ============================ \[lem:thm\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has a solution and that all coprime integer solutions $(x,y)$ to the quadratic equation $\eqref{eq:quad-eqnc}$ are given by $\eqref{eq:14c}$. Then $\eqref{eq:2}$ has at most one solution, $\left( X, Y \right)$, in coprime positive integers solutions with $Y>b^{2}/2$. Suppose that $\left( X_{1}, Y_{1} \right)$ and $\left( X_{2}, Y_{2} \right)$ are two coprime positive integer solutions to with $Y_{2}>Y_{1}>b^{2}/2$ and that the assumptions in the statement of the lemma hold. From Lemma \[lem:3.4\](a), we obtain $$\label{eq:25} X_{1}^{2}= \left( a^{2}+b^{2} \right) Y_{1}^{4}-b^{2} > \left( a^{2}+b^{2} \right) \left( Y_{1}^{4}-1 \right) > 0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4},$$ and so $$\label{eq:26} \sqrt{X_{1}^{2}+b^{2}}= \sqrt{\left(a^{2}+b^{2}\right) Y_{1}^{4}} < 1.001 X_{1}.$$ By Lemma \[lem:3.2aa\], there are integers $r_{1},s_{1},r_{2},s_{2}$ such that $$\pm X_{j} \pm bi = (a+bi) \left( r_{j} \pm s_{j} i \right)^{4}, \hspace{1.0mm} Y_{j}=r_{j}^{2}+s_{j}^{2}, \hspace{1.0mm} \gcd \left( r_{j}, s_{j} \right)=1, \hspace{1.0mm} s_{j}>r_{j}>0,$$ for $j=1,2$. We will assume that $$X_{1} \pm bi = (a+bi) \left( r_{1} + s_{1} i \right)^{4}, \quad X_{2} \pm bi = (a+bi) \left( r_{2} + s_{2} i \right)^{4},$$ as the argument is identical in the other three cases. Thus $$(a+bi) \left( r_{j} + s_{j} i \right)^{4} - (a-bi) \left( r_{j} - s_{j} i \right)^{4} =2i{\operatorname{Im}}\left( X_{j} \pm bi \right) = \pm 2bi$$ for $j=1,2$. Applying this for $j=2$ and using our expressions above for $X_{1} \pm bi$ and $Y_{1}$, we have $$\begin{aligned} & & \left( X_{1} \pm bi \right) \left( r_{1} - s_{1} i \right)^{4}\left( r_{2} + s_{2} i \right)^{4} - \left( X_{1} \mp bi \right) \left( r_{1} + s_{1} i \right)^{4}\left( r_{2} - s_{2} i \right)^{4} \\ & = & (a+bi) \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( r_{2} + s_{2} i \right)^{4} - (a-bi) \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( r_{2} - s_{2} i \right)^{4} \\ & = & \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( (a+bi) \left( r_{2} + s_{2} i \right)^{4} - (a-bi) \left( r_{2} - s_{2} i \right)^{4} \right) = 2i \left( r_{1}^{2} + s_{1}^{2} \right)^{4} {\operatorname{Im}}\left( X_{2} \pm bi \right) \\ & = & \pm 2bY_{1}^{4}i.\end{aligned}$$ Letting $$x+yi= \left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right),$$ we have $$\label{eq:27} |f(x,y)|=\left| \left( X_{1} \pm bi \right) (x+yi)^{4} - \left( X_{1} \mp bi \right)(x-yi)^{4} \right| =2bY_{1}^{4}.$$ Put $$\omega = \frac{X_{1} \pm bi}{X_{1} \mp bi}$$ and let $\zeta_{4}$ be the $4$-th root of unity such that $$\left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| = \min_{0 \leq k \leq 3} \left| \omega^{1/4} - e^{2k\pi i/4} \frac{x-yi}{x+yi} \right|.$$ From , our expressions for $x+yi$, $X_{1}$ and $Y_{1}$ and (which implies that $\left| X_{1} \pm bi \right|^{2} =X_{1}^{2}+b^{2}=\left( a^{2}+b^{2} \right)Y_{1}^{4}$), we have $$\left| \omega - \left( \frac{x-yi}{x+yi} \right)^{4} \right| = \frac{2bY_{1}^{4}}{\left| X_{1} \mp bi \right| \left| r_{1} \mp s_{1}i \right|^{4}\left| r_{2} \mp s_{2}i \right|^{4}} =\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}}$$ By Lemma \[lem:3.4\](a) and Lemma \[lem:3.3\], $Y_{2} >7.98Y_{1}^{3}>7.98 \cdot 5^{3}$. So $Y_{2} \geq 998$. Then $$\left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| < \left( 2/Y_{2}^{2} \right)^{1/4} <0.04.$$ Thus we can apply Lemma \[lem:3.2bb\] with $c_{1}=0.04$ to find that $$\label{eq:29} \frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} = \left| \omega - \left( \frac{x-yi}{x+yi} \right)^{4} \right| > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right|.$$ In what follows, we shall require a lower bound for this last quantity. To derive such a bound we shall use the lower bounds in Lemma \[lem:2.1\] with a sequence of good approximations $p_{r}/q_{r}$ obtained from the hypergeometric functions. So we collect here the required quantities. Let $u_{1}=2X_{1}$ and $u_{2}=\pm 2b$. Since $\gcd(a,b)=\gcd \left( X_{1}, Y_{1} \right)=1$, we have $g_{1}=2$. If $ab \equiv 1 \bmod 2$, then $g_{3}=2$, otherwise $g_{3}=4$. Therefore, $g=\sqrt{2}$ and $d=2b^{2}$ if $ab \equiv 1 \bmod 2$, while $g=1$ and $d=4b^{2}$ otherwise. Therefore, $${\mathcal{N}}_{d,n}=2^{\min \left( v_{2}(d)/2, v_{2}(4)+1 \right)} =2^{\min \left( v_{2}(d)/2, 3 \right)}.$$ If $ab$ is odd, then ${\mathcal{N}}_{d,4}=\sqrt{2}$. If $a$ is even and $b$ is odd, then ${\mathcal{N}}_{d,4}=2$. If $b=2$, then ${\mathcal{N}}_{d,4}=4$. If $b=2^{m}$ with $m \geq 2$, then ${\mathcal{N}}_{d,4}=8$. Thus $8 \geq |g|{\mathcal{N}}_{d,4} \geq 2$ and from Lemma \[lem:2.4\] we have $$Q=\frac{e^{1.68} \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}}\right|}{|g|{\mathcal{N}}_{d,4}} < e^{1.68} \left| X_{1}+\sqrt{X_{1}^{2}+b^{2}}\right|.$$ From , we also have $X_{1}<\sqrt{X_{1}^{2}+b^{2}}=\sqrt{a^{2}+b^{2}} Y_{1}^{2}$, so $$\label{eq:Q-UB2} Q<e^{1.68} \cdot 2 \sqrt{a^{2}+b^{2}} Y_{1}^{2} <10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2}.$$ Similarly, we have $$E = \frac{|g|{\mathcal{N}}_{d,4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{{\mathcal{D}}_{4}u_{2}^{2}} > \frac{2 \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}} \right|}{e^{1.68} \cdot 4b^{2}}.$$ From , $$\label{eq:E-LB} E > \frac{\left( 1+\sqrt{0.9984} \right)\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{e^{1.68}b^{2}} > \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}}.$$ By Lemma \[lem:3.4\](a), $Y_{1} \geq 5$, so $E>1$, as required for its use with Lemma \[lem:2.1\]. Also $Q\geq e^{1.68}\left| X_{1}+\sqrt{X_{1}^{2}+b^{2}} \right|/4>1$, again as needed for Lemma \[lem:2.1\]. Recall from that we take $k_{0}=0.89$. Since $\omega = \left( X_{1} \pm bi \right)^{2}/ \left( X_{1}^{2}+b^{2} \right)$, we have $\left| \tan \left( \varphi \right) \right| = 2b/X_{1}$. From $|\varphi| \leq |\tan (\varphi)|$, we can take $$\label{eq:ell-UB} \ell_{0}={\mathcal{C}}_{4,2}|\varphi|=0.4b/X_{1}.$$ From Lemma \[lem:3.4\](a) we have $$X_{1}^{2}=\left( a^{2}+b^{2} \right)Y_{1}^{4}-b^{2} >\left( Y_{1}^{4}-1 \right) b^{2} \geq 624b^{2},$$ which yields $|\varphi| \leq \left| \tan(\varphi) \right|=2b/X_{1}<2\sqrt{1/624}<0.081$. Therefore the condition $|\omega-1|<1$ in Lemma \[lem:2.2\] is satisfied too. Let $p=x-yi$ and $q=x+yi=\left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right)$. We are now ready to deduce the required contradiction from the assumption that there are two coprime solutions $\left( X_{1}, Y_{1} \right)$ and $\left( X_{2}, Y_{2} \right)$ to with $Y_{2}>Y_{1}>b^{2}/2$. We will consider three cases according to the value of $r_{0}$ defined in Lemma \[lem:2.1\]. [**Case 1:**]{} $r_{0}=1$ and $q_{1}\zeta_{4}p \neq qp_{1}$. In this case, by , we have $$\begin{aligned} \frac{2b}{\sqrt{a^{2}+b^{2}} \, Y_{2}^{2}} &= \left| \omega - \left( \frac{p}{q} \right)^{4} \right| > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| \\ &> \frac{3.99}{2k_{0}Q\sqrt{Y_{1}Y_{2}}} > \frac{3.99}{2 \cdot 0.89 \cdot 10.74 \sqrt{a^{2}+b^{2}} Y_{1}^{2}\sqrt{Y_{1}Y_{2}}},\end{aligned}$$ where the second inequality comes from Lemma \[lem:2.1\](b) applied with $r_{0}=1$ and $|p|=\sqrt{Y_{1}Y_{2}}$, and the last inequality comes and . Thus we obtain $$Y_{2}^{3} < 92b^{2}Y_{1}^{5}.$$ On the other hand, we have $Y_{2}^{3}>508Y_{1}^{9}$ by Lemma \[lem:3.3\], thus we get $Y_{1}^{4}<b^{2}/5$. But Lemma \[lem:3.4\](a) and our assumption that $Y_{1}>b^{2}/2$ imply that $Y_{1}^{4} > 5^{3}b^{2}/2$. Thus we cannot have two solutions with $Y_{2}>Y_{1}>1$ in this case. [**Case 2:**]{} $r_{0}=1$ and $q_{1}\zeta_{4}p= qp_{1}$. From the definitions of $p_{1}$ and $q_{1}$ in , along with Lemma \[lem:2.2\] parts (a) and (e) and $Y_{11,4,1}(\omega)=(3\omega+5)/3$, we have $$\begin{aligned} \left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| &= \frac{1}{q_{1}} \left| q_{1}\omega^{1/4} - p_{1} \right| = \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} R_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ &= \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} (\omega-1)^{3} \frac{(1/4)(5/4)}{2 \cdot 3} {} _{2}F_{1} \left( 7/4, 2; 4; 1-\omega \right) \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ & \geq \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} (\omega-1)^{3} \frac{(1/4)(5/4)}{2 \cdot 3} \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ & = \left| \frac{5(\omega-1)^{3}}{96Y_{1,4,1}(\omega)} \right| = \left| \frac{5(\omega-1)^{3}}{32(3\omega+5)} \right|.\end{aligned}$$ Since $3\omega+5=\left( 8X_{1} \mp 2bi \right)/\left( X_{1} \mp bi \right)$, we have $$\frac{(\omega-1)^{3}}{(3\omega+5)} =-\frac{4b^{3}i}{\left( 4X_{1} \mp bi \right)\left( X_{1} \mp bi \right)^{2}}.$$ so by , $$\left| \frac{(\omega-1)^{3}}{(3\omega+5)} \right| =\frac{4b^{3}}{\sqrt{16X_{1}^{2}+b^{2}}\left( X_{1}^{2}+b^{2} \right)} > \frac{b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}}.$$ Therefore, $$\left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| \geq \frac{0.156b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}},$$ so $$\frac{2b}{\sqrt{a^{2}+b^{2}} \, Y_{2}^{2}} > 3.7\frac{0.156b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}} > \frac{0.62b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}}.$$ This inequality, along with our gap principle in Lemma \[lem:3.3\] implies that $$\left( \frac{2}{0.62} \right)^{2} \frac{\left( a^{2}+b^{2} \right)^{2}}{b^{4}} Y_{1}^{12}>Y_{2}^{4} >7.98^{4} \frac{\left( a^{2}+b^{2} \right)^{4}}{b^{8}} Y_{1}^{12}.$$ This implies that $$0.0027>\left( \frac{2}{0.62 \cdot 63} \right)^{2} >\frac{\left( a^{2}+b^{2} \right)^{2}}{b^{4}}>1,$$ which is impossible. Hence we cannot have two coprime solutions with $Y_{2}>Y_{1}>b^{2}/2$ in Case 2. [**Case 3:**]{} $r_{0}>1$. Here we establish a stronger gap principle here for $Y_{1}$ and $Y_{2}$ than the one in Lemma \[lem:3.3\]. We then use this to obtain a contradiction with Lemma \[lem:2.1\](a). Here the gap principle is simpler to obtain as we can appeal to the definition of $r_{0}$ in Lemma \[lem:2.1\]. From that definition we have $$|q| \geq E^{r_{0}-1} / \left( 2\ell_{0} \right).$$ Recall too that $q=x+yi$ so that $|q|=\sqrt{Y_{1}Y_{2}}$. Thus $$\sqrt{Y_{1}Y_{2}} \geq E^{r_{0}-1} / \left( 2\ell_{0} \right).$$ From , and , $$\sqrt{Y_{1}Y_{2}} > \frac{X_{1}}{0.8b} \left( \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}} \right)^{r_{0}-1} > \frac{1.248\sqrt{a^{2}+b^{2}}Y_{1}^{2}}{b} \left( \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}} \right)^{r_{0}-1}.$$ Therefore, $$\label{eq:y2UB-case3} Y_{2} > 11.25 \cdot 0.138^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} b^{2-4r_{0}} Y_{1}^{4r_{0}-1}.$$ From equation , Lemma \[lem:2.1\](a) and $|x+yi|=\sqrt{Y_{1}Y_{2}}$, we have $$\label{eq:32} \frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| > \frac{3.99}{2k_{0}Q^{r_{0}+1}\sqrt{Y_{1}Y_{2}}}.$$ Applying and , we obtain $$\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > \frac{3.99}{2 \cdot 0.89 \left( 10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2} \right)^{r_{0}+1}\sqrt{Y_{1}Y_{2}}},$$ so $$\frac{(3.56b)^{2}}{3.99^{2}} 10.74^{2r_{0}+2} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} >Y_{2}^{3}.$$ We will simplify this to $$\label{eq:y2LB-case3} 92b^{2} \cdot 116^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} >Y_{2}^{3}.$$ We now combine and , obtaining $$92b^{2} \cdot 116^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} > 1420 \cdot b^{6-12r_{0}} \cdot 0.0026^{r_{0}} \left( a^{2}+b^{2} \right)^{3r_{0}} Y_{1}^{12r_{0}-3}.$$ Applying the assumption that $Y_{1}>b^{2}/2$ and simplifying, we have $$0.0646 \left( \frac{44,620}{\left( a^{2}+b^{2} \right)^{2}} \right)^{r_{0}} > b^{4-12r_{0}} Y_{1}^{8r_{0}-8} > b^{4r_{0}-12} 2^{-8r_{0}+8}.$$ That is, $$0.000253 \left( \frac{11,423,000}{\left( a^{2}+b^{2} \right)^{2}} \right)^{r_{0}} > b^{4r_{0}-12}.$$ For $r_{0} \geq 2$, we have $$0.000253 \frac{b^{4}}{\left( a^{2}+b^{2} \right)^{4}} \left( 11,423,000 \right)^{2} > 1.$$ This is never satisfied for $a^{2}+b^{2} \geq 181,700$. We could attempt to address the outstanding values of $a$ and $b$ by using the `IntegralQuarticPoints()` function within MAGMA, but the number of equations is quite large. Instead we proceed as follows. Suppose that $Y_{1} \geq \max \left( 1700, b^{2}/2 \right)$. From , $$X_{1}^{2}>0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4} \geq 0.9984 \left( a^{2}+b^{2} \right) 1700 \left( b^{2}/2 \right)^{3} > 212b^{6}.$$ For such $X_{1}$, we have $$E^{2}> \left( \frac{2 \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}} \right|}{e^{1.68} \cdot 4b^{2}} \right)^{2} >\frac{64X_{1}^{2}}{e^{3.36} \cdot 16b^{4}} >\frac{64 \cdot 212b^{6}}{e^{3.36} \cdot 16b^{4}}>29.4b^{2}.$$ In addition, we have $$\frac{Q}{E}<\left( 10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2} \right) \frac{b^{2}}{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}} < 29b^{2}.$$ Thus $$E^{3}>Q.$$ Therefore, $$Q^{r_{0}-2}<E^{3(r_{0}-2)} = E^{-3}E^{3(r_{0}-1)} \leq E^{-3} \left( 2\ell_{0}|q| \right)^{3} =E^{-3} \left( \frac{0.8b}{X_{1}} \sqrt{Y_{1}Y_{2}} \right)^{3}$$ From , we have $$\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > \frac{3.99}{2k_{0}Q^{r_{0}+1}\sqrt{Y_{1}Y_{2}}}. =\frac{3.99X_{1}^{3}}{1.78Q^{3}E^{-3} \left( 0.8b \sqrt{Y_{1}Y_{2}} \right)^{3}\sqrt{Y_{1}Y_{2}}}.$$ We saw above that $Q/E<29b^{2}$, so $$\frac{2b}{\sqrt{a^{2}+b^{2}}} >\frac{3.99X_{1}^{3}}{1.78 \cdot 29^{3}b^{6} \left( 0.8b \sqrt{Y_{1}} \right)^{3}\sqrt{Y_{1}}}.$$ Combining this with , we find that $$1.25 \cdot 10^{8} b^{20} Y_{1}^{4}> \left( a^{2}+b^{2} \right) X_{1}^{6} > \left( a^{2}+b^{2} \right) \left( 0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4} \right)^{3}.$$ Thus $$\label{eq:y1-UB} 1.26 \cdot 10^{8} \frac{b^{20}}{\left( a^{2}+b^{2}\right)^{4}}>Y_{1}^{8}.$$ We then calculated all coprime pairs $(a,b)$ such that $a^{2}+b^{2}<181,700$ and that had a solution $(X,Y)$ with $Y \geq 2$ and either $Y<1700$ or holding. We found 35 such pairs, $(a,b)$. Of these, for the following 12 the negative Pell equation is solvable and there is only one family of solutions of the associated quadratic equation: $(a,b)=(1,1),(1,3),(3,7),(9,7),(11,3)$, $(11,7),(18,43),(19,9)$, $(29,11)$, $(29,17),(31,5),(41,13)$. We solved each of these $12$ equations using MAGMA (version V2.23-9) and its\ `IntegralQuarticPoints()` function. No further coprime solutions were found for any of these equations. Proof of Theorem \[thm:1.1\] ---------------------------- If $\left( X_{1},Y_{1} \right)$ is a coprime positive integer solution of $x^{2} - \left( a^{2}+b^{2} \right)y^{4}=-b^{2}$ with $Y_{1}>1$, then $Y_{1}>b^{2}/2$ by Lemma \[lem:3.4\](b). Therefore, we can apply Lemma \[lem:thm\] to show that there is no other solution $(X,Y)$ with $Y>b^{2}/2$. Thus this theorem holds. 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ArXiv

--- abstract: 'Machine-designed control of complex devices or experiments can discover strategies superior to those developed via simplified models. We describe an online optimization algorithm based on Gaussian processes and apply it to optimization of the production of Bose-Einstein condensates (BEC). BEC is typically created with an exponential evaporation ramp that is approximately optimal for s-wave, ergodic dynamics with two-body interactions and no other loss rates, but likely sub-optimal for many real experiments. Machine learning using a Gaussian process, in contrast, develops a statistical model of the relationship between the parameters it controls and the quality of the BEC produced. This is an online process, and an active one, as the Gaussian process model updates on the basis of each subsequent experiment and proposes a new set of parameters as a result. We demonstrate that the Gaussian process machine learner is able to discover a ramp that produces high quality BECs in 10 times fewer iterations than a previously used online optimization technique. Furthermore, we show the internal model developed can be used to determine which parameters are essential in BEC creation and which are unimportant, providing insight into the optimization process.' author: - 'P. B. Wigley' - 'P. J. Everitt' - 'A. van den Hengel' - 'J. W. Bastian' - 'M. A. Sooriyabandara' - 'G. D. McDonald' - 'K. S. Hardman' - 'C. D. Quinlivan' - 'P. Manju' - 'C. C. N. Kuhn' - 'I. R. Petersen' - 'A. Luiten' - 'J. J. Hope' - 'N. P. Robins' - 'M. R. Hush' bibliography: - 'machinelearning.bib' title: 'Fast machine-learning online optimization of ultra-cold-atom experiments' --- Experimental research into quantum phenomena often requires the optimization of resources or processes in the face of complex underlying dynamics and shifting environments. For example, creating large Bose-Einstein condensates (BECs) with short duty cycles is one of the keys to improving the sensitivity of cold-atom based sensors [@robins_atom_2013] or for performing scientific investigation into condensed matter phases [@bloch_quantum_2012], many-body physics [@bloch_many-body_2008] and non-equilibrium dynamics [@langen_ultracold_2015]. The standard process of BEC production is evaporative cooling [@olson_optimizing_2013]; microscopic semi-classical theory exists to describe this process [@sackett_optimization_1997], but it can oversimplify the dynamics and miss more complex and effective methods of performing evaporation. For example, Shobu *et al.* [@shobu_optimized_2011] found circumventing higher order inelastic collisions can produce very large condensates. ‘Tricks’ like this are likely to exist for other species with complicated scattering processes [@altin_collapse_2011], but discovery is only possible by experimentation. We automate this process of discovery with *machine-learning* online optimization (MLOO). What distinguishes our approach from previous methods for automation is that we seek to develop a statistical model of the relationship between parameters and the outcome of the experiment. We demonstrate that MLOO can discover condensation with less experiments than a competing optimization method and provide insight into which parameters are important in achieving condensation. {width="\textwidth"} Online optimization (OO), with mostly genetic [@judson_teaching_1992; @warren_coherent_1993; @amstrup_genetic_1995; @dods_genetic_1996; @baumert_femtosecond_1997; @pearson_coherent_2001; @zeidler_evolutionary_2001; @walmsley_quantum_2003; @rohringer_stochastic_2008; @tsubouchi_rovibrational_2008; @rohringer_stochastic_2011; @starkey_scripted_2013] but also gradient [@roslund_gradient_2009] and hybrid solvers [@egger_adaptive_2014; @kelly_optimal_2014], has been used to enhance a variety of quantum experiments. Here, online means optimization is done in real time with the experiment. We apply OO based on machine learning. What distinguishes our approach is that it does not only seek to optimize the experiment, but also creates an internal model that is able to predict the performance of future experiments given any set of parameters. This is achieved by modeling the experiment using a Gaussian process (GP) [@rasmussen_gaussian_2006]. Online machine learning (OML) with GPs [@seo_gaussian_2000; @csato_sparse_2002; @rasmussen_gaussian_2006; @deisenroth_gaussian_2009; @gramacy_particle_2011] has been applied in a variety of areas including robotics [@nguyen-tuong_local_2008; @nguyen-tuong_local_2009], vision [@ranganathan_online_2011], industrial chemistry [@yu_online_2012; @li_multiple_2009] and biochemistry [@gao_gaussian_2008]. However, in all of these cases, the focus was not on optimization. Rather, the goal was the development of an accurate model. We here combine the advantages of OML with the motivation of OO. The resultant MLOO algorithm has the following beneficial features: every experimental observation from an optimization run is used to improve the GP model, and uncertainties in the measurements can be correctly accounted for; our algorithm is both deterministic and able to find global minima, instead of randomly exploring the parameter space our learner uses its estimate of the variance from the GP to pick parameter values where we are most uncertain about the value of the cost function; fast gradient-based optimization routines can not be applied directly to our experimental data as it is too noisy, but they can be employed to efficiently find optima in our smooth machine-generated model; estimation of parameter sensitivity and visualization of the functional dependence of the resource’s quality can inform experimentalists on how to best develop future optimization experiments. The experimental apparatus is described in detail in [@kuhn_Bose-condensed_2014]. Initially $^{87}$Rb atoms are cooled in a combined 2D and 3D MOT system and subsequently cooled further by RF (radio frequency) evaporation. The cloud is then loaded into a cross beam optical dipole trap for the final evaporation stage. It is this stage that is the subject of the optimization process. The cross dipole trap is formed from two intersecting $1090$nm and $1064$nm lasers with approximate waists of $100\mu$m and $60\mu$m respectively. The depth of the cross trap is determined by the intensity of the two beams. The $1064$nm beam is controlled by varying the current to the laser, while the $1090$nm beam is controlled using the current and a waveplate rotation stage combined with a polarizing beamsplitter to provide additional power attenuation while maintaining mode stability. A diagram of the experimental set up is shown in figure 1. Normally the power to these beams is ramped down over time, thereby lowering the walls of the trap and allowing the higher energy atoms to leak out. The remaining atoms rethermalize to a lower temperature, enabling cooling. Once the gas has been cooled to temperatures on the order of nK, a phase transition occurs, and a macroscopic number of atoms start to occupy the same quantum state. This transition is called Bose-Einstein condensation [@anglin_boseeinstein_2002]. We hand over control of these ramps to the MLOO. We consider two parameterizations: one simple, where we only control the start and end points of a linear interpolation; and one complex, where we add variable quadratic, cubic and quartic corrections to the simple case (see Appendix). The approach we propose is a form of supervised learning, meaning that we provide the learner with a number that quantifies the quality of the resource produced or in optimization terminology a cost that must be minimized. Naïvely one might try to use a measure based on temperature and particle number. However determining these quantities accurately near condensation is difficult when constrained to very few runs per parameter set. Instead, a semi-heuristic method is used to calculate the cost. An absorption image of the final state of the quantum gas is taken after a 30ms expansion of the cloud, with the image providing the optical depth as a function of space. The cost is calculated from all data between a lower and upper threshold optical depth. The lower threshold is determined by the noise in the system. The upper threshold is required since the absorption images are taken on resonance and hence saturate for areas with very large optical depth. The upper threshold is set slightly below this saturation level. Only data from between the bounds is used and cost is simply the average of these values. In practice this means the sharper the edges of the cloud, the lower the cost. Indeed, low quality thermal clouds have broad edges, whereas the ideal BEC has much sharper edges. Each parameter set is tested twice with the average of the two runs used for the cost. Tests of the variation in cost for a set of parameters run-to-run indicate they obey a Gaussian distribution. As such we are able to estimate the uncertainty from two runs as twice the range. In doing so, the chance we have underestimated the uncertainty will be 27%. We therefore also apply bounds to the uncertainty to eliminate outliers overly affecting the modeling process. The cost function can be evaluated as long as some atoms are present at the end of the evaporation run. In cases where the evaporation parameters produced no cloud twice for a set of parameters, we set the cost to a default high value. We treat the experiment as a stochastic process $\mathscr{C}(X)$ which is dependent on the parameters $X = (x_1,\cdots,x_M)$. When we make a measurement and determine a cost, we interpret this as a sample of this process $C(X)$ with some associated uncertainty $U(X)$. We define the set of all parameters, costs and uncertainty previously measured as $\mathcal{X} = (X_1,\cdots X_N)$, $\mathcal{C} = (C_1,\cdots, C_N)$ and $\mathcal{U} = (U_1,\cdots, U_N)$ respectively and collectively refer to these sets as our observations $\mathcal{O} = (\mathcal{X},\mathcal{C},\mathcal{U})$. The aim of OO is to use previous observations $\mathcal{O}$ to plan future experiments in order to find a set of parameters that minimize the mean cost of the stochastic process $M_{\mathscr{C}}(X)$. Unique to the MLOO approach, we first make an *estimate* of the stochastic process given our observations $\hat{\mathscr{C}}(X|\mathcal{O})$, which is then used to determine what parameters to try next. We model $\mathscr{C}(X)$ as a GP — a distribution over *functions* — with constant mean function and covariance defined by a squared exponential correlation function $K(X,X',H) = e^{-\sum_{j=1}^M (x_j- x_j')^2/h_j^2}$ where $H = (h_1, \cdots, h_M)$ is a set of correlations lengths for each of the parameters. The mean function conditional on the observations $\mathcal{O}$ and correlations lengths $H$ of our GP is: $\mu_{\hat{\mathscr{C}}}(X|\mathcal{O}, H)$, which is evaluated through a set of matrix operations [@rasmussen_gaussian_2006] (see Appendix). As we are using a GP, we can also get the variance of the functions conditioned on $\mathcal{O}$ and $H$: $\sigma_{\hat{\mathscr{C}}}(X|\mathcal{O}, H)$ [@rasmussen_gaussian_2006]. Both of these estimates depend on the correlation lengths $H$, normally referred to as the *hyperparameters* of our estimate. We assume that $H$ is not known a priori and needs to be fitted online. The correlation lengths $H$ control the sensitivity of the model to each of the parameters, and relates to how much a parameter needs to be changed before it has a significant effect on the cost (see Fig. 1). A standard approach to fit $H$ is maximum likelihood estimation [@rasmussen_gaussian_2006]. Here, the hyperparameters are globally optimized over the likelihood of the parameters $H$ given our observations $\mathcal{O}$, or $L(H | \mathcal{O})$ [@rasmussen_gaussian_2006] (see Appendix). However, when the data set is small there will often be multiple local optima for the hyperparameters whose likelihoods are comparable to the maximum. We term these hyperparameters the hypothesis set $\mathcal{H}= (H_1,\cdots,H_P)$ with corresponding likelihood set $\mathcal{L} = (L_1,\cdots,L_P)$. To produce our final estimates for the mean function and variance we treat each hypothesis as a *particle* [@gramacy_particle_2011], and perform a weighted average over $\mathcal{H}$. The weighted mean function is now defined as $M_{\hat{\mathscr{C}}}(X|\mathcal{O}, \mathcal{H}) \equiv \sum_{i=1}^P w_i \mu_{\hat{\mathscr{C}}}(X|\mathcal{O}, H_i)$ and weighted variance of the functions is $\Sigma_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, \mathcal{H}) \equiv \sum_{i=1}^P w_i(\sigma_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, H_i) + \mu_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, H_i)) - M_{\hat{\mathscr{C}}}^2(X|\mathcal{O}, \mathcal{H})$, where $w_i = L_i/\sum_{i=1}^P L_i$ are the relative weights for the hyperparameters. Now that we have our final estimate for $\mathscr{C}(X|\mathcal{O}, \mathcal{H})$, we need to determine an optimization *strategy* for picking the next set of parameters to test. {width="\columnwidth"} Consider the following homogeneous strategies: we could test parameters that minimize $M_{\hat{\mathscr{C}}}(X)$, but this strategy can get trapped in local minima; or we could test parameters that maximize $\Sigma_{\hat{\mathscr{C}}}^2(X)$ (i.e. where we are most uncertain), but this may require a large number of trials to map the space and would not prioritize refinement of the minima. We chose to implement an inhomogeneous strategy that repeatably *sweeps* between these two extremes by minimizing a biased cost function: $B_{\hat{\mathscr{C}}}(X) \equiv b M_{\hat{\mathscr{C}}}(X) - (1 - b) \Sigma_{\hat{\mathscr{C}}}(X)$, where the value for $b$ is linearly increased from $0$ to $1$ in a cycle of length $Q$. This makes the learner change strategy from getting maximum information ($b=0$), to looking for a new minima: with a high risk-seeking ($b$ small) to risk-neutral ($b=1$) preference. During testing with synthetic data, we found sweeping was more robust and efficient than the homogeneous approaches. When we minimized $B_{\hat{\mathscr{C}}}(X)$ we also put bounds, set to 20% of the parameters maximum-minimum values, on the search relative to the last best measured $X$. We call these bounds a *leash*, as it restricts how fast the learner could change the parameters but did not stop it from exploring the full space (similar to trust-regions [@conn_trust_2000; @yuan_review_2000]). This was a technical requirement for our experiment: when a set of parameters was tested that was very different from the last set, the experiment almost always produced no atoms, meaning we had to assign a default cost that did not provide meaningful gradient information to the learner. Once the next set of parameters is determined they are sent to the experiment to be tested. After the resultant cost is measured this is then added to the observation set $\mathcal{O}$ with $N\rightarrow N+1$ and the entire process is repeated. We emphasize that fitting of $H$, estimation of $M_{\hat{\mathscr{C}}}(X)$ and minimization of $B_{\hat{\mathscr{C}}}(X)$ is all done online while the experiment is being run. In a single optimization run, the learner typically performs hundreds more hypothetical experiments than the number physically run in the lab. The MLOO algorithm we developed is open source and available online at [@hush_m-loop_2015] (it uses the package scikit-learn [@pedregosa_scikit-learn:_2011] to evaluate the GPs). As a benchmark for comparison, we also performed OO using a Nelder-Mead solver [@nelder_simplex_1965], which has previously been used to optimize quantum gates [@kelly_optimal_2014]. We demonstrate the performance of machine learning online optimization in comparison to the Nelder-Mead optimizer in Fig. 2. Here we used the complex parameterization for all 3 ramps, and added an extra parameter that controlled the total time of the ramps, resulting in 16 parameters. If we were to perform a brute force search and optimize the parameters to within a $10\%$ accuracy of the parameters maximum-minimum bounds, the number of runs required would be $10^{16}$. The Nelder-Mead algorithm is able to find BEC much faster than this, in only 145 runs. The machine learning algorithm, on the other hand, is much faster. After the first 20 training runs, where the machine learning and Nelder-Mead algorithm use a common set of parameters, the machine learning algorithm converges in only 10 experiments. {width="\columnwidth"} The learner used in Fig. 2 only used the *best* hypothesis set when picking the next parameters, in other words we set $P=1$. Evaluating multiple GPs is computationally expensive with so many parameters, so to save time we made this restriction. In spite of this, the learner discovered ramps that produced BEC in very few iterations. This is because the learner consistently fitted the correlation lengths of the 3 most important parameters — the end points of the ramps — very quickly. However, we found the other correlations lengths were not estimated well and would not converge, even after a BEC was found. This meant that we were unable to make useful predictions about the cost landscape and we could not reliably determine what parameters were least important. Gramacy *et al.* [@gramacy_particle_2011] have suggested that making good online estimation of the GP correlation lengths requires multiple particles. We considered achieving this goal in a different experiment as shown in Fig. 3. Here we used a learner with many particles $P=16$, but had to use the simple parameterization for the ramps to save computational time. This resulted in a total of 7 parameters. We can see again the overall trend for the machine learner is still faster than Nelder-Mead, but less pronounced. More carefully estimating the correlation lengths has hindered the convergence rate compared to the $16$ parameter case. Nevertheless, as we now have a more reliable estimate of the correlation lengths we can take advantage of a different feature of the learner. In Fig. 4(a) we show estimates of the cost landscape as 1D cross sections about the best measured point. We plot the two most sensitive parameters and the least. We can see the least sensitive parameter appears to have no effect on the production of a BEC. This parameter corresponds to an intentionally added 7th parameter of the system that controls nothing in the experiment. Fig. 4(a) shows the learner successfully identified this, even with such a small data set. After making this observation we can then reconsider the design of the optimization process and eliminate this parameter from the experiment. {width="\columnwidth"} In Fig. 3 we plot the machine learner optimization run with $P=16$ but now with only 6 parameters. We can see the learner converges much more rapidly than the 7 parameter case, and even produces a higher quality BEC. As the learner no longer takes extra runs to determine the importance of the useless 7th parameter, it achieves BEC quite rapidly. Lastly, to give us some understanding of the complexity of the landscape, we plot a 2D cross section of the landscape against the two most sensitive parameters in Fig. 4(b) generated from the 6 parameter machine learning run. We can see there is a very sharp transition to BEC, as it exists in a very deep valley of the landscape. We have demonstrated that MLOO can discover evaporation ramps that produce high quality BECs with far fewer experiments than OO with Nelder-Mead. Rapid optimization of ultra-cold-atom experiments is not just a useful tool to overcome technical difficulties, but will be vital in the application of BECs in proposed space-based scientific investigations [@zoest_bose-einstein_2010; @arimondo_atom_2009]. Furthermore, when implemented with many particles, the learner can be used to make estimates of the cost landscape to determine what parameters most contributed to BEC production and aid better experimental design. In future work we will apply MLOO to atomic species with more exotic scattering properties [@altin_collapse_2011] in order to find novel cooling ramps that find optimal solutions in the competition between poorly characterized complex dynamical processes. Our approach is generic and available [@hush_m-loop_2015] for use on other scientific experiments, ultra-cold atom based or otherwise. MRH acknowledges funding from an Australian Research Council (ARC) Discovery Project (project number DP140101779). JJH acknowledges support of an ARC Future Fellowship (FT120100291). AL would like to thank the South Australian Government through the Premier’s Science and Research Fund for supporting this work. Appendix ======== *Gaussian process evaluation:* In practice, evaluating a Gaussian process (GP) reduces to a set of matrix operations whose derivation is given by Rasmussen *et al.* [@rasmussen_gaussian_2006] in section 2.7. Consider $N$ previous experiments have been performed with parameter sets $\mathcal{X} = (X_1,\cdots X_N)$ (each $X_j = (x_{1,j}, \cdots x_{M,j})$), measured costs $\mathcal{C} = (C_1,\cdots, C_N)$ and uncertainties $\mathcal{U} = (U_1,\cdots, U_N)$. We refer to the set of this data as our observations $\mathcal{O} = (\mathcal{X},\mathcal{C},\mathcal{U})$. We fit a GP to these observations with constant function offset $\beta$ and covariance defined by a squared exponential correlation function $K(X_p,X_q,H) = e^{-\sum_{j=1}^M (x_{j,p}- x_{j,q})^2/h_j^2}$ where $H = (h_1, \cdots, h_M)$ are the hyperparameters of the model. The mean function and variance of the functions are: $$\begin{aligned} \mu_{\hat{\mathscr{C}}}(X|\mathcal{O},H) = & \beta + r(X)^{T}\gamma \\ \sigma_{\hat{\mathscr{C}}}^2 (X|\mathcal{O},H) = & \sigma_{\mathcal{C}}^2 ( 1 - r(X)^T R^{-1} r(X) {\nonumber}\\ & + (j^T R^{-1} j)^{-1} (j^{T}R^{-1} r(X) - 1)^2 ) \label{eqn:varC}\end{aligned}$$ where $\sigma_{\mathcal{C}}^2$ is the variance of the costs $\mathcal{C}$, and we define the constant $\beta \equiv (j^{T}R^{-1} j)^{-1} j^{T}R^{-1} Y$, the $N \times 1$ vector $r(X)$ such that $\{r(X)\}_{1,i} = K(X,X_i,H)$, the $N \times 1$ vector $\gamma \equiv R^{-1}(Y - j \beta)$, the $N \times 1$ vector $Y$ of the costs defined by $\{Y\}_{1,i} = C_i$, the $N \times 1$ vector $\{j\}_{1,i} = 1$, the $N\times N$ matrix $R$ defined as $\{R\}_{i,j} = K(X_i,X_j,H) + \delta_{i,j} U_i^2$, and where $\delta_{i,j}$ is the Kronecker delta function. $\{\cdot\}_{i,j}$ is our notation for the $i$th row and $j$th column of a matrix or vector. When finding the most likely hyperparameters we maximize the likelihood function. The likelihood $L(H | \mathcal{O})$ is defined as the probability of the costs given the parameters, uncertainties and hyperparameters: $P(\mathcal{C}| \mathcal{X},\mathcal{U},H)$, the log of which is: $$\begin{aligned} \log P(\mathcal{C}| \mathcal{X},\mathcal{U},H) = & \frac{1}{2}( - \log |R| - \log j^{T}R^{-1} j {\nonumber}\\ & - (N - 1) \log 2\pi {\nonumber}- Y^{T}(R^{-1} \\ & - (j^{T}R^{-1} j)^{-1} R^{-1} j j^{T}R^{-1}) Y) \end{aligned}$$ *Parameterizations of evaporation ramps:* The simple parameterization of the evaporation ramps is $$\begin{aligned} \mathcal{R}_{s}(y_{i},y_{f},t_{f}) = y_{i}+\left(y_{f}-y_{i}\right)\frac{t}{t_{f}}\end{aligned}$$ where $y_i$ and $y_f$ specify the start and end amplitudes of the ramps and $t_f$ specifies the length in time. The complex parameterization an extension of the simple form: $$\begin{aligned} \mathcal{R}_{c}&(y_{i},y_{f},A_{1},A_{2},A_{3},t_{f}) = {\nonumber}\\ & y_{i}+\left(y_{f}-y_{i}\right)\frac{t}{t_{f}}+A_{2}t\left(t-t_{f}\right) {\nonumber}\\ & +A_{3}t\left(t-t_{f}\right)\left(t+\frac{1}{2}t_{f}\right) {\nonumber}\\ & +A_{4}t\left(t-t_{f}\right)\left(t+\frac{2}{3}t_{f}\right)\left(t+\frac{1}{3}t_{f}\right)\end{aligned}$$ where $A_1$, $A_2$ and $A_3$ correspond to the 3rd, 4th and 5th order polynomial terms respectively with each polynomial having evenly spaced roots between $t=0$ and $t=t_f$. As with the simple parametrization $t_f$ specifies the end of the ramps in time. In each of the three ramps being optimized, the parameters $y_{i}$, $y_{f}$, $A_{1}$, $A_{2}$, $A_{3}$ are independent. However, the final time $t_f$ is common.

ArXiv

--- author: - GIUSEPPE BOCCIGNONE bibliography: - 'levyeye.bib' title: A probabilistic tour of visual attention and gaze shift computational models --- This research was partially supported by the project “Interpreting emotions: a computational tool integrating facial expressions and biosignals based shape analysis and bayesian networks”, grant FIRB - *Future in Research* RBFR12VHR7\ Author’s address: G. Boccignone, Department of Computer Science, University of Milan, via Comelico 39/41, 20135 Milano, Italy; email: [email protected] Introduction ============ As the french philosopher Merleau-Ponty put it, “vision is a gaze at grips with a visible world” [@maurice1945phenomenologie]. Goals and purposes, either internal or external, press the observer to maximise his information intake over time, by moment-to-moment sampling the most informative parts of the world. In natural vision this endless endeavour is accomplished through a sequence of eye movements such as saccades and smooth pursuit, followed by fixations. Gaze shifts require visual attention to precede them to their goal, which has been shown to enhance the perception of selected part of the visual field (in turn related to the foveal structure of the human eye, see for an extensive discussion of these aspects). The computational counterpart of using gaze shifts to enable a perceptual-motor analysis of the observed world can be traced back to pioneering work on active or animate vision [@aloimonos1988active; @Ballard; @bajcsy1992active]. The main concern at the time was to embody vision in the action-perception loop of an artificial agent that purposively acts upon the environment, an idea that grounds its roots in early cybernetics [@cordeschi2002discovery]. To such aim the sensory apparatus of the organism must be active and flexible, for instance, the vision system can manipulate the viewpoint of the camera(s) in order to investigate the environment and get better information from it. Surprisingly enough, the link between attention and active vision, notably when instantiated via gaze shifts (e.g, a moving camera), was overlooked in those early approaches, as lucidly remarked by . Indeed, active vision, as it has been proposed and used in computer vision, must include attention as a sub-problem [@rothenstein2008attention]. First and foremost when it must confront the computational load to achieve real-time processing (e.g., for autonomous robotics and videosurveillance). Nevertheless, the mainstream of computer vision has not dedicated to attentive processes and, more generally, to active perception much consideration. This is probably due to the original sin of conceiving vision as a pure information-processing task, a reconstruction process creating representations at increasingly levels of abstraction, a land where action had no place: the “from pixels to predicates” paradigm [@aloimonos1988active]. To make a long story short, the research field had a sudden burst when the paper by was published. Their work provided a sound and neat computational model (and the software simulation) to contend with the problem: in a nutshell, derive a saliency map and generate gaze shifts as the result of a Winner-Take-All (WTA) sequential selection of most salient locations. Since then, proposals and techniques have flourished. Under these circumstances, a deceptively simple question arises: Where are we now? A straight answer, which is the *leitmotiv* of this paper, is that whilst early active vision approaches overlooked attention, current approaches have betrayed purposive active perception. In this perspective, here we provide a critical discussion of a number of models and techniques. It will be by no means exhaustive, and yet, to some extent, idiosyncratic. Our purpose is not to offer a review (there are excellent ones the reader is urged to consult, e.g., [@BorItti2012; @borji2014salient; @bruce2015computational; @bylinskii2015towards]), but rather to spell in a probabilistic framework the variety of approaches, so to discuss in a principled way current limitations and to envisage intriguing directions of research, e.g., the hitherto neglected link between oculomotor behavior and emotion. In the following Section we highlight critical points of current approaches and open issues. In Section \[sec:prob\] we frame such problems in the language of probability. Section \[sec:action\] discusses possible routes to reconsider the problem of oculomotor behaviour within the action/perception loop. In Section \[sec:emo\] we explore the *terra incognita* of gaze shifts and emotion. A Mini review and open issues ============================= The aim of a computational model of attentive eye guidance is to answer the question *Where to Look Next?* by providing: 1. at the *computational theory level* (following ), an account of the mapping from visual data of a natural scene, say $\mathbf{I}$ (raw image data representing either a static picture or a stream of images), to a sequence of time-stamped gaze locations $(\mathbf{r}_{F_1}, t_1), (\mathbf{r}_{F_2}, t_2),\cdots$, namely $$\mathbf{I} \mapsto \{\mathbf{r}_{F_1}, t_1; \mathbf{r}_{F_2}, t_2;\cdots \}, \label{eq:mapping}$$ 2. at the *algorithmic level*, a procedure that simulates such mapping. A simple example of the problem we are facing is shown in Figure \[Fig:variab\]. ![Scan paths eye tracked from different human observers while viewing three pictures of different information content: outdoor (top row), indoor with meaningful objects (middle row), indoor with high semantic content (person and face, bottom row). The area of yellow disks marking fixations between saccades is proportional to fixation time (images freely available from the dataset).[]{data-label="Fig:variab"}](FigVariab.jpg) Under this conceptualization, when the input $\mathbf{I}$ is a static scene (a picture), the fixation duration time and saccade (lengths and directions) sequence are the only observables of the underlying guidance mechanism. When $\mathbf{I}$ stands for a time varying scene (e.g. a video), pursuit needs to be taken into account, too. We will adopt the generic terms of gaze shifts for either pursuit, saccades and fixational movements. In the following, for notational simplicity, we will write the time series $\{\mathbf{r}_{F_1}, t_1; \mathbf{r}_{F_2}, t_2;\cdots \}$ as the sequence $\{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$, unless the expanded form is needed. Also, we will generically refer to such sequence as scan path, though this term has a historically precise meaning in the eye movement literature [@privitera2006scanpath]. The common practice of computational approaches to derive the mapping (\[eq:mapping\]) is to conceive it as a two step procedure: 1. obtaining a suitable perceptual representation $\mathcal{W}$, i.e., $\mathbf{I} \mapsto \mathcal{W}$; 2. using $\mathcal{W}$ to generate the scan path, $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$. It is important to remark that each gaze position $\mathbf{r}_{F}(t)$ sets a new field of view for perceiving the world, thus $\mathcal{W}=\{\mathcal{W}_{\mathbf{r}_{F}(1)}, \mathcal{W}_{\mathbf{r}_{F}(2)},\cdots \}$ should be a time-varying representation, even in the case of a static image input. This feedback effect of the moving gaze is hardly considered at the modelling stage [@zelinsky2008theory; @TatlerBallard2011eye]. By overviewing the field [@TatlerBallard2011eye; @BorItti2012; @bruce2015computational; @bylinskii2015towards], computational modelling has been mainly concerned with the first step: deriving a representation $\mathcal{W}$, typically in the form of a salience map. Yet, such step has recently evolved in a parallel research program, in which gaze shift prediction and simulation is not the focus, but salient object detection (for an in-depth review of this “second wave” of saliency-centered methods, see ). The second step, that is $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$, which actually brings in the question of *how* we look rather than *where*, is seldom taken into account. Surprisingly, in spite of the fact that the most cited work in the field [@IttiKoch98] clearly addressed the *how* issue (gaze shifts as the result of a WTA sequential selection of most salient locations), most models simply overlook the problem. As a matter of fact, the representation $\mathcal{W}$, once computed, is usually validated in respect of its capacity for predicting the image regions that would be explored by the overt attentional shifts of human observers (in a task designed to minimize the role of top-down factors). Predictability is assessed according to some established evaluation measures (see , and , for a recent discussion). In other cases, if needed for practical purposes, e.g. for robotic applications, the problem of oculomotor action selection is solved by adopting some simple deterministic choice procedure that usually relies on selecting the gaze position $\mathbf{r}$ as the argument that maximizes a measure on the given representation $\mathcal{W}$. We will attack on the problem related to gaze shift generation in Sections \[sec:bias\] and \[sec:var\]. In the following Section we first discuss representational problems. Levels of representation and control {#sec:levels} ------------------------------------ The guidance of eye movements is likely to be influenced by a hierarchy of several interacting control loops, operating at different levels of processing. Each processing step exploits the most suitable representation of the viewed scene for its own level of abstraction. , in a plausible portrayal, have sorted out the following representational levels: 1) *salience*, 2) *objects*, 3) *values*, and 4) *plans*. Up to this date, the majority of computational models have retained a central place for low-level visual conspicuity [@TatlerBallard2011eye; @BorItti2012; @bruce2015computational]. The perceptual representation of the world $\mathcal{W}$ is usually epitomized in the form of a spatial saliency map, which is mostly derived bottom-up (early salience) following [@IttiKoch98]. The weakness of the bottom-up approach has been largely discussed (see, e.g. ). Indeed, the effect of early salience on attention is likely to be a correlational effect rather than an causal one [@foulsham2008], [@schutz2011eye]. Few examples are provided in Fig. \[Fig:itti\], where, as opposed to human scan paths (in free-viewing conditions), the scan path generated by using a salience-based representation [@IttiKoch98] does not spot semantically important objects (faces), the latter not being detected as regions of high contrast in colour, texture and luminance with respect to other regions of the picture. Under these circumstances, early saliency can be modulated to improve its fixation prediction. has considered prior knowledge on the typical spatial location of the search target, as well as contextual information (the gist of a scene, ). Further, object knowledge can be used to top-down tune early salience. In particular, when dealing with faces, a face detection step [@cerf2008predicting], [@postma2011], [@marat2013improving] or a prior for Bayesian integration with low level features [@bocc08tcsvt], can provide a reliable cue to complement early conspicuity maps. Indeed, faces drive attention in a direct fashion [@cerf2009faces] and the same holds for text regions [@cerf2008predicting; @BocCOGN2014]. It has been argued that salience has only an indirect effect on attention by acting through recognised objects: observers attend to interesting objects and salience contributes little extra information to fixation prediction [@EinhauserSpainPerona2008]. As a matter of fact, in the real world, most fixations are on task-relevant objects and this may or may not correlate with the saliency of regions of the visual array [@canosa2009real; @rothkopfBallard2007]. Notwithstanding, object-based information has been scarcely taken into account in computational models [@TatlerBallard2011eye]. There are of course exceptions to this state of affairs, most notable ones those provided by , , the Bayesian models discussed by and . The representational problem is just the light side of the eye guidance problem. When actual eye tracking data are considered, one has to confront with the dark side: regardless of the perceptual input, scan paths exhibit both systematic tendencies and notable inter- and intra-subject variability. As put it, where we choose to look next at any given moment in time is not completely deterministic, but neither is it completely random. Biases in oculomotor behaviour {#sec:bias} ------------------------------ Systematic tendencies or “biases” in oculomotor behaviour can be thought of as regularities that are common across all instances of, and manipulations to, behavioural tasks [@tatler2008systematic; @tatler2009prominence]. In that case case useful information about how the observers will move their eyes can be found. One remarkable example is the amplitude distribution of saccades and microsaccades that typically exhibit a positively skewed, long-tailed shape [@TatlerBallard2011eye; @dorr2010variability; @tatler2008systematic; @tatler2009prominence]. Other paradigmatic examples of systematic tendencies in scene viewing are: initiating saccades in the horizontal and vertical directions more frequently than in oblique directions; small amplitude saccades tending to be followed by long amplitude ones and vice versa [@tatler2008systematic; @tatler2009prominence]. Indeed, biases affecting the manner in which we explore scenes with our eyes are well known in the psychological literature (see for a thorough review), albeit underexploited in computational models. Such biases may arise from a number of sources. have suggested the following: biomechanical factors, saccade flight time and landing accuracy, uncertainty, distribution of objects of interest in the environment, task parameters. Understanding biases in eye guidance can provide powerful new insights into the decision about where to look in complex scenes. In a remarkable study, provided striking evidence that a model based solely on these biases and therefore blind to current visual information can outperform salience-based approaches. Further, the predictive performance of a salience-based model can be improved from $56\%$ to $80\%$ by including the probability of gaze shift directions and amplitudes. Failing to account properly for such characteristics results in scan patterns that are fairly different from those generated by human observers (which can be easily noticed in the example provided in Fig. \[Fig:itti\]) and eventually in distributions of saccade amplitudes and orientations that do not match those estimated from human eye behaviour. Variability {#sec:var} ----------- When looking at natural images or movies [@dorr2010variability] under a free-viewing or a general-purpose task, the relocation of gaze can be different among observers even though the same locations are taken into account. In practice, there is a small probability that two observers will fixate exactly the same location at exactly the same time. This effect is even more remarkable when free-viewing static images: consistency in fixation locations selected by observers decreases over the course of the first few fixations after stimulus onset [@TatlerBallard2011eye] and can become idiosyncratic. Such variations in individual scan paths (as regards chosen fixations, spatial scanning order, and fixation duration) still hold when the scene contains semantically rich “objects” (e.g., faces, see Figures \[Fig:variab\] and \[Fig:itti\]). Variability is also exhibited by the same subject along different trials on equal stimuli. ![A worst case analysis of scan path generation based on early salience (top left image, via ) vs. scan paths eye tracked from human subjects. Despite of inter-subject variability concerning the gaze relocation pattern and fixation duration time, human observers consistently fixate on the two faces. The simulated scan path fails to spot such semantically important objects that have not been highlighted as regions of high contrast in colour, texture and luminance. The area of yellow disks marking human fixations is proportional to fixation time ( dataset).[]{data-label="Fig:itti"}](FigItti.jpg) Randomness in motor responses is likely to be originated from endogenous stochastic variations that affect each stage between a sensory event and the motor response: sensing, information processing, movement planning and executing [@vanBeers2007sources]. It is worth noting that uncertainty comes into play since the earliest stage of visual processing: the human retina evolved such that high quality vision is restricted to the small part of the retina (about $2^{0}-5^{0}$ degrees of visual angle) aligned with the visual axis, the fovea at the centre of vision. Thus, for many visually-guided behaviours the coarse information from peripheral vision is insufficient [@strasburger2011peripheral]. In certain circumstances, uncertainty may promote almost “blind" visual exploration strategies [@tatler2009prominence; @over2007coarse], much like the behaviour of a foraging animal exploring the environment under incomplete information; indeed when animals have limited information about where targets (e.g., resource patches) are located, different random search strategies may provide different chances to find them . Indeed, few works have been trying to cope with the variability issue, after the early work by , . The glorious WTA scheme [@IttiKoch98], or variants such as the selection of the proto-object with the highest attentional weight [@anna] are deterministic procedures. Even when probabilistic frameworks are used to infer where to look next, the final decision is often taken via the maximum a posteriori (MAP) criterion which again is a deterministic procedure (technically, an $\arg\max$ operation, see ), or variants like the robust mean (arithmetic mean with maximum value) over candidate positions [@begum2010probabilistic]. As a result, for a chosen visual input $\mathbf{I}$ the mapping $\mathcal{W} \mapsto \{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$ will always generate the same scan path across different trials. As a last remark, the variability of visual scan paths has been considered a nuisance rather than an opportunity from a modelling standpoint. Nevertheless, beside theoretical relevance for modelling human behavior, the randomness of the process can be an advantage in computer vision and learning tasks. For instance, have reported that a stochastic attention selection mechanism (a refinement of the algorithm proposed in ) enables the i-Cub robot to explore its environment up to three times faster compared to the standard WTA mechanism [@IttiKoch98]. Indeed, stochasticity makes the robot sensitive to new signals and flexibly change its attention, which in turn enables efficient exploration of the environment as a basis for action learning [@nagai2009stability; @nagai2009bottom]. There are few notable exceptions to this current state of affairs, which will be discussed in Section \[sec:prior\]. Framing models in a probabilistic setting {#sec:prob} ========================================= We contend with the above issues by stating that observables such as fixation duration and gaze shift lengths and directions are random variables (RVs) that are generated by an underlying stochastic process. In other terms, the sequence $\{\mathbf{r}_{F}(1), \mathbf{r}_{F}(2),\cdots \}$ is the realization of a stochastic process, and the ultimate goal of a computational theory is to develop a mathematical model that describes statistical properties of eye movements as closely as possible. The problem of answering the question *Where to Look Next?* in a formal way can be conveniently set in a probabilistic Bayesian framework. have re-phrased this question in terms of the posterior probability density function (pdf) $P(\mathbf{r} \mid \mathcal{W})$, which accounts for the plausibility of generating the gaze shift $\mathbf{r} = \mathbf{r}_{F}(t) - \mathbf{r}_{F}(t-1)$, after the perceptual evaluation $\mathcal{W}$. Formally, via Bayes’ rule: $$P(\mathbf{r} \mid \mathcal{W})= \frac{ P(\mathcal{W} \mid \mathbf{r} )}{P(\mathcal{W})} P(\mathbf{r}). \label{eq:BayesTatler}$$ In Eq. \[eq:BayesTatler\], the first term on the r.h.s. accounts for the likelihood $P(\mathcal{W} \mid \mathbf{r})$ of $\mathbf{r}$ when visual data (e.g., features, such as edges or colors) are observed under a gaze shift $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$, normalized by $P(\mathcal{W})$, the evidence of the perceptual evaluation. As they put it, “The beauty of this approach is that the data could come from a variety of data sources such as simple feature cues, derivations such as Itti’s definition of salience, object-or other high-level sources”. The second term is the pdf $P(\mathbf{r})$ incorporating prior knowledge on gaze shift execution. The generative model behind Eq. \[eq:BayesTatler\] is shown in Fig. \[fig:tata\] shaped in the form of a Probabilistic Graphical Model (PGM, see for an introduction). A PGM is a graph where nodes (e.g., $\mathbf{r}$ and $\mathcal{W}$) denote RVs and directed arcs (arrows) encode conditional dependencies between RVs, e.g $P(\mathcal{W} \mid \mathbf{r})$. A node with no input arcs (for example $\mathbf{r}$) is associated with a prior probability, e.g., $P(\mathbf{r})$. Technically, as a whole, the PGM specifies at a glance a chosen factorization of the joint probability of all nodes. Thus, in Fig. \[fig:tata\] we can promptly read that $P(\mathcal{W} , \mathbf{r}) = P(\mathcal{W} \mid \mathbf{r})P(\mathbf{r})$. The PGM in Fig. \[fig:tatb\] represents the PGM in Fig. \[fig:tata\], but unrolled in time. Note that now the arc $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$ makes explicit the dynamics of the gaze shift occurring with probability $P(\mathbf{r}_{F}(t+1) \mid \mathbf{r}_{F}(t))$. \[Fig:tat\] ![The dynamic PGM obtained by unrolling in time the PGM depicted in Fig. \[fig:tata\][]{data-label="fig:tatb"}](FigTatler.jpg) ![The dynamic PGM obtained by unrolling in time the PGM depicted in Fig. \[fig:tata\][]{data-label="fig:tatb"}](FigTatlerDyn.jpg) The probabilistic model represented in Fig. \[Fig:tat\] is generative in the sense that if all pdfs involved were fully specified, the attentive process could be simulated (via ancestral sampling, ) as: 1. Sampling the gaze shift from the prior: $$\mathbf{r}^{*} \sim P(\mathbf{r}); \label{eq:priorTatlersamp}$$ 2. Sampling the observation of the world under the gaze shift: $$\mathcal{W} ^{*} \sim P(\mathcal{W} \mid \mathbf{r}^{*}). \label{eq:likeTatlersamp}$$ Inferring the gaze shift $\mathbf{r}$ when $\mathcal{W}$ is known boils down to the inverse probability problem (inverting the arrows), which is solved via Bayes’ rule (Eq. \[eq:BayesTatler\]). In the remainder of this paper we will largely use PGMs to simplify the presentation and discussion of probabilistic models. We will see in brief (Section \[sec:like\]) that many current approaches previously mentioned can be accounted for by the likelihood term alone. But, crucial, and related to issues raised in Section \[sec:bias\], is the Bayesian prior $P(\mathbf{r})$. The prior, first {#sec:prior} ---------------- The prior $P(\mathbf{r})$ can be defined *prima facie* as the probability of shifting the gaze to a location *irrespective of the visual information* at that location, although the term “irrespective” should be used with some caution [@le2016introducing]. Indeed, the prior is apt to encapsulate any systematic tendency in the manner in which we explore scenes with our eyes. The striking result obtained by is that if we learn $P(\mathbf{r})$ from the actual observer’s behavior, then we can stochastically sample gaze shifts (Eq. \[eq:priorTatlersamp\]) so to obtain scan paths that, blind to visual information, out-perform feature-based accounts of eye guidance. Note that the apparent simplicity of the prior term $P(\mathbf{r})$ hides a number of subtleties. For instance, Tatler and Vincent expand the random vector $\mathbf{r}$ in terms of its components, amplitude $l$ and direction $\theta$. Thus, $P(\mathbf{r})= P(l, \theta)$. This simple statement paves the way to different options. First easy option: such RVs are marginally independent, thus, $P(l, \theta) = P(l) P(\theta)$. In this case, gaze guidance, solely relying on biases, could be simulated by expanding Eq. \[eq:priorTatlersamp\] via independent sampling of both components, i.e. at each time $t$, $l(t) \sim P(l(t)), \theta(t) \sim P(\theta(t))$. Alternative option: conjecture some kind of dependency, e.g. amplitude on direction so that $P(l, \theta) = P(l \mid \theta) P(\theta)$. In this case, the gaze shift sampling procedure would turn into the sequence $\widehat{\theta}(t) \sim P(\theta(t)), l(t) \sim P(l(t) \mid \widehat{\theta}(t) )$. Further: assume that there is some persistence in the direction of the shift, which give rise to a stochastic process in which subsequent directions are correlated, i.e., $\theta(t) \sim P(\theta(t) \mid \theta(t-1))$, and so on. To summarize, by simply taking into account the prior $P(\mathbf{r})$, a richness of possible behaviors and analyses are brought into the game. Unfortunately, most computational accounts of eye movements and visual attention have overlooked this opportunity, with some exceptions. For instance, propose a system model for saccade generation in a stochastic filtering framework. A prior on amplitude $P(l(t))$ is considered by learning a Gaussian mixture model from eye tracking data. This way one aspect of biases is indirectly taken into account. It is not clear if their model accounts for variability and whether and how oculomotor statistics compare to human data. In [@kimura2008dynamic], simple eye-movements patterns are straightforwardly incorporated as a prior of a dynamic Bayesian network to guide the sequence of eye focusing positions on videos. In a different vein, have recently addressed in-depth the bias problem and made the interesting point that viewing tendencies are not universal, but modulated by the semantic visual category of the stimulus. They learn the joint pdf $P(l, \theta)$ of saccade amplitudes and orientations via kernel density estimation; fixation duration is not taken into account. The model also brings in variability [@le2015saccadic] by generating a number $N_c$ of random locations according to conditional probability $P(\mathbf{r}_{F}(t) \mid \mathbf{r}_{F}(t-1))$ and the location with the highest saliency gain is chosen as the next fixation point. $N_c$ controls the degree of stochasticity. Others have tried to capture eye movements randomness [@keech2010eye1; @rutishauser2007probabilistic] but limiting to specific tasks such as conjunctive visual search. A few more exceptions can be found, but only in the very peculiar field of eye-movements in reading (see , for a discussion). The variability and bias issues have been explicitly addressed from first principles in the theoretical context of Lévy flights [@brockgeis; @bfpha04]. The perceptual component was limited to a minimal core (e.g., based on a bottom-up salience map) sufficient enough to support the eye guidance component. In particular in [@bfpha04] the long tail, positively skewed distribution of saccade amplitudes was shaped as a prior in the form of a Cauchy distribution, whilst randomness was addressed at the algorithmic level by prior sampling $\mathbf{r} \sim P(\mathbf{r}(t))$ followed by a Metropolis-like acceptance rule based on a deterministic saliency potential field. The degree of stochasticity was controlled via the “temperature” parameter of the Metropolis algorithm. The underlying eye guidance model was that of a random walker exploring the potential landscape (salience) according to a Langevin-like stochastic differential equation (SDE). The merit of such equation is the joint treatment of both the deterministic and the stochastic (variability) components behind eye guidance[^1]. This basic mechanism has been refined and generalized in [@BocFerAnnals2012] to composite $\alpha$-stable or Lévy random walks (the Cauchy law is but one instance of the class of $\alpha$-stable distributions), where, inspired by animal foraging behaviour, a twofold regime can be distinguished: local exploitation (fixational movements following Brownian motion) and large exploration/relocation (saccade following Lévy motion). What is interesting, with respect to the early model [@bfpha04], is that the choice between the “feed” or “fly” states is made by sampling from a Bernoulli distribution, $Bern(z \mid \pi)$, with the parameter $\pi$ sampled from the conjugate prior $Beta(\pi \mid \alpha, \beta)$. In turn, the behaviour of the Beta prior can be shaped via its hyperparameters $(\alpha, \beta)$, which, in an Empirical Bayes approximation, can be tuned as a function of the class of perceptual data at hand (in the vein of ) and of time spent in feeding (fixation duration). Most important, this approach paves the way to the possibility of treating visual exploration strategies in terms of *foraging* strategies [@wolfe2013time; @cain2012bayesian; @BocFerSMCB2013; @BocCOGN2014; @napboc_TIP2015]. We will further expand on this in Section \[sec:action\]. The unbearable lightness of the likelihood {#sec:like} ------------------------------------------ We noticed before, by inspecting Eq. \[eq:BayesTatler\] that the term $\frac{ P(\mathcal{W} \mid \mathbf{r} )} {P(\mathcal{W})} $ could be related to many models proposed in the literature. This is an optimistic view. Most of the approaches actually discard the dynamics of gaze shifts implicitly captured by the shift vector $\mathbf{r}(t)$. In practice, they are more likely to be described by a simplified version of Eq. \[eq:BayesTatler\]: $$P(\mathbf{r}_{F} \mid \mathcal{W}) = \frac{P(\mathcal{W} \mid \mathbf{r}_{F})} {P(\mathcal{W})} P(\mathbf{r}_{F}). \label{eq:BayesTatler2}$$ The difference between Eq. \[eq:BayesTatler\] and \[eq:BayesTatler2\] is subtle. The posterior $P(\mathbf{r}_{F} \mid \mathcal{W}) $ now answers the query “What is the probability of *fixating* at location $\mathbf{r}_{F}$ given visual data $\mathcal{W}$?” Further, the prior $P(\mathbf{r}_{F})$ simply accounts for the probability of spotting location $\mathbf{r}_{F}$. As a matter of fact, Eq. \[eq:BayesTatler2\] bears no dynamics. In probabilistic terms we may re-phrase this result as the outcome of an assumption of independence: $P(\mathbf{r}) = P(\mathbf{r}_{F}(t) - \mathbf{r}_{F}(t-1)) \nonumber \simeq P(\mathbf{r}_{F}(t) \mid \mathbf{r}_{F}(t-1)) = P(\mathbf{r}_{F}(t))$. To make things even clearer, let us explicitly substitute $\mathbf{r}_{F}$ with a RV $\mathbf{L}$ denoting locations in the scene, and $\mathcal{W}$ with RV $\mathbf{F}$ denoting features (whatever they may be); then, Eq. \[eq:BayesTatler2\] boils down to $$P(\mathbf{L} \mid \mathbf{F}) = \frac{P(\mathbf{F} \mid \mathbf{L})} {P(\mathbf{F})} P(\mathbf{L}). \label{eq:models}$$ The PGM underlying this inferential step is a very simple one and is represented in Figure \[fig:simple\]. A straightforward but principled use of Eq. \[eq:models\], which has been exploited by approaches that draw upon techniques borrowed from statistical machine learning [@murphy2012machine] is the following: consider $\mathbf{L}$ as a binary RV taking values in $\left[0,1\right]$ (or $\left[ -1, 1 \right]$), so that $P(\mathbf{L} = 1 \mid \mathbf{F})$ represents the probability for a pixel, a superpixel or a patch of being classified as salient. ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMsimple.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMTorralba.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMPoggio.jpg) ![An extension of the PGM by , cfr. Fig. \[fig:poggio\], adopted in [@BocCOGN2014] and [@napboc_TIP2015]. In this case $\mathcal{W}$ explicitly depends on the current gaze position $\mathbf{r}_{F}$ and goal $\mathcal{G}$.[]{data-label="fig:myPGM"}](FigPGMmy.jpg) \[Fig:models\] In the case the prior $P(\mathbf{L})$ is assumed to be uniform (no spatial bias, no preferred locations), then $ P(\mathbf{L} = 1 \mid \mathbf{F}) \simeq P( \mathbf{F} \mid \mathbf{L} = 1) $. The likelihood function $P( \mathbf{F} \mid \mathbf{L} = 1)$ can be determined in many ways; e.g., nonparametric kernel density estimation has been addressed by , who use center / surround local regression kernels for computing $\mathbf{F}$. More generally, taking into account the ratio $f(\mathbf{L})=\frac{P(\mathbf{L} = 1 \mid \mathbf{F})}{P(\mathbf{L} = 0 \mid \mathbf{F})}$ (or, commonly, the log-ratio) casts the saliency detection problem in a classification problem, in particular a discriminative one [@murphy2012machine], for which a variety of learning techniques are readily available. pioneered this approach by learning the saliency discriminant function $f(\mathbf{L})$ directly from human eye tracking data using a support vector machine (SVM). Their approach has paved the way to a relevant number of works from [@judd2009learning] – who trained a linear SVM from human fixation data using a set of low, middle and high-level features to define salient locations–, to most recent ones that wholeheartedly endorse machine learning trends. Henceforth, methods have been proposed relying on sparse representation of “feature words” (atoms) encoded in salient and non-salient dictionaries; these are either learned from local image patches [@yan2010visual; @lang2012saliency] or from eye tracking data of training images [@jiang2015image]. Graph-based learning is one other trend, from the seminal work of to (see the latter, for a brief review of this field). Crucially, for the research practice, data-driven learning methods allow to contend with large scale dynamic datasets. in the vein of and use SVM, but they remarkably exploit state-of-the art computer vision datasets (Hollywood-2 and UCF Sports) annotated with human eye movements collected under the ecological constraints of a visual action recognition task. As a general comment on (discriminative) machine learning-based methods, on the one hand it is embraceable the criticism by , who surmise that these techniques make “models data-dependent, thus influencing fair model comparison, slow, and to some extent, black-box.” But on the other hand, one important lesson of these approaches lies in that they provides a data-driven way of deriving the most relevant visual features as optimal predictors. The learned patterns can shape receptive fields (filters) that have equivalent or superior predictive power when compared against hand-crafted (and sometimes more complicated) models [@kienzle2009center]. Certainly, this lesson is at the base of the current exponentially growth of methods based on deep learning techniques [@lecun2015deep], in particular Convolutional Neural Networks (CNN, cfr. for a focused review), where the computed features seem to outperform, at least from an engineering perspective, most of, if not all, the state-of-the art features conceived in computer vision. Again, CNNs, as commonly exploited in the current practice, bring no significant conceptual novelty as to the use of Eq. \[eq:models\]: fixation prediction is formulated as a supervised binary classification problem (in some case, regression is addressed, ). For example, use a linear SVM for learning the saliency discriminant function $f(\mathbf{L})$ after a large-scale search for optimal features $\mathbf{F}$. Similarly, detect salient region via linear SVM fed with features computed from multi-layer sparse network model. [@lin2014saliency] use the simple normalization step [@IttiKoch98] to approximate $P(\mathbf{L} = 1 \mid \mathbf{F})$, where [@kruthiventi2015deepfix] use the last $1 \times 1$ convolutional layer of a fully convolutional net. Cogent here is the outstanding performance of CNN in learning and representing features that correlate well with eye fixations, like objects, faces, context. Clearly, one problem is the enormous amount of training data necessary to train these networks, and the engineering expertise required, which makes them difficult to apply for predicting saliency. However, by exploiting the well known network from [@AlexNetNIPS2012] as starting point, have given evidence that deep CNN trained on computer vision tasks like object detection boost saliency prediction. The network by has been optimized for object recognition using a massive dataset consisting of more than one million images, and results reported by on static pictures are impressive when compared to state-of-the-art methods, even to previous CNN-based proposals [@vig2014large]. Apart from the straightforward implementation via popular machine-learning algorithms, the “light” model described by Eq. \[eq:models\] is further amenable to a minimal model, which, surprisingly enough, is however capable of accounting for a large number of approaches. This can be easily appreciated by setting $P(\mathbf{F} \mid \mathbf{L}) = const., P(\mathbf{L})=const.$ so that Eq. \[eq:models\] reduces to $$P(\mathbf{L} \mid \mathbf{F}) \propto \frac{1}{P(\mathbf{F})}. \label{eq:modelItti}$$ Eq. \[eq:modelItti\] states that the probability of fixating a spatial location $\mathbf{L}= (x,y)$ is higher when “unlikely” features (unlikeliness $ \approx \frac{1} {P(\mathbf{F})}$) occur at that location. In a natural scene, it is typically the case of high contrast regions (with respect to either luminance, color, texture or motion). This is nothing but the salience-based component of the most prominent model in the literature [@IttiKoch98], which Eq. \[eq:modelItti\] re-phrases in probabilistic terms. A thorough reading of the review by is sufficient to gain the understanding that a great deal of computational models so far proposed (47 over 63 models) are much or less variations of this theme (albeit experimenting with different features, different weights for combining them, etc.) even when sophisticated probabilistic techniques are adopted to shape the distribution $P(\mathbf{F})$ (e.g., nonparametric Bayes techniques, ). Clearly, there are works that have tried to avoid weaknesses related to such a light-modelling of the perceptual input, and have tried to climb up the levels of the representation hierarchy [@schutz2011eye]. Some examples are summarized at a glance in Figure \[Fig:models\] (but see ). Nevertheless, in spite of its simplicity, Eq. \[eq:modelItti\] is apt to pave the way to interesting frameworks. For instance, by noting that $\log \frac{1} {P(\mathbf{F})}$ is nothing but Shannon’s Self- Information, information theoretic approaches become available at the algorithmic level. These approaches set computational constraints under the general assumption that saliency computation serves to maximize information sampled from the environment [@bruce2009saliency]. Keeping on with the information theory framework, and going back to Eq. \[eq:models\], a simple manipulation, $$\log P(\mathbf{L} \mid \mathbf{F}) - \log P(\mathbf{L}) = \log \frac{P(\mathbf{F} \mid \mathbf{L})} {P(\mathbf{F})} \label{eq:modelsKL}$$ sets the focus on the discrepancy, or dissimilarity, $\log P(\mathbf{L} \mid \mathbf{F}) - \log P(\mathbf{L}) = \log \frac{P(\mathbf{L} \mid \mathbf{F})}{P(\mathbf{L})}$ between the log-posterior and the log-prior. A (non-commutative) measure, formalizing this notion of dissimilarity is readily available in information theory, namely the Kullback-Leibler (K-L) divergence between two distributions $P(X)$ and $Q(X)$ [@Mackay]: $$D_{KL}(P(X) || Q(X))= \int_X \log \frac{P(x)} {Q(x)} P(x) dx \label{eq:KL}$$ Measuring differences between posterior and prior beliefs of the observers is however a general concept applicable across different levels of abstraction. For instance, one might consider the object-based model [@torralba2006contextual] in Fig. \[fig:torralba\], which can be used for inferring the joint posterior of gazing at certain kinds of objects $\mathbf{O}$ at location $\mathbf{L}$ of a viewed scene, namely, $P(\mathbf{O},\mathbf{L} \mid \mathbf{F}) \propto P(\mathbf{F} \mid \mathbf{O},\mathbf{L}) P(\mathbf{O},\mathbf{L})$. Then, $D_{KL}(P(\mathbf{O},\mathbf{L} \mid \mathbf{F}) || P(\mathbf{O},\mathbf{L}))$ is the average of the log-odd ratio, measuring the divergence between observer’s prior belief distribution on $(\mathbf{O},\mathbf{L})$ and his posterior belief distributions after perceptual data $\mathbf{F}$ have been gathered. Indeed, this is a statement that can be generalized to any model $\mathbf{M}$ in a model space $\mathcal{M}$ and new data observation $\mathbf{D}$ so to define the Bayesian surprise [@balditti2010] $\mathcal{S}(\mathbf{D},\mathbf{M})$: $\mathbf{D}$ is surprising if the posterior distribution resulting from observing $\mathbf{D}$ significantly differs from the prior distribution, i.e., $S(\mathbf{D},\mathbf{M}) = D_{KL}(P(\mathbf{M} \mid \mathbf{D} ) || P(\mathbf{M}))$. have shown that Bayesian surprise attracts human attention in dynamic natural scenes. To recap, Bayesian surprise is a measure of salience based on the K–L divergence. Eventually, note that the K-L divergence (\[eq:KL\]) is a flexible tool and can be used for different purposes. For instance, when dealing with models of perceptual evaluation such as those specified in Figs \[fig:torralba\], \[fig:poggio\], and \[fig:myPGM\], once the model has been detailed at the computational theory level via its PGM, then using the latter for learning inference and prediction brings in the algorithmic level. Indeed, for any Bayesian generative model other than trivial ones, such steps are usually performed in approximate form [@murphy2012machine]. Stochastic approximation resorting to algorithms such as Markov-chain Monte Carlo (MCMC) and Particle Filtering (PF) is one possible choice; the alternative choice is represented by deterministic optimization algorithms [@murphy2012machine] such as variational Bayes (VB) or belief propagation (BP, a message passing scheme exchanging beliefs between PGM nodes). For example, the model by , following the work of , relies upon BP message passing for inferential steps. Interestingly enough, has argued for a plausible neural implementation of BP. VB algorithms, on the other hand, are based on Eq. \[eq:KL\], where $P$ usually stands for a complete distribution and $Q$ is the approximating distribution; then, parameter (or model) learning is accomplished by minimizing the K-L divergence (as an example, the well known Expectation-Maximization algorithm, EM, can be considered a specific case of the VB algorithm, ). In Section \[sec:action\] we will also touch on a deeper interpretation of the K-L minimization / VB algorithm. But at this point a simple question arises: where have the eye movements gone? Making a step forward: back to the beginning of active vision {#sec:action} ============================================================= Visual perception coupled with gaze shifts should be considered the *Drosophila* of perception-action loops. Among the variety of active behaviors the organism can fluently engage to purposively act upon and perceive the world (e.g, moving the body, turning the head, manipulating objects), oculomotor behavior is the minimal, least energy, unit. To perform $3$-$4$ saccades per second, the organism roughly spends $300$ msecs to close the loop ($200$ msecs for motor preparation and execution, $100$ msecs left for perception). ![The perception-action loop unfolded in time as a dynamic PGM. $\mathcal{A}(t)$ denotes the ensemble of time-varying RVs defining the oculomotor action setting; $\mathcal{W}(t)$ stands for the ensemble of time-varying RVs characterising the scene as actively perceived by the observer; $\mathcal{G}$ summarizes the given goal(s), To simplify the graphics, conditional dependencies $\mathcal{G} \rightarrow \mathcal{A}(t+1)$ and $\mathcal{G} \rightarrow \mathcal{W}(t+1)$ have been omitted.[]{data-label="Fig:loop"}](FigLoop.jpg) One way to make justice of this forgotten link is going back to first principles by re-shaping the problem as the action-perception loop, which is presented in Figure \[Fig:loop\] in the form of a dynamic PGM. The model relies upon the following assumptions: - The scene that will be perceived at time $t+1$, namely $\mathcal{W}(t+1)$ is inferred from the raw data $\mathbf{I}$, gazed at $\mathbf{r}_{F}(t+1)$, under the goal $\mathcal{G}$ assigned to the observer, and is conditionally dependent on current perception $\mathcal{W}(t)$. Thus, the perceptual inference problem is summarised by the conditional distribution $P( \mathcal{W}(t+1)|\mathcal{W}(t), \mathbf{r}_{F}(t+1), \mathbf{I},\mathcal{G})$; - The external goal $\mathcal{G}$ being assigned, the oculomotor action setting at time $t+1$, $\mathcal{A}(t+1)$, is drawn conditionally on current action setting $\mathcal{A}(t)$ and the perceived scene $\mathcal{W}(t+1)$ under gaze position $\mathbf{r}_{F}(t+1)$; thus, its evolution in time is inferred according to the conditional distribution $P(\mathcal{A}(t+1) | \mathcal{A}(t), \mathcal{W}(t+1), \mathbf{r}_{F}(t+1), \mathcal{G})$. Note that the action setting dynamics $\mathcal{A}(t) \rightarrow \mathcal{A}(t+1)$ and the scene perception dynamics $\mathcal{W}(t) \rightarrow \mathcal{W}(t+1)$ are intertwined with one another by means of the gaze shift process $\mathbf{r}_{F}(t) \rightarrow \mathbf{r}_{F}(t+1)$: on the one hand next gaze position $\mathbf{r}_{F}(t+1)$ is used to define a distribution on $\mathcal{W}(t+1)$ and $\mathcal{A}(t+1)$; meanwhile, the probability distribution of $\mathbf{r}_{F}(t+1)$ is conditioned on current gaze position, $\mathcal{W}(t)$ and $\mathcal{A}(t)$, namely $P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t), \mathcal{W}(t), \mathbf{r}_{F}(t))$. We have previously discussed the perceptual evaluation component $\mathcal{W}(t)$. A general way of defining the oculomotor executive control component $\mathcal{A}(t)$ is through the following ensemble of RVs: - $\{\mathbf{V}(t), \mathbf{R}(t)\}$: $\mathbf{V}(t)$ is a spatially defined RV used to provide a suitable probabilistic representation of value; $\mathbf{R}(t)$ is a binary RV defining whether or not a payoff (either positive or negative) is returned; - $\{\pi(t), z(t), \xi(t) \}$: an *oculomotor state representation* as defined via the multinomial RV $z(t)$, occurring with probability $\pi(t)$, and determining the choice of motor parameters $\xi(z,t)$ guiding the actual gaze relocation (e.g., lenght and direction of a saccade as opposed to those driving a smooth pursuit) ; - $\mathcal{D}(t)$: a set of state-dependent statistical decision rules to be applied on a set of candidate new gaze locations $\mathbf{r}_{new}(t+1)$ distributed according to the posterior pdf of $\mathbf{r}_{F}(t+1)$. In the end, the actual shift can be summarised as the statistical decision of selecting a particular gaze location $\mathbf{r}^{\star}_{F}(t+1)$ on the basis of $P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t), \mathcal{W}(t), \mathbf{r}_{F}(t))$ so to maximize the expected payoff under the current goal $\mathcal{G}$, and the action/perception cycle boils down to the iteration of the following steps: 1. Sampling the gaze-dependent current perception: $$\mathcal{W}^{*}(t) \sim P( \mathcal{W}(t)|\mathbf{r}_{F}(t),\mathbf{F}(t), \mathbf{I}(t),\mathcal{G}); \label{eq:step1}$$ 2. Sampling the appropriate motor behavior (e.g., fixation or saccade): $$\mathcal{A}(t)^{*} \sim P(\mathcal{A}(t) | \mathcal{A}(t-1), \mathcal{W}^{*}(t),\mathcal{G}); \label{eq:step2}$$ 3. Sampling where to look next: $$\mathbf{r}_{F}(t+1) \sim P(\mathbf{r}_{F}(t+1)| \mathcal{A}(t)^{*} , \mathcal{W}^{*}(t), \mathbf{r}_{F}(t)). \label{eq:step3}$$ It is worth noticing that we have chosen to describe the observer’s action / perception cycle in terms of stochastic sampling based on the probabilistic model in Figure \[Fig:loop\]. However, one can recast the inferential problems in terms of deterministic optimization: in brief, optimising the probabilistic model $\mathcal{W}$ of how sensations are caused, so that the resulting predictions can select the optimal $\mathcal{A}$ to guide active sampling (gaze shift) of sensory data. One such approach, which is well known in theoretical neuroscience but, surprisingly, hitherto unconsidered in computer vision, relies on the free-energy principle [@friston2010free; @feldman2010attention; @friston2013anatomy]. Free-energy $\mathcal{F}$ is a quantity from statistical physics and information theory [@Mackay] that bounds the negative log-evidence of sensory data. Under simplifying assumptions, it boils down to the amount of prediction error of sensory data under a model. In such context, the action / perception cycle is the result of a dual minimization process: i) action reduces $\mathcal{F}$ by changing sensory input, namely by sampling (via gaze shifts) what one expects consistent with perceptual inferences; ii) perception reduces $\mathcal{F}$ by making inferences about the causes of sampled sensory signals and changing predictions. Friston defines this process “active inference”. To make a connection with Section \[sec:like\], by minimizing the free-energy, Bayesian surprise is maximised; indeed, in Bayesian learning, free energy minimization is a common rationale behind many optimisation techniques such as VB and BP [@Mackay; @murphy2012machine] The sampling scheme proposed is a general one and can be instantiated in different ways. For instance, in [@BocFerSMCB2013] the sampling step of Eq. \[eq:step3\] is performed through a generalization of the Langevin SDE equation used in [@BocFerAnnals2012]. Biases and variability are accounted for by the stochastic component of the equation. Since dealing with image sequences, the SDE provides operates in different dynamic modes: pursuit needs to be taken also into account in addition to saccades and fixational movements. Each mode is governed by a specific set of parameters of the $\alpha$-stable distribution estimated from eye tracking data. The choice among modes is accomplished by generalizing the method proposed in [@BocFerSMCB2013], using a Multinoulli distribution on $z(t)$ and with parameters $\pi(t)$ sampled from the conjugate prior, the Dirichlet distribution. In addition, the sampling step is used to accomplish an internal simulation step, where a number of candidates shifts is proposed and the most convenient is selected according to a decision rule. Also, the gaze dependent perception $\mathcal{W}$ can be modeled at any level of complexity (cfr. example in Figure \[Fig:levels\], Section \[sec:emo\]). In it is based on proto-objects sampled from time-varying saliency[^2]. Figure \[Fig:monica\] shows an excerpt of typical results of the model simulation, which compares human variability in gazing (top row) with that of two “simulated observers” (bottom rows) while viewing the `monica03` clip from the CRCNS eye-1 dataset. ![Top row: gaze positions (centers of the enhanced colored circles ) recorded from different observers on different frames of the `monica03` clip from the CRCNS eye-1 dataset, University of South California (freely available online). Middle and bottom rows show, on the same frames, the fovea position of two “observers” simulated by the method described in [@BocFerSMCB2013].[]{data-label="Fig:monica"}](Figmonica03.jpg) In and the perceptual evaluation component is extended to handle objects and task (external goal) levels by expanding on (cfr. Figure \[Fig:models\]), and the decision rule $\mathcal{D}(t)$ concerning the selection of the gaze is based on the expected reward according to the given goal $\mathcal{G}$ (see Section \[sec:emo\]). Meanwhile, nothing prevents to conceive more general perceptual evaluation and executive control components, by considering perceptual and action modalities other than the visual ones in the vein of , where eye movements and hand actions have been coupled with the goal of performing a drawing task. But most important, the action-perception cycle is by and large conceived in the foraging framework (see , for a thorough introduction), which at the most general level is summarized in Table \[metaphor\]. Visual foraging corresponds to the time-varying overt deployment of visual attention achieved through oculomotor actions, namely, gaze shifts. The forager feeds on patchily distributed preys or resources, spends its time traveling between patches or searching and handling food within patches. While searching, it gradually depletes the food, hence, the benefit of staying in the patch is likely to gradually diminish with time. Moment to moment, striving to maximize its foraging efficiency and energy intake, the forager should make decisions: Which is the best patch to search? Which prey, if any, should be chased within the patch? When to leave the current patch for a richer one? The spatial behavioral patterns exhibited by foraging animals (but also those detected in human mobility data) are remarkably close to those generated by gaze shifts [@viswanathan2011physics]. Figure \[Fig:spider\] presents an intriguing example in this respect. ![Monkey or human: can you tell the difference? The center image has been obtained by superimposing a typical trajectory of spider monkeys foraging in the forest of the Mexican Yucatan, as derived from [@ramos2004levy], on the “party picture” (left image) used in [@brockgeis]. The right image is an actual human scan path (modified after ) []{data-label="Fig:spider"}](FigParty.jpg) The fact that the physics underlying foraging overlaps with that of several other kinds of complex random searches and stochastic optimization problems [@viswanathan2011physics], and notably with that of visual exploration via gaze shifts [@viswanathan2011physics; @marlow2015temporal; @brockgeis], makes available a variety of analytical tools beyond classic metrics exploited in computer vision or psychology. For instance, in Figure \[Fig:ccdf\], it is shown how the gaze shift amplitude modes from human observers can be compared with those generated via simulation by using the complementary Cumulative Distribution Function (CCDF), which provides a precise description of the distribution of the gaze shift by considering its upper tail behavior. This can be defined as $\overline{F}(x)=P(| \mathbf{r}|>x)=1 - F(x)$, where $F$ is the cumulative distribution function (CDF) of amplitudes. Consideration of the upper tail, i.e. the CCDF of jump lengths is a standard convention in foraging, human mobility, and anomalous diffusion research [@viswanathan2011physics]. To sum up, foraging offers a novel perspective for formulating models and related evaluations of visual attention and oculomotor behavior [@wolfe2013time]. Unifying hypotheses such as the oculomotor continuum from exploration to fixation by can be reconsidered in the light of fundamental theorems of statistical mechanics [@weron2010generalization]. \[metaphor\] Interestingly enough, the reformulation of visual attention in terms of foraging theory is not simply an informing metaphor. It has been argued that what was once foraging for tangible resources in a physical space became, over evolutionary time, foraging in cognitive space for information related to those resources [@hills2006animal], and such adaptations play a fundamental role in goal-directed deployment of visual attention [@wolfe2013time]. Bringing value into the game: a doorway to affective modulation {#sec:emo} =============================================================== The introduction of a goal level, either exogenous (originating from outside the observer’s organism) or endogenous (internal) is not an innocent shift. From a classical cognitive perspective, the assignment of a task to the observer implicitly defines a value for every point of the space, in the sense that information in some points is more relevant than in others for the completion of the task; the shifting of the gaze on a particular point, in turn, determines the payoff that can be gained. There is a number of psychological and neurobiological studies showing the availability of value maps and loci of reward influencing the final gaze shift [@platt1999neural; @leon1999effect; @ikeda2003reward; @Hikosaka2006]. The payoff is nothing else that the value, with respect to the completion of the task, obtained by moving the fovea in a given position. Thus points associated with high values produce, when fixated, high payoffs since these fixations bring the observer closer to her/his goal. For instance, in [@BocCOGN2014] and in [@napboc_TIP2015], reward was introduced to make a choice among the candidate gaze shifts stochastically sampled according to Eq. \[eq:step3\], in terms of expected reward (e.g., tuned by the probability of finding the task-assigned object). Figure \[Fig:levels\] presents one example of the different scan paths obtained by progressively reducing the levels of representation in the perceptual evaluation component presented in Figure \[fig:myPGM\] [@BocCOGN2014; @napboc_TIP2015]. ![Different scan paths originated from the progressive reduction of representation levels in the perceptual evaluation component presented in Figure \[fig:myPGM\].[]{data-label="Fig:levels"}](FigLevels.jpg) Value set by the given goal can weight differently the objects within the scene, thus purposively biasing the scan path. Gaze is still uniformly deployed to relevant items within the scene (people, text) when object-based representation is exploited. When salience alone is used, the generated scan path fails in accounting for the relevant items and bears no relation with the semantics which can be attributed to the scene. The use of value and reward endow attentive models with the capability of handling complex task. For instance, considered the goal of text spotting in unconstrained urban environments, and its validation encompassed data gathered from a mobile eye tracking device [@clav2014]. extended the foraging framework to cope with the difficult problem of attentive monitoring of multiple video streams in a videosurveillance setting. Yet, developing eye guidance models based on reward is a difficult endeavour and computational models that use reward and uncertainty as central components are still in their infancy (but see the discussion by ). In this respect the remarkable work by Ballard and collegues counters the stream. Whilst salience, proto-objects and objects are representations that have been largely addressed in the context of human eye movements, albeit with different emphasis, in contrast, value has been neglected until recently [@schutz2011eye]. One reason is that in the real world there is seldom direct payoff (no orange juice for a primary reward) for making good eye movements or punishment for bad ones. However, the high attentional priority of ecologically pertinent stimuli can also be explained by mechanisms that do not implicate learning value through repeated pairings with reward. For example, a bias to attend to socially relevant stimuli is evident from infancy [@anderson2013value]. More generally, the selection of stimuli by attention has important implications for the survival and wellbeing of an organism, and attentional priority reflects the overall value of such selection (see for a discussion). Indeed, engaging attention with potentially harmful and beneficial stimuli guarantees that the relevant ones are selected early so to gauge the exact nature of the potential threat or opportunity and to readily initiate defensive or approach behavior. Under these circumstances, has proposed a broad definition of reward, which includes “not only the immediate primary rewards, but also other factors: the preference for a novel location or stimulus, the satisfaction of performing well or the desire to complete a given task." Such definition is consistent with the different psychological facets of reward [@berridge2003parsing]: i) learning (including explicit and implicit knowledge produced by associative conditioning and cognitive processes); ii) affect or emotion (implicit “liking” and conscious pleasure); iii) motivation (implicit incentive salience “wanting” and cognitive incentive goals). Thus, value representation level is central to both goal-driven affective and cognitive engagement with stimuli in the outside world. In this broader perspective, the effort to put value and reward into the game shows his inner worth in that, by accounting for the many aspects of “biological value" - salience, significance, unpredictability, affective content - , it paves the way to a wider dimension of information processing, as most recent results on the affective modulation of the visual processing stream advocate [@pessoa2008relationship; @pessoa2010emotion], and to the effective exploitation of computational attention models in the emerging domain of social signal processing [@vinciarelli2009social]. A number of important studies in the psychological literature (see, for a discussion, ) have addressed the relationship between overt attention behavior and emotional content of pictures. Many of them investigate specific issues related to individuals such as trait anxiety, social anxiety, spider phobia, and exploit restricted sets of stimuli such as emotional faces or spiders. In turn, the study by has exploited natural images and normal subjects, demonstrating an emotional bias both in attentional orienting and engagement among normal participants and using a wider range of emotional pictures. Results might be summarised as follows: i) emotionally pleasant and unpleasant pictures capture attention more readily than neutral pictures; ii) the emotional bias can be observed early in initial orienting and subsequent engagement of attention; iii) the early stimulus-driven attentional capture by emotional stimuli can be counteracted by goal-driven control in later stages of picture processing. Peculiarly relevant to our case, have shown that visual saliency does influence eye movements, but the effect is reliably reduced when an emotional object is present. Pictures containing negative objects were recognized more accurately and recalled in greater detail, and participants fixated more on negative objects than positive or neutral ones. Initial fixations were more likely to be on emotional objects than more visually salient neutral ones. Consistently with , the overall result suggest that the processing of emotional features occurs at a very early stage of perception. As a matter of fact, emotional factors are completely neglected in the realm of computational models of attention and gaze shifts. Some efforts have been spent in the field of social robotics, where motivational drives have an indirect influence on attention by influencing the behavioral context, which, in turn, is used to directly manipulate the gains of the attention system (e.g. by tuning the gains of different bottom-up saliency map, ). Yet, beyond these broadly related attempts, taking into account the specific influence of affect on eye-behaviour is not a central concern within this field (for a wide review, see ). Recent works in the image processing and pattern recognition community use eye tracking data for the inverse problem of the recognition of emotional content of images (e.g.,) or implicit tagging of videos [@soleymani2012multimodal]. Thus, they do not address the generative problem of how emotional factors contribute to the generation of gaze shifts in visual tasks. By contrast, neuroscience has shown that, crucially, cognitive and emotional contributions cannot be separated, as outlined in Figure \[Fig:neuro\]. ![Circuits for the processing of visual information and for the executive control [@pessoa2008relationship] show that action and perception components are inextricably linked through the mediation of the amygdala and the posterior orbitofrontal cortex (OFC). These structures are considered to be the neural substrate of emotional valence and arousal [@salzman2010emotion]. The ventral tegmental area (vTA) and the basal forebrain are responsible for diffuse neuromodulatory effects related to reward and attention. Anterior cingulate cortex (ACC) is likely to be involved in conflict detection and/or error monitoring but also in in computing the benefits and costs of acting by encoding the probability of reward. Nucleus accumbens is usually related to motivation.The lateral prefrontal cortex (LPFC) plays a central role at the cognitive level in maintaining and manipulating information, but also integrates this content with both affective and motivational information. Line thickness depicts approximate connection strength[]{data-label="Fig:neuro"}](FigNeuro.jpg) The scheme presented, which summarises an ongoing debate [@pessoa2008relationship], shows that responses from early and late visual cortex reflecting stimulus significance will be a result of simultaneous top-down modulation from fronto-parietal attentional regions (LPFC) and emotional modulation from the amygdala [@Mohanty2013]. On the one hand, stimulus’ affective value appears to drive attention and enhance the processing of emotionally modulated information. On the other hand, exogenously driven attention influences the outcome of affectively significant stimuli [@pessoa2008relationship]. As a prominent result, the cognitive or affective origin of the modulation is lost and stimulus’ effect on behaviour is both cognitive and emotional. At the same time, the cognitive control system (LPFC, ACC) guides behaviour while maintaining and manipulating goal-related information; however strategies for action dynamically incorporate value through the mediation of the nucleus accumbens, the amygdala, and the OFC. Eventually, basal forebrain cholinergic neurons provide regulation of arousal and attention [@goard2009basal], while dopamine neurons located in the vTA modulate the prediction and expectation of future rewards [@pessoa2008relationship]. It is to be noted in Figure \[Fig:neuro\] the central role of the amygdala and the OFC. It has been argued [@salzman2010emotion] that their tight interaction provides a suitable ground for representing, at the psychological level the core affect dimensions [@russell2003core] of valence (pleasure–displeasure conveyed by the visual stimuli) and arousal (activation–deactivation). From a computational standpoint, the observer’s core affect can in principle be modelled as a dynamic latent space [@vitale2014affective], which we surmise might be readily embedded within the loop as proposed in Fig. \[Fig:newloop\]. This way, gaze shifts would benefit from the crucial emotional mediation between the action control and perceptual evaluation components. ![The perception-action loop unfolded in time as a dynamic PGM as in Figure \[Fig:loop\], but integrated with the core affect latent dimension abstracted as the RV $\mathcal{E}(t)$. $\mathcal{C}(t)$ stands for a higher cognitive control RV. To simplify the graphics, conditional dependencies $\mathcal{G} \rightarrow \mathcal{A}(t+1)$ and $\mathcal{G} \rightarrow \mathcal{W}(t+1)$ have been omitted.[]{data-label="Fig:newloop"}](FigAffectLoop2.jpg) Conclusion ========== Time is ripe to abandon the marshland of mass production and evaluation of bottom-up saliency techniques. Such boundless effort is partially based on a fatally flawed assumption [@santini2008context]: that visual data have a meaning *per se*, which can be derived as a function of a certain representation of the data themselves. Meaning is an outcome of an interpretative process rather than a property of the viewed scene. It is the act of perceiving, contextual and situated, that gives a scene its meaning [@wittgenstein2010philosophical]. The course of modelling can be more fruitfully directed not only to climb the hierarchy of representation levels and to cope with overlooked aspects of eye guidance, but to eventually reappraise the observer within his natural setting: an active observer compelled to purposively exploit visual attention for accomplishing real-world tasks [@maurice1945phenomenologie]. [^1]: Matlab simulation is available for download at <http://www.mathworks.com/matlabcentral/fileexchange/38512-visual-scanpaths-via-constrained-levy-exploration-of-a-saliency-landscape> [^2]: Matlab simulation is available for download at <https://www.researchgate.net/publication/290816849_Ecological_sampling_of_gaze_shifts_Matlab_code>

ArXiv

--- abstract: 'Within the framework of Ginzburg-Landau theory we study the rich variety of interfacial phase transitions in twinning-plane superconductors. We show that the phase behaviour strongly depends on the transparency of the twinning plane for electrons measured by means of the coupling parameter $\alpha_{{\rm TP}}$. By analyzing the solutions of the Ginzburg-Landau equations in the limit of perfectly transparent twinning planes, we predict a first-order interface delocalization transition for all type-I materials. We further perform a detailed study of the other limit in which the twinning plane is opaque. The phase diagram proves to be very rich and fundamentally different from the transparent case, recovering many of the results for a system with an external surface. In particular both first-order and critical delocalization transitions are found to be possible, accompanied by a first-order depinning transition. We provide a comparison with experimental results and discuss the relevance of our findings for type-II materials.' author: - 'F. Clarysse' - 'J.O. Indekeu' title: 'Interfacial phase transitions in twinning-plane superconductors' --- [^1] Introduction {#sec:intro} ============ In recent years, local enhancement of superconductivity has been predicted to provide the mechanism to induce several intriguing interfacial phase transitions in type-I superconductors [@IND; @IND1; @CJB; @BAC; @MON]. Typical phase diagrams are calculated using the Ginzburg-Landau (GL) theory in which the enhancement is accounted for by allowing the extrapolation length $b$ to be negative. The microscopic origin of this parameter remains an unsolved problem making the experimental verification of the theoretical results non-trivial. So far, the most feasible realization of a negative extrapolation length seems to originate from the concept of twinning-plane superconductivity (TPS), a well understood phenomenon that occurs, e.g., in Sn, In, Nb, Re and Tl [@BUZ]. A twinning plane (TP) is a defect plane representing the boundary between two single-crystal regions or twins and, consequently, the physics encountered in the behaviour of a superconducting/normal interface near TP’s is the natural analogy to grain-boundary wetting or interface depinning, a topic which has been well studied in magnetic systems [@ABR; @SEV; @IGL]. The characteristic feature of the original GL approach of TPS is the a priori assumption that the TP is perfectly transparent for electrons at the microscopic level which implies that the superconducting order parameter $\psi$ is continuous at the TP [@BUZ]. Subsequent extensions of the theory relax this assumption allowing a discontinuity in $\psi$ [@AND; @GES; @MIN; @SAM]. More specifically, a second phenomenological parameter, $\alpha_{{\rm TP}}$, is introduced to describe the coupling between the twins such that, by means of $\alpha_{{\rm TP}}$, one can mimic the effect of microscopically tuning the TP from completely transparent to completely opaque for electrons. In this paper we present an overview of the variety of interfacial phase transitions in the two limiting cases to develop a thorough understanding of the influence of the transparency. Earlier work [@BAC] has focused on the case of mixed bulk boundary conditions with the bulk normal (N) phase on one side and the bulk superconducting (SC) phase on the other side of the TP, at bulk two-phase coexistence. This is appropriate for the study of the proper depinning transition of an interface that is initially pinned at the TP. Here we choose to settle for the configuration of equal bulk conditions, that is, we impose the bulk N phase on both sides of the TP. In so doing we are no longer restricted to the case of bulk two-phase coexistence and this allows us to establish the complete magnetic field versus temperature phase diagram for a given material. This type of diagram is accessible to experimental verification and is relevant for comparing the present results with known TPS phase diagrams [@BUZ; @MIN; @SAM]. The solutions of the GL equations strongly depend on the boundary conditions imposed at the TP itself which in turn relates to the level of transparency, i.e. the value of $\alpha_{{\rm TP}}$. For highly transparent planes, corresponding to the limit $\alpha_{{\rm TP}} \rightarrow 0$, it is natural to consider fully symmetric profiles for the order parameter. In the opposite limit, $\alpha_{{\rm TP}} \rightarrow \infty$, the TP is completely opaque for electrons and both sides are largely independent. In this case there is a wide range of possible solutions, including profiles with $\psi$ identically zero at one side of the TP. The latter are refered to as *wall* solutions, since they are equivalent to the ones found in a type-I superconductor with an external surface or wall characterized by a negative extrapolation length $b$ [@IND; @IND1]. Therefore we anticipate that, in the opaque limit, we will recover to a great extent the results of a wall system. This is very different from the case of complete transparency, for which drastic qualitative modifications are predicted compared to the case with a wall. The outline of the paper is as follows. In the next section we collect the main ideas of the GL theory applied to twinning-plane superconductors. Section \[sec:transp\] covers the results for perfectly transparent TP’s. We calculate in detail the phase diagrams and provide a comparison with the predicted TPS diagrams as described in Ref. [@BUZ]. The fully opaque system is the subject of Section \[sec:opaque\]. We present a classification of the various solutions and establish their stability to derive the phase behaviour. We summarize our main results and discuss the experimental relevance in Section \[sec:conc\]. Ginzburg-Landau theory for twinning-plane superconductors {#sec:gl} ========================================================= We consider a type-I superconductor with a TP located at $x=0$ and impose on both sides the N phase, with $\psi =0$, as the bulk condition. The GL free-energy functional has the form $$\Gamma[\psi,{\bf A}]=\int_{-\infty}^{+\infty}{\cal G}[\psi,{\bf A}]{\rm d}x+ \Gamma_{\rm TP}(\psi_-,\psi_+), \label{eq:gammatp1}$$ with the free-energy density ${\cal G}$ given by $${\cal G}= \epsilon|\psi|^2+\frac{\beta}{2}|\psi|^4+\frac{1}{2m}\left| \left( \frac{\hbar}{i}{\bf \nabla}-2e{\bf A} \right) \psi\right|^2+\frac{[{\bf \nabla}\times{\bf A} - \mu_0{\bf H}]^2}{2\mu_0}. \label{eq:enedens}$$ As usual, $\epsilon \propto T-T_c$, where $T_c$ is the bulk critical temperature which must be distinguished from the second critical temperature in the system, $T_{{\rm c, TP}}$, below which local superconductivity sets in at the TP in zero magnetic field. Since $T_{{\rm c,TP}}$ was experimentally [@BUZ] proved to be only slightly higher than $T_c$, the use of the GL theory is justified. Further, $\beta >0$ is a stabilizing parameter and ${\bf A}$ is the vector potential. We choose the applied magnetic field ${\bf H}={\rm H}{\bf e}_z$ parallel to the TP. Using the notation $\psi_-\equiv \psi(0^-)$ and $\psi_+ \equiv \psi(0^+)$, the local contribution $\Gamma_{\rm TP}$ in (\[eq:gammatp1\]) reads $$\Gamma_{\rm TP}(\psi_-,\psi_+)=\frac{\hbar^2}{2mb}(|\psi_+|^2+|\psi_-|^2)+ \frac{\hbar^2}{2m\alpha_{{\rm TP}}}\left|\psi_+ - \psi_-\right|^2. \label{eq:localenetp}$$ The first term, with $b<0$, describes the enhancement of superconductivity and was introduced by Khlyustikov and Buzdin [@BUZ] to reproduce theoretically the observed TPS phase diagrams. The phenomenological parameter $b$ is the extrapolation length and can be related to the temperature difference $T_c-T_{{\rm c,TP}}$. In addition, we have followed others [@BAC; @AND; @GES; @MIN; @SAM] by adding a second term in (\[eq:localenetp\]) to describe the coupling between the twins. In so doing, we allow the SC wave function to be discontinuous across the TP, hence in general $\psi_- \neq \psi_+$. The coupling constant $\alpha_{{\rm TP}}$ can be expressed in terms of the Fermi velocity and either the transmission or reflection coefficient for electrons, thus fully in terms of microscopic properties [@GES]. We note that for $\alpha_{{\rm TP}}> 0$, the phase of the wave function is continuous at the TP, while for $\alpha_{{\rm TP}}<0$ a phase jump of $\pi$ can occur [@AND]. We omit the latter possibility and restrict our attention to $\alpha_{{\rm TP}} >0$. In what follows we assume translational invariance in the $y$- and $z$-directions and choose the gauge so that ${\bf A}=(0,A(x),0)$. It proves to be convenient to adopt the rescaling introduced in earlier work [@IND1] using the two basic length scales of the superconductor, i.e. the zero-field coherence length $\xi$ and the magnetic penetration depth $\lambda$ defined by $$\xi^2=\frac{\hbar^2}{2m|\epsilon|} \ \ \ , \ \ \ \lambda^2=\frac{m\beta}{\mu_0q^2|\epsilon|}. \label{eq:xilam}$$ The ratio of $\lambda$ to $\xi$ gives the GL parameter $\kappa$, with $\kappa< 1/ \sqrt2$ for type-I materials. We use $\xi$ to scale the distances but, for simplicity, retain the notation $x$ for the dimensionless coordinate $x/\xi$ perpendicular to the TP. For the magnetic quantities $A$ and $H$ we introduce the dimensionless $a$ and $h$ defined by $$a=\frac{2e\lambda}{\hbar}A \ \ \ , \ \ \ h=\frac{2e \lambda^2 \mu_0}{\hbar}H, \label{eq:aenh}$$ and rescale the wave function $\psi$ according to $$\varphi=\psi/\psi_{eq},$$ where $\psi_{eq}=\sqrt{|\epsilon|/\beta}$ is the equilibrium value of the SC order parameter for $T<T_c$. Clearly, $\varphi$ attains the value 1 in the bulk SC phase. Finally, we rescale the free energy $\Gamma$ divided by the surface area $S$ such that $\gamma=\Gamma\beta/(\epsilon^2\xi S)$, yielding $$\begin{aligned} \gamma[\varphi,a] = \int_{-\infty}^{+\infty} dx\left\{ \pm\varphi^2+\frac{\varphi^4}{2}+\dot{\varphi}^2 + \frac{a^2\varphi^2}{\kappa^2}+(\dot a -h )^2 \right\} \nonumber \\ + \frac{\xi}{b}(\varphi_-^2+\varphi_+^2) + \frac{\xi}{\alpha_{{\rm TP}}}(\varphi_--\varphi_+)^2. \label{eq:gammatp2}\end{aligned}$$ The $\pm$ refers to the sign of $T-T_c$. Minimization of $\gamma$ with respect to $\varphi$ and $a$ yields the well-known GL equations $$\ddot{\varphi}=\pm\varphi+a^2\varphi/\kappa^2+\varphi^3, \label{eq:gleqtp1}$$ and $$\ddot{a}=a\varphi^2/\kappa^2 . \label{eq:gleqtp2}$$ In addition, two coupled boundary conditions are obtained from stationarity with respect to $\varphi_-$ and $\varphi_+$ $$\dot{\varphi}_-= -\frac{\xi}{b}\varphi_- +\frac{\xi}{\alpha_{{\rm TP}}}(\varphi_+-\varphi_-), \label{eq:boundtp1}$$ $$\dot{\varphi}_+= \frac{\xi}{b}\varphi_+ +\frac{\xi}{\alpha_{{\rm TP}}}(\varphi_+-\varphi_-), \label{eq:boundtp2}$$ while the vector potential and its first derivative must be continuous at $x=0$, $$a(x=0^-)=a(x=0^+) \ \ \ , \ \ \ \dot a(x=0^-)=\dot a(x=0^+). \label{eq:a0}$$ In the subsequent sections we aim at solving the above equations for both the transparent and the opaque limit. Transparent twinning planes {#sec:transp} =========================== Boundary conditions and solutions {#subsec:introtra} --------------------------------- For highly transparent TP’s, $\alpha_{{\rm TP}} \rightarrow 0$ and we recover the original description of TPS [@BUZ] with a continuous order parameter at the TP, thus $\varphi_+=\varphi_-$. Consequently, it is natural to look for fully symmetric solutions for $\varphi(x)$ of the differential equation (\[eq:gleqtp1\]). In practice this is done by initially restricting ourselves to one half space, say $x>0$. The solution in the other half space $x<0$ can then be constructed using $\varphi(x)=\varphi(-x)$, with $\varphi(x)$ the solution for the semi-infinite system. Note that this makes the problem very similar to the one of a semi-infinite system with an external surface [@IND; @IND1], the only difference emerging from the behaviour at $x=0$ which is manifested most clearly in the boundary condition for the vector potential. Indeed, in the presence of an external surface, stationarity of $\gamma$ with respect to $a(0)$ results in $\dot a(0)=h$. In the present situation, however, owing to the assumed symmetry in the profile for $\varphi(x)$ the vector potential $a(x)$ will be anti-symmetric with respect to $x$, i.e. $a(-x)=-a(x)$ and thus obeys the condition $a(0)=0$. The boundary conditions for the wave function reduce in this limit to $$\dot\varphi_+=-\dot\varphi_-=\frac{\xi}{b}\varphi_+ . \label{eq:boundtptrans}$$ In the following analysis, we will be interested in two types of solutions distinguished by their asymptotic behaviour for $|x| \rightarrow \infty$. The enhancement of superconductivity near the TP will typically induce a SC sheath (with the bulk of the system prepared in the N phase) and it is precisely the thickness of this surface sheath that characterizes the solution. Clearly, if this thickness is finite superconductivity disappears for $|x| \rightarrow \infty$ and the magnetic field penetrates such that $$\varphi(\pm \infty)=0 \ \ \ , \ \ \ \dot a(\pm\infty)=h. \label{eq:bulktranstp1}$$ Adapting the terminology of [@IND; @IND1] which follows from the analogy to wetting transitions in adsorbed fluids [@DIE], this solution is referred to as a *partial wetting* state. We remark that this class of solutions also includes the so-called null solution without any SC phase in the system (thus describing a SC sheath with zero thickness). On the other hand, in the case of *complete wetting* the sheath can be macroscopically thick so that it fully separates the N phase from the TP. These solutions are possible only when the magnetic field equals the thermodynamic critical field $H_c$ at which there is bulk two-phase coexistence between the N and SC phase. Instead of Eq.(\[eq:bulktranstp1\]) they obey the asymptotic conditions $$\varphi(\pm \infty)=1 \ \ \ , \ \ \ a(\pm \infty)=0. \label{eq:bulktranstp2}$$ Remarkably, the calculation of this class of solutions is very simple and analytic results can be obtained for the entire type-I regime. Indeed, the asymptotic condition for the vector potential along with the boundary condition at $x=0$ imply that $a\equiv 0$ and we are left with a single differential equation for $\varphi(x)$ taking the simplified form, for $T<T_c$, $$\ddot{\varphi}=-\varphi+\varphi^3, \label{eq:gleq1tpmscl}$$ with the solution $\varphi(x)=\coth[(x+\delta)/\sqrt 2]$, for $x>0$, and where $\delta$ is determined by the boundary condition (\[eq:boundtptrans\]) [@CJB]. This solution must then be combined with a SC/N interface in the limit of $|x| \rightarrow \infty$ as necessary to obey the bulk N condition as initially imposed. This combination defines a macroscopic SC layer. For general values of $\kappa$, the SC/N interface, as well as the solutions describing a finite SC sheath, must be determined numerically. At bulk two-phase coexistence the two solutions described above may occur yielding an interface delocalization transition (or a wetting transition). This transition describes the delocalization of the SC/N interface from the TP into the bulk of the material away from the TP. We now address the analysis of this phase transition. Phase diagram at bulk coexistence {#subsec:pdtranstp} --------------------------------- In Fig.\[fig:twinfig1\] we show the phase diagram for a type-I superconductor with a transparent TP as a function of the parameters $\kappa$ and $\xi/b$. The magnetic field $h$ is fixed to its coexistence value $h_c=1/\sqrt 2$ [@FN1]. Varying the temperature for a given material, i.e. for fixed $\kappa$, corresponds to traveling horizontally in the diagram. Moving along coexistence towards $T_c$ corresponds to decreasing $\xi/b$ towards $-\infty$, since $\xi\propto|T-T_c|^{-1/2}$ and $b$ is a negative TP constant. We distinguish two regions according to the thickness $l$ of the SC sheath near the TP. In the first region $l=0$ indicating that the stable solution is the null solution without a SC sheath near the TP, while in the other region, the stable solution describes a macroscopic sheath on both sides of the TP and thus $l=\infty$. Note that a solution with a strictly finite thickness, i.e. $0<l<\infty$, is never stable and plays no significant role. The two regions are separated by the equilibrium surface phase transition line FD (first-order delocalization) while the dashed line ML marks the metastability limit of the N state of the TP, when $T$ is increased towards $T_c$. To identify the loci of the first-order phase boundary FD we can apply the technique of phase portraits which is a well-known concept in the study of wetting phase transitions and has proven to be equally instructive for investigating interfacial phase transitions in superconductors [@IND; @IND1; @BLO]. In practice, however, it is simpler to calculate $\gamma$ from Eq. (\[eq:gammatp2\]) for a macroscopic SC sheath and to see where it equals the surface free energy of the null solution. Since $\gamma=0$ in the latter state the condition for the delocalization transition reads $$\gamma_{TP/SC}+\gamma_{SC/N}=0, \label{eq:deloctranstp}$$ where $\gamma_{TP/SC}$ is the surface free energy of the bulk SC phase against the TP and $\gamma_{SC/N}$ the surface tension of the SC/N interface. The former can be calculated by considering the first integral of Eq.(\[eq:gleq1tpmscl\]) (for $h=h_c$) $$\dot{\varphi}=\frac{1}{\sqrt 2}(1-\varphi^2), \label{eq:fiint}$$ which leads, by inserting into the integral (\[eq:gammatp2\]), using also (\[eq:boundtptrans\]) and considering the half space $x>0$, to $$\gamma_{TP/SC}=\frac{2\sqrt 2}{3}-\varphi^2_+\left[\frac{2\sqrt 2}{3}\varphi_+ +\frac{\xi}{b} \right]. \label{eq:gammatpsc}$$ Here, $\varphi_+$ is given by the solution of $$\varphi_+^2+\sqrt 2 \xi\varphi_+/b-1=0.\label{eq:varphi+}$$ This equation follows from combining the first integral with the boundary condition (\[eq:boundtptrans\]). Concerning the surface tension $\gamma_{SC/N}$ we can use the accurate analytic expression pioneered in [@MIS] and improved in [@BOU], $$\gamma_{SC/N}=\frac{2\sqrt 2}{3}-1.02817\sqrt \kappa -0.13307\kappa\sqrt\kappa + {\cal O}(\kappa^2\sqrt\kappa). \label{eq:gammascnanalyt}$$ Due to the high accuracy of this expansion when truncated at order $\kappa\sqrt\kappa$ in the entire type-I regime, we immediately find, by inserting (\[eq:gammatpsc\]) and (\[eq:gammascnanalyt\]) into the condition (\[eq:deloctranstp\]), an accurate analytic result for the first-order phase boundary FD. The deviation from the numerical results lies within the thickness of the solid line in Fig. \[fig:twinfig1\], even at $\kappa=1/\sqrt 2$. For $\kappa=0$ the transition occurs at $(\xi/b)^*=-0.6022$ and expanding the phase boundary about this point reveals that it approaches the $\kappa=0$ axis in a parabolic manner $\kappa(\xi/b)\sim a(\xi/b - (\xi/b)^*)^2$, with $a\approx 4.95$. This demonstrates that in the low-$\kappa$ regime the system behaves both qualitatively and quantitatively precisely like the semi-infinite system with a wall [@IND; @IND1; @CJB]. A final aspect of the phase diagram relates to the calculation of the metastability limit ML for which it is justified to use the linearized version of the GL theory. In this approximation the non-linear terms in the GL equations are omitted so that Eq. (\[eq:gleqtp1\]) reads, for $T<T_c$, $$\ddot\varphi_0=-\varphi_0+a_0^2\varphi_0/\kappa^2, \label{eq:gleqtplin}$$ while the second one becomes trivial with the general solution $$a(x)=a_0(x)=h(x+x_0), \label{eq:gleqtp2lin}$$ where the boundary condition (\[eq:a0\]) immediately gives $x_0=0$. Eq. (\[eq:gleqtp2lin\]) expresses that the magnetic field is at no point expelled by the SC sheath that nucleates at ML. Thus we need only solve Eq. (\[eq:gleqtplin\]), subject to the boundary conditions $$\dot{\varphi}_0(0^+)=\frac{\xi}{b}\varphi_0(0^+) \ \ \ , \ \ \ \varphi_0(\infty)=0. \label{eq:boundtplin}$$ To find $\varphi_0(x)$ we follow Ref. [@IND1] and reduce (\[eq:gleqtplin\]) to a first-order (non-linear) differential equation by introducing the function $q_0(x)=\dot{\varphi}_0(x)/\varphi_0(x)$ obeying the equation $$\dot{q}_0+q_0^2=-1+a_0^2/\kappa^2, \label{eq:eqqtp}$$ with the boundary condition $q_0(0)=\xi/b$. This equation must be solved such that $q_0(x)$ has the acceptable asymptotic behaviour $q_0(x) \sim -h x/\kappa$, implying a Gaussian decay for $\varphi_0(x)$. This is done by performing (backwardly) the numerical integration of the auxiliary equation $$\dot{q}_0+q_0^2=-1+(hx/\kappa)^2, \label{eq:eqqtp2}$$ starting from $q_0(x)=-h x/\kappa$ for large $x$, down to $x=0$. For given $h/\kappa$, the ML value for $\xi/b$ then simply follows from $q_0(0)=\xi/b$. Moreover, using the result for the function $q_0(x)$, we can construct explicitly a solution to (\[eq:gleqtplin\]) of the form $$\varphi_0(x)=\varphi_0(0)\exp \left( \int_0^x q_0(x')dx' \right). \label{eq:solutlintp}$$ The line ML is obtained by taking a value of $\kappa$ and applying the procedure for the coexistence value $h_c=1/\sqrt 2$. Further, the same method applies away from coexistence (as well as to $T>T_c$) where it serves to obtain the critical nucleation field as a function of temperature (see below). The phase diagram discussed above demonstrates that for all type-I materials with an *internal transparent* TP there exists an interface delocalization transition at some temperature $T_D$ strictly below $T_c$. This transition can be interpreted as a genuine wetting transition such that for temperatures $T>T_D$ the SC state completely wets the TP and this occurs on both sides of the TP. For a *transparent* TP this phase transition is of *first order* regardless of the value of $\kappa$. This is very different from the analogous phase diagram for wetting at the surface of the material for which *both first-order and critical transitions* are predicted [@IND; @IND1]. A TP displaying TPS is characterized by a negative extrapolation length, hence it is not relevant to consider here positive $b$ values. Explicit surface free-energy calculations further reveal that for $b<0$ the SC phase is preferred by the TP such that $\gamma_{TP/SC}<\gamma_{TP/N}$, with $\gamma_{TP/N}$ the surface free energy of the N phase in bulk, according to (\[eq:bulktranstp1\]). This equality is reversed for all $\kappa$ when $\xi/b > 0$ so that the question of wetting by the SC phase is only relevant for $b<0$. In other words, the line of reversal of preferential adsorption in the $(\kappa,\xi/b)$-plane is located at $\xi/b=0$. Also in this respect the wetting phase diagram differs from that for an external surface or wall [@IND1]. Off-of-coexistence phase behaviour {#subsec:offcxtra} ---------------------------------- Now we turn our attention to the issue of phenomena outside coexistence which can most easily be clarified by inspecting the magnetic field versus temperature phase diagram for a given material. These diagrams are much more accessible to experimental verification than the global diagram in Fig.\[fig:twinfig1\] and provide a means of comparing our results with the TPS phase diagrams of Ref. [@BUZ]. Fig.\[fig:twinfig2\] shows a typical example for $\kappa=0.3$ where we employ units based on the delocalization field $H_D=H_c(T_D)$ and the delocalization temperature $T_D$, i.e., we take the ratios $H/H_D$ and $t=(T-T_c)/(T_c-T_D).$ A first ingredient in this diagram is the bulk coexistence line CX with two anchor points, the delocalization point D at $H/H_D=1$ and $t=-1$ and the bulk critical point at $H/H_D=0$ and $t=0$. Since the delocalization transition is first-order the point D is the starting point for a *prewetting* line FN (first-order nucleation) which marks a first-order transition between the null solution with $l=0$ and a finite sheath. The line FN is tangential to the coexistence line CX at D and changes into a continuous or critical nucleation line CN at a tricritical point TCP. The short stretch CN can be computed using the technique discussed earlier to determine the line ML in Fig. \[fig:twinfig1\]. It is important to stress that this transition is critical since the sheath will appear with an infinitesimal amplitude. Concerning the spatial extension in the direction perpendicular to the TP, however, it turns out that the sheath always has a thickness of the order of $\xi$ even just below the transition. Note that at the tricritical point FN and CN meet with common tangents and, to the left of TCP, CN continues as the metastability limit ML of the N state of the TP (dashed line). Finally, the line CN ends at zero field at the temperature $T_{{\rm c,TP}}$ which obeys the equation [@IND; @IND1] $$\frac{\xi(T_{{\rm c,TP}})}{b}=-1, \label{eq:tptcsb}$$ showing that the extrapolation length $b$ can be found in principle from experimental determination of $T_{{\rm c,TP}}$. We obtain the first-order nucleation line FN by numerically solving the GL equations for a sheath-type solution and finding the point in the phase diagram at which the free energy of this solution is zero. The location of TCP can be determined accurately by extending the zeroth-order solution of the linear GL equations (see above) a little further. In order to obtain a non-vanishing solution of the non-linear theory we need a correction to $a_0(x)$ and $\varphi_0(x)$. In view of Eq. (\[eq:solutlintp\]) $\varphi_0(x)$ is the small quantity through its small amplitude and thus it suffices to expand $a(x)=a_0(x)+a_1(x)+\ldots.$ Using the first integral of the equations and working to second order in $\varphi_0$ we immediately get $$\dot a_1= \frac{1}{2h}(\frac{a_0^2\varphi_0^2}{\kappa^2}- \dot \varphi_0^2 \pm \varphi_0^2), \label{eq:correcatp}$$ or, using (\[eq:gleqtplin\]), $$\dot a_1= \frac{1}{2h}\dot q_0(x) \varphi_0^2(x). \label{eq:correcatp2}$$ This result is particularly useful when we use it in an alternative expression for $\gamma$ evaluated in the extrema (see [@IND1]) $$\gamma=\int_{-\infty}^{+\infty} {\rm d}x \left\{-\frac{\varphi^4}{2}+(\dot a -h) ^2\right\}. \label{eq:alterntp1}$$ The advantage of this expression is twofold, firstly the boundary term is absorbed and secondly it shows that both terms are small, of order $\varphi_0^4$ for small $\varphi_0$. Along the line CN, $\gamma=0$ because $\varphi_0=0$ and $\dot a_0=h$. Another mechanism, however, to make $\gamma$ vanish is the compensation of the two terms in (\[eq:alterntp1\]), which can be put in the form $$2h^2=\int_{-\infty}^{+\infty}{\rm d}x \dot q_0^2(x) \varphi_0^4(x) / \int_{-\infty}^{+\infty}{\rm d}x \varphi_0^4(x). \label{eq:tricritp}$$ and which is of use in determining TCP as follows. For each value of $\kappa$ a continuous set of $(h,\xi/b)$ pairs can be found from the linear theory, but only one pair will satisfy the condition (\[eq:tricritp\]) for the onset of the nucleation of a finite (in contrast with infinitesimal) sheath, which is then by definition the TCP. The nucleation transitions discussed above indicate the point at which local superconductivity sets in near the TP, which obviously corresponds to the phenomenon of TPS. Consequently the results presented here should agree with the predicted TPS phase diagrams for a parallel magnetic field, at least with those pertaining to strongly transparent TP’s [@BUZ]. We have checked this for various $\kappa$ values and indeed found an excellent agreement for all the transitions. Furthermore, we may interpret the TPS transition as a genuine *prewetting transition* and thus by reinterpreting the existing theoretical and experimental TPS phase diagrams we stress that prewetting has since long been observed in classical superconductors. Intriguingly, we recall that only in more recent years has clear evidence of prewetting transitions been found in experiments on classical binary liquid mixtures [@KEL]. An important feature of the prewetting line found here is that it does not end at a surface critical point as typically predicted and found in liquid mixtures [@DIE; @KEL] but becomes, via a tricritical point, a continuous nucleation line. Further, our results are particularly interesting with respect to the behaviour near the wetting point D, since from the experimental TPS diagram it is unclear what happens to the TPS transition near the bulk critical field $H_c$. We now understand that the surface phase transition at the TP becomes a bulk transition precisely at the delocalization transition. Opaque twinning planes {#sec:opaque} ====================== Classification of solutions and terminology {#subsec:classol} ------------------------------------------- In this section we focus on the second limiting case, i.e., that of completely *opaque* TP’s for which $\alpha_{{\rm TP}} \rightarrow \infty$. In this case the two sides of the plane are, to some extent, independent since the requirement of a continuous order parameter at $x=0$ is no longer applicable, hence for a general solution $\varphi_+\neq\varphi_-$. However, there remains some coupling between the two half-spaces due to the vector potential and the magnetic induction which must both be continuous at the TP. Moreover, the number of possible solutions is further restricted by the assumption that the extrapolation length $b$ is equal on both sides of the TP. This means that the critical temperature $T_{{\rm c,TP}}$ (which is defined for zero magnetic field) is assumed to be the same for both sides. Nevertheless, a large number of different solutions can still be found among which several are characteristic for the opaque limit and deserve special attention here. For a neatly arranged description of the phases and phase transitions it is appropriate at this stage to present a classification and a schematic overview of the various solutions at bulk coexistence with equal bulk (N) conditions on the two sides of the TP. A first (trivial) solution is the null solution $\varphi(x)=0$ without local superconductivity. For solutions with at least one finite SC sheath we distinguish three scenarios as depicted in Fig. \[fig:twinfig3\]. Interestingly, due to the presumed opacity we can consider imposing $\varphi(x)\equiv 0$ for one half-space, say $x<0$, leading to a solution as shown in Fig. \[fig:twinfig3\](a) and referred to as “wall solution" since it corresponds exactly to the solution found in a system with a wall obeying the boundary condition $\dot a(0)=h$. In the case of a double SC sheath we can have either a continuous solution with $\varphi_+=\varphi_-$ and $a(0)=0$ (Fig. \[fig:twinfig3\](b)) corresponding to the sheath solution for a transparent TP, or a more general discontinuous solution with $\varphi_+\neq\varphi_-$ and $a$ and $\dot a$ continuous at $x=0$ (Fig. \[fig:twinfig3\](c)). Arguing along the same lines we also find a series of solutions with a macroscopic SC layer on only one or on both sides of the TP. For the former, in the other half-space, we can either have the wall solution with $\varphi(x)\equiv 0$ or a finite sheath with a continuous or discontinuous order parameter at the TP. The order parameter profiles for these solutions are schematically drawn in Fig. \[fig:twinfig4\]. For a double macroscopic layer, on the other hand, we should in principle take into account both the continuous and discontinuous solution. It can be shown, however, by a simple argument (see below) that the latter is impossible and therefore unnecessary to consider. As outlined in the previous section a double macroscopic layer will completely expel the magnetic field such that $a=0$ over the entire region of the SC layer. Consequently expression (\[eq:fiint\]) applies also here and at $x=0^{+,-}$ this yields $$\dot\varphi_{+,-}=\frac{1}{\sqrt 2}(1-\varphi^2_{+,-}), \label{eq:fasepoptp1}$$ While it may appear that any combination of $\varphi_+$ and $\varphi_-$ obeying these equations is a possible solution, this is only true if the boundary conditions at the TP are omitted. Indeed, by applying the boundary conditions which in this case are again given by Eq. (\[eq:boundtptrans\]) it is straightforward to see that, since $b$ is assumed to take the same value on both sides of the TP, $\varphi_+$ must equal $\varphi_-$ for a solution with two macroscopic SC layers as exemplified in Fig. \[fig:twinfig5\]. Note that this solution is the one we found for the case of a transparent TP. From the above classification of solutions it is obvious that an opaque TP is in a sense a combination of a system with a wall and one with a transparent TP, a feature which gives rise to additional phase transitions. Before further embarking on the phase behaviour it is convenient to comment here briefly on the terminology used below. The transition from the null solution to a state with either one or two finite sheaths as well as the transition from a single sheath to a pair of sheaths is referred to as a *nucleation* phenomenon. Likewise, the transition from a “wall state" with one macroscopic SC layer to a state that is the combination of such a layer and a finite sheath is called *nucleation*. Further, the appearance of one macroscopic SC layer, either from the null solution or from one finite sheath, is a *delocalization* or *wetting transition*, while a *depinning transition* refers to the transition between one and two macroscopic layers. The latter describes the depinning of a SC/N interface that is initially pinned at the TP and is analogous to the one investigated in Ref. [@BAC]. We start with a discussion of the TP states at bulk two-phase coexistence. Phase diagram at bulk coexistence {#subsec:pdopaque} --------------------------------- The fundamental diagram for interfacial phase transitions in the case of opaque TP’s is presented in Fig. \[fig:twinfig6\]. The different regions are distinguished according to the values of the two length scales $l_1$ and $l_2$ representing the thicknesses of the sheaths on the two sides of the TP. We assume here, without loss of generality, that $l_1 \geq l_2$. It is striking how different this phase diagram is compared to that for the case of perfect transparency, see Fig. \[fig:twinfig1\]. As we will demonstrate below, the main part of the diagram for the opaque case simply recovers the results of the system with a wall [@IND]. It is important to stress at this stage that we are only interested in the wall results for negative $b$ values since this is the relevant region for TP’s that show TPS. One ingredient of Fig. \[fig:twinfig6\] concerns the stability of the sheath solutions given in Fig. \[fig:twinfig3\]. From our analysis in Section \[sec:transp\] we know that at bulk coexistence the double symmetric sheath (Fig. \[fig:twinfig3\](b)) is never stable, and hence plays no significant role. Furthermore, explicit free energy calculations reveal that the discontinuous double sheath (Fig. \[fig:twinfig3\](c)) has an even higher free energy than the continuous one and thus is irrelevant. The single sheath solution (Fig. \[fig:twinfig3\](a)) is stable for a certain interval of $\xi/b$ values provided that $\kappa>0.374$ as can be inferred from the results for a system with a wall [@IND; @IND1]. Hence, in the upper right corner of the diagram we retrieve a feature of the wall diagram, with a finite sheath on only one side of the TP while the other side is still in the N phase, thus $l_1>0$ and $l_2=0$. In the lower half of the diagram we have on the low-$\xi/|b|$ side a region where no stable sheath solution exists other than the null solution ($l_1=0$ and $l_2=0$). The two regions are separated either by the line CN (critical nucleation) ending in a tricritical point TCP, or by the short stretch FN (first-order nucleation) between TCP and a critical end point CEP. To the left of TCP the line CN continues as the metastability limit ${\rm ML_1}$ of the null solution. Next we need to consider the solutions of Fig. \[fig:twinfig4\] and \[fig:twinfig5\] and to investigate their stability. Our calculations suggest that when the temperature is increased towards $T_c$ the system first enters the state given in Fig. \[fig:twinfig4\](a), either from one finite sheath or from the null solution. These transitions are indicated by the line CD (critical delocalization) and FD (first-order delocalization) respectively, meeting each other at the CEP. By further increasing the temperature the system undergoes a *first-order depinning transition* towards the double symmetric solution, i.e. a transition from the solution given in Fig. \[fig:twinfig4\](a) towards the solution of Fig. \[fig:twinfig5\]. Thus the regions $l_1=\infty$, $l_2=0$ and $l_1=l_2=\infty$ are separated by the first-order phase boundary FDP (first-order depinning). Lastly, the line ${\rm ML_2}$ represents the metastability limit of the state with $l_1=\infty$, $l_2=0$, with the SC/N interface pinned at the TP. Note that the solutions represented in Fig. \[fig:twinfig4\](b) and (c) are never stable. An important conclusion that we draw from this phase diagram is that for relatively low temperatures an opaque TP behaves as a system with a wall, with on one side of the TP the different wall solutions while the other side remains fully in the N phase. In other words, to the right of the line FDP in Fig. \[fig:twinfig6\] we immediately obtain the phase diagram by copying the results for a wall without additional computations. All the details concerning the determination of the transition lines for this part of the diagram can be found in Ref. [@IND1] and [@CJB]. In particular, the critical delocalization condition reads $$\gamma_{W/N}=\gamma_{W/SC}+\gamma_{SC/N}, \label{eq:critopaqtp}$$ while the first-order delocalization condition is simply given by $$\gamma_{W/SC}+\gamma_{SC/N}=0, \label{eq:foopaqtp}$$ where $\gamma_{W/N}$ ($\gamma_{W/SC}$) is the surface free energy of the bulk N (SC) phase against a wall. The novel feature in Fig. \[fig:twinfig6\] is the FDP line arising from the possibility of having a macroscopic SC layer also in the other region leading to a depinning transition at higher temperatures along with the associated spinodal ${\rm ML_2}$. It is precisely these two transition lines that we will be concentrating on in the remainder of this section. The condition for the first-order depinning transition is obtained by equating the free energy of the solution with one macroscopic layer at $x>0$ and no sheath at $x<0$ to that of the solution with the double symmetric macroscopic SC layer. The former can be written conveniently as the sum $\gamma_{W/SC}+\gamma_{SC/N}$, thus the depinning condition reads $$\gamma_{W/SC}+\gamma_{SC/N}=2(\gamma_{TP/SC}+\gamma_{SC/N}). \label{eq:depinopaqtp}$$ In general the depinning phase boundary has to be calculated numerically. An approximate analytic result can be obtained using the powerful expansions in $\kappa$ for the surface free energies. For the surface tension $\gamma_{SC/N}$ we use the result (\[eq:gammascnanalyt\]) while a similar expansion has been derived for $\gamma_{W/SC}$ in Ref. [@CJB]. Using these earlier results in the depinning condition (\[eq:depinopaqtp\]) provides a very accurate approximation for the phase boundary across the complete type-I range with an error less than $1\%$. For $\kappa=0$ the depinning and the delocalization transitions coincide at $(\xi/b)^*=-0.6022$. Both phase boundaries have a parabolic foot near $\kappa=0$, i.e. $\kappa(\xi/b)\sim a(\xi/b-(\xi/b)^*)^2$ with $a\approx 27.0$ for depinning and $a\approx 4.95$ for delocalization [@CJB]. From a mathematical point of view it is interesting to examine whether the extensions of the lines FDP and CD meet in the type-II regime, for $\kappa > 1/\sqrt{2}$. The condition for a line crossing is given by combining (\[eq:critopaqtp\]) and (\[eq:depinopaqtp\]). Explicit calculations using an expansion in $\kappa$ for the line CD obtained in [@VLHA] reveal indeed an intersection at $\kappa \approx 0.815$, $\xi/b \approx -0.251$. The line ${\rm ML_2}$ marks the nucleation of an infinitesimal sheath on one side of the TP under the condition that on the other side of the TP a macroscopic sheath is stable. To obtain this line we first compute numerically a solution for a macroscopic layer in one half space, $x>0$ say, for a given value of $\xi/b$ subject to the condition $\dot a(0)=h$. To proceed we combine it with a solution of the linear GL theory applied to the region $x<0$ using the scheme introduced in Section \[sec:transp\]. The zeroth-order solution for the vector potential is again given by Eq. (\[eq:gleqtp2lin\]) with in this case $x_0$ determined by the value of $a(0)$ which we get from the numerical calculation in $x>0$. Finally we have to compare $q_0(x_0)$ with the given value of $\xi/b$ and iterate the procedure such that the two become equal. This defines the critical nucleation of the sheath in the region $x<0$. The line ${\rm ML_2}$ is now formed by applying this method for the coexistence field $h_c$. The procedure is equally applicable for any other field appropriate for the study of off-of-coexistence phenomena, an issue which is addressed below. TPS phase diagrams {#subsec:pdopaque2} ------------------ From the properties of an opaque TP at bulk coexistence elucidated above we anticipate that the onset of local superconductivity near this plane generally happens in two steps. This is a natural consequence of the stability of “wall solutions" in these systems with $\varphi(x)=0$ on one side of the TP while on the other side the SC phase has already nucleated. We now wish to concentrate on the various nucleation phenomena, the order of which depends on the parameter $\kappa$, as a function of magnetic field and temperature. For $\kappa<0.374$ the delocalization transition on one side of the TP is first-order and thus will have a prewetting extension into the region of the phase diagram where the N phase is stable in bulk. This prewetting line FN is the first-order nucleation of a sheath on one side of the TP, and hence corresponds to the transition between the null solution and a solution as given in Fig. \[fig:twinfig3\](a). This line will be tangential to the bulk coexistence line CX at the first-order delocalization point D and changes at the tricritical point TCP into a continuous nucleation line CN. The latter describes the nucleation of one infinitesimal sheath and continues as the metastability limit ${\rm ML_1}$ of the null solution to the left of TCP. So far the surface phase diagram, an example of which is given in Fig. \[fig:twinfig7\] for $\kappa=0.3$, resembles the diagram for the system with a wall [@IND; @IND1]. Note that we use the same units as in Fig. \[fig:twinfig2\]. As we have argued above, however, another distinct point exists at bulk coexistence at higher temperatures but still below $T_c$, namely the first-order depinning point DP. The presence of the depinning transition leads to a variety of additional transitions compared to the wall system. Indeed, due to the first-order character of this point a second first-order nucleation line ${\rm FN}_2$ appears, attached to the point DP. This phase boundary can be interpreted as a *predepinning* line. For the sake of clarity this line is omitted in the main figure, with all the details near the depinning transition given in the inset of Fig. \[fig:twinfig7\]. At the line ${\rm FN}_2$ the SC phase nucleates on the other side of the TP, hence it is the transition between one sheath and a double sheath. The line ${\rm FN}_2$ is tangential to the line CX at DP and meets a continuous nucleation line ${\rm CN}_2$ at a second tricritical point ${\rm TCP'}$. To the left of ${\rm TCP'}$, ${\rm CN}_2$ continues as the metastability limit ${\rm ML_2}$ of the state with one sheath. Obviously ${\rm CN}_2$ denotes the critical nucleation of the second SC sheath. The two critical nucleation lines end in zero field at the same point $t_{{\rm c,tp}}$ which follows from the assumption that the TP is characterized by a single value of the parameter $b$. For materials with $\kappa >0.374$ for which the delocalization transition is critical, the field-temperature phase diagram will undergo qualitative changes as regards the wetting-like transitions, i.e., the transitions related to the nucleation of the SC phase on *one* side of the TP. We refer interested readers to Ref. [@IND1] for examples of $(H,T)$-phase diagrams containing these nucleation transitions and restrict ourselves here to a brief discussion of the pertinent features relevant for the present study. For $\kappa=0.374$ the point D and the starting point of the first-order nucleation line FN separate and by further increasing $\kappa$ this first-order transition disappears. The critical nucleation line CN then extends to low temperatures without intersection with the bulk coexistence line CX. Hence in this case the nucleation of the first sheath is always critical. For the nucleation of the second sheath there is still the possibility of first-order as well as second-order transitions. In fact, we know that the depinning transition is first-order for all type-I materials so that the situation that we sketched for $\kappa=0.3$ near the depinning point DP is representative for the entire type-I regime. Thus for an *opaque* TP there are always two distinct nucleation transitions, i.e. the SC phase never appears simultaneously on both sides of the TP. Discussion and concluding remarks {#sec:conc} ================================= In this paper we have analyzed the phase behaviour of the SC/N interface near internal TP’s in type-I superconductors. Our calculations reveal that the results are highly sensitive to the boundary conditions imposed at the TP, which in turn depend on the degree of transparency of the TP for electrons. For perfectly transparent TP’s the order parameter is continuous such that only symmetric profiles need to be considered and at first sight the analysis resembles that of an external surface or wall. The only differences originate from the boundary condition for the vector potential and, surprisingly, this small technical modification turns out to have dramatic consequences for the order of the delocalization transition. In particular, *for a transparent TP only first-order transitions are found for the entire type-I regime* while the possibility of critical transitions that are predicted for the wall system [@IND; @IND1] are suppressed. Our results for the magnetic field versus temperature phase diagrams are in excellent agreement with earlier experimental and theoretical results obtained in the context of TPS [@BUZ]. We comment further that our approach for the *nucleation* of the SC phase near the TP equally applies to type-II materials (with $\kappa>1/\sqrt 2$). Specifically, we have considered the case of Nb with $\kappa\approx 1$ for which the experimental results suggest that the properties of the TP are very close to the limiting case of perfect transparency. The nucleation transition in this case is always of second-order and our results again perfectly agree with earlier work [@BUZ]. We remark that in this case the nucleation lines are only relevant at sufficiently high fields above their intersection with the upper critical field $H_{c2}$. The situation drastically changes when considering opaque TP’s in which case a discontinuity in the order parameter is allowed at the TP. The decoupling of the two sides of the TP makes it possible to consider wall solutions with $\varphi(x)=0$ on one side of the TP. Moreover, from free-energy considerations, we have shown that these solutions are stable in a large region of the phase space and, as a result, the system precisely undergoes the transitions predicted for a wall system. In this case only one side of the TP will be wetted by the SC phase at the delocalization transition which can be either first-order or critical depending on the value of $\kappa$. By further increasing the temperature at bulk coexistence a first-order depinning transition is predicted for all type-I materials. Consequently, the field-temperature phase diagrams for opaque TP’s fundamentally differ from their counterparts for the transparent limit. A characteristic feature of an opaque TP is the existence of two distinct nucleation lines which in principle should be measurable and thus can *provide an experimental means of distinguishing between transparent and opaque TP’s*. Related to this we remark that experiments in Sn ($\kappa \approx 0.13$) have revealed only one nucleation transition, although it is assumed that the twin boundary in this material has a low transmission for electrons [@BUZ; @MIN; @SAM]. The apparent absence of a second nucleation transition can in this case be attributed to the low-$\kappa$ value of the material, since in the low-$\kappa$ regime the various transition lines lie extremely close together (see Fig. \[fig:twinfig6\]) and they would be difficult to distinguish in an experiment. Note that in the limit $\kappa\rightarrow0$ the differences between a transparent TP, an opaque TP and an external surface vanish. Finally, for type-II materials with opaque TP’s our calculations show that the nucleation can be either first-order or critical and this is at variance with the conclusion of earlier work [@AVE] stating that the TPS transition is always second-order for type-II materials. Our study demonstrates that this surmise is correct only for the case of transparent TP’s. The reason for this is that the tricritical nucleation point TCP merges with the delocalization transition at $\kappa=1/\sqrt 2$ for transparent TP’s. This can be seen also from the merging of the spinodal ML with the line FD in Fig. \[fig:twinfig1\]. In contrast, for opaque TP’s the TCP of nucleation remains well off of coexistence, at $H>H_c$, and in the type-II regime at $H > H_{c2}$, so that there is room for first-order nucleation. This is further demonstrated by the fact that, in Fig. \[fig:twinfig6\], the lines ${\rm ML_2}$ and FD are still far apart at $\kappa=1/\sqrt 2$. One remarkable implication of our work is that, in general, with the exception of perfectly transparent TP’s, there exist stable states of local superconductivity which are *asymmetric* about the TP. The possibility that a SC sheath or a macroscopic SC layer can occur on one side of the TP while the other side is in the normal state is indeed noteworthy and has been met with scepticism. It has been suggested that, since our analysis is essentially one-dimensional and neglects states which are inhomogeneous in the direction(s) parallel to the TP, there may exist modulated states , e.g., composed of a linear array of soft vortices parallel to the TP [@SAM] or states with a local field penetration and a change of phase of the wave function, which may have a lower free energy than the states we have considered [@BUZP]. Although we cannot rule out this possibility at present, we would like to point out that asymmetric wetting, followed by symmetric depinning, has been found previously in the context of grain-boundary wetting [@EBHA], in the framework of a real scalar order parameter model of Ising type. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Chris Boulter, Todor Mishonov and Alexander Buzdin for stimulating discussions. This research has been supported by the Belgian Fund for Scientific Research (FWO-Vlaanderen), the Inter-University Attraction Poles (IUAP) and the Concerted Action Research Programme (GOA). Most of the results of this paper have been obtained in the framework of two theses [@FC]. [99]{} J.O. Indekeu and J.M.J. van Leeuwen, Phys. Rev. Lett. [**75**]{}, 1618 (1995); For a tutorial, see Physica A [**236**]{}, 114 (1997). J.O. Indekeu and J.M.J. van Leeuwen, Physica C [**251**]{}, 290 (1995). C.J. Boulter and J.O. Indekeu, Physica C [**271**]{}, 94 (1996); Int. J. Thermophys. [**19**]{}, 857 (1998). G. Backx and J.O. Indekeu, Physica C [**274**]{}, 55 (1997). E. Montevecchi and J.O. Indekeu, Phys. Rev. B [**62**]{}, 14359 (2000). I.N. Khlyustikov and A.I. Buzdin, Adv. Phys. [**36**]{}, 271 (1987). D.B. Abraham, in [*Phase Transitions and Critical Phenomena*]{} Vol.10, edited by C. Domb and J.L. Lebowitz (Academic, London, 1986). A. Sevrin and J.O. Indekeu, Phys. Rev. B [**39**]{}, 4516 (1989). F. Igloi and J.O. Indekeu, Phys. Rev. B [**41**]{}, 6836 (1990). A.F. Andreev, JETP Lett. [**46**]{}, 584 (1987). V.B. Geshkenbein, Sov. Phys. JETP [**67**]{}, 2166 (1988). V.P. Mineev and K.V. Samokhin, JETP Lett. [**57**]{}, 383 (1993). K.V. Samokhin, JETP [**78**]{}, 909 (1994). For a review of wetting see, e.g., S. Dietrich, in [*Phase Transitions and Critical Phenomena*]{} Vol.12, edited by C. Domb and J.L. Lebowitz (Academic, London 1988). This value for the thermodynamic field $h_c$ is easily inferred from the functional for $\gamma$, Eq.(\[eq:gammatp2\]). Indeed, the free energy density approaches in the bulk SC state the value $-1/2+h^2$ and thus $\gamma$ would diverge unless $h=h_c=1/\sqrt 2$. R. Blossey and J.O. Indekeu, Phys. Rev. B [**53**]{}, 8599 (1996). T.M. Mishonov, J. Phys. (France) [**51**]{}, 447 (1990). C.J. Boulter and J.O. Indekeu, Phys. Rev. B [**54**]{}, 12407 (1996). H. Kellay, D.Bonn and J. Meunier, Phys. Rev. Lett. [**71**]{}, 2607 (1993); ibid., [**73**]{}, 3560 (1994). J.M.J. van Leeuwen and E.H. Hauge, J. Stat. Phys. [**87**]{}, 1335 (1997). V.V. Averin, A.I. Buzdin and L.N. Bulaevskii, Sov. Phys. JETP [**57**]{}, 426 (1983). A.I. Buzdin, private communication. C. Ebner. F. Hayot and J. Cai, Phys. Rev. B [**42**]{}, 8187 (1990). F. Clarysse, B.Sc. thesis (KULeuven, 1996) and Ph.D. thesis (KULeuven, 2000). [^1]: Present address: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, United Kingdom.

ArXiv

--- abstract: | Let $p$ be a prime and let $x$ be a $p$-adic integer. We provide two supercongruences for truncated series of the form $$\sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{}\cdots j_r^{}}\quad\mbox{and}\quad \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{1}{k}\sum_{1\le j_1\le\cdots\le j_r\le k}\frac{1}{j_1^{2}\cdots j_r^{2}}.$$ address: 'Dipartimento di Matematica, Università di Roma “Tor Vergata”, via della Ricerca Scientifica,00133 Roma, Italy' author: - Roberto Tauraso title: Two supercongruences related to multiple harmonic sums --- Introduction and main results ============================= In [@Ta:10 Theorem 1.1] and [@Ta:12 Theorem 7] we showed that for any prime $p\not=2$, $$\sum_{k=1}^{p-1} \frac{\binom{2k}{k}}{k 4^k} \equiv_{p^3} -H_{(p-1)/2}\quad\text{and}\quad \sum_{k=1}^{p-1} \frac{\binom{2k}{k}^2}{k 16^k} \equiv_{p^3} -2H_{(p-1)/2}$$ where $H_n^{(t)}=\sum_{j=1}^n\frac{1}{j^t}$ is the $n$-th harmonic number of order $t\geq 1$. Here we present two extensions of such congruences which involves the (non-strict) multiple harmonic sums $$S_n(t_1,\dots,t_r):=\sum_{1\le j_1\le\cdots\le j_r\le n}\frac{1}{j_1^{t_1}\cdots j_r^{t_r}}$$ with $t_1,t_2,\dots, t_r$ positive integers. For the sake of brevity, if $t_1 = t_2 = \dots = t_r = t$ we write $S_n(\{t\}^r)$. Let $(x)_n:=x(x+1)\cdots (x+n-1)$ be the Pochhammer symbol, and let $B_n(x)$ be the $n$-th Bernoulli polynomial. For any prime $p$, $\mathbb{Z}_p$ denotes the ring of all $p$-adic integers and $\langle \cdot\rangle_p$ is the least non-negative residue modulo $p$ of the $p$-integral argument. \[MT\] Let $p$ be a prime, $x\in\mathbb{Z}_p$ and $r\in\mathbb{N}$. Let $s:=(x+\langle-x\rangle_p)/p$. i\) If $p>r+3$ then $$\begin{aligned} \label{SI1} \sum_{k=1}^{p-1} \frac{(x)_k}{(1)_k}\cdot \frac{S_k(\{1\}^r)}{k} &\equiv_{p^2} -H^{(r+1)}_{\langle-x\rangle_p}-(-1)^rsp B_{p-r-2}(x).\end{aligned}$$ ii\) If $p>2r+3$ then $$\begin{aligned} \label{SI2} \sum_{k=1}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{S_k(\{2\}^r)}{k} &\equiv_{p^3} -2H^{(2r+1)}_{\langle-x\rangle_p}-2(2r+1)sp H^{(2r+2)}_{\langle-x\rangle_p} \nonumber \\&\qquad\qquad +\frac{2s(1+3sr+2sr^2)}{2r+3}\,p^2 B_{p-2r-3}(x).\end{aligned}$$ Note that, when $r=0$, both and have been established by Zhi-Hong Sun in [@Sunzh:15]. Moreover, for the special value $x=1/2$, and yield $$\label{CI1} \sum_{k=1}^{p-1} \frac{\binom{2k}{k}}{k 4^k}\cdot S_k(\{1\}^r) \equiv_{p^2} \begin{cases} -H_{(p-1)/2}^{(r+1)} &\text{if $r\equiv_2 0$},\vspace{3mm}\\ \frac{2^{r+2}-1}{2(r+2)}\,p B_{p-r-2} &\text{if $r\equiv_2 1$,} \end{cases}$$ and $$\label{CI2} \sum_{k=1}^{p-1} \frac{\binom{2k}{k}^2}{k 16^k}\cdot S_k(\{2\}^r) \equiv_{p^3} -2H_{(p-1)/2}^{(2r+1)} -\frac{r(2^{2r+3}-1)}{2}\,p^2B_{p-2r-3}.$$ For $r=1$, the congruence proves the conjecture [@Sunzw:15 Conjecture 5.3]. In the last section we provide $q$-analogs of two binomial identities related to the congruences and . Proof of in Theorem \[MT\] ========================== By taking the partial fraction expansion of the rational function $$x\to \frac{(x)_k}{(x)_n}$$ with $0\leq k<n$, we find $$\label{PDF1} \sum_{k=0}^{n-1} \frac{(x)_k}{(1)_k}\cdot a_k= (x)_n\sum_{j=0}^{n-1}\frac{(-1)^j T_j}{j!(n-1-j)!}\cdot \frac{1}{x+j}$$ where $T_j$ is the binomial transform of the sequence $a_k$, $$T_j:=\sum_{k=0}^j(-1)^k\binom{j}{k}\cdot a_k.$$ It is easy to see from that if $a_0,\dots,a_{p-1},x\in \mathbb{Z}_p$ then $$\label{A1} \sum_{k=0}^{p-1} \frac{(x)_k}{(1)_k}\cdot a_k\equiv_{p} T_{{\langle-x\rangle_p}}.$$In order to show we introduce the function $$G_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k}{(1)_k}\cdot S_k(\{1\}^r).$$ We have that $$G_n^{(0)}(x)=\frac{(1+x)_{n}}{(1)_{n}}-1$$ and $S_{k}(\{1\}^r)= S_{k-1}(\{1\}^r)+S_{k}(\{1\}^{r-1})/k$ implies $$\label{G1} G_n^{(r)}(x)=\frac{(1+x)_{n}}{(1)_{n}}\cdot S_{n}(\{1\}^r)-\frac{G_n^{(r-1)}(x)}{x}.$$ Moreover $$F_n^{(r)}(x+1)-F_n^{(r)}(x)=\frac{G_n^{(r)}(j)}{x}$$ where $$F_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k}{(1)_k}\cdot \frac{S_k(\{1\}^r)}{k}.$$ Then, for any positive integer $m$, $$\label{F1} F_n^{(r)}(x+m)-F_n^{(r)}(x)= \sum_{j=0}^{m-1}\frac{G_n^{(r)}(x+j)}{x+j}.$$ By , for $u=1,\dots,n$ $$G_n^{(r)}(-u)=\frac{G_n^{(r-1)}(-u)}{u}=\cdots=\frac{G_n^{(0)}(-u)}{u^r}=-\frac{1}{u^r}.$$ Hence by letting $x=-n$ and $m=n$ in we obtain the known identity (see [@He:99]) $$\label{He} \sum_{k=1}^n (-1)^k\binom{n}{k}\frac{S_k(\{1\}^r)}{k}=-H_n^{(r+1)}.$$ Thus, for $a_k=\frac{S_k(\{1\}^r)}{k}$, we have that $T_j=-H_j^{(r+1)}$, and by , we already have the modulo $p$ version of . Since $sp=x+\langle-x\rangle_p$ it follows that $$G_{p-1}^{(0)}(x) =\frac{(1+x)_{p-1}}{(1)_{p-1}}-1 \equiv_{p^2} \frac{sp}{x}-1.$$ By [@Zh:07 Theorem 1.6], $S_{p-1}(\{1\}^r)\equiv_p 0$ and therefore $$G_{p-1}^{(r)}(x)\equiv_{p^2} -\frac{G_{p-1}^{(r-1)}(x)}{x}\equiv_{p^2}\cdots \equiv_{p^2} (-1)^r\frac{G_{p-1}^{(0)}(x)}{x^{r}} \equiv_{p^2} \frac{(-1)^r sp}{x^{r+1}}-\frac{(-1)^r}{x^{r}}.$$ Moreover $$\begin{aligned} F_{p-1}^{(r)}(sp)&= \sum_{k=1}^{p-1} \frac{(sp)_k}{(1)_k}\cdot \frac{S_k(\{1\}^r)}{k}\equiv_{p^2} \sum_{k=1}^{p-1} \frac{sp}{k}\cdot \frac{S_k(\{1\}^r)}{k}\\ &=spS_{p-1}(\{1\}^{r},2)) \equiv_{p^2} spB_{p-r-2}\end{aligned}$$ where we used $S_{p-1}(\{1\}^r,2)\equiv_p B_{p-r-2}$ (see [@HHT:14 Theorem 4.5]). Finally, by , $$\begin{aligned} F_{p-1}^{(r)}(x)&\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1} \left(\frac{(-1)^r}{(x+j)^{r+1}}-\frac{(-1)^rsp}{(x+j)^{r+2}}\right)+spB_{p-r-2}\\ &\equiv_{p^2}-\sum_{j=1}^{\langle-x\rangle_p} \frac{1}{(j-sp)^{r+1}}-sp\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{r+2}}+spB_{p-r-2}\\ &\equiv_{p^2}-H_{\langle-x\rangle_p}^{(r+1)}- (r+2)spH_{\langle-x\rangle_p}^{(r+2)}+spB_{p-r-2}\\ &\equiv_{p^2}-H_{\langle-x\rangle_p}^{(r+1)}-(-1)^r spB_{p-r-2}\\\end{aligned}$$ where the last step uses the following congruence: for $2\leq t<p-1$ $$\label{Hp} H_{\langle-x\rangle_p}^{(t)} \equiv_p \sum_{j=1}^{\langle-x\rangle_p} j^{p-1-t}= \frac{B_{p-t}(\langle-x\rangle_p+1)-B_{p-t}}{p-t} \equiv_p (-1)^t\frac{B_{p-t}(x)-B_{p-t}}{t}$$ which is an immediate consequence of [@Sunzh:00 Lemma 3.2]. Proof of in Theorem \[MT\] ========================== We follow a similar strategy as outlined in the previous section. We start by considering the partial fraction decomposition of the rational function $$x\to\frac{(x)_k(1-x)_k}{(x)_n(1-x)_n}$$ with $0\leq k<n$. We have that $$\label{PDF2} \sum_{k=0}^{n-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k= (x)_n(1-x)_n \sum_{j=0}^{n-1}\frac{(-1)^j A_j}{(n+j)!(n-1-j)!}\left(\frac{1}{x+j}+\frac{1}{1-x+j}\right)$$ where $$A_j:=\sum_{k=0}^j(-1)^k\binom{j}{k}\binom{j+k}{k}\cdot a_k.$$ For $n\to \infty$, if the series is convergent, the identity becomes $$\sum_{k=0}^{\infty} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k= \frac{\sin(\pi x)}{\pi} \sum_{j=0}^{\infty}(-1)^j A_j\left(\frac{1}{x+j}+\frac{1}{1-x+j}\right).$$ In many cases the transformed sequence $A_j$ has a *nice* formula. For example if $a_k=1/(k+z)$ then $$A_j=\frac{(1-z)_j}{(z)_{j+1}}$$ and for $x=z=1/2$ we recover this series representations the Catalan’s constant $G=\sum_{j=0}\frac{(-1)^j}{(2j+1)^2}$: $$\sum_{k=0}^{\infty} \frac{\binom{2k}{k}^2}{(2k+1)16^k}=\frac{1}{2}\sum_{k=0}^{\infty} \frac{(1/2)_k^2}{(1)_k^2(k+1/2)}= \frac{1}{2\pi} \sum_{j=0}^{\infty}(-1)^j \frac{4}{(1/2+j)^2}= \frac{8G}{\pi}.$$ As regards congruences we have the following result. Let $p$ be a prime with $a_0,\dots,a_{p-1},x\in \mathbb{Z}_p$. Then $$\label{A2} \sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k \equiv_{p^2} A_{\langle-x\rangle_p}+s(A_{p-1-\langle-x\rangle_p}-A_{\langle-x\rangle_p})$$ For $x=1/2$ and $p>2$ then $$\sum_{k=0}^{p-1} \frac{\binom{2k}{k}^2}{16^k}\cdot a_k \equiv_{p^2} A_{(p-1)/2}.$$ Rearranging in a convenient way, we have $$\sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot a_k= \frac{(x)_p(1-x)_p}{(1)_p^2}\binom{2p-1}{p-1}^{-1} \sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} A_j\left(\frac{p}{x+j}+\frac{p}{1-x+j}\right).$$ If $0\leq k\leq j\leq p-1$ then $A_{p-1-j}\equiv_p A_{j}$ because $$\begin{aligned} \binom{p-1-j}{k}\binom{p-1-j+k}{k} &=\frac{(p-1-j)\cdots (p-j-k)(p-1-j+k)\cdots (p-j) }{(k!)^2}\\&\equiv_p \frac{(j+1)\cdots (j+k)(j-k+1)\cdots j }{(k!)^2} =\binom{j}{k}\binom{j+k}{k}.\end{aligned}$$ Thus, since $\langle -x\rangle_p+\langle-(1-x)\rangle_p=p-1$, it follows that $$\begin{aligned} \sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} \frac{pA_j}{x+j} &\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1} \frac{pA_j}{x+j}+(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \frac{A_{\langle-x\rangle_p}}{s}+\sum_{j=\langle-x\rangle_p+1}^{p-1} \frac{pA_j}{x+j}\\ &\equiv_{p^2}\sum_{j=0}^{\langle-x\rangle_p-1} \frac{pA_j}{x+j}+(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \frac{A_{\langle-x\rangle_p}}{s} -\!\!\!\!\!\!\sum_{j=0}^{\langle-(1-x)\rangle_p-1}\!\!\! \frac{pA_{p-1-j}}{1-x+j}.\end{aligned}$$ Therefore $$\begin{aligned} \sum_{j=0}^{p-1}(-1)^j\binom{2p-1}{p+j} A_j\left(\frac{p}{x+j}+\frac{p}{1-x+j}\right) &\equiv_{p^2}(-1)^{\langle-x\rangle_p}\binom{2p-1}{p+{\langle-x\rangle_p}} \left(\frac{A_{\langle-x\rangle_p}}{s} +\frac{A_{\langle-(1-x)\rangle_p}}{1-s}\right).\end{aligned}$$ Finally, by using $$\begin{aligned} &\binom{2p-1}{p-1} \equiv_{p^3} 1,\\ &\binom{2p-1}{p+j} \equiv_{p^2} (-1)^{j}\left(1-2p H_{j}\right),\\ &\frac{(x)_p(1-x)_p}{(1)_p^2}\equiv_{p^2} s(1-s)\left(1+2p H_{\langle-x\rangle_p}\right),\end{aligned}$$ we are done. For $x=1/2$ it suffices to note that $$\langle-x\rangle_p=(p-1)/2= p-1-\langle-x\rangle_p.$$ As an application of the previous theorem, we note that when $a_k=1$ then $A_j=(-1)^j$, and, by , it follows that $$\sum_{k=0}^{p-1} \frac{(x)_k(1-x)_k}{(1)_k^2} \equiv_{p^2} (-1)^{\langle -x\rangle_p}$$ which has been established in [@Sunzh:14 Corollary 2.1]. Another example worth to be mentioned is $a_k=1/k^r$ for $k\geq 1$ (and $a_0=0$). Then by [@Pr:10 Theorem 1] $$A_j=-\sum_{1\cdot k_1+3\cdot k_3+\dots= r} \frac{2^{k_1+k_3+\dots} (H_j^{(1)})^{k_1}(H_j^{(3)})^{k_3}\cdots}{1^{k_1}3^{k_3}\cdots k_1!k_3!\cdots}.$$ Now we consider the case $a_k=S_k(\{2\}^r)/k$. Let $$G_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k(-x)_k}{(1)_k^2}\cdot S_k(\{2\}^r).$$ We have that $$G_n^{(0)}(x)=\frac{(1+x)_{n}(1-x)_{n}}{(1)_{n}^2}-1,$$ and $S_{k}(\{2\}^r)= S_{k-1}(\{2\}^r)+S_{k}(\{2\}^{r-1})/k^2$ implies $$\label{G2} G_n^{(r)}(x)=\frac{(1+x)_{n}(1-x)_{n}}{(1)_{n}^2}\cdot S_{n}(\{2\}^r)+\frac{G_n^{(r-1)}(x)}{x^2}.$$ Moreover $$F_n^{(r)}(x+1)-F_n^{(r)}(x)=\frac{2G_n^{(r)}(x)}{x}$$ where $$F_n^{(r)}(x):=\sum_{k=1}^{n} \frac{(x)_k(1-x)_k}{(1)_k^2}\cdot \frac{S_k(\{2\}^r)}{k}.$$ Hence $$\label{F2} F_n^{(r)}(x+m)-F_n^{(r)}(x)= 2\sum_{j=0}^{m-1}\frac{G_n^{(r)}(x+j)}{x+j}.$$ The next identity is a variation of and it appears to be new. For any integers $n\geq 1$ and $r\geq 0$, $$\label{Ta} \sum_{k=1}^{n} (-1)^k\binom{n}{k}\binom{n+k}{k}\frac{S_k(\{2\}^r)}{k}=-2H_{n}^{(2r+1)}.$$ By , for $u=1,\dots,n$, $$G_n^{(r)}(-u)=\frac{G_n^{(r-1)}(-u)}{u^2}=\dots=\frac{G_n^{(0)}(-u)}{u^{2r}}=-\frac{1}{u^{2r}}.$$ Hence by letting $x=-n$ and $m=n$ in $$\begin{aligned} \sum_{k=1}^{n} (-1)^k\binom{n}{k}\binom{n+k}{k}\frac{S_k(\{2\}^r)}{k}&=F^{(r)}_n(-n)=F_n^{(r)}(0)-2\sum_{j=0}^{n-1}\frac{G_n^{(r)}(-n+j)}{-n+j}\\ &=2\sum_{j=0}^{n-1}\frac{1}{(-n+j)^{2r+1}}=-2H_{n}^{(2r+1)}.\end{aligned}$$ Thus by applying $\eqref{A2}$ we find a modulo $p^2$ version of $\eqref{SI2}$. A more refined reasoning will lead us to the $p^3$ congruence. Since $sp=x+\langle-x\rangle_p$, $$G_{p-1}^{(0)}(x) =\frac{(1+x)_{p-1}(1-x)_{p-1}}{(1)_{p-1}^2}-1 \equiv_{p^3} -\frac{s(1-s)p^2}{x^2}-1$$ By [@Zh:07 Theorem 1.6], $S_{p-1}(\{2\}^r)\equiv_p 0$ and therefore $$G_{p-1}^{(r)}(x)\equiv_{p^3} \frac{G_{p-1}^{(r-1)}(x)}{x^{2}}\equiv_{p^3}\cdots \equiv_{p^3} \frac{G_{p-1}^{(0)}(x)}{x^{2r}} \equiv_{p^3} -\frac{s(1-s)p^2}{x^{2r+2}}-\frac{1}{x^{2r}}.$$ It follows that $$\begin{aligned} F_{p-1}^{(r)}(sp)-F_{p-1}^{(r)}(x) &\equiv_{p^3} 2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{G_{p-1}^{(0)}(x+j)}{(x+j)^{2r+1}}\\ &\equiv_{p^3} -2s(1-s)p^2\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{2r+3}}-2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}}.\end{aligned}$$ By $$\sum_{j=1}^{\langle-x\rangle_p}\frac{1}{j^{2r+3}}=H_{\langle-x\rangle_p}^{(2r+3)} \equiv_p -\frac{B_{p-2r-3}(x)-B_{p-2r-3}}{2r+3}.$$ Moreover $$\begin{aligned} F_{p-1}^{(r)}(sp)&= \sum_{k=1}^{p-1} \frac{(sp)_k(1-sp)_k}{(1)_k^2}\cdot \frac{S_k(\{2\}^r)}{k}\\ &\equiv_{p^3} \sum_{k=1}^{p-1} \frac{sp(k-sp)}{k^2}\cdot \frac{S_k(\{2\}^r)}{k}\\ &=sp\sum_{k=1}^{p-1} \frac{S_k(\{2\}^r)}{k^2} -p^2s^2\sum_{k=1}^{p-1} \frac{S_k(\{2\}^r)}{k^3}\\ &=spS_{p-1}(\{2\}^{r+1})) -p^2s^2S_{p-1}(\{2\}^{r},3))\\ &\equiv_{p^3} sp\frac{2pB_{p-2r-3}}{2r+3} +p^2s^22rB_{p-2r-3}\\ &\equiv _{p^3} \frac{2sp^2(1+sr(2r+3))B_{p-2r-3}}{2r+3}\end{aligned}$$ where we used $$\frac{(sp)_k(1-sp)_k}{(1)_k^2} =\frac{sp(k-sp)}{k^2}\cdot \frac{(1+sp)_{k-1}(1-sp)_{k-1}}{ (1)_{k-1}^2}\equiv_{p^3}\frac{sp(k-sp)}{k^2}$$ and the congruences $$S_{p-1}(\{2\}^r)\equiv_{p^2} \frac{2pB_{p-2r-1}}{2r+1}\quad\text{and}\quad S_{p-1}(\{2\}^r,3)\equiv_p -2rB_{p-2r-3}.$$ which have been established in [@Zh:07 Theorem 1.6] in [@HHT:14 Theorem 4.1] respectively. Finally, $$\begin{aligned} F_{p}^{(r)}(x) &\equiv_{p^3} \frac{2sp^2(1+sr(2r+3))B_{p-2r-3}}{2r+3} -\frac{2s(s-1)p^2(B_{p-2r-3}(x)-B_{p-2r-3})}{2r+3}\\ &\qquad\qquad+2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}}\\ &\equiv_{p^3} 2\sum_{j=0}^{\langle-x\rangle_p-1}\frac{1}{(x+j)^{2r+1}} +\frac{2s(1-s)}{2r+3}p^2B_{p-2r-3}(x) \\&\qquad\qquad +\frac{2s^2(r+1)(2r+1)}{2r+3}p^2 B_{p-2r-3}\end{aligned}$$ We observe that follows by letting $x=1/2$. Then $\langle-x\rangle_p-1=(p-1)/2$, $B_{2n}(1/2)=(2^{1-2n}-1)B_{2n}$ and for $p-4>t>1$ $$H^{(t)}_{(p-1)/2}\equiv \begin{cases} \frac{t(2^{t+1}-1)}{2(t+1)}\,p B_{p-t-1} \pmod{p^2} &\text{if $t\equiv_2 0$},\vspace{3mm}\\ -\frac{(2^{t}-2)}{t}\, B_{p-t} \qquad \pmod{p} &\text{if $t\equiv_2 1$}. \end{cases}$$ see [@Sunzh:00 Theorem 5.2]. Final remarks: $q$-analogs of and ================================== It is interesting to note that identities and have both a $q$-version (the first one appears in [@Pr:00]). For any integers $n\geq 1$ and $r\geq 0$, $$\label{Heq} \sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{n}{k}_q}q^{\binom{k}{2}-(n-1)k}\cdot \frac{S_k(\{1\}^r;q)}{1-q^k} =-\sum_{k=1}^n\frac{q^{rk}}{(1-q^k)^{r+1}}$$ and $$\label{Taq} \sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{n}{k}_q}{\genfrac{[}{]}{0pt}{}{n+k}{k}_q} q^{\binom{k}{2}-(n-1)k} \cdot \frac{S_k(\{2\}^r;q)}{1-q^k} =-\sum_{k=1}^n\frac{(1+q^k)q^{rk}}{(1-q^{k})^{2r+1}}$$ where ${\genfrac{[}{]}{0pt}{}{m}{k}_q}$ is the Gaussian binomial coefficient $${\genfrac{[}{]}{0pt}{}{m}{k}_q}=\left\{ \begin{array}{ll} \frac{(1-q^m)(1-q^{m-1})\cdots (1-q^{m-k+1})}{(1-q^k)(1-q^{k-1})\cdots (1-q)} &\mbox{if $0\leq k\leq m$},\\[3pt] 0 &\mbox{otherwise}, \end{array}\right.$$ and $$S_n(t_1,\dots,t_r;q):=\sum_{1\le j_1\le\cdots\le j_r\le n}\frac{q^{j_1+\dots +j_r}}{(1-q^{j_1})^{t_1}\cdots (1-q^{j_r})^{t_r}}.$$ We show and we leave the proof of other one to the interested reader. The procedure is quite similar to the one given for the corresponding ordinary identity . Let $$G_n^{(r)}(u):=\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{u}{k}_q}{\genfrac{[}{]}{0pt}{}{u+k-1}{k}_q} q^{\binom{k}{2}-(u-1)k} \cdot S_k(\{2\}^r;q).$$ Then for $u=1,\dots,n$, $G_n^{(0)}(u)=-1$ and $$G_n^{(r)}(u)=\frac{q^u G_n^{(r-1)}}{(1-q^u)^2}=\dots=\frac{q^{ru }G_n^{(0)}(u)}{(1-q^u )^{2r}}=-\frac{q^{ru }}{(1-q^u)^{2r}}.$$ Moreover $$F_n^{(r)}(u)-F_n^{(r)}(u-1)=\frac{(1+q^u)G_n^{(r)}(u)}{(1-q^u)} =-\frac{(1+q^u)q^{ru }}{(1-q^u)^{2r+1}}$$ where $$F_n^{(r)}(u):=\sum_{k=1}^n(-1)^k{\genfrac{[}{]}{0pt}{}{u}{k}_q}{\genfrac{[}{]}{0pt}{}{u+k}{k}_q} q^{\binom{k}{2}-(u-1)k} \cdot S_k(\{2\}^r;q).$$ Thus, since $F_n^{(0)}(n)=0$, $$\begin{aligned} F_n^{(r)}(n)&=\sum_{u=1}^n\frac{(1+q^u)G_n^{(r)}(u)}{(1-q^u)}+F_n^{(0)}(n) =-\sum_{u=1}^n\frac{(1+q^u)q^{ru}}{(1-q^{u})^{2r+1}}\end{aligned}$$ and the proof is complete. [99]{} V. 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Sun, [*A new series for $\pi^3$ and related congruences*]{}, Internat. J. Math. [**26**]{} (2015), ID 1550055, 23 pp. R. Tauraso, [*Congruences involving alternating multiple harmonic sum*]{}, Electron. J. Combin., \#R16 (2010). R. Tauraso, [*Supercongruences for a truncated hypergeometric series*]{}, Integers **12** (2012), \#A45, 12 pp. J. Zhao, [*Bernoulli numbers, Wolstenholme’s theorem, and $p^5$ variations of Lucas’ theorem*]{}, J. Number Theory [**123**]{} (2007), 18–26.

ArXiv

--- abstract: 'We have assembled a sample of high spatial resolution far-UV (Hubble Space Telescope Advanced Camera for Surveys Solar Blind Channel) and H$\alpha$ (Maryland-Magellan Tunable Filter) imaging for 15 cool core galaxy clusters. These data provide a detailed view of the thin, extended filaments in the cores of these clusters. Based on the ratio of the far-UV to H$\alpha$ luminosity, the UV spectral energy distribution, and the far-UV and H$\alpha$ morphology, we conclude that the warm, ionized gas in the cluster cores is photoionized by massive, young stars in all but a few (Abell 1991, Abell 2052, Abell 2580) systems. We show that the extended filaments, when considered separately, appear to be star-forming in the majority of cases, while the nuclei tend to have slightly lower far-UV luminosity for a given H$\alpha$ luminosity, suggesting a harder ionization source or higher extinction. We observe a slight offset in the UV/H$\alpha$ ratio from the expected value for continuous star formation which can be modeled by assuming intrinsic extinction by modest amounts of dust (E(B-V) $\sim$ 0.2), or a top-heavy IMF in the extended filaments. The measured star formation rates vary from $\sim$ 0.05 M$_{\odot}$ yr$^{-1}$ in the nuclei of non-cooling systems, consistent with passive, red ellipticals, to $\sim$ 5 M$_{\odot}$ yr$^{-1}$ in systems with complex, extended, optical filaments. Comparing the estimates of the star formation rate based on UV, H$\alpha$ and infrared luminosities to the spectroscopically-determined X-ray cooling rate suggests a star formation efficiency of 14$^{+18}_{-8}$%. This value represents the time-averaged fraction, by mass, of gas cooling out of the intracluster medium which turns into stars, and agrees well with the global fraction of baryons in stars required by simulations to reproduce the stellar mass function for galaxies. This result provides a new constraint on the efficiency of star formation in accreting systems.' author: - 'Michael McDonald, Sylvain Veilleux, David S. N. Rupke, Richard Mushotzky, and Christopher Reynolds' title: Star Formation Efficiency in the Cool Cores of Galaxy Clusters --- Introduction ============ The high densities and low temperatures of the intracluster medium (hereafter ICM) in the cores of some galaxy clusters suggests that massive amounts (100–1000 M$_{\odot}$ yr$^{-1}$) of cool gas should be deposited onto the central galaxy. The fact that this gas reservoir is not observed has been used as prime evidence for feedback-regulated cooling (see review by Fabian 1994). By invoking feedback, either by active galactic nuclei (hereafter AGN) (e.g., Guo [et al. ]{}2008; Rafferty [et al. ]{}2008; Conroy [et al. ]{}2008), mergers (e.g., Gómez [et al. ]{}2002; ZuHone 2010), conduction (e.g., Fabian [et al. ]{}2002; Voigt [et al. ]{}2004), or some other mechanism, theoretical models can greatly reduce the efficiency of ICM cooling, producing a better match with what is observed in high resolution X-ray grating spectra of cool cores (0–100 M$_{\odot}$ yr$^{-1}$, Peterson [et al. ]{}2003). However, these modest cooling flows had remained unaccounted for at low temperatures until only recently. The presence of warm, ionized gas in the form of H$\alpha$ emitting filaments has been observed in the cores of several cooling flow clusters to date (e.g., Hu [et al. ]{}1985, Heckman [et al. ]{}1989, Crawford [et al. ]{}1999, Jaffe [et al. ]{}2005, Hatch [et al. ]{}2007). More recently, it has been shown by McDonald [et al. ]{}(2010, 2011; herafter M+10 and M+11, respectively) that this emission is intimately linked to the cooling ICM and may be the result of cooling instabilities. However, while it is possible that the warm gas may be a byproduct of ICM cooling, the source of ionization in this gas remains a mystery. A wide variety of ionization mechanisms are viable in the cores of clusters (see Crawford [et al. ]{}2005 for a review), the least exotic of which may be photoionization by massive, young stars. ------------ ------------- -------------- -------- -------- ------ ----------- -------------- Name RA Dec z E(B-V) M F$_{1.4}$ Proposal No. (1) (2) (3) (4) (5) (6) (7) (8) Abell 0970 10h17m25.7s -10d41m20.3s 0.0587 0.055 – $<$ 2.5 11980 Abell 1644 12h57m11.6s -17d24m33.9s 0.0475 0.069 3.2 98.4 11980 Abell 1650 12h58m41.5s -01d45m41.1s 0.0846 0.017 0.0 $<$ 2.5 11980 Abell 1795 13h48m52.5s +26d35m33.9s 0.0625 0.013 7.8 924.5 11980, 11681 Abell 1837 14h01m36.4s -11d07m43.2s 0.0691 0.058 0.0 4.8 11980 Abell 1991 14h54m31.5s +18d38m32.4s 0.0587 0.025 14.6 39.0 11980 Abell 2029 15h10m56.1s +05d44m41.8s 0.0773 0.040 3.4 527.8 11980 Abell 2052 15h16m44.5s +07d01m18.2s 0.0345 0.037 2.6 5499.3 11980 Abell 2142 15h58m20.0s +27d14m00.4s 0.0904 0.044 1.2 $<$ 2.5 11980 Abell 2151 16h04m35.8s +17d43m17.8s 0.0352 0.043 8.4 2.4 11980 Abell 2580 23h21m26.3s -23d12m27.8s 0.0890 0.024 – 46.4 11980 Abell 2597 23h25m19.7s -12d07m27.1s 0.0830 0.030 9.5 1874.6 11131 Abell 4059 23h57m00.7s -34d45m32.7s 0.0475 0.015 0.7 1284.7 11980 Ophiuchus 17h12m27.7s -23d22m10.4s 0.0285 0.588 0.0 28.8 11980 WBL 360-03 11h49m35.4s -03d29m17.0s 0.0274 0.028 – $<$ 2.5 11980 ------------ ------------- -------------- -------- -------- ------ ----------- -------------- *-0.2 in (1): Cluster name, (2–4): NED RA, Dec, redshift of BCG (<http://nedwww.ipac.caltech.edu>), (5): Reddening due to Galactic extinction from Schlegel [et al. ]{}(1998), (6): Spectroscopically-determined X-ray cooling rates (M$_{\odot}$ yr$^{-1}$) from McDonald [et al. ]{}(2010), (7): 1.4 GHz radio flux (mJy) from NVSS (<http://www.cv.nrao.edu/nvss/>) (8) HST proposal number for FUV data. Proposal PIs are W. Jaffe (\#11131), W. Sparks (\#11681), S. Veilleux (\#11980).\ $^a$: No available *Chandra* data. \[sample\]* The identification of star-forming regions in cool core clusters has a rich history in the literature. Early on, it was noted by several groups that brightest cluster galaxies (hereafter BCGs) in cool core clusters have higher star formation rates than non-cool core BCGs (Johnstone [et al. ]{}1987; Romanishin [et al. ]{}1987; McNamara and O’Connell 1989; Allen 1995; Cardiel [et al. ]{}1995). These studies all found evidence for significant amounts of star formation in cool cores, but the measured star formation rates were orders of magnitudes smaller than the X-ray cooling rates (e.g. McNamara and O’Connell 1989). In recent history, two separate advances have brought these measurements closer together. First, as mentioned earlier in this section, the X-ray spectroscopically-determined cooling rates are roughly an order of magnitude lower than the classically-determined values based on the soft X-ray luminosity. Secondly, large surveys in the UV (e.g., Rafferty [et al. ]{}2006; Hicks [et al. ]{}2010), optical (e.g., Crawford [et al. ]{}1999; Edwards [et al. ]{}2007; Bildfell [et al. ]{}2008; McDonald [et al. ]{}2010), mid-IR (e.g., Hansen [et al. ]{}2000; Egami [et al. ]{}2006; Quillen [et al. ]{}2008; O’Dea [et al. ]{}2008; hereafter MIR), and sub-mm (e.g., Edge 2001; Salomé and Combes 2003) have allowed a much more detailed picture of star formation in BCGs. The typical star formation rates of $\sim$ 1–10 M$_{\odot}$ yr$^{-1}$ (O’Dea [et al. ]{}2008) imply that gas at temperatures of $\sim10^{6-7}$ K is being continuously converted into stars with an efficiency on the order of $\sim$ 10%. The fact that most of these studies consider the *integrated* SF rates makes it difficult to determine the exact role of young stars in ionizing the extended warm gas observed at H$\alpha$, since the two may not be spatially coincident or the measurements may be contaminated by the inclusion of a central AGN. In order to understand both the role of star formation in ionizing the warm gas and the efficiency with which the cooling ICM is converted into stars, we have conducted a high spatial resolution far-UV survey of BCGs in cooling and non-cooling clusters. We describe the collection and analysis of the data from this survey in §2. In §3 we desribe the results of this survey, while in §4 we discuss the implications of these results in the context of our previous work (M+10,M+11). Finally, in §5 we summarize our findings and discuss any outstanding questions. Throughout this paper, we assume the following cosmological values: H$_0$ = 73 km s$^{-1}$ Mpc$^{-1}$, $\Omega_{matter}$ = 0.27, $\Omega_{vacuum}$ = 0.73. Data Collection and Analysis ============================ To study the far-UV (hereafter FUV) emission in cluster cores, we selected 15 galaxy clusters from the larger samples of M+10 and M+11, which have deep, high spatial resolution (FWHM $\sim$ 0.6$^{\prime\prime}$) H$\alpha$ imaging from the Maryland-Magellan Tunable Filter (hereafter MMTF; Veilleux [et al. ]{}2010) on the Baade 6.5-m telescope at Las Campanas Observatory. Additionally, most of these systems have deep Chandra X-ray Observatory (hereafter CXO) spectroscopic imaging, as well as Two-Micron All Sky Survey (hereafter 2MASS; Skrutskie [et al. ]{}2006) and NRAO VLA Sky Survey (hereafter NVSS; Condon [et al. ]{}1998) fluxes. This broad energy coverage provides an excellent complement to a FUV survey, allowing for the source of emission to be carefully identified. A summary of these 15 clusters can be found in Table 1. For further information about the reduction and analysis of the H$\alpha$ and X-ray data, see M+10. FUV imaging was acquired using the Advanced Camera for Surveys Solar Blind Channel (hereafter ACS/SBC) on the Hubble Space Telescope (hereafter HST) in both the F140LP and F150LP bandpasses whenever possible, with a total exposure time of $\sim$ 1200s each (PID \#11980, PI Veilleux). The pointings were chosen, based on the results of our MMTF survey, to include all of the H$\alpha$ emission in the field of view. Exposures with multiple filters are required to properly remove the known ACS/SBC red leak, which has a non-negligible contribution due to the fact that the underlying BCG is very luminous and red. Since the aforementioned filters are long-pass filters, they have nearly identical throughputs at longer wavelengths. Thus, by subtracting the F150LP exposure from the F140LP exposure we can effectively remove the red leak and consider only a relatively small range in wavelength, from 1400Å–1500Å. Due to the small bandpass, it is possible for line emission to dominate the observed flux – we investigate this possibility in §4. We have carried out this subtraction for 13/15 of the BCGs in our sample which have both F140LP and F150LP imaging. For Abell 1795 and Abell 2597, we are unable to remove the red leak contribution due to the lack of paired exposures, but we point out that, conveniently, these two systems have the brightest FUV flux in our sample and, thus, are largely unaffected by the inclusion of a small amount of non-FUV flux. Following the red-leak subtraction, we also bin the images $8\times8$ and smooth the images with a 1.5 pixel smoothing radius, yielding matching spatial resolution at FUV and H$\alpha$. This process is also necessary in order to increase the signal-to-noise of the FUV image, allowing us to identify interesting morphological features. The final pixel scale for both the H$\alpha$ and FUV images is 0.2$^{\prime\prime}$/pixel. All FUV and H$\alpha$ fluxes were corrected for Galactic extinction following Cardelli [et al. ]{}(1989) using reddening estimates from Schlegel [et al. ]{}(1998). Results ======= {width="90.00000%"} In the Appendix, we show the stellar continuum, H$\alpha$ and FUV images for each of the 15 BCGs in our sample. At a glance there does not appear to be consistent agreement between the H$\alpha$ and FUV morphologies. We observe systems having H$\alpha$ filaments without accompanying FUV emission (Abell 1991, Abell 2052), systems with complex FUV emission without accompanying H$\alpha$ (Abell 1837, Abell 2029), and systems with coincident H$\alpha$ and FUV extended emission (Abell 1644, Abell 1795). Thus, it is obvious that a single explanation (e.g., star formation) is unable to account for the variety of FUV and H$\alpha$ emission that we observe. As we did with the H$\alpha$ emission in M+10, the FUV morphology can be classified as either nuclear or extended. We find, in the FUV, 7/15 systems have extended emission, 5/15 have nuclear emission, while 3/15 have no emission at all. In order to quantitatively examine both the nuclear and extended emission, we extract FUV and H$\alpha$ fluxes in several regions, as shown in the Appendix. ![Distribution of the FUV to H$\alpha$ luminosity ratios for the 10 clusters in our sample with high S/N detections in either the FUV or H$\alpha$. Each panel considers a different region, from top to bottom: the nucleus, the central 3$^{\prime\prime}$, excluding the nucleus, the filaments, and the entire system. The red vertical band represents the region consistent with shocks (Dopita & Sutherland 1996), while the blue band represents the region consistent with continuous star formation (Leitherer [et al. ]{}1999). In the nucleus, the majority of the observed H$\alpha$ emission is consistent with having been shock-heated, while in the outer regions, including filaments, the FUV/H$\alpha$ ratio lies between the two highlighted regions. In the bottom panel, the dotted histogram shows how the distribution would be altered if we corrected the fluxes for an i intrinsic extinction of E(B-V)=0.2 (dust screen model), while the blue dashed line represents the expected value for O8V stars (symbolic of a top-heavy IMF).[]{data-label="uvha_hists"}](uvha_hists.pdf){width="48.00000%"} In Figure \[uvha\]a, we show the correlation between the FUV and H$\alpha$ luminosity for the regions identified in the Appendix. We find a significant amount of FUV emission in all 5 of the systems for which we do not detect any H$\alpha$ emission. Additionally, we see that at least 3 systems are consistent with being shock-heated (Dopita & Sutherland 1996) – a point we will return to later in this section. As discussed by Hicks [et al. ]{}(2010), a significant fraction of the FUV emission may be due to old stellar populations (e.g., horizontal branch stars) in the BCG. To proceed, we must isolate the FUV excess due to young, star-forming regions. In order to remove the contribution from old stars, we consider the inner 3$^{\prime\prime}$ and plot the K-band (from 2MASS; Skrutskie [et al. ]{}2006) versus the FUV luminosity (Figure \[uvha\]b). Hicks [et al. ]{}(2010) show that the FUV luminosity from old stars is highly concentrated in the central region, thus removing this contribution in the inner region will act as a suitable first-order correction. In order to calibrate this correction for our sample, which lacks a control sample of confirmed non-star-forming galaxies, we opt to fit a line which is chosen to pass through the four points with the lowest $L_{FUV}$/$L_{K^{\prime}}$ ratio. We make the assumption that these four galaxies with the lowest $L_{FUV}/L_{K^{\prime}}$ ratio are non-star-forming, which is supported by non-detections at H$\alpha$. The equation for this relation is: log$_{10}(L_{FUV,3^{\prime\prime}}) = 2.35$log$_{10}(L_{K^{\prime},3^{\prime\prime}}) - 42.98$. The fact that four points with non-detections at H$\alpha$ lie neatly along the same line suggests that this correction is meaningful. Figure \[uvha\]c shows the FUV excess due to young stars versus the H$\alpha$ luminosity for the total, nuclear and extended regions in our complete sample of BCGs. With the contribution from old stellar populations removed, we find a tight correlation between $L_{FUV}$ and $L_{H\alpha}$ over four orders of magnitude. The majority of systems in our sample are consistent with the continuous star formation scenario (Kennicutt 1998, Leitherer [et al. ]{}1999), suggesting that much of the warm gas found in cluster cores may be photoionized by young stars. Two systems, Abell 0970 and Abell 2029 have anomolously high FUV/H$\alpha$ ratios, suggesting that star formation may be proceeding in bursts. As a starburst ages, the UV/H$\alpha$ ratio will climb quickly due to the massive stars dying first. This means that, by 10 Myr after the burst, the UV/H$\alpha$ ratio can already be an order of magnitude higher than the expected value for continuous star formation (see M+10 for further discussion). We find that the filaments in Abell 1991 and Abell 2052 are consistent with being heated by fast shocks, along with the nuclei of Abell 2052 and Abell 2580. In the case of Abell 2052, there exist high quality radio and X-ray maps which show that the observed H$\alpha$ emission is coincident with the inner edge of a radio-blow bubble. In Abell 1991, the H$\alpha$ morphology is reminiscent of a bow shock, and is spatially coincident with a soft X-ray blob which is offset from the cluster core. Much of the FUV and H$\alpha$ data is clustered between the regions depicting continuous star formation and shock heating, as shown in the zoomed-in portion of Figure \[uvha\]. These regions may indeed be heated by a combination of processes, or they may simply be reddened due to intrinsic extinction. Based on their FUV/H$\alpha$ ratios, H$\alpha$ morphology, disrupted X-ray morphology, and high radio luminosity, we propose that the optical emission in Abell 1991, Abell 2052, and Abell 2580 is the product of shock heating, while the remaining 12 systems are experiencing continuous or burst-like star formation. We will return to this classification in the discussion. The SF rates that we measure in the systems with nuclear emission only range from 0.01–0.1 M$_{\odot}$ yr$^{-1}$, which are typical of normal red-sequence ellipticals (Kennicutt 1998). For the systems with extended emission, excluding those that are obviously shock-heated, the measured star formation rates range from 0.1–5 M$_{\odot}$ yr$^{-1}$ which is similar to the rates of 0.008–3.6 M$_{\odot}$  yr$^{-1}$ observed in “blue early-type galaxies” (Wei [et al. ]{}2010). -------------------------------------- ------------------------------------------ {width="49.00000%"} {width="47.00000%"} -------------------------------------- ------------------------------------------ In Figure \[uvha\_hists\] we present the distribution of the FUV/H$\alpha$ ratio in various regions for the 10 systems with detections ($>$1-$\sigma$) at either H$\alpha$ or FUV. In the innermost region ($r<0.8^{\prime\prime}$), the warm gas appears to be shock-heated in 60% of systems – these shocked nuclei may be associated with AGN-driven outflows. Due to the small radial extent of this bin, the 2MASS data is of insufficient spatial resolution to remove any contribution from old stars. Thus, these FUV/H$\alpha$ ratios are upper limits. Of the remaining 4 systems, 2 are consistent with continuous star formation or a young starburst, while the remaining two are consistent with an aged starburst. At larger radius ($0.8^{\prime\prime}<r<3.0^{\prime\prime}$) the FUV/H$\alpha$ ratio is slightly larger, with the distribution peaking in between the regions describing shocks and star formation (see inset of Figure \[sfr\]). These data have had the contribution from old stellar populations removed, as described earlier in this section. The FUV/H$\alpha$ ratio at this radius is similar to what we observe in the filaments, as is seen in the third panel of Figure \[uvha\_hists\]. The fact that the distribution of FUV/H$\alpha$ peaks between the values for shocks and star formation supports a number of scenarios, including a mixture of the two processes, dusty star formation and star formation with an IMF skewed towards high-mass stars (see right-most panel of Figure \[uvha\_hists\]). We will investigate these scenarios in §4. In M+10 and M+11, we showed that the H$\alpha$ emission observed in the cool cores of galaxy clusters is intimately linked to the cooling ICM. In general, the thin, extended filaments observed in many of these clusters are found in regions where the ICM is cooling most rapidly, suggesting that this warm gas may be a byproduct of the ongoing cooling. If this is the case, it is relevant to ask what fraction of the cooling ICM is turning into stars. We address this question in Figure \[sfr\] by comparing the star formation rate with the X-ray cooling rate (dM/dt) for 32 galaxy clusters. In order to compute the star formation rate, we use the prescriptions in Kennicutt (1998). For the systems observed with HST, we use the average of the FUV- and H$\alpha$-determined star formation rates (filled blue circles). For an additional 10 clusters from M+10 and M+11 we make use of archival GALEX data for 5 clusters (open blue circles) and assume that both the UV and H$\alpha$ emisison trace star forming regions. These data have also had the contribution from old stellar populations removed, as described in M+10. In the remaining 5 cases where there is no accompanying UV data, we assume that the H$\alpha$ emission is the result of photoionization by young stars, and convert the H$\alpha$ luminosity into a continuous star formation rate (green triangles). For the three shock-heated systems (red crosses) mentioned above, we determine the SF rate based on the FUV luminosity alone – this represents an upper limit on the amount of continuous star formation. Finally, we also include 10 clusters from the sample of O’Dea [et al. ]{}(2008), who compute SF rates based on [*Spitzer*]{} data, and dM/dt in a similar manner to us (open purple circles). We note that the SFRs derived from FUV data are systematically lower than those derived from 24$\mu m$ [*Spitzer*]{} data. This is most likely due to our lack of an intrinsic extinction correction for these FUV data. We find that the “efficiency” of star formation, defined as the current ratio of stars formed to gas cooling out of the ICM, can range from 1% to 50% over the full sample. This spread is independent of whether we only consider systems classified as star-forming based on their FUV/H$\alpha$ ratios in Figure \[uvha\]. For the majority of the 32 clusters in Figure \[sfr\], there is a tight correlation between SF rate and dM/dt with a typical efficiency between 10–50%. In the right panel of Figure \[sfr\], we show that the efficiency is nearly independent of the cluster temperature, with only a weak dependence which is primarily driven by highly extincted systems. This suggests that the temperature of the surrounding ICM does not hamper the BCG’s ability to form stars. While not shown here, we also investigated the distribution of star formation efficiencies with the central entropy, $K_0$, from Cavagnolo [et al. ]{}(2009) and, similarly to $T_X$, find no correlation. In the following section we will discuss possible interpretations of this efficiency measure. Discussion ========== Star Formation as an Ionization Source -------------------------------------- ------------------------------------------- ------------------------------------------- {width="45.00000%"} {width="44.50000%"} ------------------------------------------- ------------------------------------------- In Crawford [et al. ]{}(2005), a variety of ionization sources are discussed which could produce the observed H$\alpha$ emission in the cool cores of galaxy clusters. The purpose of this HST survey was to investigate one of the most plausible ionization sources: photoionization by young stars. In Figures \[uvha\] and \[uvha\_hists\] we showed that, once the contribution to the FUV emission from old stellar populations is removed, the majority of the H$\alpha$ and FUV emission that we observe in cluster cores is roughly consistent with the star formation scenario. Based on the FUV/H$\alpha$ ratios, we identify three different types of system: 1. FUV/H$\alpha$ $\gtrsim$ $10^{-12}$ Hz$^{-1}$. Suggests a starburst that has aged by at least 10 Myr. Two systems, Abell 0970 and Abell 2029, fulfill this criteria, while several others may fall into this category if their H$\alpha$ luminosity is significantly less than the measured upper limits. 2. FUV/H$\alpha$ $\sim$ $10^{-13}$ Hz$^{-1}$. The FUV/H$\alpha$ ratios of these systems are consistent with continuous star formation or a recent (0–5 Gyr ago) burst of star formation. The filaments in Abell 1644, Abell 1795, Abell 2597, and part of Abell 2052, along with the nuclei of Abell 1795, Abell 1991, Abell 2151, and Abell 2597 appear to be star-forming. 3. FUV/H$\alpha$ $\lesssim$ $10^{-14}$ Hz$^{-1}$. Suggests heating by fast shocks or some other source of hard ionization (e.g., cosmic rays, AGN). The filaments of Abell 2052 and Abell 1991, and the nuclei of Abell 2052 and Abell 2580 have FUV/H$\alpha$ ratios which are consistent with this picture, in the absence of internal reddening. Figures \[uvha\] (inset) and \[uvha\_hists\] show that a large fraction of the systems which we observe fall between the regions describing shock heating and star formation. However, these data have not been corrected for intrinsic reddening due to dust. Correcting for a very modest reddening (E(B-V) $\sim$ 0.2) would boost the FUV luminosity of these systems such that the FUV/H$\alpha$ ratio is consistent with star formation (see Figure \[uvha\]c and the lower panel of Figure \[uvha\_hists\]). Unfortunately, the amount of reddening in the filaments and nuclei of these systems is currently not well constrained for very many systems, but typical values of E(B-V) can range from 0–0.4 in the cores of galaxy clusters (Crawford [et al. ]{}1999). In the case of Abell 2052, for which we measure FUV/H$\alpha$ ratios indicative of shock-heating *and* have an estimate of the amount of intrinsic reddening from Crawford ($E(B-V)=0.22^{+0.36}_{-0.21}$), we can investigate whether correcting for this extinction would provide FUV/H$\alpha$ ratios consistent with star-forming regions. Assuming a simple dust-screen model, correcting for a reddening of $E(B-V)=0.22$ would transform a FUV/H$\alpha$ ratio of $4.4\times10^{-15} Hz^{-1}$ in the filaments of Abell 2052 to $1.2\times10^{-14} Hz^{-1}$, which is consistent with the upper limit for shock-heated systems (see Figure \[uvha\_hists\]). However if, contrary to expectations, the filaments have a slightly higher reddening than the nucleus, the FUV/H$\alpha$ ratio may be even higher. Thus, it is certainly possible that the systems which we classify as shock-heated, or those which have ambiguous FUV/H$\alpha$ ratios, may in fact be highly-obscurred star-forming systems. We will address this possibility in significantly more detail in an upcoming paper which will include long-slit spectroscopy of the H$\alpha$ filaments providing, for the first time, reddening estimates away from the nucleus in these systems. An alternative explanation for the intermediate FUV/H$\alpha$ ratios is that the IMF in the filaments is top-heavy ($\alpha \ll 2.35$). Again, the lower panel of Figure \[uvha\_hists\] shows that the peak of the FUV/H$\alpha$ distribution is consistent with the value expected for O8V stars. There is a substantial amount of literature providing evidence for a top-heavy IMF in various environments including the Galactic center (Maness [et al. ]{}2007) and disturbed galaxies (Habergham [et al. ]{}2010). Thus, regardless of whether there is a small amount of dust or a slightly altered IMF, we suspect that the majority of the systems with intermediate FUV/H$\alpha$ ratios are in fact star-forming, with the exception of Abell 1991, Abell 2052, and Abell 2580, which have low FUV/H$\alpha$ ratios *and* morphologies which resemble bow-shocks and/or jets. Due to our use of the F150LP filter to remove red leak contamination, we are considering only a very small wavelength range from 1400–1500Å. In this region, there may be line emission from \[OIV\] and various ionization states of sulfur due to gas cooling at $\sim 10^5$ K. In order to establish that we are indeed observing continuum emission from young stars, we have computed UV spectral energy distributions (SEDs) for 6 BCGs which have deep GALEX, XMM-OM, and HST UV data. These data are presented in the left panel of Figure \[sed\]. We see that, in general, the UV SED follows a powerlaw over the range of 1500–3000Å. The new HST data, depicted as colored stars in this plot, agree well with the extrapolation of the continuum to shorter wavelengths, suggesting that there is very little contamination from line emission. This also suggests that there is little contribution from a diffuse UV component. This is further emphasized in the right panel of Figure \[sed\] where we show the residuals from the continuum fit for each of the 6 BCGs. Our measured FUV fluxes from these new HST data are consistent with the measurement of a UV continuum from archival GALEX and XMM-OM data. The idea that massive, young stars may be responsible for heating the majority of the warm, ionized filaments observed in cool core clusters is certainly not a new one (see e.g., Hu [et al. ]{}1985; Heckman [et al. ]{}1989; McNamara and O’Connel 1989). Most recently, O’Dea [et al. ]{}(2008), Hicks [et al. ]{}(2010) and McDonald [et al. ]{}(2010a) conducted MIR, UV and H$\alpha$ surveys, respectively, of cool core clusters and found a strong correlation between the SF rate and the cooling properties of cluster cores. However, this work extends these findings to include spatially-resolved SF rates, which the previous studies have been unable to provide. This allows us to say conclusively that the young stars and the warm, ionized gas are in close proximity ($\lesssim 1^{\prime\prime}$) in the vast majority of systems, offering a straightforward explanation for the heating of these filaments. Star Formation Efficiencies in Cooling Flows -------------------------------------------- In §3, we provide estimates of the efficiency with which the cooling ICM is converted into stars, assuming that this is indeed the source of star formation. This assumption is based on the results of M+10 and M+11, which provided several strong links between the X-ray cooling properties and the warm, ionized gas. These estimates of star formation efficiency represent a constraint on the so-called “accreting box model” of star formation. The simplified model that we propose is that the ICM is allowed to cool rapidly in regions where cooling *locally* dominates over feedback (Sharma [et al. ]{}2010). Our estimates of the ICM cooling rate, based on medium-resolution CXO spectra, are consistent with estimates based on high-resolution XMM grating spectroscopy by Peterson [et al. ]{}(2003) for the five overlapping systems. Once the gas reaches temperatures of $\sim$10$^{5.5}$K, it can continue to cool rapidly via UV/optical/IR line emission without producing fluxes that are inconsistent with what are observed. In the standard way, star formation will proceed once the gas reaches low enough temperature and high enough density. Observations by Edge (2001) and Salomé & Combes (2003) show evidence for molecular gas in the cool cores of several galaxy clusters, consistent with this picture. ![Distribution of measured star formation efficiencies from Figure \[sfr\] for 26 systems. Systems which are likely shock-heated (Abell 1991, Abell 2052, Abell 2580) have not been included. The additive contribution to the total histogram (solid black line) from UV+H$\alpha$, H$\alpha$ and MIR data are shown in different colors following the color scheme in Figure \[sfr\]. The dotted line shows a Gaussian fit to this histogram, which peaks at $14^{+18}_{-8}$% efficiency.[]{data-label="sfre"}](sfre_hist.pdf){width="49.00000%"} If the above scenario is correct, our estimate of the star formation efficiency provides a constraint on the fraction of hot gas that will be converted to stars, assuming a steady inflow of gas. In Figure \[sfre\] we provide a histogram of SF efficiencies for all of the systems in Figure \[sfr\] with non-zero X-ray cooling rates (dM/dt). The peak of this distribution is well defined at an efficiency of $14^{+18}_{-8}$%, regardless of which SF indicator (UV, H$\alpha$, MIR) is used. We note that, while the distribution for UV- and MIR-determined SF rates both peak at roughly the same value, the UV-determined SF histogram extends to much lower values. The low-efficiency tail of this distribution may be an artifact produced by intrinsic extinction due to dust, to which the UV will be most sensitive, or may be indicative of a selection bias in the MIR sample. If we measure the peak efficiency based on the subsamples excluding the MIR and MIR+H$\alpha$ (with no accompanying UV) data, we get 10$^{+25}_{-7}$% and 14$^{+20}_{-8}$%, respectively. Thus, the peak value of 14% is not solely driven by the inclusion of MIR data. The average efficiency of 14$^{+18}_{-8}$%, based on MIR, H$\alpha$ and UV data, is consistent with the estimates of star formation efficiency over the lifetime of a typical molecular cloud (20–50%; Kroupa [et al. ]{}2001, Lada & Lada 2003). This large variance in star formation efficiency may be due to differences in the ICM cooling and star formation timescales. Naturally, one would expect that there is some delay between the ICM cooling and the formation of stars, so that a reservoir of cold gas can accumulate and the formation of stars can be triggered. If this is indeed the case, one would expect to observe cooling-dominated periods (low SFE) followed by periods of strong star formation (high SFE) once the cold gas reservoir has reached some critical mass. Over an ensemble of systems, the average SFE is then an estimate of the time-averaged efficiency of an accreting system in converting a steady stream of cooling gas into stars. An alternative explanation for the spread of observed efficiencies is that the source of feedback is episodic (e.g., AGN). In this scenario, an episode of strong feedback from the AGN would re-heat the reservoir of cool gas, severely reducing the potential for star formation. This may indeed be the case, since two of the three systems with the highest 1.4 GHz luminosity (Abell 2052, Abell 2597, and Perseus A) have SFE $\lesssim$ 0.1. The fact that Figure \[sfre\] shows a well-defined peak suggests that the fraction of stars formed in an accreting system is constant over long enough timescales. Our estimate of an average efficiency indicates that, for a steady-state system accreting hot gas which is then allowed to cool, roughly 4 M$_{\odot}$ of gas will either be re-heated or expelled via winds for every 1 M$_{\odot}$ of stars formed. This fraction of baryons in stars is consistent with the global fraction of $\sim$ 20–30% required by simulations to reproduce the observed stellar mass function of galaxies (Somerville [et al. ]{}2008). Unlike measurements of SF efficiency for giant molecular clouds, this estimate does not require the use of a specific timescale, since we are assuming that stars are forming out of the inflow of hot gas and that the reservoir for this hot gas is inexhaustible. Summary and Future Prospects ============================ We have assembled a unique set of high spatial resolution far-UV and H$\alpha$ images for 15 cool core galaxy clusters. These data provide an unprecedented view of the thin, extended filaments in the cores of galaxy clusters. Based on the ratio of the far-UV to H$\alpha$ luminosity, the UV SED, and the far-UV and H$\alpha$ morphology, we conclude that the warm, ionized gas in the cluster cores is photoionized by massive, young stars in all but a few (Abell 1991, Abell 2052, Abell 2580) systems. We show that the extended filaments, when considered separately, appear to be forming stars in the majority of cases, while the nuclei tend to have slightly lower FUV/H$\alpha$ ratios, suggesting either a harder ionization source or higher extinction. The slight deviation from expected FUV/H$\alpha$ ratios for continuous star formation (Leitherer [et al. ]{}1999) may be due to the fact that we have made no attempt to correct for intrinsic extinction due to dust or due to a top-heavy ($\alpha \ll 2.35$) IMF. We note that modest amounts of dust (E(B-V) $\sim$ 0.2) in the most dense regions of the ICM can account for this deviation. Ideally, one would like spatially-resolved optical spectra of the filaments in order to constrain the heat source and intrinsic reddening of the filaments. We plan on addressing this issue in upcoming studies. Comparing the estimates of the star formation rates based on FUV, H$\alpha$ and MIR luminosities to the spectroscopically-determined X-ray cooling rate suggests a star formation efficiency of 14$^{+18}_{-8}$%. This value represents the time-averaged fraction, by mass, of gas cooling out of the ICM which turns into stars and agrees well with the stars-to-gas fraction of $\sim$20–30% required by simulations to reproduce the observed stellar mass function. This result provides a new constraint for studies of star formation in accreting systems. Many aspects of this simplified scenario are still not well understood, including whether the star formation is similar to that seen in nearby spirals or vastly different. We intend to investigate such differences via an assortment of star formation indicators from the UV to radio in future work. Acknowledgements {#acknowledgements .unnumbered} ================ Support for this work was provided to M.M. and S.V. by NSF through contracts AST 0606932 and 1009583, and by NASA through contract HST GO-1198001A. We thank E. Ostriker and A. Bolatto for useful discussions. We also thank the technical staff at Las Campanas for their support during the ground-based observations, particularly David Osip who helped in the commissioning of MMTF. 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ArXiv

--- abstract: | Random variables equidistributed on convex bodies have received quite a lot of attention in the last few years. In this paper we prove the negative association property (which generalizes the subindependence of coordinate slabs) for generalized Orlicz balls. This allows us to give a strong concentration property, along with a few moment comparison inequalities. Also, the theory of negatively associated variables is being developed in its own right, which allows us to hope more results will be available. Moreover, a simpler proof of a more general result for $\ell_p^n$ balls is given. author: - 'Marcin Pilipczuk ([email protected])' - | Jakub Onufry Wojtaszczyk[^1] ([email protected])\ Department of Mathematics, Computer Science and Mechanics\ University of Warsaw\ ul. Banacha 2, 02-097 Warsaw, Poland\ title: The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball --- Introduction ============ Notation -------- We shall begin by introducing the notation used throughout the paper. For any set $A$ by $\1_A$ we shall denote the characteristic function of $A$. As usually, $\R$ and ${\R_{+}}$ will denote the reals and the non-negative reals respectively. By $\R^k$ we shall mean the $k$-dimensional Euclidean space equipped with the standard scalar product $\is{\cdot}{\cdot}$, the Lebesgue measure denoted by $\lambda$ or $\lambda_k$ and a system of orthonormal coordinates $x_1,x_2,\ldots,x_k$. By ${\R_{+}}^k$ we mean the generalized positive quadrant, that is the set $\{(x_1,\ldots,x_k) \in \R^k : \forall_i\ x_i \geq 0\}$. For a given set $K\subset \R^k$ by ${K_{+}}$ we shall denote the positive quadrant of $K$, that is $K \cap {\R_{+}}^k$. For a given set $A$ by ${{\bar{A}}}$ we will denote the complement of $A$. For a measure $\mi$ on $\R^n$ and an affine subspace $H \subset \R^n$, by the [*projection of $\mi$ onto $H$*]{} we mean the measure $\mi_H$ defined by $\mi_H(C) = \mi(\{x \in \R^n : P(x) \in C\})$, where $P$ is the orthogonal projection onto $H$. If $\mi$ is given by a density function $m$ and $K \subset H \subset \R^n$, then by the [*restriction of $\mi$ to $K$*]{} we mean the measure $\mi_{|K}$ on $H$ given with the density $m \cdot \1_K$. By the support of a function $m:X \to \R$, denoted ${{\rm supp}}m$, we mean ${{\rm cl}}\{x \in X : m(x) \neq 0\}$. If $\mi$ is a measure, then by ${{\rm supp}}\mi$ we mean the smallest closed set $A$ such that $\mi(\bar{A}) = 0$. In the cases we consider, when $\mi$ will be given by a density $m$, we will always have $supp \mi = {{\rm supp}}m$. We shall call a set $K \subset \R^n$ a [*symmetric body*]{} if it is convex, bounded, central-symmetric (i.e. if $x \in K$ then $-x \in K$) and has a non-empty interior. A body $K \subset \R^n$ is called [*1-symmetric*]{} if for any $(\eps_1,\ldots,\eps_n) \in \{-1,1\}^n$ and any $(x_1,\ldots,x_n) \in K$ we have $(\eps_1 x_1, \ldots, \eps_n x_n) \in K$. Such a body is sometimes called [*unconditional*]{}. A function $f : {\R_{+}}{{\rightarrow}}{\R_{+}}\cup \{\infty\}$ is called a [*Young function*]{} if it is convex, $f(0) = 0$ and $\exists_x : f(x) \neq 0$, $\exists_{x\neq 0} : f(x) \neq \infty$. If we have $n$ Young functions $f_1,\ldots,f_n$, then the set $$K = \{(x_1,\ldots,x_n) : \sum_{i=1}^n f_i(|x_i|) \leq 1\}$$ is a 1-symmetric body in $\R^n$. Such a set is called a [*generalized Orlicz ball*]{}, also known in the literature as a modular sequence space ball. We shall call a Young function $f$ [*proper*]{} if it does not attain the $+\infty$ value and $f(x) > 0$ for $x > 0$. A generalized Orlicz ball is called [*proper*]{} if it can be defined by proper Young functions. If the coordinates of the space $\R^n$ are denoted $x_1,x_2,\ldots,x_n$, the appropriate Young functions will be denoted $f_1,f_2,\ldots,f_n$, with the assumption $f_i$ is applied to $x_i$. If some of the coordinates are denoted $x,y,z,\ldots$, the appropriate Young functions will be denoted $f_x, f_y, f_z, \ldots$, with the assumption that $f_x$ is applied to $x$, $f_y$ to $y$ and so on. A function $f: \R {{\rightarrow}}\R$ is called increasing (decreasing) if $x \geq y$ implies $f(x) \geq f(y)$ ($f(x) \leq f(y)$) — we do not require a sharp inequality. A function $f:\R^k {{\rightarrow}}\R$ or $f:{\R_{+}}^k {{\rightarrow}}\R$ is called [*coordinate-wise increasing (decreasing)*]{}, if for $x_i \geq y_i$, $i = 1,2,\ldots,n$ we have $f(x_1,\ldots,x_k) \geq f(y_1,\ldots,y_n)$ ($f(x_1,\ldots,x_k) \leq f(y_1,\ldots,y_n)$). A set $A \subset {\R_{+}}^k$ is called a [*c-set*]{}, if for $x_i \geq y_i \geq 0$, $i = 1,2,\ldots,n$ and $(x_1,\ldots,x_n) \in A$ we have $(y_1,\ldots,y_n) \in A$. For a coordinate-wise increasing function $f : {\R_{+}}^k {{\rightarrow}}\R$ the sets $f^{-1}((-\infty,t])$ are c-sets, and conversely the characteristic function of a c-set is a coordinate-wise decreasing function on ${\R_{+}}^k$. Similarily a function $f : {\R_{+}}^k {{\rightarrow}}\R$ is [*radius-wise increasing*]{} if $f(tx_1,tx_2,\ldots,tx_n) \geq f(x_1,x_2,\ldots,x_n)$ for $t > 1$, and a set $A$ is a [*radius-set*]{} if its characteristic function is radius-wise decreasing. We say a function $f : \R^n {{\rightarrow}}\R_+$ is [*log-concave*]{} if $\ln f$ is concave. A measure $\mi$ on $\R^n$ is called log-concave if for any nonempty $A, B \subset \R^n$ and $t \in (0,1)$ we have $\mi(tA + (1-t)B) \geq \mi(A)^t\mi(B)^{1-t}$. A classic theorem by Borell (see [@Bo]) states that any log-concave density not concentrated on any affine hyperplane has a density function, and that function is log-concave. A random vector in $\R^n$ is said to be log-concave if its distribution is log-concave. A sequence of random variables $(X_1,\ldots,X_n)$ is said to be [*negatively associated*]{}, if for any coordinate-wise increasing bounded functions $f,g$ and disjoint sets $\{i_1,\ldots,i_k\}$ and $\{j_1,\ldots,j_l\} \subset \{1,\ldots,n\}$ we have $$\label{NegAss} {{\rm Cov}}\big(f(X_{i_1},\ldots,X_{i_k}), g(X_{j_1},\ldots,X_{j_l})\big) \leq 0.$$ We say that the sequence $(X_j)$ is [*weakly negatively associated*]{} if inequality (\[NegAss\]) holds for $l = 1$, and [*very weakly negatively associated*]{} if (\[NegAss\]) holds for $l = k = 1$. For a 1-symmetric body $K \subset \R^n$ we can treat the body, or its positive quadrant, as a probability space, with the normalized Lebesgue measure as the probability. Formally, we consider $\Omega = K$, the Borel subsets of $K$ as the $\sigma$-family and $\P = \frac{1}{\lambda(K)}\lambda$ as the probability measure. We do similarly for ${K_{+}}$. We also define $n$ random variables $X_1, \ldots, X_n$, with $X_i$ being the $i$-th coordinate of a point $\omega \in {K_{+}}$ or $K$. Results ------- Our main subject of interest is to prove negative associacion type properties for some classes of symmetric bodies in $\R^n$. An straightforward approach is bound to fail due to the following proposition: If for a 1-symmetric body $K$ we consider the random vectors uniformly distributed on $K$ (not just on ${K_{+}}$) and the coordinate variables are very weakly negatively associated, then they are pairwise independent, and thus $K$ is a rescaled cube. Take any $i, j \in \{1,\ldots,n\}$ and any increasing functions $f, g : \R {{\rightarrow}}\R$. Then $f^\circ(x) = -f(-x)$ is increasing too. $K$ is 1-symmetric, so $(X_i,X_j)$ has the same joint distribution as $(-X_i,X_j)$, so $${{\rm Cov}}(f^\circ(X_i), g(X_j)) = {{\rm Cov}}\big(-f(-X_i), g(X_j)) = -{{\rm Cov}}(f(X_i), g(X_j)\big).$$ If both ${{\rm Cov}}(f(X_i),g(X_j))$ and $-{{\rm Cov}}(f(X_i),g(X_j))$ are non-positive, then ${{\rm Cov}}(f(X_i),g(X_j)) = 0$. This holds for every $i,j,f,g$. In particular for every $a,b$ we have $$\P (X_i \in [a,\infty) \cap X_j \in [b,\infty)) - \P (X_i \in [a,\infty)) \cdot \P(X_j \in [b,\infty)) = {{\rm Cov}}(\1_{[a,\infty)},\1_{[b,\infty)}) = 0.$$ A standard argument shows that $X_i$ and $X_j$ are independent, thus the density of $\1_K$ is a product density, so $K$ has to be a product of intervals. Thus, even very weak negative associacion for coordinate variables occurs only in the trivial case. The problem becomes more interesting if we look at the variables $|X_i|$ (or, equivalently, restrict ourselves to $X_i \geq 0$). K. Ball and I. Perissinaki in [@BP] prove the subindependence of coordinate slabs for $\ell^p$ balls, from which very weak negative association of $(|X_1|,\ldots,|X_n|)$ is a simple consequence. In the paper [@ja] Corollary 3.2 states that the sequence of variables $(|X_1|, \ldots, |X_n|)$ is very weakly negatively associated for generalized Orlicz balls. In this paper we shall prove that for a generalized Orlicz ball the sequence of variables $(|X_1|, \ldots, |X_n|)$ is negatively associated: \[orliczglownetw\] Let $K$ be an generalized Orlicz ball, and let $X_i$ be the coordinates of a random vector uniformly distributed on $K$. Then the sequence $|X_i|$ is negatively associated. We shall also prove an even stronger property of $\ell_p^n$ balls: \[lpglownetwierdzenie\] Take any $p \in [1,\infty)$ and any $n \in \N$. Let $m: \R_+ {{\rightarrow}}\R_+$ be any log-concave function and let $\mi$ be the measure on $\R^n$ with the density at $x$ equal to $m(\|x\|_p^p)$ normalized to be a probability measure. Let $I = \{i_1,\ldots,i_k\}, J = \{j_1,\ldots,j_l\}$ be two disjoint subsets of $\{1,2,\ldots,n\}$, and let $f : {\R_{+}}^{k} {{\rightarrow}}\R$, $g : {\R_{+}}^{l} {{\rightarrow}}\R$ be any radius-wise increasing functions bounded on ${{\rm supp}}\mi$. Let $X = (X_1,X_2,\ldots,X_n)$ be the vector distributed according to $\mi$. Then $${{\rm Cov}}(f(|X_{i_1}|,|X_{i_2}|,\ldots,|X_{i_k}|),g(|X_{j_1}|,|X_{j_2}|,\ldots,|X_{j_l}|)) \leq 0.$$ This is an equivalent of the above theorem, but the uniform distribution is replaced by the class of distribution with the density being a log-concave function of the $p$-th power of the $p$-th norm, and the coordinate-wise increasing function replaced by radius-wise decreasing functions. Let us comment on the organization of the paper. In the following subsection we shall state the main results and show a few corollaries which motivate these results. Section 2 is a collection of general lemmas, which allow us to reformulate the problem in a simpler fashion. In Section 3 a simple proof for the $\ell_p^n$ result is given. Section 4 introduces the definitions used in dealing with the generalized Orlicz ball case and investigates the basic properties of the defined objects. Section 5 states the $\theta$-theorem, which is the main tool of the proof, and gives a part of the proof. Section 6 contains the second part of the proof, which is a large transfinite inductive construction. Finally Section 7 applies the $\theta$-theorem to obtain the result for generalized Orlicz balls. Motivations ----------- This study was motivated by a desire to link the results achieved in convex geometry in [@ABP] for $\ell_p$ balls and in [@ja] for generalized Orlicz balls with an established theory, which will hopefully allow us to avoid repeating proofs already made in a more general case. For example, a form of the Central Limit Theorem for negative associated variables was already known in 1984 (see [@newman]). We also hope some new observations can be made using this approach. The negative association property is stronger then the sub-independence of coordinate slabs, which has been studied in the context of the Central Limit Theorem (see [@ABP], [@BP]). The statement of Theorem \[lpglownetwierdzenie\] was motivated by Theorem 6 of [@4a], where a proof of subindependence of coordinate slabs is given for a different class of measures with density dependent on the $p$-th norm, also including the uniform measure and the normalized cone measure on the surface. An example that can prove useful for applications in convex geometry is a pair of comparison inequalities due to Shao (see [@shao]). First, notice that as $|X_i|$ are negatively associated, they remain negatively associated when multiplied by any non-negative scalars (which amounts to multiplying $X_i$ by any scalars) and after the addition of any constant scalars. Thus the vectors $|a_i X_i| - c_i$ are negatively associated for any $a_i, c_i \in \R$. Shao’s inequalities, when applied to our case it will state the following: Let $K \subset \R^n$ be a generalized Orlicz ball, $(a_i)_{i=1}^n$ be any sequence of reals and $(X_i)_{i=1}^n$ be the coordinates of the random vector uniformly distributed on $K$. Then for any convex function $f: \R {{\rightarrow}}\R$ we have $${{\mathbb{E}}}f\Big(\sum_{i=1}^n |a_i X_i|\Big) \leq {{\mathbb{E}}}f\Big(\sum_{i=1}^n |a_i X_i^\star|\Big),$$ where $X_i^\star$ denote independent random variables with $X_i$ and $X_i^\star$ having the same distribution for each $i$. Additionally, if $f$ is increasing, then for any sequence of reals $(c_i)_{i=1}^n$ we have $${{\mathbb{E}}}f\Big(\max_{k = 1,2,\ldots,n} \sum_{i=1}^k |a_i X_i| - c_i \Big) \leq {{\mathbb{E}}}f\Big(\max_{k = 1,2,\ldots,n} \sum_{i=1}^k |a_i X_i^\star| - c_i \Big).$$ A more direct consequence is a moment comparision theorem suggested by R. Lata[ł]{}a (note we compare the moments of the sums of variables, and not their absolute values): Let $K \subset \R^n$ be a generalized Orlicz ball, $(a_i)_{i=1}^n$ be a sequence of reals and $(X_i)_{i=1}^n$ be the coordinates of the random vector uniformly distributed on $K$. Then for any even positive integer $p$ we have $${{\mathbb{E}}}\Big(\sum_{i=1}^n a_i X_i\Big)^p \leq {{\mathbb{E}}}\Big(\sum_{i=1}^n a_i X_i^\star\Big)^p,$$ with $X_i^\star$ defined as before. When we open the brackets in $(\sum a_i X_i)^p$ the summands in which at least one $X_i$ appears with an odd exponent average out to zero, as $K$ is 1-symmetric. Thus what is left is a sum of elements of the form $$(a_i X_1)^{2\alpha_1} (a_2 X_2)^{2\alpha_2} \ldots (a_n X_n)^{2\alpha_n} = |a_1 X_1|^{2\alpha_1} |a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n}.$$ If we put $f(a_1 x_1) = (a_1 x_1)^{2\alpha_1}$ and $g(a_2 x_2,\ldots,a_n x_n) = (a_2 x_2)^{2\alpha_2}\cdot \ldots\cdot (a_n x_n)^{2\alpha_n}$, applying negative association we get $$\begin{aligned} {{\mathbb{E}}}|a_1 X_1|^{2\alpha_1} |a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n} & \leq {{\mathbb{E}}}|a_1 X_1|^{2\alpha_1} {{\mathbb{E}}}|a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n} \\ & = {{\mathbb{E}}}|a_1 X_1^\star|^{2\alpha_1} {{\mathbb{E}}}|a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n} \\ & = {{\mathbb{E}}}|a_1 X_1^\star|^{2\alpha_1} |a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n} .\end{aligned}$$ Repeating this process inductively we separate all the variables and get $$\begin{aligned} {{\mathbb{E}}}\Big(\sum_{i=1}^n a_i X_i\Big)^p & = \sum_{\alpha_1 + \ldots + \alpha_n = p \slash 2} C_{\alpha_1,\ldots,\alpha_n} {{\mathbb{E}}}|a_1 X_1|^{2\alpha_1} |a_2 X_2|^{2\alpha_2} \ldots |a_n X_n|^{2\alpha_n} \leq \\ &\leq \sum_{\alpha_1 + \ldots + \alpha_n = p \slash 2} C_{\alpha_1,\ldots,\alpha_n} {{\mathbb{E}}}|a_1 X_1^\star|^{2\alpha_1} |a_2 X_2^\star|^{2\alpha_2} \ldots |a_n X_n^\star|^{2\alpha_n} = \\ & = {{\mathbb{E}}}\Big(\sum_{i=1}^n a_i X_i^\star\Big)^p.\end{aligned}$$ Finally, we can apply Shao’s maximal inequality to get a exponential concentration of the euclidean norm. Theorem 3 in [@shao] states: Let $(X_i)_{i=1}^n$ be a sequence of negatively associated random variables with zero means and finite second moments. Let $S_k = \sum_{i=1}^k X_i$ and $B_n = \sum_{i=1}^n {{\mathbb{E}}}X_i^2$. Then for all $x>0$, $a > 0$ and $0 < \alpha < 1$ $$\P \Big(\max_{1 \leq k \leq n} |S_k| \geq x\Big) \leq 2\P(\max_{1\leq k \leq n} |X_k| > a) + \frac{2}{1-\alpha} \exp\Bigg(-\frac{x^2 \alpha}{2(ax + B_n)} \cdot \Big(1+\frac{2}{3}\ln\Big(1 + \frac{ax}{B_n}\Big)\Big)\Bigg).$$ We say $K \subset \R^n$ is [*in isotropic position*]{} if $\lambda_n(K) = 1$ and ${{\mathbb{E}}}X_i^2 = L_K^2$ for some constant $L_K$ (any bounded convex set with a non-empty interior can be moved into isotropic position by an affine transformation, for more on this subject see e.g. [@MS]). Notice that if $|X_i|$ are negatively associated and $f_i$ are increasing, then $f_i(|X_i|)$ are also negatively associated. Thus the sequence $(X_i^2 - L_K^2)_{i=1}^n$ for $X = (X_i)_{i=1}^n$ uniformly distributed on a generalized Orlicz ball is also negatively associated. The moments of log-concave variables are comparable (see for instance [@klo], Section 2, remark 5), thus we have $${{\mathbb{E}}}(X_i^2 - L_K^2)^2 = {{\mathbb{E}}}X_i^4 + L_K^4 - 2 L_K^2 {{\mathbb{E}}}X_i^2 = {{\mathbb{E}}}X_i^4 - L_K^4 \leq 5 L_K^4.$$ If we put $\alpha = 1\slash 2$ and $x = nt$ in Shao’s inequality and apply the bound we got above for the variance we get \[shao1\]Let $K \subset \R^n$ be a generalized Orlicz ball in isotropic position, and $(X_i)_{i=1}^n$ be the coordinates of the random vector uniformly distributed on $K$. Then for any $t > 0$, $a > 0$ we have: $$\begin{aligned} \P\Big(\max_{1 \leq k \leq n} \Big|\sum_{i=1}^k (X_i^2 - L_K^2)\Big| > n t\Big) & \leq 2\P \Big(\max_{1\leq k \leq n} |X_k^2 - L_K^2| > a\Big) + \\ & + 4 \exp\Bigg(-\frac{nt^2}{4(at + 5 L_K^4)} \cdot \bigg(1+\frac{2}{3}\ln\Big(1 + \frac{at}{5L_K^4}\Big)\bigg)\Bigg).\end{aligned}$$ To apply this result probably an idea on what order of convergence is possible to achieve with this formula would be needed. To this end we give the following corollary: \[shao2\]Let $K \subset \R^n$ be a generalized Orlicz ball in isotropic position, and $(X_i)_{i=1}^n$ be the coordinates of the random vector uniformly distributed on $K$. Then for any $t > 0$ we have: $$\P\Big(\Big|\frac{\sum_{i=1}^n X_i^2}{n} - L_K^2\Big| > t\Big) \leq C e^{-cnt^2} + C n e^{-c\sqrt[3]{nt}},$$ where $C$ and $c$ are universal constants independent of $t$, $n$ and $K$. For $t > t_0$ a better bound (of the order of $e^{-t\sqrt{n}}$) is due to Bobkov and Nazarov (see [@BN]). However, frequently a bound for $t {{\rightarrow}}0$ is needed — for instance the proof of the Central Limit Theorem for convex bodies uses bounds for the concentration of the second norm for small $t$ (see for instance [@ABP]). In full generality (ie. for an arbitrary log-concave isotropic measure and for arbitrary $t$) such a result is given in a very recent paper by Klartag (see [@nowyKlartag]) with worse exponents — the bound for the probabilty is of the order of $e^{t^{ 3.33} n^{0.33}}$. Previous proofs of such results (see [@fgp], [@staryKlartag]) gave a logarithmic dependence of the exponent on $n$. The bound given in the corollary above is very rough, and in any particular case it is very likely it may be improved. However, we give it in order to show an explicit exponential bound in the concentration inequality which is uniform for all generalized Orlicz balls in a given dimension and applies for any $t > 0$. Obviously $$\P\Bigg(\Bigg|\frac{\sum_{i=1}^n X_i^2}{n} - L_K^2\Bigg| > t\Bigg) \leq \P\Bigg(\max_{1 \leq k \leq n} \Bigg|\sum_{i=1}^k (X_i^2 - L_K^2) \Bigg| > nt\Bigg),$$ so we have only to bound the right hand side in Corollary \[shao1\]. Put $a = \sqrt[3]{n^2t^2}$. We know (see [@isotropic]) that $L_K^2$ is bounded by some universal constant $L$ independent of $n$ and $K$ for any 1-symmetric body in $\R^n$. If $c$ is small enough and $C$ large enough, then for $a < L_K^2$ we have $$C n e^{-c\sqrt[3]{nt}} = Cn e^{-c\sqrt{a}} \geq 1.$$ Thus we may consider only the case $a > L_K^2$. In this case $$\begin{aligned} \P(\max_{1\leq k \leq n} |X_k^2 - L_K^2| > a) & \leq n \max_{1 \leq k \leq n} \P(|X_k^2 - L_K^2| > a) = n \max_k \P(X_k^2 > a + L_K^2) \\ & \leq n \max_k \P(X_k^2 > a) = n \max_k \P(|X_k| > \sqrt{a}).\end{aligned}$$ Due to the Brunn-Minkowski inequality $X_k$ is log-concave (see for instance [@ff]), we know that $Var(X_k) \leq L_K^2 < C$ and ${{\mathbb{E}}}X_k = 0$, and thus $P(|X_k| > t) \leq c_1 e^{-c_2t}$ for some universal constants $c_1$ and $c_2$ independent of the distribution of $X_k$ and of $t$ (Borell’s Lemma, see for instance [@MS]). Thus we get $$\P\Big(\max_{1\leq k \leq n} |X_k^2 - L_K^2| > a\Big) \leq c_1 e^{-c_2\sqrt{a}} = c_1 e^{-c_2 \sqrt[3]{nt}}.$$ In the second part we shall simply bound $$\Bigg(1+\frac{2}{3}\ln\Big(1 + \frac{at}{5L_K^4}\Big)\Bigg) \geq 1.$$ Then $$4 \exp\Bigg(-\frac{nt^2}{4(at + 5 L_K^4)} \cdot \bigg(1+\frac{2}{3}\ln\Big(1 + \frac{at}{5L_K^4}\Big)\bigg)\Bigg) \leq 4 \exp\big(-\frac{nt^2}{4n^{2\slash 3}t^{5\slash 3} + 20 L_K^4}\big) \leq C e^{-c\sqrt[3]{nt}} + C e^{-cnt^2}.$$ Acknowledgements ---------------- We would very much like to thank Rafa[ł]{} Lata[ł]{}a, who encouraged us to write the paper, was the first person to read it and check the reasoning, and helped improve the paper in innumerable aspects. He also taught us most of what we know in the subject. We would also like to thank prof. Stanis[ł]{}aw Kwapie[ń]{}, who first suggested to us the idea of searching for negative-association type properties for convex bodies. Easy facts ========== Simplifying ----------- We want to prove inequality (\[NegAss\]) for various classes of functions (coordinate-wise increasing in the case of Theorem \[orliczglownetw\] and radius-wise increasing in the case of Theorem \[lpglownetwierdzenie\]). We may assume $k+l = n$ by putting $\tilde{g}(x_{j_1},\ldots,x_{j_l},x_{r_1},\ldots,x_{r_{n-l-k}}) = g(x_{j_1},\ldots,x_{j_l})$. For convienience we shall assume that the Lebesgue volume of ${K_{+}}$ is 1 (inequality (\[NegAss\]) is invariant under homothety). It will be more convienient to work with c-sets or radius-sets than with functions, which motivates the following Lemma: \[prelim\] Let $\mi$ be any probability measure on ${\R_{+}}^n$ and let $X = (X_1,X_2,\ldots,X_n)$ be the random vector distributed according to $\mi$. Assume that for given $0 \leq k,l \leq n$ we have two families of bounded functions $\mathcal{F}$ on ${\R_{+}}^k$ and $\mathcal{G}$ on ${\R_{+}}^l$. Let $\mathcal{A} = \{f^{-1}(-\infty,t] : f \in \mathcal{F}, t\in \R\}$, and similarly $\mathcal{B}$ for $\mathcal{G}$. If for any $A \in \mathcal{A}$ and $B \in \mathcal{B}$ we have $$\label{wzorek_mi2} \mi(A\times B) \mi({{\bar{A}}}\times {{\bar{B}}}) \leq \mi(A \times {{\bar{B}}}) \mi({{\bar{A}}}\times B),$$ then inequality (\[NegAss\]) holds for $X$ and any $f \in \mathcal{F}, g \in \mathcal{G}$. In particular, if inequality (\[wzorek\_mi2\]) holds for any $k$ and for any c-sets $A,B$, then the random variables $X_1,X_2,\ldots,X_n$ are negatively associated. Let us take any two functions $\mathcal{F} \ni f : {\R_{+}}^k {{\rightarrow}}\R$ and $\mathcal{G} \ni g: {\R_{+}}^l {{\rightarrow}}\R$. As covariance is bilinear and is 0 if one of the functions is constant, we may assume without loss of generality that $f$ and $g$ are non-negative. For non-negative functions we have $$f(x) = \int_0^\infty \1_{f^{-1}[t,\infty)} (x)\ dt.$$ Thus (again, by the bilinearity of the covariance) we can restrict ourselves to functions $f$ and $g$ of the form $1 - \1_A$ and $1 - \1_B$, where $A \in \mathcal{A}$ and $B \in \mathcal{B}$. Since ${{\rm Cov}}(1-\1_A, 1-\1_B) = {{\rm Cov}}(\1_A, \1_B)$, we have to prove that ${{\rm Cov}}(\1_A, \1_B) \leq 0$. Let us denote by [[$\mathbf{X}$]{}]{} the $k$-dimensional vector $(X_{i_1},\ldots,X_{i_k})$ on which $f$ is taken, and by [[$\mathbf{Y}$]{}]{}  the $l$-dimensional vector on which $g$ is taken. Then $$\begin{aligned} {{\rm Cov}}\big(\1_A({{\ensuremath{\mathbf{X}}}}),\1_B({{\ensuremath{\mathbf{Y}}}})\big) &= {{\mathbb{E}}}\1_A({{\ensuremath{\mathbf{X}}}})\1_B({{\ensuremath{\mathbf{Y}}}}) - {{\mathbb{E}}}\1_A({{\ensuremath{\mathbf{X}}}}) {{\mathbb{E}}}\1_B({{\ensuremath{\mathbf{Y}}}}) = \mi(A \times B) - \mi(A \times \R^l) \mi(\R^k \times B) \\ &= \mi(A \times B) \mi\big((A \cup {{\bar{A}}}) \times (B \cup {{\bar{B}}})\big) - \mi\big(A \times (B \cup {{\bar{B}}})\big) \mi \big((A \cup {{\bar{A}}}) \times B\big) \\ &= \mi(A\times B) \mi({{\bar{A}}}\times {{\bar{B}}}) - \mi (A \times {{\bar{B}}}) \mi({{\bar{A}}}\times B),\end{aligned}$$ which is non-positive by (\[wzorek\_mi2\]). Simple proportion lemmas ------------------------ During the course of further proofs we shall frequently need to compare two ratios of integrals of the same functions over different sets. In this subsection we will demonstrate some simple properties of ratios of integrals. \[obvi\] Let $a,b \geq 0$ and $c,d > 0$. Then the following are equivalent: - $\frac{a}{c} \geq \frac{b}{d}$, - $\frac{a}{c} \geq \frac{a+b}{c+d}$, - $\frac{a+b}{c+d} \geq \frac{b}{d}$. Whenever there is equality in one of the inequalities, all aforementioned fractions are equal. \[rosncalk\]\[rosnmix\]\[gencheb\] Let $\mi$ be a non-negative measure on $\R$ supported on the (possibly unbounded) interval $[l_\mi,r_\mi]$. Suppose that $f, g, h:\R {{\rightarrow}}{\R_{+}}$ are functions bounded on ${{\rm supp}}\mi$, positive on the interior of their supports, satisfying: 1. The support of any function $u \in \{f,g,h\}$ is an interval $[l_u,r_u]$ (possibly unbounded), 2. $\frac{f}{g}$ is a decreasing function where defined, and $r_f \leq r_g$, 3. $h$ is an increasing function, Then: 1. For any $a<b<c$, $b \in (l_\mi, r_\mi) \cap (l_g,r_g)$ we have $$\frac{\int_a^b f(x)d\mi}{\int_a^b g(x)d\mi} \geq \frac{f(b)}{g(b)} \hbox{ and } \frac{f(b)}{g(b)} \geq \frac{\int_b^c f(x)d\mi}{\int_b^c g(x)d\mi}$$ whenever both sides of an inequality are defined. 2. Moreover, if for some $a < b < c$ we have two equalities in inequality (1a) then $\frac{f(x)}{g(x)}$ is constant on $(a,c) \cap {{\rm supp}}g \cap {{\rm supp}}\mi$ and for any $a \leq s < t \leq c$ $$\frac{\int_s^t f(x) d\mi}{\int_s^t g(x) d\mi}$$ is equal to $f(b) \slash g(b)$ if defined. 3. For any points $a,b,c,d$ satisfying $a < b \leq d$ and $a \leq c < d$ we have: $$\frac{\int_a^b f(x)d\mi}{\int_a^b g(x)d\mi} \geq \frac{\int_c^d f(x)d\mi}{\int_c^d g(x)d\mi}$$ whenever both sides are defined. 4. Moreover, if this inequality is an equality and either $\int_a^c g(x) d\mi(x)$ or $\int_b^d g(x) d\mi(x)$ is strictly positive, then $\frac{f}{g}$ is constant on $[a,d]$ where defined, and we have an equality for any $a \leq a' \leq b' \leq d' \leq d$ and $c' \in [a',d']$ if both sides are defined. 5. If $l_g = l_f$ the following inequality occurs for any interval $I$: $$\frac{\int_I f(x) d\mi(x)}{\int_I g(x) d\mi(x)} \geq \frac{\int_I f(x) h(x) d\mi(x)}{\int_I g(x) h(x) d\mi(x)}$$ if both sides are defined. <!-- --> 1. Consider the first inequality. Let $a' = \max\{l_\mi,l_g,a\}$ . We have $a \leq a' < b$ (otherwise the denominator of the left-hand side would be undefined). Also $\int_a^b g(x) d\mi(x) = \int_{a'}^b g(x) d\mi(x) > 0$ and $g > 0$ on $(a',b]$ (it has to be positive in $b$ or the right-hand side would be undefined). Thus $$\frac{\int_a^b f(x)d\mi(x)}{\int_a^b g(x)d\mi(x)} \geq \frac{\int_{a'}^b f(x)}{\int_{a'}^b g(x)} = \frac{\int_{a'}^b g(x) \frac{f(x)}{g(x)}}{\int_{a'}^b g(x)} \geq \frac{\int_{a'}^b g(x) \frac{f(b)}{g(b)}}{\int_{a'}^b g(x)} = \frac{f(b)}{g(b)},$$ A similar reasoning with $c' = \min \{r_\mi,r_g,c\}$ proves the second inequality (note $r_f \leq r_g$, so the first inequality in the reasoning above becomes an equality). 2. If equality occurs, then $\frac{f(x)}{g(x)} = \frac{f(b)}{g(b)}$ for almost all $x \in (a',c')$ as $g$ is strictly positive on $(a',c')$. As $\frac{f}{g}$ is decreasing, if it is constant on almost whole $(a',c')$, it is constant on the whole interval and thus $$\frac{\int_s^t f(x) d\mi(x)}{\int_s^t g(s) d\mi(x)} = \frac{f(b)}{g(b)}$$ if defined for any $s,t \in (a',c')$. We know $\int_a^{a'} g(x) d\mi(x) = \int_{c'}^c g(x) d\mi(x) = 0$, so to have equalities we also have to have $\int_a^{a'} f(x) d\mi(x) = \int_{c'}^c f(x) d\mi(x) = 0$, thus $\int_s^t f(x)d\mi(x) = \int_{(s,t) \cap (a',c')} f(x) d\mi(x)$ and similarly for $g$, thus the thesis. 3. Let $F(x,y) = \int_x^y f(t)$ and $G(x,y) = \int_x^y g(t)$. As the left-hand side is defined, $G(a,b) > 0$ and thus $G(a,d) > 0$. We apply [*(1a)*]{} to get: $$\label{insideeqrosnmix} \frac{F(a,b)}{G(a,b)} \geq \frac{F(b,d)}{G(b,d)}$$ if the right-hand side is defined and from Fact \[obvi\] we have $$\frac{F(a,b)}{G(a,b)} \geq \frac{F(a,b) + F(b,d)}{G(a,b) + G(b,d)} = \frac{F(a,d)}{G(a,d)}.$$ If the right-hand side in (\[insideeqrosnmix\]) was not defined, $G(b,d) = 0$ and thus $F(b,d) = 0$ as $r_f \leq r_g$, so $\frac{F(a,b)}{G(a,b)} \geq \frac{F(a,d)}{G(a,d)}$. Similarly from [*(1a)*]{} $$\frac{F(a,c)}{G(a,c)} \geq \frac{F(c,d)}{G(c,d)}$$ if the left-hand side is defined, and thus from Fact \[obvi\] $$\frac{F(a,d)}{G(a,d)} \geq \frac{F(c,d)}{G(c,d)}.$$ If the left-hand side was undefined, $G(a,d) = G(c,d)$ and obviously $F(a,d) \geq F(c,d)$, so we get the same inequality. Linking the two inequalities we get the thesis. 4. Suppose $G(b,d) > 0$. As $$\frac{F(a,b)}{G(a,b)} \geq \frac{F(a,d)}{G(a,d)} \geq \frac{F(c,d)}{G(c,d)}$$ and the first and last expressions are equal, all inequalities are in fact equalities. Thus from the first one of them and Fact \[obvi\] we get $$\frac{F(a,b)}{G(a,b)} = \frac{F(b,d)}{G(b,d)},$$ and applying [*(1b)*]{} we get the thesis. 5. Let $I' = I \cap {{\rm supp}}g$. As ${{\rm supp}}f \subset {{\rm supp}}g$ all integrals in the thesis over $I$ are equal to the appropriate integrals over $I'$. Consider the functions $h$ and $\frac{f}{g}$ on the interval ${{\rm Int}}I'$ (note $\frac{f}{g}$ is defined on ${{\rm Int}}I'$) taken with a measure with density $\frac{g(x)}{\int_{I'} g(t) d\mi(t)}d\mi$ (this is defined as the left-hand side in the thesis was defined, so $\int_{I'} g(t) d\mi(t) > 0$). From the continuous Chebyshev sum inequality (that is, if $F$ is increasing and $G$ is decreasing, then $\int F\int G \geq \int FG \int 1$) we know $$\begin{aligned} \int_{I'} h(x) \frac{g(x)}{\int_{I'} g(t) d\mi(t)} d\mi(x) & \int_{I'} \frac{f(x)}{g(x)} \frac{g(x)}{\int_{I'} g(t) d\mi(t)} d\mi(x) \\ \geq & \int_{I'} h(x) \frac{f(x)}{g(x)} \frac{g(x)}{\int_{I'} g(t) d\mi(t)}d\mi(x) \int_{I'} \frac{g(x)}{\int_{I'} g(t) d\mi(t)}d\mi(x).\end{aligned}$$ Multiplying both sides by $[\int_{I'} g(t) d\mi(t)]^2$ we get the thesis. \[rosdolp\] Let $\mi$ be a non-negative measure on $I \subset \R$. Suppose $f,g,p,q: I {{\rightarrow}}{\R_{+}}$ are functions satisfying $f(x) g(y) \geq f(y) g(x)$ for $x \geq y$ and $p(x) q(y) \leq p(y) q(x)$ for $x \geq y$. Then $$\int_I p(x) f(x) d\mi(x) \int_I q(x) g(x) d\mi(x) \leq \int_I p(x) g(x) d\mi(x) \int_I q(x) f(x) d\mi(x).$$ Using Fubini’s theorem we have to prove $$\int_I \int_I p(x)f(x)q(y)g(y)\ d\mi(y)\ d\mi(x) \leq \int_I \int_I p(y)f(x)q(x)g(y)\ d\mi(y)\ d\mi(x).$$ Multiplying sides by two and changing names $x$ and $y$: $$\int_I \int_I \big[p(x)f(x)q(y)g(y) + p(y)f(y)q(x)g(x) - p(x)f(y)q(y)g(x) - p(y)f(x)q(x)g(y)\big] \ d\mi(y)\ d\mi(x)\leq 0$$ $$\int_I \int_I \big(p(x)q(y) - p(y)q(x)\big)\big(f(x)g(y)-f(y)g(x)\big) \ d\mi(y) \ d\mi(x) \leq 0,$$ which follows from the assumptions, as the integrand is always non-positive. \[dividesets\]\[dividepoints\] Suppose $f,g : X {{\rightarrow}}{\R_{+}}$ are defined on any set $X$ with a measure $\mi$. Let $\{ D_i\}_{i\in I}$ be a family of disjoint subsets of $X$. If $$t\int_{D_i} g(x) d\mi(x) \geq \int_{D_i} f(x) d\mi(x) \geq s\int_{D_i} g(x) d\mi(x)$$ for some $t,s \in \R \cup \{-\infty,\infty\}$, then $$t\int_{\bigcup_i D_i} g(x) d\mi(x) \geq \int_{\bigcup_i D_i} f(x) d\mi(x)\geq s\int_{\bigcup_i D_i} g(x) d\mi(x).$$ If $X = X_1 \times X_2$ and $\mi = \mi_1 \otimes \mi_2$, and for some set $D\subset X_1 \times X_2$ and any $x_1 \in X_1$ we have $$t\int_{(\{x_1\} \times X_2) \cap D} g(x) d\mi_2(x) \geq \int_{(\{x_1\} \times X_2) \cap D} f(x) d\mi_2(x) \geq s\int_{(\{x_1\} \times X_2) \cap D} g(x) d\mi_2(x),$$ then $$t\int_D g(x) d\mi(x) \geq \int_D f(x) d\mi(x) \geq s\int_D g(x) d\mi(x).$$ In the first case, we should add all the inequalities by sides. In the second case, we should not sum but integrate using Fubini’s theorem. The $\ell_p^n$ ball case ======================== First we shall give the proof for $\ell_p^n$ balls. Recall the $\ell_p^n$ ball is the generalized Orlicz ball defined by the Young functions $f_i(x) = |x|^p$. We include this case for two reasons: first, it is much simpler than the Orlicz ball case, and serves as a good illustration of what is happening, and second, because we are able to achieve a stronger result, namely prove Theorem \[lpglownetwierdzenie\]. Note that in particular we can take $m$ to be $c_r \1_{[0,r]}$ to get the result for the uniform measure on the $\ell_p^n$ ball. As any coordinate-wise increasing function is radius-wise increasing, this result is stronger than the negative associacion property we prove for generalized Orlicz balls. By a simple approximation argument we can also get the result above for $\mi$ being the cone measure on the surface of $\ell_p^n$. Let $B_p^n$ denote the $\ell_p^n$ ball. Let $M(x_1,x_2,\ldots,x_n) = (|x_1|,|x_2|,\ldots,|x_n|)$ and let $\tilde{\mi}$ be defined by $\tilde{\mi}(A) = \mi(M^{-1}(A))$. Notice $\tilde{\mi}$ describes the distribution of $(|X_1|,|X_2|,\ldots,|X_n|)$. As $\mi$ is 1-symmetric, we may equivalently define $\tilde{\mi}$ as $2^n$ times the restriction of $\mi$ to ${\R_{+}}^n$. Recall that the cone measure on $\partial B_p^n$ (that is, the boundary of $B_p^n$), which we shall denote $\nu$, is defined for $A \subset \partial B_p^n$ by $$\nu_n(A) = \frac{\lambda_n(ta : t \in \R, a \in A, ta \in B_p^n)}{\lambda_n(B_p^n)}.$$ For this measure we have the polar integration formula: $$\int_{\R^n} f(x) dx = n \lambda_n(B_p^n) \int_{R_+} r^{n-1} \int_{\partial B_p^n} f(r\theta) d\nu_n(\theta) dr.$$ Let $C_n = n \lambda_n(B_p^n)$. Due to Lemma \[prelim\] we only need to prove inequality $\tilde{\mi}(A \times B) \tilde{\mi}({{\bar{A}}}\times {{\bar{B}}}) \leq \tilde{\mi}(A\times {{\bar{B}}}) \tilde{\mi} ({{\bar{A}}}\times B)$ for any radius-sets $A,B$, which is equivalent to $\mi(A \times B)\mi({{\bar{A}}}\times {{\bar{B}}}) \leq \mi(A\times {{\bar{B}}})\mi({{\bar{A}}}\times B)$. We have: $$\begin{aligned} \mi(A\times B) &= \int_{\R^k} \int_{\R^{n-k}} \1_{A}(x) \1_B(y) m(\|x\|_p^p + \|y\|_p^p) dx dy = \\ &= \int_{\R_+} \int_{\partial B_p^k} \int_{\R^{n-k}} C_k r^{k-1} \1_{A}(r\theta) \1_B(y) m(r^p + \|y\|_p^p) d\nu_k(\theta) dr dy \\ &= \int_{\R_+} \bigg[\int_{\R^{n-k}} \1_B(y) m(r^p + \|y\|_p^p) dy \bigg] \bigg[\int_{\partial B_p^k} \1_A(r\theta) d\nu_k(\theta)\bigg] C_k r^{k-1} dr.\end{aligned}$$ Denote $f_B(r) = \int_{\R^{n-k}} \1_B(y) m(r^p + \|y\|_p^p) dy$ and $g_A(r) = \int_{\partial B_p^k} \1_A(r\theta) d\nu_k(\theta)$. Let $\sigma_1$ be the measure on $\R_+$ with density $C_k r^{k-1}$. We can perform similar operations for the other three expressions in inequality (\[wzorek\_mi2\]). What we have to prove becomes the inequality $$\int_{\R_+} f_B(r) g_A(r) d\sigma_1(r) \int_{\R_+} f_{{{\bar{B}}}}(r) g_{{{\bar{A}}}}(r) d\sigma_1(r) \leq \int_{\R_+} f_{{{\bar{B}}}}(r) g_A(r) d\sigma_1(r) \int_{\R_+} f_B(r) g_{{{\bar{A}}}}(r) d\sigma_1(r).$$ Due to lemma \[rosdolp\] it is enough to prove the following two inequalities: $$\begin{aligned} f_B(r_1) f_{{{\bar{B}}}}(r_2) \geq f_B(r_2) f_{{{\bar{B}}}}(r_1) \hbox{ for } r_1 \geq r_2, \label{fffdd} \\ g_A(r_1) g_{{{\bar{A}}}}(r_2) \leq g_A(r_2) g_{{{\bar{A}}}}(r_1) \hbox{ for } r_1 \geq r_2. \label{dddff}\end{aligned}$$ Inequality (\[dddff\]) is simple — $\1_A(r\theta)$ is decreasing as a function of $r$ for any fixed $\theta$, while $\1_{{{\bar{A}}}}(r\theta)$ is increasing, as $A$ is a radius-set. Thus $g_A(r)$ is decreasing, $g_{{{\bar{A}}}}$ is increasing, so $g_A(r_1) \leq g_A(r_2)$ and $g_{{{\bar{A}}}}(r_2) \leq g_{{{\bar{A}}}}(r_1)$. Inequality (\[fffdd\]) will require a bit more work. We have: $$\begin{aligned} f_B(r_1) &= \int_{\R^{n-k}} \1_B(y) m(r_1^p + \|y\|_p^p) dy \\ &= \int_{\R_+} \int_{\partial B_p^{n-k}} C_{n-k} s^{n-k-1} \1_B(s\xi) m(r_1^p + s^p) d\nu_{n-k}(\xi) dr \\ &= \int_{\R_+} \bigg[m(r_1^p + s^p)\bigg] \bigg[\int_{\partial B_p^{n-k}} \1_B(s\xi) d\nu_{n-k}(\xi)\bigg] C_{n-k} s^{n-k-1} ds.\end{aligned}$$ We are going to use Lemma \[rosdolp\] once again. Let $p_{r_1}(s) = m(r_1^p + s^p)$ and $q_B(s) = \int_{\partial B_p^{n-k}} \1_B(s\xi) d\nu_{n-k}(\xi)$ and $\sigma_2$ the measure with density $C_{n-k} s^{n-k-1}$. We do the similar calculation for the other three expressions in inequality (\[fffdd\]), and it becomes $$\int_{\R_+} p_{r_1}(s) q_B(s) d\sigma_2(s) \int_{\R_+} p_{r_2}(s) q_{{{\bar{B}}}}(s) d\sigma_2(s) \geq \int_{\R_+} p_{r_2}(s) q_B(s) d\sigma_2(s) \int_{\R_+} p_{r_1}(s) q_{{{\bar{B}}}}(s) d\sigma_2(s).$$ Applying Lemma \[rosdolp\] we have to prove $$\begin{aligned} p_{r_1}(s_1) p_{r_2}(s_2) \leq p_{r_2}(s_1) p_{r_1}(s_2) \hbox{ for } s_1 \geq s_2, \label{fdfd} \\ q_B(s_1) q_{{{\bar{B}}}}(s_2) \leq q_{{{\bar{B}}}}(s_1) q_B(s_2) \hbox{ for } s_1 \geq s_2. \label{dfdf}\end{aligned}$$ Inequality (\[dfdf\]) is proved in the same way as inequality (\[dddff\]) — $q_B$ is decreasing and $q_{{{\bar{B}}}}$ is increasing. Inequality (\[fdfd\]) means $$m(r_1^p + s_1^p) m(r_2^p + s_2^p) \leq m(r_2^p + s_1^p) m(r_1^p + s_2^p),$$ which follows from the log-concavity of $m$. As we saw, this proof was quite simple. Unfortunately, it takes advantage of the fact that the Young function of the $\ell_p^n$ ball scales well with the radius, that is, that $f_i(tx_i) = \phi(t)f_i(x_i)$ for some function $\phi$. Of all Orlicz ball only the $\ell_p$ balls have this property, which makes it impossible to apply the same proof to the generalized Orlicz ball case. The generalized Orlicz ball case — preliminaries, the proper measure, lens sets =============================================================================== Idea of the proof ----------------- We would like to transfer the result given above for $\ell_p^n$ balls to the more general case of generalized Orlicz balls. In the generalized Orlicz ball cas the Young function does not, unfortunately, scale with the radius, and this creates the need for a different approach. Again by Lemma \[prelim\] we can restrict ourselves to characteristic functions of c-sets. As generalized Orlicz balls are 1-symmetric, we can restrict ourselves to the positive quadrant of our generalized Orlicz ball. We shall proceed in two steps. The first will be to prove that generalized Orlicz balls satisfy inequality (\[NegAss\]) if one of the functions, say $g$, is univariate — in other words, to begin by proving weak negative association. This is equivalent to proving \[wzorek\_mi2\] for one of the sets, say $B$, being one-dimensional. Due to Lemma \[rosncalk\], part 1, we will simply need to prove that the function $\frac{\lambda_{n-1}(A \times \{z\} \cap K)}{\lambda_{n-1}({{\bar{A}}}\times \{z\} \cap K)}$ is decreasing with $z$. Thus, we take any $z_2 > z_1 \geq 0$ and concentrate on them. We want to prove $$\frac{\lambda_{n-1}(A \times \{z_1\} \cap K)}{\lambda_{n-1}({{\bar{A}}}\times \{z_1\} \cap K)} \leq \frac{\lambda_{n-1}(A \times \{z_2\} \cap K)}{\lambda_{n-1}({{\bar{A}}}\times \{z_2\} \cap K)}.$$ Switching the right denominator with the left numerator we get $$\frac{\lambda_{n-1} ((\{z_2\} \times A) \cap K)}{\lambda_{n-1} ((\{z_1\} \times A) \cap K)} \geq \frac{\lambda_{n-1} ((\{z_2\} \times {{\bar{A}}}) \cap K)}{\lambda_{n-1} ((\{z_1\} \times {{\bar{A}}}) \cap K)}$$ as the inequality we need to prove. We shall denote the proportion of the measure of $K_{z_2}$ to the measure of $K_{z_1}$ on a given set $D$ by $\theta(D)$. The second step will be to pass from the univariate case to the general case. It turns out that a very similar argument, using the proportion $\frac{\lambda(D\cap {{\bar{B}}})}{\lambda(D \cap B)}$ as $\theta(D)$ will allow us to do that. Thus, to avoid repetition (as the argument is quite long), we shall take the properties of both of these functions which make the similar arguments possible and call any function with such properties a $\Theta$-function, then attempt to prove $$\label{ttt}\theta(K \cap A) \geq \theta(K) \geq \theta(K \cap {{\bar{A}}})$$ for any $\Theta$-function $\theta$. Section 4 is devoted to defining the concepts used in the proof (subsection 4.2) and proving general lemmas about those concepts (subsections 4.3, 4.4 and 4.5). In particular, the properties defining a $\Theta$-function are given. Section 7 assumes inequality \[ttt\] and proves Theorem \[orliczglownetw\]. Sections 5 and 6 are devoted to the proof of inequality \[ttt\]. The idea of Section 7 is quite simple — a Brunn-Minkowski argument and a few approximations are enough to verify that the appropriate functions considered for generalized Orlicz balls are in fact $\Theta$-functions. The main line of the reasoning is similar to [@ja]. To prove inequality \[ttt\] we shall attempt to divide the set ${K_{+}}$ into appropriately small convex subsets $D$ for which $\theta(D) = \theta(K)$. On each of these sets we will prove inequality (\[ttt\]) with $D$ substituted for $K$, which proves the thesis ($\theta$ is a proportion, so if it is attains some value on a family of disjoint sets, it attains the same value on the sum of this family). The problem, of course, is to prove the inequality (\[ttt\]) for any set $D$ (this is the aim of Section 5) and to construct a division into suitable sets $D$ (this is the aim of Section 6). For Section 5, the sets $D$ will have to be of the form $\tilde{D} \times \R^{n-2}$, where $\tilde{D}$ is 2-dimensional. Moreover, we will need $\tilde{D}$ to be “long and narrow”. This will allow us to take one direction (the one in which $\tilde{D}$ is “long”) to be a new coordinate, replacing the two coordinates of $\tilde{D}$, and to approximate the set $A$ and the function $\theta$ on $D$ with their approximations constant in the other, “narrow”, variable. If the approximation is good enough (and it turns out to be), we can inductively use the inequality (\[ttt\]) for the $n-1$ dimensional case for the approximating functions and then transfer the result to the original functions. We cannot reasonably expect the sets $D$ to have constant width in the “narrow” coordinate. This means that in the inductive step we shall have to consider weighted measures to take this into account. This motivates us to consider a more general theorem, in which the Lebesgue measure on $K$ will be replaced by a proper weighted measure. The argument in Section 6 is somewhat similar to the Kanaan–Lovasz–Simonovits localization lemma. However, we need the sets $D$ to satisfy additional assumptions, in particular to be “positively inclined” (this roughly means that the “long” coordinate axis has to be of the form $y = ax + b$, where $a$ is positive). We were unable to fit this into the localization lemma scheme, so the division is done by hand. We prove in Section 5 we can cut off a “good” set $D$ from our ball. Unfortunately, we have no control of the measure of the set we cut off (apart from the fact it is positive). Thus inductive cutting off good sets does not necassarily cover the whole $K$. This leads us to a transfinite inductive reasoning, where we cut off “good” sets in a transfinite fashion (that is, after cutting off countably many we see what is left and continue cutting). This approach leads to a number of technical problems associated with the limit step, and Section 6 is devoted to dealing with these problems and following through with the transfinite induction. Definitions ----------- For the convienience of the reader all the basic definitions have been gathered in one place. So here we will just introduce the concepts required in the proof, and the next sections will be devoted to gaining a deeper understanding of those concepts. We shall usually consider a generalized Orlicz ball $K \subset \R_x \times \R_y \times \R^{n-2}$. By $K_{x = u}$ we shall mean the section of ${K_{+}}$ with the hyperplane $x = u$, similarly for any other variable in $\R^n$. For a given set $D \subset \R_x \times \R_y \times \R^{n-2}$ by ${{\tilde{D}}}$ we shall denote the projection of $D$ to $\R_x \times \R_y$. If not said otherwise, we shall assume $D = {{\tilde{D}}}\times \R^{n-2}$. A function $f:\R^n {{\rightarrow}}[0,\infty)$ is called [*$1\slash m$-concave*]{} if its support is a convex set and the function $f^{1\slash m}$ is concave on its support. \[propermeasure\] Let $K \subset \R^n$ be a generalized Orlicz ball. A measure $\mi$ on $\R^n$ is called a [*proper measure*]{} with respect to $K$ for $\R^n = \R_x \times \R_y \times \R^{n-2}$ ($n \geq 2$) if the following conditions are satisfied: - $\mi$ is a non-negative measure with density $f(x) g(y) \1_{{K_{+}}}$. - The functions $f$ and $g$ are $1\slash m$-concave for some $m > 0$. - If $K_{x = x_0} = \emptyset$ for a given $x_0$ then $f(x_0) = 0$, and if $K_{y = y_0} = \emptyset$ for a given $y_0$ then $g(y_0) = 0$. In the case $n = 1$ a proper measure is a non-negative measure with a $1\slash m$-concave density $f$ for some $m > 0$, satisfying ${{\rm supp}}f \subset {K_{+}}$. This definition describes the “proper weighted measures” which we will have to analyze in the subsequent induction steps of the proof outlined above. We shall denote the support of $f$ by $[x_-,x_+]$ and the support of $g$ by $[y_-,y_+]$. Of course $0 \leq x_- \leq x_+$ and similarly for $y$. If we have a proper measure on $\R^n$ with respect to $K$ we can define a [*lens set*]{}. This definition describes the shape of a set, which will be one of the conditions of “not losing too much on approximation” and also will be a condition under which further dividing will be possible. A set $D \subset \R_x \times \R_y \times \R^{n-2}$ is called a [*lens set*]{} if: - $D$ is a convex set, - $D = \tilde{D} \times \R^{n-2}$, - for some $x_- \leq x_1 < x_2 \leq x_+$ and $y_- \leq y_1 < y_2 \leq y_+$, we have ${{\tilde{D}}}\subset [x_1,x_2] \times [y_1, y_2]$ and $(x_1, y_1) \in {{\tilde{D}}}$ and $(x_2, y_2) \in {{\tilde{D}}}$, - $\mi(D) > 0$. A lens set is said to be a [*strict lens set*]{} if $x_- < x_1 < x_2 < x_+$, $y_- < y_1 < y_2 < y_+$ and points $(x_1, y_1)$ and $(x_2, y_2)$ are the only points of ${{\rm cl}}{{\tilde{D}}}$ belogning to the boundary of the rectangle $[x_1, x_2] \times [y_1, y_2]$. Note that the boundary of the projection of a strict lens set onto $\R_x \times \R_y$ consists of an upper part, which is a graph of an concave, strictly increasing function, and a lower part, which is the graph of a convex, strictly increasing function. The boundary of a (non-strict) lens set may additionaly contain horizontal and vertical intervals adjacent to $(x_1,y_1)$ and $(x_2,y_2)$. We shall speak of the upper-left border and the lower-right border of a lens set. For a lens set $D$ we define the [*extremal points*]{} of ${{\tilde{D}}}$ to be two points $(x_1,y_1)$ and $(x_2,y_2)$. From the definition of a lens set, the extremal points belong to ${{\tilde{D}}}$. The [*extremal line*]{} of a lens set is the line connecting extremal points. By the [*width*]{} of a lens set we shall mean the length of its projection upon the line perpendicular to its extremal line in the plane $\R_x \times \R_y$. For a line $L$ in $\R_x \times \R_y$ the [*inclination*]{} of $L$ will denote measure of the angle between $\R_x$ and $L$ oriented so that the inclination of the line $\{x = y\}$ is $\pi \slash 4$. A line is said to have [*positive inclination*]{} if its inclination belongs to $(0,\pi \slash 2)$, and [*non-negative inclination*]{} if the inclination belongs to $[0,\pi\slash 2]$. The inclination of a lens set $D$ is simply the inclination of its extremal line. By a [*positively inclined hyperplane*]{} in $\R^n$ we mean a hyperplane $H$ defined by $x_i = \lambda x_j + c$, where $\lambda \geq 0$. For a given convex set $D$ and a proper measure $\mi$ by the relevant diameter of $D$ we mean the diameter of $D \cap {{\rm supp}}\mi$. For a given generalized Orlicz ball $K \subset \R^n$ by its [*restriction to a positively inclined hyperplane $H$*]{} we mean such a generalized Orlicz ball $K' \subset \R^{n-1}$ such that ${K_{+}}\cap H$ is isometric to ${K_{+}}'$. By Lemma \[sekcjajest\] there exists such a generalized Orlicz ball $K'$. For a given generalized Orlicz ball $K \subset \R^n$ by its [*restriction to an interval $I \subset {\R_{+}}$ with respect to the coordinate $x_i$*]{} we mean such a generalized Orlicz ball $K' \subset \R^n$ that ${K_{+}}'$ is isometric to $K \cap \{x_i \in I\}$. By Lemma \[obciacboki\] there exists such a generalized Orlicz ball $K'$. When it is obvious in which coordinate the interval $I$ is taken we shall simply write that $K'$ it a restriction of $K$ to $I$. For a given generalized Orlicz ball $K \subset \R^n$ and a generalized Orlicz ball $K' \subset \R^m$ we say that $K'$ is a [*derivative*]{} of $K$ if there exists a sequence $K = K_0, K_1,\ldots,K_n = K'$ of generalized Orlicz balls such that for each $i \in \{1,\ldots,n\}$ the ball $K_i$ is either a restriction of $K_{i-1}$ to some positively inclined hyperplane or a restriction of $K_{i-1}$ with respect to some variable $x_k$ to some interval $I \subset {\R_{+}}$. We can embed isometrically the positive quadrant of any derivative of $K$ into the positive quadrant of $K$. We shall identify without notice the positive quadrant of the derivative with the image of this embedding in the positive quadrant of $K$. In particular for a function $f$ defined on ${K_{+}}$ we shall speak of its restriction to ${K_{+}}'$, meaning such a function $\tilde{f}$ that $\tilde{f}(x) = f(\phi(x))$, where $\phi$ is the embedding of ${K_{+}}'$ into ${K_{+}}$. For the space $\R^n$ with a fixed orthonormal system $e_1,\ldots,e_n$ by a [*coordinate-wise decompostion*]{} of $\R^n$ we mean a decompostion $\R^n = \R^k \times \R^l$, where $\R^k = {{\rm span}}\{e_{i_1},\ldots,e_{i_k}\}$ and $\R^l = {{\rm span}}\{e_{j_1},\ldots,e_{j_l}\}$, with $i_p \neq j_q$ for any $p,q$. The main tool used in this proof will be the $\Theta$ functions. We define the $\Theta$ functions as follows: For a given generalized Orlicz ball $K \subset \R^n$ and two functions $\eta_1, \eta_2$ defined on ${K_{+}}$ we say that $\eta_1$ and $\eta_2$ define a $\Theta$ function on $K$ if the following properties are satisfied: \[t-1\] The functions $\eta_1$ and $\eta_2$ are bounded. \[t0\] The functions $\eta_1$ and $\eta_2$ are coordinate-wise non-increasing. \[t1\] We have $\eta_1 \geq \eta_2 \geq 0$. \[t2\]\[t3\] For any derivative $K' \subset \R^m$ of $K$, any proper measure $\mi$ on $K'$ and any coordinate-wise decomposition $\R^m = \R^k \times \R^{m-k}$ the function $$\theta_k^\mi(\mathbf{x}) = \frac{\int_{{\R_{+}}^k} \eta_2((\mathbf{y},\mathbf{x})) d\mi_{|\R^k}(\mathbf{y})}{\int_{{\R_{+}}^k} \eta_1((\mathbf{y},\mathbf{x})) d\mi_{|\R^k}(\mathbf{y})}$$ is a coordinate-wise non-increasing function of $\mathbf{x} = (x_{j_1},\ldots,x_{j_{m-k}})$ where defined. Recall $\mi_{|\R^k}$ denotes the restriction of $\mi$ to $\R^k$. For a fixed proper measure $\mi$ on $K$ we define the function $\theta^{\mi}$ by $$\theta^{\mi}(A) = \frac{\int_A \eta_2(x) d\mi(x)}{\int_A \eta_1(x) d\mi(x)}$$ for any Borel set $A$ with $\mi(A) > 0$. We shall say that $\theta^{\mi}$ is the $\Theta$ function defined for the measure $\mi$ by $\eta_1$ and $\eta_2$. For a fixed proper measure $\mi$ on $K$ and a fixed coordinate-wise decomposition $\R^n = \R^k \times \R^{n-k}$ we shall also define $$\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k;A) = \frac{\int_A \eta_2(a_1,a_2,\ldots,a_k,x_{k+1},x_{k+2},\ldots,x_n) d\mi_{|\{(a_1,a_2,\ldots,a_k)\} \times \R^{n-k}}}{\int_A \eta_1(a_1,a_2,\ldots,a_k,x_{k+1},x_{k+2},\ldots,x_n) d\mi_{|\{(a_1,a_2,\ldots,a_k)\} \times \R^{n-k}}}$$ for such sets $A$ and number $a_1,a_2,\ldots,a_k$ for which the denominator is positive. If $A = \R^{n-k}$ we shall omit it and write $\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k)$ for $\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k;\R^{n-k})$, and if $\mathbf{a} = (a_1,a_2,\ldots,a_k)$, we will write $\theta_{n-k}^\mi(\mathbf{a};A)$ or $\theta_{n-k}^\mi(\mathbf{a})$ for $\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k;A)$ and $\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k;\R^{n-k})$ respectively, which is consistent with the notation above. If $A \subset \R^n$ by $\theta^\mi_{n-k}(\mathbf{a};A)$ we mean $\theta^\mi_{n-k}(\mathbf{a};A \cap \{\mathbf{a}\} \times \R^{n-k})$. If there could be doubts as to what coordinate-wise decomposition is taken, we may write $\theta_{n-k}^\mi(x_1 = a_1,x_1 = a_2,\ldots,x_k = a_k;A)$ for $\theta_{n-k}^\mi(a_1,a_2,\ldots,a_k;A)$. \[wlasnoscitheta\] If $\eta_1$ and $\eta_2$ define a $\Theta$ function $\theta^\mi$ for a proper measure $\mi$ on a generalized Orlicz ball $K$, then the following are true: \[t6\] The function $\theta^\mi$ is continuous with respect to the symmetric difference distance, that is if $\theta^\mi$ is defined for all $C_i$ and $\mi(C_0 \bigtriangleup C_i) {{\rightarrow}}0$, then $\theta^\mi(C_i) {{\rightarrow}}\theta^\mi(C_0)$. \[t7\] If $D' \subset D \subset \R^n$, $\theta^\mi(D) = \theta^\mi(D')$ and $\theta^\mi$ is defined for $D \setminus D'$, then $\theta^\mi (D\setminus D') = \theta^\mi(D)$. \[t8\] If $K'$ is a derivative of $K$, then the restrictions of $\eta_1$ and $\eta_2$ to $K'$ define a $\Theta$ function on $K'$. Further on, as the proper measure taken rarely changes, we omit the $\mi$ in the upper index and simply write $\theta$ for $\theta^\mi$. Note that as $\eta_1$ is positive on ${K_{+}}$ from property (\[t1\]) and ${{\rm supp}}\mi \subset {K_{+}}$, we know that $\theta^\mi(D)$ is well defined if and only if $\mi(D) > 0$. Functions $\eta_1$ and $\eta_2$ defining a $\Theta$ function on a generalized Orlicz ball $K$ are said to define a strict $\Theta$ function if the following extra conditions are satisfied: 1. \[s2\] ${{\rm supp}}\eta_2 \subset {{\rm Int}}_{{\R_{+}}} {{\rm supp}}\eta_1$, where ${{\rm Int}}_{{\R_{+}}}$ denotes the interior taken with respect to the space ${\R_{+}}$. 2. \[s3\] The generalized Orlicz ball $K$ is proper. 3. \[s4\] For any coordinate-wise decomposition $\R^n = \R^k \times \R^{n-k}$ with $n > k$ the functions ${{\tilde{\eta}}}_i : \R^k {{\rightarrow}}\R$ defined by $x \mapsto \int_{\R^{n-k}} \eta_i(x,y) d\lambda_{n-k}(y)$ are continuous. 4. \[t4\] $\eta_1 > 0$ on ${{\rm Int}}K_+$. For $\eta_1$ and $\eta_2$ defining a $\Theta$ function $\theta$ on a generalized Orlicz ball $K$ by a [*derivative*]{} of $\theta$ we mean the function defined on a derivative $K'$ of $K$ by the restrictions of $\eta_1$ and $\eta_2$ to $K'$. Note that the derivatives of a $\Theta$ function are $\Theta$ functions. For a given generalized Orlicz ball $K$ we say that $\eta_1$ and $\eta_2$ define a [*weakly non-degenerate $\Theta$ function*]{} on $K$ if for every $\eps > 0$ there exists a generalized Orlicz ball $K' \subset K$ with $\lambda(K \setminus K') \leq \eps \lambda(K)$ and functions $\eta_1'$ and $\eta_2'$ defining a strict $\Theta$ function on $K'$ with $\int|\eta_i - \eta_i'| d\lambda \leq \eps$. A $\Theta$ function is called [*non-degenerate*]{} if it is weakly non-degenerate and all its derivatives are weakly non-degenerate. Note that as the density of any proper measure is bounded, in all the bounds in the definition above we can replace $\lambda$ by any proper measure $\mi$. Frequently we shall take the same collection of assumptions for our theorems. To make reading the paper easier, we will use the following notation: We shall speak of - [*Standard assumptions*]{} if $K \subset \R_x \times \R_y \times \R^{n-2}$ is a generalized Orlicz ball, $\mi$ is a proper measure for $K$, $\eta_1$ and $\eta_2$ define a $\Theta$ function $\theta = \theta^\mi$ on $K$ for $\mi$ and $A$ is a c-set in ${\R_{+}}^n$, - [*Non-degenerate assumptions*]{} if additionally we require the $\Theta$ function defined by $\eta_1$ and $\eta_2$ to be non-degenerate, and - [*Strict assumptions*]{} if $K$ is a proper generalized Orlicz ball and $\eta_1$ and $\eta_2$ define a strict non-degenerate $\Theta$ function. \[appropriate\] Under standard assumptions a set $D \subset \R^n$ will be called [*appropriate*]{}, if - $\theta(D)$ is defined, - $\theta(D \cap A) \geq \theta(K) \geq \theta(D \cap {{\bar{A}}})$ if the left-hand side and the right-hand side are defined, - $\theta(D) = \theta(K)$. \[epsappropriate\] Under standard assumptions let $\mi_2$ be the restriction of $\mi$ to $\R_x \times \R_y \times \{0\}$. For any $\eps > 0$ a set ${{\tilde{D}}}\times \R^{n-2} = D$ is called [*$\eps$-appropriate*]{}, if - $\theta(D)$ is defined, - $\theta(D) = \theta(K)$, - For each $U \in \{A, {{\bar{A}}}\}$ and each $i \in \{1,2\}$ there exists a number $C_{U,i}$ such that $$\bigg|\int_{D\cap U} \eta_i(t) d\mi(t) - C_{U,i}\bigg| \leq \eps \mi_2({{\tilde{D}}})$$ and $$\frac{C_{A,2}}{C_{A,1}} \geq \theta(D) \geq \frac{C_{{{\bar{A}}},2}}{C_{{{\bar{A}}},1}}.$$ The definition of an appropriate set describes the properties we desire for the set into which we divide ${K_{+}}$. In fact, due to the approximation, we shall divide ${K_{+}}$ into $\eps$-appropriate sets to prove it is $\eps$-appropriate, and then take $\eps {{\rightarrow}}0$. The generalized Orlicz ball lemmas ---------------------------------- In this subsection we will prove a few lemmas about the structure generalized Orlicz balls. They show that the class of generalized Orlicz balls is closed under taking derivatives, and that proper generalized Orlicz balls are, in a sense, dense in the class of generalized Orlicz balls. These lemmas are the main reason the whole reasoning in this paper has to be done for generalized Orlicz balls, and not simply Orlicz balls — the class of Orlicz balls does not enjoy the same closedness propeties. A product of intervals $\prod_{i=1}^n [a_i,b_i]$ is isometric to the positive quadrant of the Orlicz ball $K \subset \R^k$ defined by the functions $$\begin{aligned} f_i(x_i) = \begin{cases} 0 & \hbox{ if $x_i \leq b_i - a_i$} \\ \infty & \hbox{ if $x_i > b_i - a_i$}\end{cases}\end{aligned}$$ for $b_i > a_i$. \[obciacboki\] If ${K_{+}}\subset \R^n$ is a generalized Orlicz ball positive quadrant and $0 \leq x_a < x_b$, then ${K_{+}}\cap \{x_1 \in [x_a,x_b]\}$ is isometric to a generalized Orlicz ball positive quadrant or empty Let $f_1, f_2, \ldots, f_n$ be the Young functions defining $K$. Let ${K_{+}}' = {K_{+}}\cap \{x_1 \in [x_a,x_b]\}$. Let $c = f_1(x_a)$. If $c = 1$, then ${K_{+}}' = \{x : x_1 = x_a, \forall_{i > 1} f_i(x_i) = 0\}$, which is a product of intervals and thus isometric to a generalized Orlicz ball positive quadrant. If $c > 1$ then ${K_{+}}'$ is empty. If $c < 1$ we define $\bar{f}_1$ by $$\begin{aligned} \bar{f}_1(x_1) = \begin{cases} \frac{f_1(x_1 + x_a) - f_1(x_a)}{1-c} & \hbox{ for $x_1 < x_b$} \\ \infty & \hbox{ for $x_1 > x_b$,}\end{cases}\end{aligned}$$ and $\bar{f}_i$ for $i > 1$ by $$\bar{f}_i(x_i) = \frac{f_i(x_i)}{1-c}.$$ Now $(x_1, x_2, \ldots, x_n) \in \bar{K}_+$ iff $(x_1 + x_a, x_2, \ldots, x_n) \in {K_{+}}'$, where $\bar{K}_+$ is the positive quadrant of the Orlicz ball defined by $\bar{f}_i$. \[sekcjajest\] If $K \subset \R^n$ is a generalized Orlicz ball and $H = \{x\in \R^n : x_1 = \lambda x_2 + c\}$ is a positively inclined hyperplane in $\R^n$, then ${K_{+}}\cap H$ is the positive quadrant of some generalized Orlicz ball $L$ or an empty set. As $H$ is positively inclined, $\lambda \geq 0$. If $\lambda = 0$ and $c < 0$ we have ${K_{+}}\cap H = \emptyset$. If $c < 0$ and $\lambda > 0$ we can transform the equation giving $H$ to $H = \{x \in \R^n : x_2 = \frac{1}{\lambda}x_1 - \frac{c}{\lambda}\}$. Thus we can assume $c \geq 0$. For $x_2 \geq 0$ we have $x_1 \geq c$ in $H$. Thus if $f_1(c) > 1$, then for $x_2 \geq 0$ we have $f_1(x_1) > 1$ for $x \in H, x_1 \geq 0$, thus $H \cap {K_{+}}= \emptyset$. If $f_1(c) = 1$ and $\lambda > 0$, then $H \cap {K_{+}}$ is the set $\{x : x_1 = c, x_2 = 0, f_i(x_i) = 0$ for $i > 2\}$. This set is a cartesian product of intervals, and isometric to a generalized Orlicz ball positive quadrant. If $f_1(c) = 1$ and $\lambda = 0$, the situation is the same, except $x_2 = 0$ is replaced by $f_2(x_2) = 0$. Now we may assume $f_1(c) < 1$. Let $x_3,x_4,\ldots,x_{n+1}$ be the coordinates on $H$, with $x_1 = \lambda x_{n+1} + c$, $x_2 = x_{n+1}$. Let us take $f_{n+1}(t) = f_1(\lambda t + c) + f_2(t) - f_1(c)$, then $f_1(x_1) + f_2(x_2) = f_{n+1}(x_2) + f_1(c)$. The function $f_{n+1}$ is a sum of three convex functions, thus it is convex, and $f_{n+1}(0) = 0$. The set $\{(x_i)_{i=3}^{n+1} : f_i(x_i) < 1 -f_1(c)\} \cap {\R_{+}}$ is equal to ${K_{+}}\cap H$. If we consider Young functions $\tilde{f}_i(t) = \frac{f_i(t)}{1 - f_1(c)}$ for $i = 3,4,\ldots,n+1$ we get the generalized Orlicz ball $L \subset H$ such that ${K_{+}}\cap H = L \cap {\R_{+}}^{n}$. \[approrlicz\] For any generalized Orlicz ball $K\subset \R^n$ and any $\eps > 0$ there exists a proper generalized Orlicz ball $K' \subset K$ with $\lambda(K \setminus K') < \eps$. Furthermore if any Young function $f_i$ of $K$ is already a proper Young function, the same $f_i$ will be the appropriate Young function of $K'$. This lemma is easy to believe in, but somewhat technical to prove. An impatient reader might be well advised to skip the next two proofs (or prove the Lemmas her$\slash$himself, if desired) and go to the more crucial parts of the paper. As any generalized Orlicz ball is 1-symmetric, it suffices to prove $\lambda({K_{+}}\setminus K'_{\geq 0}) \leq \eps \slash 2^n$. We shall thus consider only the points in ${\R_{+}}^n$ and decrease $\eps$ to be $2^n$ times smaller. Recall that a proper Young function is such a Young function that $f(x) = 0$ only for $x = 0$ and $f(x) < \infty$. Thus we have to get rid of superfluous zeroes and of infinity values. First we shall take care of the zeroes. Let $f_i$ be Young functions defining $K$. Let $M$ be the largest of the $(n-1)$-dimensional measures of the projections of $K$ onto the hyperplanes $x_i = 0$. Let $t_i = \inf\{x_i : f_i(x_i) > 1\slash 2n\}$. Let $c = \inf_i \{f_i'(t_i)\}$. We shall prove that for $\delta < 1\slash 2n$ the set $U_\delta = \{x : \sum f_i(x_i) \in [1-\delta,1]\}$ has measure no larger than $\frac{M n \delta}{c}$. First note that $U_\delta = \bigcup U_i$, where $U_i = U_\delta \cap \{x : f_i(x_i) > 1\slash 2n\}$, as at least one of $f_i(x_i)$ has to be large for the sum to be large. We shall bound the measure of each $U_i$ separately. For each point $x = (x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n)$ the set of those $x_i$ that $(x,x_i) \in U_i$ has length at most $\frac{\delta}{c}$. Thus, from Fubini’s theorem, the measure of $U_i$ can be bounded by $\frac{M \delta}{c}$, and summing over all $i$ we get the desired bound for $U_\delta$. Let us take $\delta = \frac{c\eps}{2Mn^2}$. For each $i$ for which we have superfluous zeroes let us take $s_i = \inf\{x_i : f_i(x_i) > \delta\}$ and replace $f_i$ by $g_i$ defined by $$g_i(x_i) = \begin{cases} f_i(x_i) &\hbox{ for }x_i \geq s_i \\ x_i \delta \slash s_i &\hbox{ for }x_i < s_i.\end{cases}$$ We have $g_i(x_i) \geq f_i(x_i)$ and $g_i(x_i) - f_i(x_i) \leq \delta$. Thus if $K'$ is the generalized Orlicz ball defined by $g_i$, we have $K' \subset K$ and $K \setminus K' \subset U_{n\delta}$, and thus $\lambda(K \bigtriangleup K') \leq \frac{n^2 M \delta}{c} = \eps \slash 2$. Now we shall deal with the $\infty$ values. Let $\delta = \frac{\eps}{2nM}$. Note that the shape of $K'$ is determined by the values of $g_i$ only up to $g_i(x_i) = 1$. Thus we have to make some corrections to $g_i$ up to $g_i(x_i) = 1$, and then extend $g_i$ anyhow, say linearly. For each $i$ such that $g_i$ attains the $\infty$ value let $r_i = \inf \{x_i : g_i(x_i) = \infty\}$, and let $v_i = \lim_{x_i {{\rightarrow}}v_i^-} g_i(x_i)$. If $v_i \geq 1$, then all we have to do is to extend $g_i$ in a different way after $r_i$, and that does not change the ball $K'$ defined by $g_i$. If, however, $v_i < 1$, we define $h_i$ as follows: $$h_i(x_i) = \begin{cases} g_i(x_i) &\hbox{ for }x_i < r_i - \delta \\ 2 &\hbox{ for }x_i = r_i \\ \hbox{linear continuous extension} &\hbox{ otherwise.} \end{cases}$$ Let $K''$ be the ball defined by $h_i$. Again, $K'' \subset K'$, as $h_i \geq g_i$ on the set where $g_i \geq 1$, from the convexity of $g_i$. The difference, however, is obviously contained in $\bigcup_i K' \cap \{x_i \in [r_i - \delta,r_i]\}$, thus $\lambda(K' \setminus K'') \leq n M \delta = \eps \slash 2$. Adding the two estimates together we get $\lambda(K \setminus K'') \leq \eps$. \[apprsection\] With the assumptions of Lemma \[approrlicz\] if we take any $y_0$ (where $y$ is any coordinate in $\R^n$), then we can take such a $K'$ as before and $y_1$ that $\lambda_{n-1} ((K \cap \{y = y_0\}) \bigtriangleup (K' \cap \{y = y_1\})) < \eps$. This, again, is easy to believe in, and actually simple if $f_y(y_0) \neq 1$. The special case where $f_y(y_0) = 1$ could arguably be ignored (as it happens only on a set of measure zero), but to avoid omitting a set of measure zero in all other places of the proof, we shall go through the technicalities here. If $f_y(y_0) > 1$, we can simply take $y_1 = y_0$, and $\lambda_{n-1} (K \cap \{y = y_0\}) = \lambda_{n-1} (K' \cap \{y = y_1\}) = 0$. If $f_y(y_0) < 1$, we need to control the Orlicz ball $\sum f_i(x_i) = 1 - f_y(y_0)$. This Orlicz ball $L$ is given by Young functions $f_i \slash (1- f_y(y_0))$. For this Orlicz ball we also calculate values of $M$ and $c$, and apply the reasoning in the proof of Lemma \[approrlicz\] taking the larger $M$ and the smaller $c$ of those calculated for the two balls. We thus get good approximations $K'$ and $L'$ of both $K$ and $L$. Now take such a $y_1$ that $f_y(y_0) = h_y(y_1)$, this can be done as $h_y$ is continuous. Now $K' \cap \{y = y_1\} = L'$, which proves the thesis. In the case $f_y(y_0) = 1$ if any of the other $f_i$ do not have superfluous zeroes, the measure of $K \cap \{y = y_0\}$ is 0, and thus taking $y_1 = y_0 + 1$ we get the thesis. If, however, all the other $f_i$ have superfluous zeroes, the intersection $K \cap \{y = y_0\}$ is the cube ${{\prod}}_i f_i^{-1}(0)$. In this case we shall need a better approximation. Let $z_i = \sup \{x_i : f_i(x_i) = 0\}$. Let us, as before for $\eps$, define $\delta' = \frac{c\eps'}{2Mn^2}$ and $s_i' = \inf\{x_i : f_i(x_i) > \delta'\}$. We need $\eps'$ to be so small that $$s_i' \slash z_i \leq 1 + \frac{(1 + \eps \slash M)^{1\slash n} - 1}{n}$$ and smaller than $\eps$. Note that as $\eps' {{\rightarrow}}0$ we have $\delta' {{\rightarrow}}0$ and $s_i' {{\rightarrow}}z_i$, so taking $\eps'$ small enough we can achieve the desired inequality for all $i$. Conduct the proof of Lemma \[approrlicz\] taking $\eps'$ instead of $\eps$. Take $y_1$ such that $h_y(y_1) = 1 - n\delta'$. Note that if $x_i \leq s_i'$ for all $i$, then $\sum h_i(x_i) \leq n \delta'$, and thus $x = (x_i) \in K' \cap \{y = y_1\}$. On the other hand if for any $i$ we have $x_i > s_i' + (n-1) (s_i' - z_i)$, then $h_i(x_i) \geq f_i(x_i)$, and as $f_i(z_i) = 0$, $f_i(s_i') \geq \delta'$ and $f_i$ is convex, we have $f_i(x_i) > n\delta'$, and thus $\sum h_i(x_i) > n\delta'$ and $x \not\in K' \cap \{y = y_1\}$. Thus $$K \cap \{y = y_0\} = {{\prod}}_i f_i^{-1}(0) \subset K' \cap \{y = y_1\} \subset {{\prod}}_i [0,z_i + n (s_i' - z_i)].$$ Now we have the following inequalities: $$\begin{aligned} \frac{s_i'}{z_i} &\leq 1 + \frac{(1 + \eps \slash M)^{1\slash n} - 1}{n} \\ \frac{n(s_i' - z_i)}{z_i} & \leq (1 + \eps \slash M)^{1\slash n} - 1 \\ \frac{z_i + n(s_i' - z_i)}{z_i} &\leq (1 + \eps \slash M)^{1\slash n} \\ \prod_i \frac{z_i + n(s_i' - z_i)}{z_i} &\leq 1 + \eps \slash M \\ \lambda ({{\prod}}_i [0,z_i + n(s_i' - z_i)]) &\leq \lambda ({{\prod}}[0,z_i]) + \frac{\eps \lambda ({{\prod}}[0,z_i])}{M} \\ \lambda (K' \cap \{y = y_1\}) - \lambda(K \cap \{y = y_0\}) &\leq \eps.\end{aligned}$$ The last inequality follows as ${{\prod}}[0,z_i]$ is a subset of the projection of $K$ onto $y = 0$, and thus its measure is no bigger than $M$. This, along with the fact that $K \cap \{y = y_0\} \subset K' \cap \{y = y_1\}$ gives the thesis. This reasoning can be extended to approximate any finite number of sections of $K$ along with $K$. $1\slash m$-concave functions and proper measures ------------------------------------------------- Here we give a few elementary facts about $1\slash m$-concave functions and proper measures. Most facts are easily proved and quite a few are well known, so we skip some of the proofs. If a function $f$ is $1\slash m$-concave for some $m > 0$, then it is also $1 \slash m'$-concave for any $m' > m$. \[prodncon\] The product of $1\slash m$-concave functions is $1\slash 2m$-concave. \[convconc\]From the Brunn-Minkowski inequality, if $K \subset \R^n$ is a convex set, then $(y_1,y_2,\ldots,y_k) \mapsto \lambda_{n-k}(K \cap \{\forall_{1\leq i \leq k} x_i = y_i\})$ is a $1\slash (n-k)$-concave function, where $x_i$ are the coordinates on $\R^n$. Conversely, if we have a $1\slash m$-concave function on $\R^n$, then there exists a convex set $K \subset \R^{n+m}$ such that $f$ is the projection of the Lebesgue measure restricted to $K$ onto $\R^n$. As a corollary of these two facts the projection of a $1\slash m$ concave function on $\R^n$ onto $\R^k$ is a $1\slash(n+m-k)$-concave function. The restriction of the Lebesgue measure to ${K_{+}}$ is a proper measure with respect to $K$. The support of a proper measure $\mi$ is a convex set. \[sekcjami\] If $\mi$ is a proper measure on $\R^n$ and $H = \{x\in \R^n : x_1 = \lambda x_2 + c\}$, with $\lambda \geq 0$ is a hyperplane in $\R^n$, then $\mi$ restricted to $H$ with coordinates $(v,x_3,\ldots,x_n)$ is a proper measure. The density of $\mi$ is equal to $f(x) g(y) \1_K$ for some $(1\slash m)$-concave functions $f$ and $g$ and some generalized Orlicz ball $K$. From Lemma \[sekcjajest\] the set ${K_{+}}\cap H$ is the positive quadrant of some Orlicz ball $K'$ with coordinates $(v,x_3,\ldots,x_n)$. The product $f \cdot g$ on $H$ varies with at most two variables, and is from, Fact \[prodncon\], a $(1\slash 2n)$-concave function with respect to $v$. \[cieciemi\] If $\mi$ is a proper measure on $\R_x \times \R_y \times R^{n-2}$, then the restriction of $\mi$ to an interval $I$ with respect to any variable is also a proper measure. Due to Lemma \[obciacboki\] if ${K_{+}}$ is the Orlicz ball quadrant for which $\mi$ is defined, ${K_{+}}' = {K_{+}}\cap \{t \in I\}$ is also an Orlicz ball quadrant. Let $f$ and $g$ be the functions defining the density of $\mi$. To make them define a proper measure on ${K_{+}}'$ we simply have to restrict them to the set $\{x_0 : \lambda_{n-1} (K'_{x = x_0}) > 0\}$ for $f$ and similarly for $g$, and additionaly to the interval $I$ if it was taken in $x$ or $y$ respectively. Both functions will have a convex support after this restriction, and as they were $1\slash m$-concave on a larger domain, they will still be $1\slash m$-concave. Lens sets and $\Theta$ functions -------------------------------- Let $D$ be a lens set with extremal points $(x_1,y_1)$ and $(x_2,y_2)$. Let $x^-(y) = \inf \{x : (x,y) \in D\}$, $x^+(y) = \sup \{x : (x,y) \in D\}$ for $y \in [y_1,y_2]$. Then $x^-$ and $x^+$ are increasing function on their domains, $x^-$ is convex, and $x^+$ is concave. \[moveinterval\] Under standard assumptions consider a fixed $y_0$ and two intervals $[x_a,x_b], [x_c,x_d]$ with $x_a \leq x_c$ and $x_b \leq x_d$. Then we have $$\theta^\mi_{n-1}(y_0; [x_a,x_b] \times \R^{n-2}) \geq \theta^\mi_{n-1}(y_0; [x_c,x_d] \times \R^{n-2})$$ if both sides are defined. The same applies when $x$ is exchanged with $y$. From property (\[t2\]) we know that $\theta^\mi_{n-2}(x,y_0)$ is a decreasing function of $x$. The domain of this function is a convex set, so its intersections with $[x_a,x_b]$ and $[x_c,x_d]$ are both intervals (they are non-empty, for $\theta^\mi_{n-1}$ is defined for both intervals). Applying Lemma \[rosnmix\], part 2, to $\int_{\R^{n-2}} \eta_2(x,y,t_1,\ldots,t_{n-2}) d\mi_{|(x,y)\times \R^{n-2}} (t_1,\ldots,t_{n-2})$ and $\int_{\R^{n-2}} \eta_1(x,y,t_1,\ldots,t_{n-2}) d\mi_{|(x,y)\times \R^{n-2}} (t_1,\ldots,t_{n-2})$ and the shortened intervals we get the thesis. \[raiseinterval\] Under standard assumptions for a given interval $I = [x_a,x_b]$ the function $\theta^\mi_{n-1}(y;I \times \R^{n-2})$ is a decreasing function of $y$ on its domain. The same applies when $x$ is exchanged with $y$. Take any $0 \leq y_1 \leq y_2$ in the domain. $K' = K \cap \{x \in I\}$ is a derivative of $K$, and $\theta^\mi_{n-1}(y;I\times \R^{n-2}) = \bar{\theta}_{n-1}^{\mi}(y)$, where $\bar{\theta}^\mi$ is defined by the restrictions of $\eta_1$ and $\eta_2$ to $K'$. Thus from property (\[t3\]) we get the thesis. \[incthetay\] Under standard assumptions for a given lens set $D$ the domain of the function $y \mapsto \theta^\mi_{n-1}(y;D)$ is an interval and the function is decreasing. Let $(x_1,y_1)$ and $(x_2,y_2)$ be the extremal points of ${{\tilde{D}}}$. Take any $y_4$ such that $\theta_{n-1}^\mi(y_4;D)$ is defined and take any $y_3 \in (y_1,y_4]$. Thus $\theta_{n-2}^\mi(x,y_4)$ is defined for more than one $x$ such that $(x,y_4) \in {{\tilde{D}}}$ (actually, for a set of positive Lebesgue measure), let $x_4$ be any such $x$ except the smallest. We want to prove $\theta_{n-1}^\mi(y_3;D)$ is defined. Note that ${{\rm supp}}\mi$ is a c-set on $x > x_-$, $y > y_-$ and ${{\rm supp}}\eta_1$ is also a c-set, thus their intersection is a c-set. Thus $\theta_{n-2}^\mi(x,y)$ is defined for any $x_- < x \leq x_4$ and $y_- < y \leq y_4$. As $D$ is a lens set, the set of $x \leq x_4$ such that $(x,y_3) \in {{\tilde{D}}}$ has positive Lebesgue measure, thus $\theta_{n-1}^\mi(y_3;D)$ is defined, which means that $\theta_{n-1}^\mi(y;D)$ is defined on some interval $(y_1,y_0)$ and undefined outside. Now we shall prove $\theta_{n-1}^\mi(y;D)$ is decreasing. Take $y_3 \leq y_4$ from the domain. Let $[x_3^-,x_3^+]$ be the interval ${{\tilde{D}}}\cap \{y = y_3\}$ and $[x_4^-,x_4^+]$ the interval ${{\tilde{D}}}\cap \{y=y_4\}$. From the definition of a lens set $x_3^- \leq x_4^-,x_3^+ \leq x_4^+$. From Lemmas \[raiseinterval\] and \[moveinterval\] (twice) we have $$\begin{aligned} \theta_{n-1}^\mi(y_3;D) & = \theta_{n-1}^\mi(y_3;[x_3^-,x_3^+] \times \R^{n-2}) \geq \theta_{n-1}^\mi(y_3;[x_3^-,x_4^+] \times \R^{n-2}) \\ & \geq \theta_{n-1}^\mi(y_4;[x_3^-,x_4^+]\times \R^{n-2}) \geq \theta_{n-1}^\mi(y_4;[x_4^-,x_4^+] \times \R^{n-2}) = \theta_{n-1}^\mi(y_4;D).\end{aligned}$$ Note that the last expression in the first line and the first in the second line are well defined, for the second argument is a superset of the second argument for $\theta_{n-1}^\mi(y_3;D)$ and $\theta_{n-1}^\mi(y_4;D)$ respectively. \[thetamain\] Under standard assumptions for a given lens set $D$ and a given $y_0$ in the domain of $\theta_{n-1}^\mi(y;D)$ we have $$\theta^\mi(D \cap \{y \leq y_0\}) \leq \theta_{n-1}^\mi(y_0;D) \leq \theta^\mi(D \cap \{y \geq y_0\})$$ and $$\theta^\mi(D \cap \{y \leq y_0\}) \leq \theta^\mi(D) \leq \theta^\mi(D \cap \{y \geq y_0\})$$ if the left and right hand sides are defined. Moreover, if $\theta^\mi(D \cap \{y \leq y_0\}) = \theta^\mi(D \cap \{y \geq y_0\})$ for any $y_0 \in (y_1,y_2)$, then $\theta^\mi_{n-1}(y_3;D), \theta^\mi(D \cap \{y \geq y_3\})$ and $\theta^\mi(D \cap \{y \leq y_3\})$ are all constant where defined and equal $\theta^\mi(D)$ for $y_3 \in (y_1,y_2)$. From Lemma \[incthetay\] the function $\theta_{n-1}^\mi(y;D)$ is decreasing as a function $y$ on its domain, and its domain is an interval. We know that ${{\rm supp}}\eta_2 \subset {{\rm supp}}\eta_1$, so we can apply Lemma \[rosncalk\], part 1a, to the appropriate integrals of $\eta_2$ and $\eta_1$ to get the first part of the thesis. The second part follows from the first and Fact \[obvi\]. The third follows again from Lemma \[rosncalk\], part 1b. \[epstotot\] Under standard assumptions if $D$ is $\eps$-appropriate for any $\eps > 0$, then $D$ is appropriate. The third and first condition in Definition \[appropriate\] follows from the definition of $\eps$-appropriate for any $\eps$. We have to check the second condition. Let $C_{U,i}^\eps$ denote the numbers $C_{U,i}$ which show $D$ is and $\eps$-appropriate set. We have $$\theta^\mi(D) = \frac{C_{A,2}^\eps}{C_{A,1}^\eps} \leq \frac{\int_{D\cap A} \eta_2(t) d\mi(t) + \eps \mi_2({{\tilde{D}}})}{\int_{D \cap A} \eta_1(t) d\mi(t) - \eps \mi_2({{\tilde{D}}})} \rightarrow_{\eps {{\rightarrow}}0} \theta^\mi(D\cap A),$$ and similarly for the second inequality. \[epssum\] Under standard assumptions if $D_k$ are $\eps$-appropriate sets for $k \in K$, then $D = \bigcup_{k\in K} D_k$ is $\eps$-appropriate. We have $\theta^\mi(D) = \theta^\mi(K)$ from Lemma \[dividesets\]. For the third condition in Definition \[epsappropriate\] we take $C_{U,i}^D = \sum_{k\in K} C_{U,i}^{D_k}$. These are good approximations as $\mi_2({{\tilde{D}}}) = \sum_{k} \mi_2({{\tilde{D}}}_k)$, and obviously satisfy the proportion inequality. The $\Theta$ theorem ==================== Preparations for divisibility ----------------------------- In this section we shall prove the main theorem concerning $\Theta$ functions. Under standard assumptions, we shall consider $\mi$ to be a fixed proper measure on $\R_x \times \R_y \times \R^{n-2}$. By $\mi_2$ we shall denote the restriction of $\mi$ to $\R_x \times \R_y \times \{0\}$. Note that as the support of $\mi$ is a c-set with respect to the $n-2$ variables of $\R^{n-2}$, the support of $\mi_2$ is the projection of the support of $\mi$. As we fix $\mi$, we shall omit the upper index when writing the $\Theta$ function and write $\theta$ or $\theta_k$ instead of $\theta^\mi$ or $\theta_k^\mi$. The main theorem we want to prove is: \[main\] Under non-degenerate assumptions $\theta(A) \geq \theta(K)$ and $\theta(K) \geq \theta({{\bar{A}}})$, whenever both sides of an inequality are defined. This looks like a quite simple theorem, and we suspect there is a simpler proof than the one we present here. However, we were not able to find it (and would be interested to learn if anyone does). Notice that if $\theta(A)$ is undefined, then $\int_A \eta_1 d\mi = 0$, which implies $\int_A \eta_2 d\mi = 0$ from property (\[t1\]). Thus $\theta({{\bar{A}}}) = \theta(K)$ and the Theorem is satisfied. Thus we assume $\theta(A)$ is defined. Similarly we may assume $\theta({{\bar{A}}})$ is defined. From Fact \[obvi\] it is enough to prove $\theta(A) \geq \theta(K)$ and the second inequality will follow. Thus, we concentrate on the first inequality. First, for technical reasons, we shall deal with the low-dimensional case: \[lowdimmain\] Under standard assumptions with $n \leq 2$ (that is, $K \subset \R$ or $K \subset \R^2$) we have $\theta(A) \geq \theta(K)$ and $\theta(K) \geq \theta({{\bar{A}}})$, whenever both sides of an inequality are defined. For $n = 1$ the set $A$ is one-dimensional, and thus (being a c-set) is an interval of the form $[0,a)$. We apply property (\[t2\]) to $K' = K$, the measure $\mi$ and the decomposition $\R \times \{0\}$ and get that $\frac{\eta_2}{\eta_1}$ is a decreasing function. Thus from Lemma \[rosncalk\], part 1a, $\theta(A) \geq \theta({{\bar{A}}})$ and the thesis follows from Fact \[obvi\]. For $n = 2$ we shall approximate the set $A$ by a $l$-stair set. The $l$-stair set is defined as follows: A $1$-stair set defined by $x_1 = 0$ and $a_1 \geq 0$ (denoted $A(x_1;a_1)$) is the empty set. A $l$-stair set defined by $0 = x_1 \leq x_2 \leq \ldots \leq x_l$ and $a_1 \geq a_2 \geq \ldots \geq a_l \geq 0$, denoted $A(x_1,x_2,\ldots,x_l;a_1,a_2,\ldots,a_l)$ is defined by $$A(x_1,x_2,\ldots,x_l;a_1,a_2,\ldots,a_l) = \Big(A(x_1,x_2,\ldots,x_{l-1};a_1,a_2,\ldots,a_{l-1}) \cap \{x \leq x_l\}\Big) \cup A(0,a_l).$$ That means that a $l$-stair set consists of $l$ steps, the $k$-th step goes from $x_k$ to $x_{k+1}$ (the last one goes all the way to infinity) at height $a_k$. A proper $l$-stair set is a $l$-stair set with $a_l = 0$ Notice that $\theta(A) = \theta(K \cap A)$ as ${{\rm supp}}\mi \subset K$. Thus we may assume $A \subset K$, and thus $A$ is bounded. Take $A_n = \{(x,y) : ([xn]\slash n,y) \in A\}$, where $[xn]$ denotes the integer part of $xn$. This is a proper stair set defined by $0,1\slash n, 2\slash n, \ldots$ (a finite sequence as $A$ is bounded) and the sequence $a_k = \sup\{y : (k\slash n, y) \in A\}$. Notice also $A_{2^n} \supset A_{2^{n+1}} \supset A$ and $\mi (A_{2^n} \setminus A) {{\rightarrow}}0$. Thus $\theta(A_{2^n}) {{\rightarrow}}\theta(A)$, so it is enough to prove $\theta(A_{2^n}) \geq \theta(K)$ and go to the limit. Thus, instead of considering all c-sets we may restrict ourselves to proper $l$-stair sets. The proof for $A$ being a proper $l$-stair set will be an induction upon $l$. For $l = 1$ the set $A$ is empty and the thesis is obvious. For $l = 2$ let $I = [0,x_2]$. From Lemma \[raiseinterval\] the function $\theta_1(y;I)$ is decreasing where defined. Thus from Lemma \[rosncalk\] we have $\theta(I \times [0,a_1]) \geq \theta_1(a_1;I) \geq \theta(I\times [a_1,\infty))$. Note that $A = I \times [0,a_1]$. Thus if $\theta(A) \geq \theta(K)$ or $\theta(A)$ is undefined, the thesis is satisfied. Otherwise, as $\theta(K) > \theta(I \times [0,a_1])$ we have $\theta(K) > \theta(I \times [a_1,\infty))$ if defined, and thus from Lemma \[dividesets\] $\theta(K) > \theta(I \times \R_y)$. Now apply property (\[t2\]) to $K' = K$, the decomposition $\R_x \times \R_y$ and the measure $\mi$ to get that $\theta_1(x)$ is a decreasing function. Again from Lemma \[rosncalk\] and Lemma \[dividesets\] this implies $\theta(I \times \R_y) \geq \theta(K)$, a contradiction. Thus the thesis is satisfied for $l=2$. For larger $l$ let $I = [x_{l-1},x_l]$. Again from Lemma \[raiseinterval\] the function $\theta_1(y;I)$ is decreasing. Thus $$\label{numerequa} \theta(I \times [0,a_{l-1}]) \geq \theta(I \times [a_{l-1},a_{l-2}])$$ if both are defined. Note $$A(x_1,x_2,\ldots,x_l;a_1,a_2,\ldots,a_l) \setminus (I \times [0,a_{l-1}]) = A(x_1,x_2,\ldots,x_{l-1}; a_1,a_2,\ldots,a_{l-2},0)$$ and $$A(x_1,x_2,\ldots,x_l;a_1,a_2,\ldots,a_l) \cup (I \times [a_{l-1},a_{l-2}]) = A(x_1,x_2,\ldots,x_{l-2},x_l; a_1,a_2,\ldots,a_{l-2},a_l).$$ Suppose $\theta(A) < \theta(K)$. If $\theta(I \times [0,a_{l-1}]) > \theta(A)$ or is undefined, then from Lemma \[dividesets\] we have $\theta(A \setminus (I \times [0,a_{l-1}])) \leq \theta(A)$, but from the inductive assumption for $l-1$ we have $\theta(A \setminus (I \times [0,a_{l-1}])) \geq \theta(K)$, from which $\theta(A) \geq \theta(K)$. If, on the other hand, $\theta(I \times [0,a_{l-1}]) \leq \theta(A)$, then from (\[numerequa\]) $\theta(I \times [a_{l-1},a_{l-2}]) \leq \theta(A)$ or is undefined, thus from Lemma \[dividesets\] $\theta(A) \geq \theta(A \cup (I \times [a_{l-1},a_{l-2}]))$, and again from the inductive assumption $\theta(A) \geq \theta(K)$. Thus for any $l$ and for any $A$ being a proper $l$-stair set we have $\theta(A) \geq \theta(K)$, which ends the proof. The proof will proceed by induction upon $n$. For $n \leq 2$ we use Theorem \[lowdimmain\]. For greater $n$ let $K \subset \R_x \times \R_y \times \R^{n-2}$. Assume the thesis is true for all cases with $n' < n$. If the theorem holds for strict $\Theta$ functions, then for any non-degenerate $\theta$ we take a sequence $\theta_i$ of strict $\Theta$ functions for $\eps = 1\slash i$. For any set $C$ for which $\theta(C)$ is defined, we have $\theta_i(C) {{\rightarrow}}\theta(C)$, so as we had $\theta_i(A) \geq \theta_i(K_i) \geq \theta_i({{\bar{A}}})$, we get the thesis when $i$ tends to infinity. Thus it is enough to restrict ourselves to strict assumptions. Note that under strict assumptions $\theta(U)$ is defined for any set $U$ with $\mi(U) > 0$ as ${{\rm supp}}\mi \subset {{\rm supp}}\eta_1$. In particular, if $\mi_2({{\tilde{U}}}) > 0$, then $\theta(U \times \R^{n-2})$ is well defined. Also note that if $\theta({K_{+}}) = 0$, then $\eta_2$ has to be zero $\mi_2$-almost everywhere, which means $\theta(U) = 0$ for any $U$ such that it is defined, thus Theorem \[main\] holds. Thus we can assume $\theta({K_{+}}) > 0$. We want to prove that for any $\eps > 0$ the quadrant ${K_{+}}$ is an $\eps$-appropriate set. We shall frequently require the following property from various sets $D$: $$\label{eqthetacon} \theta(D) = \theta(K),$$ or (for lower-dimensional sets) $$\label{eqthetaconpt} \theta_k(\mathbf{a};D) = \theta(K).$$ We shall need to bound the diameter of the constructed sets from below. To this end consider the following sets: ${{\tilde{S}}}^0 = \{ (x,y) : \theta_{n-2}(x,y) = \theta(K)\}$, ${{\tilde{S}}}^+ = \{ (x,y) : \theta_{n-2}(x,y) \geq \theta(K)\}$, ${{\tilde{S}}}^- = \{ (x,y) : \theta_{n-2}(x,y) \leq \theta(K)\}$ and ${{\tilde{S}}}= {{\rm cl}}{{\tilde{S}}}^+ \cap {{\rm cl}}{{\tilde{S}}}^-$. We take a $\delta$-neighbourhood ${{\tilde{S}}}_\delta$ of ${{\tilde{S}}}$ with $\delta$ so small that $$\mi({{\tilde{S}}}_\delta \setminus {{\tilde{S}}}) \leq \eps \mi_2({{\tilde{K}}}) (\lambda_{n-2}(K \cap \{x = 0, y=0\}) \sup_K \eta_1)^{-1} \slash 3.$$ Note that as from property (\[t2\]) the function $\theta_{n-2}(x,y)$ is coordinate-wise decreasing, the set ${{\tilde{S}}}\setminus {{\tilde{S}}}^0$ has measure 0. \[sdelta\]Note that any ${{\tilde{D}}}\subset \R_x \times \R_y$ having property (\[eqthetacon\]) must, from Fact \[dividepoints\], have a non-empty intersection both with ${{\tilde{S}}}^+$ and ${{\tilde{S}}}^-$ in some points where the density of $\mi_2$ is positive. Thus any convex set ${{\tilde{D}}}$ with property \[eqthetacon\] will satisfy ${{\tilde{D}}}\cap {{\rm supp}}\mi_2 \cap {{\tilde{S}}}\neq \emptyset$, as the set ${{\tilde{D}}}\cap {{\rm supp}}\mi_2$ is convex, and thus connected. Thus either ${{\tilde{D}}}\cap {{\rm supp}}\mi_2$ is contained in ${{\tilde{S}}}_\delta$ or it has diameter at least $\delta$. The main part of the proof will be an transfinite inductive construction of subsequent $\eps$-appropriate strict lens sets by the following Theorem: \[divide\] Let $n > 2$. Assume Theorem \[main\] holds under non-degenerate assumptions for any $n' < n$. Then under strict assumptions if $\theta({K_{+}}) > 0$ for any ordinal $\gamma$ there exists a division of the set ${\tilde{K}_{+}}$ into $\gamma + 2$ sets $U(\gamma,\beta)$ for $0 \leq \beta \leq \gamma+1$ satisfying: - The set $U(\gamma,\gamma +1)$ is of $\mi_2$ measure at most $\eps' \mi_2({{\tilde{K}}}) (\lambda_{n-2} (K \cap \{x = 0, y=0\}) \sup_K \eta_1)^{-1}$. - The set $U(\gamma,\gamma)$ is either an appropriate set, a strict lens set satisfying condition (\[eqthetacon\]) or has $\mi_2$ measure 0. - All sets $U(\gamma,\beta)$ for $\beta < \gamma$ are either $\eps'$-appropriate sets, empty, or have non-zero $\mi_2$ measure and satisfy $U(\gamma,\beta) \cap {\tilde{K}_{+}}\subset {{\tilde{S}}}_\delta$ - If any $U(\gamma,\beta)$ is empty for $\beta < \gamma$, then $U(\gamma,\gamma)$ has measure 0. If we prove this Theorem, we can apply it to prove Theorem \[main\]. By the inductive assumption we assume Theorem \[main\] holds for $n' < n$. We take $\gamma = \omega_1$ and $\eps' = \eps \slash 3$. As the measure of ${\tilde{K}_{+}}$ is finite, it cannot have $\omega_1$ disjoint subsets of non-zero measure, thus some of $U(\omega_1,\beta)$ for $\beta < \omega_1$ are empty. Thus $U(\omega_1,\omega_1)$ has measure 0. Let ${{\tilde{T}}}$ be the sum of those $U(\omega_1, \beta)$ which are subsets of ${{\tilde{S}}}_\delta$. For every point $(x,y)$ in ${{\tilde{T}}}\cap {{\tilde{S}}}^0$ we apply Theorem \[main\] to the restrictions of $K,A,\theta$ and $\mi$ to $(x,y) \times \R^{n-2}$. The conditions are satisfied — the restiction of $\theta$ is a derivative of $\theta$ and thus non-degenerate, the restriction of $K$ is an generalized Orlicz ball due to Lemma \[sekcjajest\] and the restriction of $\mi$ is a proper measure due to \[sekcjami\], the restriction of a c-set is obviously a c-set. Thus for all $(x,y) \in {{\tilde{S}}}^0$ we have $$\theta_{n-2}(x,y;A) \geq \theta_{n-2}(x,y) \geq \theta_{n-2} (x,y;{{\bar{A}}}),$$ and as $\theta_{n-2}(x,y) = \theta(K)$ from the definition of ${{\tilde{S}}}^0$, from Lemma \[dividepoints\] we get $$\theta \big((({{\tilde{T}}}\cap {{\tilde{S}}}^0) \times \R^{n-2}) \cap A\big) \geq \theta(K) \geq \theta\big((({{\tilde{T}}}\cap {{\tilde{S}}}^0) \times \R^{n-2}) \cap {{\bar{A}}}\big),$$ and also $\theta(({{\tilde{T}}}\cap {{\tilde{S}}}^0) \times \R^{n-2}) = \theta(K)$, if only $\mi ({{\tilde{T}}}\cap {{\tilde{S}}}^0) > 0$. Thus ${{\tilde{T}}}\cap {{\tilde{S}}}^0$ either has measure 0, or is an appropriate set. Meanwhile ${{\tilde{T}}}\setminus {{\tilde{S}}}^0$ has measure at most $\eps (\lambda_{n-2} (K \cap \{x = 0, y=0\}) \sup_K \eta_1)^{-1} \mi_2({{\tilde{K}}}) \slash 3$ from the definition of ${{\tilde{S}}}_\delta$. We therefore have a division of ${\tilde{K}_{+}}$ except a set of measure $2\eps (\lambda_{n-2} (K \cap \{x = 0, y=0\}) \sup_K \eta_1)^{-1} \mi_2({{\tilde{K}}}) \slash 3$ into $(\eps \slash 3)$-appropriate sets. The sum of all the $(\eps \slash 3)$-appropriate sets is by Remark \[epssum\] an $(\eps \slash 3)$-appropriate set. As the integral of $\eta_i$ over the remaining set is at most $2 \eps \mi_2({{\tilde{K}}}) \slash 3$, the whole ${\tilde{K}_{+}}$ is an $\eps$-appropriate set with the same $C_{U,i}$. As we can do this for any $\eps > 0$, by Lemma \[epstotot\] $K$ is an appropriate set, which is the thesis of Theorem \[main\] Almost horizontal divisions {#aux3} --------------------------- We shall prove that if we can divide a lens set with a horizontal, or even almost horizontal (under strict assumptions) line into two sets with equal $\theta$, then the lens set is appropriate. \[horline\] Assume Theorem \[main\] holds for $n' < n$. Under strict assumptions if for a given lens set $D \subset \R^n$ satisfying (\[eqthetacon\]) there exists a horizontal or vertical line $L$ in $\R_x \times \R_y$ dividing ${{\tilde{D}}}$ into two sets ${{\tilde{D}}}_-$ and ${{\tilde{D}}}_+$ of non-zero $\mi_2$-measure with $\theta({{\tilde{D}}}_- \times \R^{n-2}) = \theta({{\tilde{D}}}_+ \times \R^{n-2})$, then $D$ is an appropriate set. Suppose, without loss of generality, the line is horizontal given by $y = a_0$. From Corollary \[thetamain\] we know that for any $a$ we also have $\theta_{n-1}(y = a;D) = \theta(D)$ if defined. From Lemma \[sekcjami\] and property (\[t8\]) we know that the restriction of $\mi$ to $\{y = a\}$ is a proper measure and the restriction of $\theta$ is a non-degenerate $\Theta$ function. From the assumption we can apply Theorem \[main\], thus $$\theta_{n-1}(y = a; D \cap A) \geq \theta_{n-1}(y = a; D) \geq \theta_{n-1}(y=a;D \cap {{\bar{A}}}).$$ As $\theta_{n-1}(y=a;D) = \theta(D)$, which does not depend on $a$, we can apply Lemma \[dividesets\] to get $\theta(D\cap A) \geq \theta(D) \geq \theta(D \cap {{\bar{A}}})$, and as $\theta(D) = \theta(K)$ this means $D$ is appropriate. \[almhorline\] Assume Theorem \[main\] holds for $n' < n$. Under strict assumptions if for a given strict lens set $D$ satisfying (\[eqthetacon\]) for every $\beta > 0$ there exists a line $L_\beta$ with inclination between $0$ and $\beta$ (i.e. almost horizontal) or between $\frac{\pi}{2} -\beta$ and $\frac{\pi}{2}$ (i.e. almost vertical) dividing ${{\tilde{D}}}$ into two sets ${{\tilde{D}}}_-$ and ${{\tilde{D}}}_+$ of non-zero $\mi_2$-measure with $\theta(D_-) = \theta(D_+) = \theta(D)$, then $D$ is an appropriate set. Assume $\theta(D) > 0$ (otherwise the thesis is trivial). We choose a sequence of such lines $L_i$ with $\beta {{\rightarrow}}0$. We choose a subsequence such that all lines are almost vertical or all are almost horizontal (we shall assume without loss of generality that all are almost horizontal). From the compactness of the set of lines intersecting the closure of ${{\tilde{D}}}\cap {{\rm supp}}(\mi_2)$ we can choose a subsequence of lines converging to some line $L$, which, of course, will be horizontal. If $L$ cuts off a non-zero $\mi_2$ measure both above and below it, then both the sets into which ${{\tilde{D}}}$ is divided have the same $\theta = \theta(D)$ from the continuity of $\theta$ with respect to the set and the thesis follows from Lemma \[horline\]. The case left to examine is when $L_i$ approaches the lowest or highest point $p$ of ${{\tilde{D}}}\cap {{\rm supp}}(\mi_2)$. From the definition of a lens set we know that the only points of ${{\tilde{D}}}$ on which $\mi_2$ vanishes lie outside ${\tilde{K}_{+}}$. Thus the lowest point of ${{\tilde{D}}}\cap {{\rm supp}}(\mi_2)$ is the lower extremal point of ${{\tilde{D}}}$. The highest point can be either the upper extremal point of ${{\tilde{D}}}$ or can lie on the boundary of ${{\rm supp}}\mi_2$. First consider the second, simpler case. As $D$ is a strict lens set and $K$ is proper, for any neighbourhood ${{\tilde{U}}}\subset \R_x \times \R_y$ of the highest point $p$ if we take a sufficiently horizontal line passing sufficiently close to $p$, the set it will cut off from ${{\tilde{D}}}$ will be a subset of ${{\tilde{U}}}$ (${{\tilde{D}}}\cap {{\rm supp}}\mi_2$ has no horizontal edges). We know that ${{\rm supp}}\eta_2 \subset {{\rm Int}}\ {{\rm supp}}\mi$, so ${{\rm cl}}\ \widetilde{{{\rm supp}}\eta_2} \subset {{\rm supp}}\mi_2$. As $p$ lies on the boundary of ${{\rm supp}}\mi_2$, it lies outside ${{\rm Int}}\ {{\rm supp}}\mi_2$ and thus outside ${{\rm cl}}\ \widetilde{{{\rm supp}}\eta_2}$, so we can choose an open neighbourhood ${{\tilde{U}}}$ of $p$ on which $\eta_2$ is 0. This neighbourhood has non-zero $\mi_2$ measure, and as $\mi_2({{\tilde{D}}}) > 0$, $\mi_2({{\tilde{D}}}\cap {{\tilde{U}}}) > 0$. But $\eta_2$ on the whole set ${{\tilde{U}}}$ is zero, thus any line cutting off only a part of ${{\tilde{U}}}$ cannot satisfy $\theta(D_+) = \theta(D) > 0$. In the first case $(L_i)$ approaches one of the extremal points of ${{\tilde{D}}}$. Assume it is the lower point. For any line $L_i$ the set ${{\tilde{D}}}^{L_i}_-$ is a lens set. From Lemma \[dividepoints\] there has to be a point $p_i \in D^{L_i}_-$ with $\theta(\{p_i\} \times \R^{n-2}) \leq \theta(D^{L_i}_-) = \theta(D)$. The lines $L_i$ tend to the horizontal line through $(x_1,y_1)$, the lower extremal point of $D$. Thus, the vertical coordinate of $p_i$ tends to $y_1$, and as ${{\tilde{D}}}$ has no horizontal edges, the horizontal coordinate of $p_i$ tends to $x_1$, meaning $p_i {{\rightarrow}}(x_1,y_1)$. From property (\[t2\]), as $\theta_{n-2}(\{p_i\} \times \R^{n-2}) \leq \theta(D)$, for all points $p \in D$ except for $(x_1,y_1)$ we have $\theta_{n-2}(\{p\} \times \R^{n-2}) \leq \theta(D)$. This, however, from Lemma \[dividepoints\] implies in particular, that for any horizontal line $M$ dividing ${{\tilde{D}}}$ into two sets of non-zero $\mi$-measure we have $\theta(D_+) \leq \theta(D)$, which from Lemma \[thetamain\] implies $\theta(D_+) = \theta(D)$, which from Lemma \[horline\] implies that $D$ is appropriate. $\eps$-appropriateness of lens sets ----------------------------------- This subsection puts down precisely what we meant by “long and narrow” in the idea of the proof, and show how to go from the “longness and narrowness” to $\eps$-appropriateness. \[convexproj\] Let $C \subset \R^n$ be a convex set with $\lambda(C) > 0$, let $I \subset C$ be an interval of length $a$, and let $L$ be the line containing $I$. Let $f : C {{\rightarrow}}(0,\infty)$ be a $1\slash m$-concave function. Let $P : \R^m {{\rightarrow}}L$ be the orthogonal projection onto $L$. Let $J \subset I$ be an subinterval of length $b$. Let $C' \subset C$ be such a set that $P(C') \subset J$. Then $$\int_{C'} f(x) dx \leq \Big(\Big(\frac{a+b}{a-b}\Big)^{n+m} - 1\Big) \int_C f(x) dx$$ and also $$\int_{C'} f(x) dx \leq \frac{2^{n+m+2} b}{a} \int_{C} f(x) dx.$$ Let $p(y) = \int_{x : P(x) = y} f(x) dx$ and let $I' = \{y \in L : p(y) > 0\}$. From Fact \[convconc\] the function $p(y) = \int_{x : P(x) = y} f(x) dx$ is a $(1 \slash m+n-1)$-concave function on $L$, thus $I'$ is an interval. As $f$ is positive and $C$ is convex and has positive measure, $p$ is positive on ${{\rm Int}}I$, thus the length of $I'$ is at least $a$. If $J \cap I' = \emptyset$, then $\int_{C'} f(x) dx = 0$ and the thesis is satisfied, so assume $J \cap I' \neq \emptyset$. Then $I' \setminus J$ is a sum of two intervals (one may be empty) of total length at least $a-b$. Thus it contains an interval $I''$ of length at least $\frac{a-b}{2}$, let $\{y_1\} = {{\rm cl}}I''\cap {{\rm cl}}J$ and $y_2$ be the other end of $I''$. Let $y_2$ and $y_3$ be the ends of $I'$ and let $T$ be such that $y_3 = Ty_1 + (1-T)y_2$ (as $y_1$ lies between $y_2$ and $y_3$ we know $T \geq 1$). As $p$ is $1\slash (n+m-1)$-concave, $$p^{1\slash (n+m-1)}(ty_1 + (1-t)y_2) \geq tp^{1\slash (n+m-1)}(y_1) + (1-t)p^{1\slash (n+m-1)}(y_2) \geq tp^{1\slash (n+m-1)}(y_1)$$ for $t \in [0,1]$, which means $$\int_{I''} p(y)dy \geq |I''| \int_{[0,1]} t^{n+m-1} p(y_1)dt = |I''| \frac{1}{n+m} p(y_1).$$ Similarly for $T \geq t \geq 1$ we have $$p^{1\slash (n+m-1)}(ty_1 + (1-t)y_2) \leq tp^{1\slash (n+m-1)}(y_1) + (1-t)p^{1\slash (n+m-1)}(y_2) \leq tp^{1\slash (n+m-1)}(y_1)$$ for $t \in [1,T]$, which gives $$\int_J p(y) \leq |I''| \int_1^{(|J| + |I''|)\slash |I''|} t^{n+m-1} p(y_1) = |I''| \frac{1}{n+m}\Big(\Big(\frac{a+b}{a-b}\Big)^{n+m} - 1\Big) p(y_1).$$ Thus $$\frac{\int_{C'} f(x) dx}{\int_C f(x) dx} \leq \Big(\frac{a+b}{a-b}\Big)^{n+m} - 1,$$ which proves the first part of the Lemma. For the second part note that if $a \slash b \leq 2^{m+n+2}$, then the thesis is true, as $\int_{C'} f(x) dx \leq \int_C f(x) dx$ because $C' \subset C$. For $b \slash a \leq 2^{-(n+m+1)}$ we have $$\Big(\frac{a+b}{a-b}\Big)^{n+m} - 1 = \Big(\frac{1+b \slash a}{1-b \slash a}\Big)^{n+m} - 1 \leq \frac{1 + 2^{n+m} b\slash a}{1 - 2^{n+m} b\slash a} - 1 \leq \frac{2^{n+m+1} b\slash a}{1 \slash 2} = 2^{n+m+2} b \slash a.$$ \[2dim\] Let $\eps > 0$. Let $\mi$ be a measure on $\R^2$ with a $1\slash m$ concave density. Let ${{\tilde{D}}}\subset \R^2$ be a lens set. Let $L$ be the extremal line of ${{\tilde{D}}}$ and $p : \R^2 {{\rightarrow}}L$ the orthogonal projection onto $L$. Let $A$ be a c-set in $\R^2$. Let $A' = p^{-1}(A \cap L)$. Assume the relevant length of ${{\tilde{D}}}$ (that is, the length of $L \cap {{\tilde{D}}}\cap {{\rm supp}}\mi$) is at least $d > 0$, the inclination of ${{\tilde{D}}}$ between $\beta$ and $\pi\slash 2 - \beta$ with $\beta > 0$ and width at most $w = \frac{1}{2 \max(\cot \beta ,\tan \beta)} 2^{-m-3} \eps d $. Then $\mi((A \bigtriangleup A') \cap {{\tilde{D}}}) \leq \eps \mi({{\tilde{D}}})$. Let $p=(x_p,y_p)$ be the rightmost point on $L \cap A$ (and at the same time on $L \cap A'$, from the definition of $A'$). As both $A$ and $A'$ are c-sets, we have $A \bigtriangleup A' \subset \{(x,y) : x > x_p, y < y_p\} \cup \{(x,y) : x < x_p, y > y_p\}$. As $D$ has width at most $w$, the projection of $(A \bigtriangleup A') \cap D$ onto $L$ has length at most $2w \max(\tan \beta , \cot \beta)$. From Lemma \[convexproj\] we know that as $2w \max(\tan \beta , \cot \beta) < 2^{-m-3} \eps d$, we have $\mi((A \bigtriangleup A') \cap D) \leq \eps \mi(D)$. \[ndim\] Let $K \subset \R_x \times \R_y \times \R^{n-2}$ be a generalized Orlicz ball with a proper measure $\mi$ and $D$ be a lens set of relevant length at least $d$, inclination between $\beta$ and $\pi \slash 2 - \beta$ and width at most $w = \frac{1}{2 \max(\cot\beta , \tan\beta)} 2^{-m-3} \eps d (\lambda_{n-2}(K \cap \{x = 0,y=0\}))^{-1}$, where $m$ is such that the density of $\mi$ is $1\slash m$ concave. Let $A$ be a c-set in $\R^n$ and let $A'$ be defined as before. Then $$\mi((A \bigtriangleup A') \cap D) \leq \eps \mi_2 ({{\tilde{D}}}).$$ For each $t \in \R^{n-2}$ we may apply Corollary \[2dim\], and integrate over $K \cap \{x = 0, y = 0\}$ to get a bound for the Lebesgue measure. Note the same argument works if $A$ is the complement of a c-set. \[fundim\] Let $K \subset \R_x \times \R_y \times \R^{n-2}$ be a generalized Orlicz ball with a proper measure $\mi$. Let $D$ be a lens set of relevant length at least $d$, inclination between $\beta$ and $\pi \slash 2 - \beta$ and width at most $w = \frac{1}{2 \max (\cot\beta , \tan\beta)} 2^{-m-3} d M^{-1} \eps (\lambda_{n-2}(K \cap \{x = 0,y=0\}))^{-1}$ and $\phi: \R^n {{\rightarrow}}[0,M]$ be a coordinate-wise decreasing function with $\bar{\phi}(t) := \phi(p(t))$, where $p$ is the orthogonal projection onto $L \times \R^{n-2}$, $L$ being the extremal line of ${{\tilde{D}}}$. Then for any $U \subset D$ we have $$\big|\int_U \phi(t) d\mi(t) - \int_U \bar{\phi}(t) d\mi(t)\big| < \eps \mi_2 ({{\tilde{D}}}).$$ As $\phi$ is coordinate-wise decreasing, the sets $\phi^{-1} ([s,\infty))$ are c-sets. By the integration by parts, $$\int_U \phi(t) d\mi(t) = \int_0^M \mi(\phi^{-1}([s,\infty) \cap U) ds.$$ The sets $(\bar{\phi})^{-1}([s,\infty))$ are formed from the sets $\phi^{-1}([s,\infty))$ as in Corollary \[ndim\]. Thus for each $s$ we have $$\begin{aligned} \Bigg|\mi\Big(\phi^{-1}([s,\infty) \cap U\Big) - \mi\Big((\bar{\phi})^{-1}([s,\infty) \cap U\Big)\Bigg| & \leq \mi\Big(\big(\phi^{-1}([s,\infty) \cap U\big) \bigtriangleup \big((\bar{\phi})^{-1}([s,\infty) \cap U\big)\Big) \\ & \leq \mi\Big(\big(\phi^{-1}([s,\infty) \cap D\big) \bigtriangleup \big((\bar{\phi})^{-1}([s,\infty) \cap D\big)\Big) \leq M^{-1} \eps \mi_2({{\tilde{D}}}),\end{aligned}$$ which integrated over $[0,M]$ gives the thesis. \[lensapp\] Consider a generalized Orlicz ball $K \subset \R_x \times \R_y \times \R^{n-2}$ with a proper measure $\mi$ with both its defining functions $1\slash m$ concave, a strict non-degenerate $\Theta$ function, any $\eps > 0$ and any c-set $A$. Assume Theorem \[main\] holds for $n' < n$. Let $D$ be a lens set satisfying $\theta(D) = \theta(K)$ of relevant length at least $\delta$, inclination between $\beta$ and $\frac{\pi}{2} - \beta$ and width at most $$w = \frac{1}{2 \max (\cot\beta, \tan\beta)} 2^{-3m-4} d \min\{1,(\sup_K \eta_1)^{-1}\} \eps (\lambda_{n-2}(K \cap \{x = 0,y=0\}))^{-1}.$$ Then $D$ is an $8\eps$-appropriate set. Let $L$ be the extremal line of $D$. We switch coordinates in the plane $\R_x \times \R_y$ to orthogonal coordinates $(u,v)$ such that $L = \{v = 0\}$ and $u > 0$ on the positive quadrant of $\R_x \times \R_y$. Define for any set $U \in \{A,{{\bar{A}}}\}$ the set $U'$ by $U \cap \{v = 0\}$ and $U''$ by $U \times \R_v$. Let $K'$ be a generalized Orlicz ball in $\R_u \times \R^{n-2}$ such that ${K_{+}}' = {K_{+}}\cap \{v = 0\}$ given by Lemma \[sekcjajest\] and $K'' = K' \times \R_v$. Let $\eta_i'$ be the restriction of $\eta_i$ to $\{v = 0\}$ and $\eta_i''(u,v,t) = \eta_i(u,0,t)$. Let $\mi'$ be the measure on $K'$ with density $h(u) = \int_{\R_v} \1_{(u,v) \in {{\tilde{D}}}} f(u,v) g(u,v)$, where $f$ and $g$ are the density functions defining $\mi$, and $\mi''$ be the measure on $\R^n$ with density $f(x)g(y)$ (without restricting to $K$). We want to prove that $\int_{U' \cap K'} \eta_i' d\mi'$ is a good approximation of $\int_{U \cap D} \eta_i d\mi$, then check the assumptions for Theorem \[main\] on $K'$ and apply it for $A'$. First note that $$\int_{K'} \phi(u,t) d\mi'(u,t) = \int_{K'' \cap D} \phi(u,0,t) d\mi''(u,v,t)$$ for any function $\phi$ defined on $K'$. This follows directly from the definitions of $K''$, $\mi'$ and $\mi''$. Let $M = \min\{1,(\sup_K \eta_1)^{-1}\}$. We know ${K_{+}}$ is a c-set, thus $\mi((K \bigtriangleup K'') \cap D) \leq \eps \mi_2({{\tilde{D}}})$ by Corollary \[ndim\]. Thus for any $\phi$ we have $$\begin{aligned} \Bigg|\int_{K'} \phi(u,t) d\mi'(u,t) & - \int_{K \cap D} \phi(u,0,t) d\mi(u,v,t)\Bigg| = \\ &\Bigg|\int_{K'' \cap D} \phi(u,t) d\mi''(u,v,t) - \int_{K \cap D} \phi(u,0,t) d\mi''(u,v,t)\Bigg| < M \eps \mi_2({{\tilde{D}}}) \sup|\phi|.\end{aligned}$$ We repeat the same trick for $U \in \{A'', {{\bar{A}}}''\}$, putting $\phi' = \phi \cdot \1_U$ in the above inequality and applying Corollary \[ndim\] again to get $$\begin{aligned} \Bigg|\int_{K' \cap A'} \phi(u,t) d\mi'(u,t) - \int_{K \cap D \cap A} \phi(u,0,t) d\mi(u,v,t) \Bigg| < M \eps \mi_2({{\tilde{D}}}) \sup |\phi|.\end{aligned}$$ Finally, we insert $\eta_i$ for $\phi$ and apply Corollary \[fundim\] to get $$\Bigg|\int_{K' \cap A'} \eta_i'(u,t) d\mi'(u,t) - \int_{K \cap D \cap A} \eta_i(u,v,t) d\mi(u,v,t) \Bigg| \leq 3 \eps \mi_2({{\tilde{D}}}),$$ and the same for integration over $K \cap D \cap {{\bar{A}}}$ and $K \cap D$. Now we want to check assumptions for Theorem \[main\]. $K'$ is a generalized Orlicz ball due to Lemma \[sekcjajest\]. $A'$ is a c-set in $\R_u \times \R^{n-2}$ because $L$ is positively inclined, thus an increase in $u$ translates to an increase in both $x$ and $y$. $\mi'$ is a projection of the measure with the density $f(x) g(y) \1_D$. The first two functions are $1\slash m$ concave, the third is $1 \slash 1$ concave as $D$ is convex. Thus from Facts \[prodncon\] and \[convconc\] the density $h(u)$ of $\mi'$ is a $1\slash (3m + 1)$ concave function. Thus $\mi'$ is a proper measure on $K'$ (recall $\mi'$ is restricted to $K'$, thus the third point of the Definition \[propermeasure\] is satisfied). $\eta_1'$ and $\eta_2'$ are restrictions of $\eta_1$ and $\eta_2$ to $K'$, which is a derivative of $K$, thus they define a non-degenerate $\Theta$-function on $K'$. Let us apply Theorem \[main\]. We get $$\frac{\int_{K' \cap A'} \eta_2'(u,t) d\mi'(u,t)}{\int_{K' \cap A'} \eta_1'(u,t) d\mi'(u,t)} \geq \frac{\int_{K'} \eta_2'(u,t) d\mi'(u,t)}{\int_{K'} \eta_1'(u,t) d\mi'(u,t)} \geq \frac{\int_{K' \cap {{\bar{A}}}'} \eta_2'(u,t) d\mi'(u,t)}{\int_{K' \cap {{\bar{A}}}'} \eta_1'(u,t) d\mi'(u,t)}.\label{4listopada}$$ We need to make the middle expression equal to $\theta(D)$, so for any $u_0, t_0$ we define $$\bar{\eta}_i'(u_0,t_0) = \frac{\int_{K \cap D} \eta_i(u,v,t) d\mi(u,v,t)}{\int_{K'} \eta_i'(u,t) d\mi'(u,t)} \eta_i'(u_0,t_0).$$ As $\bar{\eta}_i' = C_i \eta_i'$, we have inequalities (\[4listopada\]) for functions $\bar{\eta}_i'$ (although they do not necessarily define a $\Theta$ function on $K'$). To bound the error we have $$\begin{aligned} \int_{K'} \big|\bar{\eta}_i'(u,t) - \eta_i'(u,t)\big|d\mi'(u,t) &= \int_{K'} \eta_i'(u,t) \Big| \frac{\int_{K \cap D} \eta_i(u,v,t) d\mi(u,v,t)}{\int_{K'} \eta_i'(u,t) d\mi'(u,t)} - 1 \Big| d\mi'(u,t) \\ &= \Bigg|\int_{K \cap D} \eta_i(u,v,t) d\mi(u,v,t) - \int_{K'} \eta_i'(u,t) d\mi'(u,t)\Bigg| \leq 3 \eps \mi_2({{\tilde{D}}}).\end{aligned}$$ As we bounded the integral of errors, the error on $K' \cap A'$ and $K' \cap {{\bar{A}}}'$ is no larger than $3 \eps \mi_2({{\tilde{D}}})$. We can now for $U \in \{A',{{\bar{A}}}'\}$ and $i \in \{1,2\}$ put $C_{U,i} = \int_{K' \cap U} \bar{\eta}_i' d\mi'$. Applying inequalities (\[4listopada\]) to $\bar{\eta}_i'$ we get $$\frac{C_{A,2}}{C_{A,1}} \geq \frac{\int_{K'} {\bar{\eta}}_2'(u,t) d\mi'(u,t)}{\int_{K'} \bar{\eta}_1'(u,t) d\mi'(u,t)} = \frac{\int_{K\cap D} \eta_2(u,v,t) d\mi(u,v,t)}{\int_{K\cap D} \eta_1(u,v,t) d\mi(u,v,t)} = \theta(D) = \theta(K) \geq \frac{C_{{{\bar{A}}},2}}{C_{{{\bar{A}}},1}},$$ and putting together all the estimates we made we get $|C_{U,i} - \int_{K \cap D \cap U} \eta_i d\mi| \leq 6 \eps \mi_2({{\tilde{D}}})$. The transfinite induction ========================= What is left to prove is Theorem \[divide\]. We will prove by transfinite induction an extended version of Theorem \[divide\], which will allow us to carry the information we need through the induction steps. The sets $U(\gamma, \beta)$ will have to satisfy the conditions of Theorem \[divide\], and furthermore the following conditions: - For any $\gamma > \beta$ we have $U(\gamma,\beta) = U(\beta+1,\beta)$. - For any $\gamma$ we have $U(\gamma,\gamma+1) = U(0,1)$. - If $\gamma$ is a successor ordinal and $U(\gamma,\gamma)$ has positive $\mi_2$ measure, the sets $U(\gamma,\gamma-1)$ and $U(\gamma,\gamma)$ are formed by dividing $U(\gamma-1,\gamma-1)$ with a straight line of positive inclination. - If $\gamma$ is a limit ordinal, $U(\gamma,\gamma) = \bigcap_{\beta < \gamma} U(\beta,\beta)$. - For any $\gamma$ if $U(\gamma,\gamma)$ has positive $\mi_2$ measure, then for all $\beta < \gamma$ the sets $U(\beta,\beta)$ are strict lens sets. Remark that this in fact means we carry out a transfinite inductive construction. The sets $U(\gamma,\beta)$ for $\beta < \gamma$ depend only on the second argument, once constructed. The set $U(\gamma,\gamma+1)$ is equal to $U(0,1)$. The set $U(\gamma,\gamma)$ in each step has a part cut off to make a new set $U(\gamma+1,\gamma+1)$. Note that if $\theta(K) = 0$, then $K$ is appropriate (as any $U \subset K$ with $\mi_2(U) > 0$ has $\theta(U) = 0$). Thus by putting $U(\gamma,0) = {\tilde{K}_{+}}$ for any $\gamma$ and $U(\gamma,\beta) = \emptyset$ for $\gamma + 1 \geq \beta > 0$ we satisfy the conditions of Theorem \[divide\]. Thus, further on, we assume $\theta(K) > 0$. Starting the transfinite induction ---------------------------------- First we need to define the sets $U(0,0)$ and $U(0,1)$ to start the induction. If we take $D = [x_-,x_+] \times [y_-,y_+] \times \R^{n-2}$, then $D$ is a lens set and satisfies condition (\[eqthetacon\]). It is not, however, a strict lens set. The idea is to take two almost vertical lines — one close to the left edge of ${{\tilde{D}}}$ and the other close to the right edge, then look at the $\theta$ of the set they cut off. If $\theta$ is too large, we move the left line closer to the edge, if too small, we move the right line closer to the edge. When we have balanced $\theta$, we repeat the same for horizontal lines. By cutting off a bit from each edge we shall also ensure $[x_1,x_2] \subset (x_-,x_+)$ and similarly for $y$. Below is a formalization of the argument. If ${\tilde{K}_{+}}$ is appropriate to begin with, we take $U(0,1) = \emptyset$ and $U(0,0) = {\tilde{K}_{+}}$. Thus we assume ${\tilde{K}_{+}}$ is not appropriate. Denote by $L^-(x,\beta)$ the line through $(x,y_-)$ with inclination $\pi\slash 2 - \beta$ and by $L^+(x,\beta)$ the line through $(x,y_+)$ with inclination $\pi\slash 2 - \beta$. Denote by ${{\tilde{D}}}^-(x,\beta)$ the subset of $[x_-,x_+]\times [y_-\times y_+]$ to the left of $L^-(x,\beta)$ and by ${{\tilde{D}}}^+(x,\beta)$ the subset to the right of $L^+(x,\beta)$. Note that for $\beta \in (0,\pi \slash 2)$ those sets have positive $\mi_2$ measure by the definition of a proper measure. Let $\phi^-(x,\beta) = \theta({{\tilde{D}}}^-(x,\beta) \times \R^{n-2}) - \theta(K)$ and $\phi^+(x,\beta) = \theta({{\tilde{D}}}^+(x,\beta) \times \R^{n-2})-\theta(K)$. From property \[t6\] these functions are continuous in both arguments. From Lemma \[thetamain\] and Lemma \[almhorline\] there is a $\beta_0 > 0$ such that for $\beta < \beta_0$ we have $\phi^-(x,\beta) > 0$ and $\phi^+(x,\beta) < 0$ for $x \in (x_-,x_+)$. Now start with any $x_l$, $x_u$ and $0 < \beta_l, \beta_u < \beta_0$ such that the sets ${{\tilde{D}}}^-(x_l,\beta_l)$ and ${{\tilde{D}}}^+(x_u,\beta_l)$ have measure no larger than $\eps' (\lambda_{n-2} (K \cap \{x = 0, y=0\}) \sup_K \eta_1)^{-1} \mi(K) \slash 4$ and do not intersect. Now if we fix $x_u$ and $\beta_u$ while letting $x_l$ tend to $x_-$ and $\beta_l$ to 0, then $\theta$ of the sum of the two sets will tend to $\theta({{\tilde{D}}}^+(x_u,\beta_u)\times \R^{n-2})$, which is strictly smaller than $\theta(K)$. If, on the other hand, we fix $x_l$ and $\beta_l$ and let $x_u$ tend to $x_+$ and $\beta_u$ to 0, the $\theta$ of the two sets will approach $\theta({{\tilde{D}}}^-(x_l,\beta_l)\times \R^{n-2})$, which is strictly greater than $\theta(K)$. Thus, from the Darboux property, for some values $x_- < x_l < x_u < x_+$ and $\beta_l$ and $\beta_u$ we have the function $$\theta \Big({{\tilde{D}}}^+(x_u,\beta_u)\times \R^{n-2} \cup {{\tilde{D}}}^-(x_l,\beta_l)\times \R^{n-2}\Big) = \theta(K).$$ The set that remains is a lens set with no vertical boundaries and satisfies property (\[eqthetacon\]). If it is appropriate, we have found our $U(0,0)$ and define $U(0,1) = {{\tilde{D}}}^-(x_l,\beta_l) \cup {{\tilde{D}}}^+(x_u,\beta_u)$. If not, then we can repeat the same trick for $y$ (we needed the non-appropriateness to use Lemma \[horline\]), and achieve a lens set with no horizontal and no vertical boundaries and separated from $x_-$ and $x_+$, i.e. a strict lens set. Thus we define $U(0,1) = {{\tilde{D}}}^-(x_l,\beta_l) \cup {{\tilde{D}}}^+(x_u,\beta_u) \cup {{\tilde{D}}}^-(y_l,\alpha_l) \cup {{\tilde{D}}}^+(y_u,\alpha_u)$ and and $U(0,0) = ([x_-,x_+] \times [y_-,y_+]) \setminus U(0,1)$. \[boundden\] Assume $U(0,0)$ is a strict lens set (otherwise the induction will be trivial). Recall $f$ and $g$ are $1\slash m$-concave functions defining the proper measure $\mi$. As $U(0,0)$ is a strict lens set, it is separated from the boundary of the support of $f \cdot g$. Thus (as $f$ and $g$ are continuous on the interior of their support), they both attain positive minimal values $f_L$ and $g_L$. Also, as they are continuous on their support and $1\slash m$ concave, they are bounded from above by some $f_U$ and $g_U$. Thus for any set $T \subset U(0,0)$ we have $$f_U g_U \lambda_2(T) \geq \mi_2(T) \geq f_L g_L \lambda_2(T),$$ and for any function $t$ on $T$ we have $$f_U g_U \int_T t(p) d\lambda_2(p) \geq \int_T t(p) d\mi_2(p) \geq f_L g_L \int_T t(p) d\mi_2(p).$$ The induction step for successor ordinals ----------------------------------------- For a successor ordinal $\gamma + 1$ we have a division of ${\tilde{K}_{+}}$ for $\gamma$. We put $U(\gamma+1,\gamma+2) = U(\gamma,\gamma+1)$. If $U(\gamma,\gamma)$ is appropriate of positive measure, we put $U(\gamma+1,\gamma) = U(\gamma,\gamma)$ (as an appropriate set is an $\eps$-appropriate set) and $U(\gamma+1,\gamma+1) = \emptyset$. If $U(\gamma,\gamma)$ has measure 0, we put $U(\gamma+1,\gamma) = \emptyset$ and $U(\gamma+1,\gamma+1) = U(\gamma,\gamma)$. The difficult case to deal with will be when $U(\gamma,\gamma)$ is a non-appropriate strict lens set. For brevity denote $U(\gamma,\gamma)$ by ${{\tilde{D}}}$. In this case from Lemma \[almhorline\] there exists an angle $\alpha' > 0$ such that any positively inclinated line dividing ${{\tilde{D}}}$ into two sets of non-zero $\mi$-measure with equal $\theta$ has inclination greater than $\alpha'$ and smaller than $\frac{\pi}{2} - \alpha'$. If the inclination of ${{\tilde{D}}}$ is $\alpha''$, let $\alpha = \min\{\alpha',\alpha'',\frac{\pi}{2} - \alpha''\}$. We shall attempt to cut off a “long and narrow” lens set $U(\gamma+1,\gamma)$ from $U(\gamma,\gamma)$. We shall cut off a narrow set satisfying (\[eqthetacon\]). From Remark \[sdelta\] it will either be long, or be a subset of ${{\tilde{S}}}_\delta$, both of which satisfy us. Take a sufficiently small $w$ ($w < \frac{1}{2 \max (\cot\alpha , \tan\alpha)} 2^{-3m-4} \delta \min\{1,(\sup_K \eta_1)^{-1}\} \frac{\eps}{8} (\lambda_{n-2}(K \cap \{x = 0,y=0\})^{-1}$, where $m$ is such that the density functions of $\mi$ are $1\slash m$-concave, will suffice). For any angle $\xi \in [0,\frac{\pi}{2}]$ we can find continuously a line $L_\xi$ of inclination $\xi$ such that the part ${{\tilde{D}}}_+(\xi)$ of ${{\tilde{D}}}\cap {{\rm supp}}\mi$ lying above and to the left of $L_\xi$ has width no larger than $w$. From Lemma \[thetamain\] we have $\theta(D_+(0)) \geq \theta(D)$ and $\theta(D_+(\pi\slash2)) \leq \theta(D)$. From the Darboux property for some $\xi$ we have $\theta(D_+(\xi)) = \theta(D)$. We take $U(\gamma+1,\gamma) = {{\tilde{D}}}_+(\xi)$. Let $I_\xi$ denote the segment of $L_\xi$ intersecting ${{\tilde{D}}}$. The set $U(\gamma + 1, \gamma + 1) = U(\gamma,\gamma) \setminus U(\gamma+1,\gamma)$ is, of course, a strict lens set, satisfying condition (\[eqthetacon\]), because the new edge has inclination between $\alpha$ and $\frac{\pi}2 - \alpha$, and all the other edges come from the old set ${{\tilde{D}}}$. It remains to check that $U(\gamma+1,\gamma)$ satisfies the conditions of the transfinite induction. First let us check what is the inclination of $U(\gamma+1,\gamma)$. If both the ends $I_\xi$ fall upon the upper-left border of $U(\gamma,\gamma)$, then they are the extremal points of $U(\gamma+1,\gamma)$, and thus the inclination of $U(\gamma+1,\gamma)$ is the inclination of the segment, which is between $\alpha'$ and $\frac{\pi}{2} - \alpha'$. If one of them falls upon the lower-right border, then the extremal points of $U(\gamma+1,\gamma)$ are the end of $I_\xi$ on the upper-left border and one of the extremal points of $U(\gamma,\gamma)$, and the inclination of $U(\gamma+1,\gamma)$ is between the inclination of $U(\gamma+1,\gamma)$ and the inclination of the segment, which means it is between $\alpha$ and $\frac{\pi}{2} - \alpha$. If both ends fall upon the lower-right border, the extremal points of $U(\gamma+1,\gamma)$ are the extremal points of $U(\gamma,\gamma)$, which means $U(\gamma+1,\gamma)$ has inclination $\alpha''$. Thus, the inclination of $U(\gamma,\gamma)$ is between $\alpha$ and $\frac{\pi}{2} - \alpha$. If $U(\gamma+1,\gamma) \subset {{\tilde{S}}}_\delta$, the induction thesis is satisfied. Thus we may assume $U(\gamma+1,\gamma)$ sticks outside ${{\tilde{S}}}_\delta$. Note that as $\theta_{n-2}(p)$, $p \in \R_x \times \R_y$, is a coordinate-wise increasing function from property (\[t2\]), one of the extremal points of $U(\gamma+1,\gamma)$ has to lie outside ${{\tilde{S}}}_\delta$, and at least one point of ${{\tilde{S}}}$ lies on the extremal line of $U(\gamma+1,\gamma)$. Thus, the length of the segment of the extremal line contained in ${\tilde{K}_{+}}$ is at least $\delta$. Thus $U(\gamma+1,\gamma)$ has relevant length at least $\delta$, width at most $w$ and inclination between $\alpha$ and $\frac{\pi}{2} - \alpha$. Thus from Lemma \[lensapp\] we know that $U(\gamma+1,\gamma)$ is $\eps$-appropriate, which means we completed the induction step. The induction step for limit ordinals ------------------------------------- For limit ordinals $\gamma$ the set $U(\gamma,\gamma + 1) = U(0,1)$, the sets $U(\gamma,\beta)$ for $\beta < \gamma$ are defined by $U(\gamma,\beta) = U(\beta+1,\beta)$, and from the inductive assumption the conditions for $U(\gamma,\beta)$ are met. We define $U(\gamma,\gamma)$ as the intersection $\bigcap_{\beta < \gamma} U(\beta,\beta)$. We have to check that $U(\gamma,\gamma)$ thus defined satisfies the induction thesis. If any of the sets $U(\gamma',\beta), \beta < \gamma'$ was empty, then from the inductive assumption $U(\gamma'+1,\gamma'+1)$ has $\mi_2$ measure 0 and thus $U(\gamma,\gamma)$ has $\mi_2$ measure 0, which satisfies the conditions. If $U(\beta,\beta)$ was not a strict lens set for some $\beta < \gamma$, then $U(\gamma,\gamma)$ has measure 0, again satisfying the conditions. The case to worry about is when $U(\gamma,\gamma)$ is a intersection of a descending family of strict lens sets satisfying condition (\[eqthetacon\]) and has a positive $\mi_2$ measure. A descending intersection of lens sets is a lens set — the circumscribed rectangle is the intersection of circumscribed rectangles, the extremal points belong to the intersection, and the intersection is convex. A descending intersection of sets satisfying (\[eqthetacon\]) with positive $\mi_2$ measure satisfies (\[eqthetacon\]) by property (\[t6\]). We have to prove that the intersection is either a strict lens set, or appropriate. As $U(\gamma,\gamma) \subset U(0,0)$, it is separated from $x_-,x_+,y_-$ and $y_+$. Thus we only have to check it does not have a horizontal or vertical edge. Suppose $U(\gamma,\gamma)$ has a horizontal or vertical edge $I$. We may assume, without loss of generality, that $I$ is a horizontal edge. We will assume it is an upper horizontal edge. In the case of the lower one, the proof goes very similarily: every construction of new points is done centrally-symetric, and every inequality is opposite. In one place, where the proof significantly changes, we will say it explicitly. Let $(x_0, y_0)$ be the left end of $I$ and $(x_1, y_0)$ the right end. First we shall prove the following Lemma: With the notation as previously we have ${{\rm cl}}I \cap {{\rm supp}}\eta_2 \neq \emptyset$. \[5.9\] We shall prove the Lemma by contradiction. Suppose that ${{\rm cl}}I \cap {{\rm supp}}\eta_2 = \emptyset$. The idea of the proof is that at some moment, a line dividing some $U(\beta,\beta)$ into $U(\beta+1,\beta+1)$ and $U(\beta+1,\beta)$ lies above $I$ and cuts off only points that are above and to the right of the left end of $I$, or almost so, and thus only cuts off points, which do not belong to ${{\rm supp}}\eta_2$. Thus $\eta_2$ is zero on the set $U(\beta+1,\beta)$ which was cut off, $\theta(U(\beta+1,\beta)) = 0$, a contradiction. Now for a formal proof: As $\theta(U(\gamma,\gamma)) > 0$, some point of $U(\gamma,\gamma)$ has to lie inside ${{\rm supp}}\eta_2$, thus (as ${{\rm supp}}\eta_2$ is a c-set), the lower left extremal point of $U(\gamma,\gamma)$ lies in ${{\rm supp}}\eta_2$. Note, that as $\eta_2 = 0$ on $I$, $I$ has to be an upper edge, the lower edge case is trivial here. Let $x_2 < x_0$ be such that $(x_2,y_0) \not\in {{\rm supp}}\eta_2$. Then let $y_2 < y_0$ be a number so close to $y_0$ that $(x_2,y_2) \not\in {{\rm supp}}\eta_2$ and $(x_2,y_2) \not\in U(\gamma,\gamma)$. Take a $\beta < \gamma$ such that $(x_2,y_2) \not\in U(\beta,\beta)$. As $U(\beta,\beta)$ is a lens set, no points $(x_2,y)$ with $y > y_2$ belong to $U(\beta,\beta)$. As $U(\beta,\beta)$ is a strict lens set, and $I \subset U(\beta,\beta)$, there exists a $y_3 > y_0$ such that $(x_1,y_3) \in U(\beta,\beta)$. Take $y_3$ to be so small that $$\label{nachyl} \frac{y_3 - y_0}{x_1-x_0} < \frac{y_0 - y_2}{x_0 - x_2}.$$ Let $\beta'$ be the smallest such ordinal that $(x_1,y_3) \not\in U(\beta',\beta')$. Of course $\beta'> \beta$ and from the inductive assumption $\beta'$ is a successor ordinal. Let $L$ be the line which divides $U(\beta'-1,\beta'-1)$ into $U(\beta',\beta')$ and $U(\beta',\beta'-1)$. $L$ intersects the interval $[(x_1,y_0),(x_1,y_3)]$ and does not intersect $I$, so, from (\[nachyl\]), $L$ intersects the line $x = x_2$ at some point above $(x_2,y_2)$. $U(\beta',\beta'-1) \subset U(\beta'-1,\beta'-1) \subset U(\beta,\beta)$, thus $U(\beta',\beta'-1)$ contains no points $(x_2,y)$ with $y > y_2$. Thus all points from $U(\beta',\beta'-1)$ lie above and to the right of $(x_2,y_2)$. As ${{\rm supp}}\eta_2$ is a c-set and $(x_2,y_2) \not\in {{\rm supp}}\eta_2$, we have $U(\beta',\beta'-1) \cap {{\rm supp}}\eta_2 = \emptyset$, thus $\theta(U(\beta',\beta'-1)) = 0$. But as we assumed $\theta(K) > 0$ this means that $U(\beta',\beta'-1)$ is empty, a contradiction. Thus we know that ${{\rm cl}}I \cap {{\rm supp}}\eta_2 \neq \emptyset$, and as ${{\rm Int}}\ {{\rm supp}}\eta_1 \supset {{\rm supp}}\eta_2$, there is an interval $I' \subset I \cap {\tilde{K}_{+}}$ of positive length, which means $\theta_{n-1}(y_0;I\times \R^{n-2})$ is defined. The idea of the proof in this case is to prove that $\theta_{n-1}(y_0;I\times\R^{n-2}) = \theta(K)$, which from Lemma \[thetamain\] and Lemma \[horline\] will imply $U(\gamma,\gamma)$ is appropriate. We prove this by selecting a moment at which the set $U(\beta+1,\beta)$ which is being cut off lies above $I$, and comparing its $\theta$ (which we know to be $\theta(K)$) to $\theta_{n-1}(x_0; I\times\R^{n-2})$. The formal proof goes as follows: We assume $I$ is an upper horizontal edge. In the case of $I$ being a lower horizontal edge, the below construction works centrally-symetrically.Recall $(x_0, y_0)$ be the left end of $I$ and $(x_1,y_0)$ the right end. Take any $0 < \eps < |I|$. Take $x_2 = x_0 - \eps$ and $y_2 < y_0$ and close enough that $(x_2,y_2) \not\in U(\gamma,\gamma)$. Take $\beta_1 < \gamma$ such that $(x_2,y_2) \not\in U(\beta_1,\beta_1)$. Next take a point $(x_1,y_3)$ with $y_3 > y_0$ such that (\[nachyl\]) is satisfied, and take $\gamma > \beta_2 > \beta_1$ such that the upper extremal point of $U(\beta_2,\beta_2)$ lies below $y_3$. Again, as in the proof of Lemma \[5.9\], any line dividing some $U(\beta,\beta)$ for $\beta > \beta_2$ and crossing $x = x_1$ between $y_3$ and $y_0$ will exit $U(\beta,\beta)$ at some $x > x_2$. For $\gamma > \beta > \beta_2$ any line cutting off the upper extremal point $p$ of $U(\beta,\beta)$ will cross $x = x_1$ between $y_3$ and $y_0$ because $p$ will lie below $y_3$ (as $U(\beta,\beta) \subset U(\beta_2,\beta_2)$ and to the right of and above $(x_1,y_0)$ as $U(\beta,\beta) \subset U(\gamma,\gamma)$ and the line has to go below $p$ and above $(x_1,y_0)$ as $(x_1,y_0) \in U(\beta,\beta)$. Let us consider the functions ${{\tilde{\eta}}}_i(x,y) = \int_{\R^{n-2}} \eta_i(x,y,t) dt$ for $i = 1,2$. The set $[x_-,x_+] \times [y_-,y_+]$ is compact and ${{\tilde{\eta}}}_i$ are continuous from property (\[t4\]) (recall $n > 2$), thus we can find a $\tilde{\delta} > 0$ such that $$\|p_1 - p_2\| < \tilde{\delta}\ \ {{\Rightarrow}}\ \ |{{\tilde{\eta}}}_i(p_1) - {{\tilde{\eta}}}_i(p_2)| < \eps$$ for $i = 1,2$. Also, as $g$ (the density of $\mi$ with respect to $y$) is $1\slash m$-concave, it is continuous on the interior of its support, and thus we can take $\tilde{\delta}$ such that also $|g(p_1) - g(p_2)| < \eps$. If ${{\tilde{\eta}}}_1((x_1,y_0)) > 0$ take $\delta = \tilde{\delta}$. If not, then as ${{\rm Int}}{\tilde{K}_{+}}\supset {{\rm supp}}{{\tilde{\eta}}}_2$, there exists an interval $J' \subset I \cap ({{\rm Int}}{\tilde{K}_{+}}\setminus {{\rm supp}}{{\tilde{\eta}}}_2)$ of positive length $c$. As $\R_x \times \R_y \setminus {{\rm Int}}{\tilde{K}_{+}}$ and ${{\rm supp}}{{\tilde{\eta}}}_2$ are closed, we may take $\delta \leq \tilde{\delta}$ small enough, that there exists an interval $J \subset I$ of length at least $c\slash 2$, such that $$J \times [y_0-\delta,y_0 + \delta] \subset {{\rm Int}}{\tilde{K}_{+}}\setminus {{\rm supp}}{{\tilde{\eta}}}_2.$$ Take $\gamma > \beta_3 > \beta_2$ such that the whole set $U(\beta_3,\beta_3)$ lies below the line $y = y_0 + \delta$. Now let $(x_4,y_4)$ be the upper right extremal point of $U(\beta_3,\beta_3)$. Let $\beta_4$ be the first $\beta$ such that $(x_4,y_4) \not\in U(\beta_4,\beta_4)$. The ordinal $\beta_4$ has to be a successor, let $L'$ be the line dividing $U(\beta_4 - 1, \beta_4 -1)$ into $U(\beta_4,\beta_4 - 1)$ and $U(\beta_4,\beta_4)$, and let $l$ be the inclination of $L'$. Any tangent to the upper-left border of $U(\beta_4,\beta_4)$ has inclination no smaller than $l$. Let $\beta_5$ be the first ordinal greater than $\beta_4$ for which some tangent to the upper left edge of $U(\beta_5, \beta_5)$ has inlination strictly smaller than $l$. Again, $\beta_5$ has to be a successor ordinal. Let $L$ be the line dividing $U(\beta_5 - 1,\beta_5 - 1)$ into $U(\beta_5,\beta_5 - 1)$ and $U(\beta_5,\beta_5)$. This line has to go above $I$, to become a part of the upper edge of $U(\beta_5,\beta_5)$. As the inclination of this line is smaller than the inclination of any tangent to the upper left edge of $U(\beta_5 - 1,\beta_5 - 1)$, the right end of $L \cap U(\beta_5 - 1,\beta_5 - 1)$ lies on the lower right edge of $U(\beta_5 - 1,\beta_5 - 1)$. It lies above $y_0$, as it goes above $I$ and has positive inclination, and lies to the right of $x_1$, as the lower right edge of $u(\beta_5 - 1,\beta_5 - 1)$ above $y_0$ lies to the right of $x_1$. Now we will prove some inequalities on $\theta$. In the case of $I$ being lower edge, the inequalities are simply reversed. Let ${{\tilde{D}}}= {{\tilde{D}}}(\eps)$ be the part of $U(\beta_5,\beta_5 - 1)$ that lies to the left of $x = x_1$. As usual, $D = D(\eps) = {{\tilde{D}}}(\eps) \times \R^{n-2}$. As $U(\beta_5, \beta_5 - 1)$ is a lens set, from Lemma \[thetamain\] we know $$\theta({{\tilde{D}}}\times \R^{n-2}) \leq \theta(U(\beta_5,\beta_5 - 1) \times \R^{n-2}) = \theta(K).$$ Remark that the line $L''$ that cut $(\beta_5,\beta_5 - 1)$ off contains the whole lower edge of ${{\tilde{D}}}$. Thus as the inclination of $L''$ is smaller than the inclination of the upper edge of ${{\tilde{D}}}$ the function $x \mapsto \lambda_1 ({{\tilde{D}}}_x)$, where ${{\tilde{D}}}_x$ is the section of ${{\tilde{D}}}$ at $x$, is strictly increasing. If ${{\tilde{D}}}$ has $\mi_2$ measure 0, then the lower extremal point $p_5$ of $U(\beta_5,\beta_5 - 1)$ lies above and to the right of any point of $U(\gamma,\gamma)$. However, from property (\[t2\]) $$\theta_{n-2} (p_5) \leq \theta(U(\beta_5,\beta_5 - 1) \times \R^{n-2}) = \theta(K),$$ which means that from property (\[t2\]) for any point $p \in U(\gamma,\gamma)$ we have $$\theta_{n-2}(p) \leq \theta_{n-2}(p_5) \leq \theta(K).$$ However, we know $\theta(U(\gamma,\gamma) \times \R^{n-2}) = \theta(K)$, which, from Fact \[dividesets\] implies that for almost all points in $U(\gamma,\gamma)$ we have $\theta_{n-2}(p) = \theta(K)$. Thus any horizontal line divides $U(\gamma,\gamma)$ into two sets with equal $\theta$, which from Lemma \[horline\] implies $U(\gamma,\gamma)$ is appropriate. Hereafter we shall assume $\mi_2({{\tilde{D}}}) > 0$. Note that the whole set ${{\tilde{D}}}$ lies in the rectangle $[x_2,x_1] \times [y_0 - \delta,y_0 + \delta]$. It lies to the left of $x_1$ from its definition. To the right of $x_2$ as $\beta_5 > \beta_2$. Below $y_0 + \delta$ because $\beta_5 > \beta_3$. Above $y_0 - \delta$ because its lower edge is the line $L''$ which passes above $(x_0,y_0)$, so if it dipped below $y_0 - \delta$, it would also (as $\eps < |I|$) have to reach above $y_0 + \delta$. Now we want to estimate $\theta_{n-1}(y_0;I\times \R^{n-2})$ by $\theta({{\tilde{D}}}\times \R^{n-2})$. This will, unfortunately, involve quite a lot of technicalities. We begin with a lemma: \[compla\] There exist two numbers $c_1, c_2 > 0$ independent of $\eps$ such that for sufficiently small $\eps > 0$ and a set ${{\tilde{D}}}$ constructed as above for this $\eps$ we have $$\lambda_2({{\tilde{D}}}\cap \{(x,y) : {{\tilde{\eta}}}_i(x,y) > c_1\}) > c_2 \lambda_2({{\tilde{D}}}),$$ for $i = 1,2$. The proof for this lemma is a bit different for $I$ being a lower edge. First, let us prove it for an upper edge. First we prove the thesis for ${{\tilde{\eta}}}_1$. Suppose ${{\tilde{\eta}}}_1(x_1,y_0) > 0$. Then supposing $\eps < \frac{1}{2} {{\tilde{\eta}}}_1(x_1,y_0)$ for any $(x,y) \in {{\tilde{D}}}$ we have $${{\tilde{\eta}}}_1(x,y) \geq {{\tilde{\eta}}}_1(x_1,y) \geq {{\tilde{\eta}}}_1(x_1,y_0) - \eps > \frac{1}{2}{{\tilde{\eta}}}_1(x_1,y_0),$$ as $|y - y_0| < \delta$ and ${{\tilde{\eta}}}_1$ is decreasing as $\eta_1$ is decreasing, thus it is enough to have $c_1 < \frac{1}{2} {{\tilde{\eta}}}_1(x_1,y_0)$ and $c_2 < 1$. In the case ${{\tilde{\eta}}}_1(x_1,y_0) = 0$ let $b_1 = \sup \{x : {{\tilde{\eta}}}_1(x,y_0) > 0\}$. Recall that we constructed an interval $J$ of length $c$ (independent of $\eps$) such that $J \times [y_0 - \delta, y_0 + \delta] \subset {{\rm supp}}{{\tilde{\eta}}}_1 \setminus {{\rm supp}}{{\tilde{\eta}}}_2$. Let $J = [j_0,j_1]$. Now as ${{\tilde{D}}}\subset [x_2,x_1] \times [y_0 - \delta,y_0 + \delta]$ for $x \in J$ and $(x,y) \in {{\tilde{D}}}$ we have ${{\tilde{\eta}}}_2(x,y) = 0$ and ${{\tilde{\eta}}}_1(x,y) > 0$, which means $j_2 \leq b_1$. On the other hand $\theta({{\tilde{D}}}) \geq \theta(K) > 0$, thus ${{\tilde{D}}}$ contains points with positive $\eta_2$, and thus for these points $(x,y)$ we have $x < j_0$. Note that as $\lambda_1({{\tilde{D}}}_x)$ is strictly increasing, so if ${{\tilde{D}}}$ condains some point to the left of $j_0$, then for every $x \in J$ the set ${{\tilde{D}}}_x$ has positive Lebesgue measure. Let $j = \frac{j_0 + j_1}{2}$ be the midpoint of $J$. If $\eps < \frac{1}{2} {{\tilde{\eta}}}_1(j,y_0)$ we have $$\begin{aligned} \lambda_2\Bigg({{\tilde{D}}}\cap \Big\{(x,y) : {{\tilde{\eta}}}_1(x,y) > \frac{1}{2}{{\tilde{\eta}}}_1(j, y_0)\Big\}\Bigg) &\geq \lambda_2 \Bigg(\Big\{(x,y) \in {{\tilde{D}}}: {{\tilde{\eta}}}_1(x,y_0) \geq {{\tilde{\eta}}}_1(j,y_0)\Big\}\Bigg) \\ &\geq \lambda_2\Big(\big\{(x,y) \in {{\tilde{D}}}: x < j\big\}\Big).\end{aligned}$$ Now we perform a similar operation as in Lemma \[convexproj\]. The function $p(x) = \lambda({{\tilde{D}}}_x)$ is concave on its support, $p(j_0) \geq 0$, thus for every $t \in [0,1]$ we have $p((1-t)j_0 + tj) \geq t p(j)$ and for $t > 1$ we have $p((1-t)j_0 + tj) \leq t p(j)$. Thus $$\lambda_2 \Big(\{(x,y) \in {{\tilde{D}}}: x < j\}\Big) = \int_{x < j} p(x) \geq |j - j_0| \int_0^1 t p(j) = \frac{j - j_0}{2} p(j).$$ In a similar vein $$\lambda_2 \Big(\{x,y) \in {{\tilde{D}}}: x \geq j\}\Big) = \int_{x \geq j} p(x) \leq |j - j_0| \int_1^{\frac{x_1 - j_0}{j - j_0}} t p(j) = \frac{j - j_0}{2} \Bigg(\frac{(x_1 - j_0)^2}{(j-j_0)^2} - 1\Bigg) p(j),$$ which gives us: $$\frac{\lambda_2({{\tilde{D}}})}{\lambda_2({{\tilde{D}}}\cap \{(x,y) : {{\tilde{\eta}}}_i(x,y) > \frac{1}{2}{{\tilde{\eta}}}_1(j,y_0)\})} \leq 1 + \frac{\lambda_2 (\{x,y) \in {{\tilde{D}}}: x \geq j\})}{\lambda_2 (\{x,y) \in {{\tilde{D}}}: x < j\})} \leq 1 + \frac{(x_1 - j_0)^2}{(j-j_0)^2} - 1 = \frac{(x_1 - j_0)^2}{(j-j_0)^2},$$ which gives the thesis for $c_1 \leq \frac{1}{2}{{\tilde{\eta}}}_1(j,y_0)$ and $c_2 \leq \frac{(j-j_0)^2}{(x_1 - j_0)^2}$. To deal with ${{\tilde{\eta}}}_2$ first use Remark \[boundden\] to get $$\int_{{\tilde{D}}}{{\tilde{\eta}}}_2(x,y) d\mi_2(x,y) = \theta({{\tilde{D}}}\times \R^{n-2}) \int_{{{\tilde{D}}}} {{\tilde{\eta}}}_1(x,y) d\mi_2(x,y) \geq \theta(K) c_1 c_2 f_L g_L \lambda_2({{\tilde{D}}}).$$ On the other hand ${{\tilde{\eta}}}_2$ is bounded from above on ${{\rm supp}}{{\tilde{\eta}}}_2$ by $M = {{\tilde{\eta}}}_2(0,0)$, as it is continuous. We have $$\begin{aligned} f_L g_L c_1 c_2 \theta(K) \lambda_2({{\tilde{D}}}) &\leq \int_D {{\tilde{\eta}}}_2(x,y) d\mi_2(x,y) \leq f_U g_U \int_D {{\tilde{\eta}}}_2(x,y) d\lambda_2 \\ &\leq f_U g_U \big(M \lambda_2({{\tilde{D}}}\cap \{ {{\tilde{\eta}}}_2(x,y) > a\}) + a \lambda({{\tilde{D}}})\big).\end{aligned}$$ The above holds for any $a$. Let us take $2a = \frac{c_1 c_2 \theta(K) f_L g_L}{f_U g_U}$. Then we have $$\lambda_2({{\tilde{D}}}\cap \{{{\tilde{\eta}}}_2(x,y) > a \}) \geq a \lambda_2({{\tilde{D}}}) \slash M,$$ which implies (with the assumption $\eps < a \slash 2$) $$\lambda({{\tilde{D}}}\cap \{{{\tilde{\eta}}}_2(x,y_0) > a \slash 2\}) \geq \lambda({{\tilde{D}}}\cap \{{{\tilde{\eta}}}_2(x,y_0) > a - \eps\}) \geq (a \slash M) \lambda(D).$$ Now, let us assume that $I$ is a lower horizontal edge. The proof is much easier in that case. Since $\theta(U(\gamma, \gamma)) > 0$, there is a segment $I' \subset I$ starting at lower left end of $I$, such that $I' \subset {{\rm supp}}{{\tilde{\eta}}}_2$. Moreover, we can take such $I'' \subset I'$, that on $I''$ we have ${{\tilde{\eta}}}_2 > c$ for some $c$. Since $x \to \lambda({{\tilde{D}}}_x)$ is decreasing on $I$, we have $$\lambda_2({{\tilde{D}}}\cap \{(x,y): {{\tilde{\eta}}}(x,y)>c \}) \geq \lambda_2({{\tilde{D}}}\cap I'' \times \R) \geq \lambda_2({{\tilde{D}}})\frac{|I''|}{|I|}.$$ \[calkateta\]There exists a constant $c_3$ such that for all sufficiently small $\eps$ we have $$\int_{{\tilde{D}}}{{\tilde{\eta}}}_i(x,y) d\mi_2(x,y) \geq c_3 \mi_2({{\tilde{D}}}).$$ $$\int_{{\tilde{D}}}{{\tilde{\eta}}}_i(x,y) d\mi_2 \geq \int_{{\tilde{D}}}{{\tilde{\eta}}}_i(x,y) \1_{{{\tilde{\eta}}}_i(x,y) > c_1} d\mi_2 \geq c_1 c_2 \lambda_2({{\tilde{D}}}) \geq c_1 c_2 f_L g_L\mi_2({{\tilde{D}}}).$$ The rest of the proof is independent of the fact, whether $I$ is lower or upper edge, we simply use already proven facts. Now to estimate $\theta(D(\eps))$. As $\beta_5 > \beta_2$ we know $\|(x,y) - (x,y_0)\| < \delta$, thus $|{{\tilde{\eta}}}_i(x,y) - {{\tilde{\eta}}}_i(x,y_0)| < \eps$. Thus we get: $$\begin{aligned} \theta(D(\eps)) &= \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y) d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y) d\mi_2(x,y)} \leq \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) + \eps d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) - \eps d\mi_2(x,y)} \\ &= \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) + \eps d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) d\mi_2(x,y)} \cdot \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) - \eps d\mi_2(x,y)} \cdot \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) d\mi_2(x,y)}.\end{aligned}$$ The first and second fraction will both be bounded by 1 as $\eps {{\rightarrow}}0$ from Corollary \[calkateta\]: $$\begin{aligned} \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) + \eps\ d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) d\mi_2(x,y)} - 1 = \frac{\eps \int_{{{\tilde{D}}}(\eps)} d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) d\mi_2(x,y)} \leq \frac{\eps \mi_2({{\tilde{D}}}(\eps))}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y) - \eps\ d\mi_2(x,y)} = \frac{\eps}{c_3 - \eps},\end{aligned}$$ and (here we prove that the lower bound for the reciprocal converges to 1, which is equivalent) $$\begin{aligned} \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) - \eps\ d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) d\mi_2(x,y)} - 1 = \frac{-\eps \int_{{{\tilde{D}}}(\eps)} d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) d\mi_2(x,y)} \geq \frac{-\eps \mi_2({{\tilde{D}}}(\eps))}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y) - \eps\ d\mi_2(x,y)} = \frac{-\eps}{c_3 - \eps}.\end{aligned}$$ The third fraction is the one that should converge to (or at least, for very small $\eps$, be bounded by) $\theta_{n-1}(y_0;I\times \R^{n-2})$. Let $I_\eps = [x_0 - \eps,x_1] = [x_2,x_1]$. As $\|(x,y) - (x,y_0)\| < \delta$, we have: $$\begin{aligned} \frac{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_2(x,y_0) d\mi_2(x,y)}{\int_{{{\tilde{D}}}(\eps)} {{\tilde{\eta}}}_1(x,y_0) d\mi_2(x,y)} = \frac{\int_{I_\eps} \int_{{{\tilde{D}}}_x(\eps)} {{\tilde{\eta}}}_2(x,y_0) f(x) g(y) dx dy)}{\int_{I_\eps} \int_{{{\tilde{D}}}_x(\eps)} {{\tilde{\eta}}}_1(x,y_0) f(x) g(y) dx dy)} \leq \frac{g(y_0) + \eps}{g(y_0) - \eps} \cdot \frac{\int_{I_\eps} {{\tilde{\eta}}}_1(x,y_0) f(x) \lambda({{\tilde{D}}}_x) dx}{\int_{I_\eps} {{\tilde{\eta}}}_2(x,y_0) f(x) \lambda({{\tilde{D}}}_x) dx}.\end{aligned}$$ The first of these fractions obviously tends to $1$ as $g(y_0) \geq g_L > 0$. The second can be bounded using Lemma \[gencheb\], part 3: $$\begin{aligned} \frac{\int_{I_\eps} {{\tilde{\eta}}}_1(x,y_0) f(x) \lambda({{\tilde{D}}}_x) dx}{\int_{I_\eps} {{\tilde{\eta}}}_2(x,y_0) f(x) \lambda({{\tilde{D}}}_x) dx} \leq \frac{\int_{I_\eps} {{\tilde{\eta}}}_1(x,y_0) f(x) dx} {\int_{I_\eps} {{\tilde{\eta}}}_2(x,y_0) f(x) dx} = \frac{\int_{I_\eps} {{\tilde{\eta}}}_1(x,y_0) f(x) g(y_0) dx} {\int_{I_\eps} {{\tilde{\eta}}}_2(x,y_0) f(x) g(y_0) dx} = \theta_{n-1}(y_0;I_\eps \times \R^{n-2})\end{aligned}$$ From property (\[t6\]) used for restrictions to $y = y_0$ we have $\theta_{n-1}(y_0;I_\eps \times \R^{n-2}) {{\rightarrow}}\theta_{n-1}(y_0;I\times \R^{n-2})$ when $\eps {{\rightarrow}}0$. Putting all the estimates together we get $\theta(K) \leq \theta(D(\eps)) \leq c(\eps) \theta_{n-1}(y_0;I \times \R^{n-2})$, where $c(\eps) {{\rightarrow}}1$. Thus we can go with $\eps$ to 0 to get $\theta(K) \leq \theta_{n-1}(y_0;I\times \R^{n-2})$. On the other hand from Lemma \[thetamain\] we have $\theta_{n-1}(y_0;I\times \R^{n-2}) \leq \theta(U(\gamma,\gamma)) = \theta(K)$, which means $\theta_{n-1}(y_0;I\times \R^{n-2}) = \theta(K)$. From Lemma \[incthetay\] this means that for any horizontal line $L$ intersecting $U(\gamma,\gamma)$ we have $\theta(I) \leq \theta(U(\gamma,\gamma) \cap L)$, which, from Lemma \[thetamain\] implies that any horizontal line divides $U(\gamma,\gamma)$ into two sets with equal $\theta$. Thus, from Lemma \[horline\], $U(\gamma,\gamma)$ is appropriate. This finishes the proof of the inductive step in the limit ordinal case: the assumption $U(\gamma,\gamma)$ has positive measure and is not a strict lens set led us to the conclusion it is appropriate. $\Theta$ functions on Orlicz balls ================================== Our main target is proving Theorem \[orliczglownetw\]: Due to Lemma \[prelim\] we need to prove inequality (\[wzorek\_mi2\]) for any c-sets $A \subset \R^k$ and $B\subset \R^{n-k}$. We shall attempt to prove (\[wzorek\_mi2\]) using Theorem \[main\]. The one-dimensional case — the $\phi$ functions ----------------------------------------------- First we need to apply the Brunn-Minkowski theorem to get a $\Theta$-like condition: \[BMlemma\] Let $K \subset \R_x \times \R_y \times \R^{n-2}$ be a generalized Orlicz ball. Let $0 \leq x_1 \leq x_2 \in \R_x$, $0 \leq y_1 \leq y_2 \in \R_y$. Let $K_{x_i,y_j} = K \cap (\{(x_i,y_j)\} \times \R^{n-2})$ for $i,j \in \{1,2\}$. Let $\nu$ be a log-concave measure on $\R^{n-2}$. Then $$\nu(K_{x_1,y_1}) \cdot \nu(K_{x_2,y_2}) \leq \nu(K_{x_1,y_2}) \cdot \nu(K_{x_2,y_1}).$$ Let $f_i$, $i = 1,2,\ldots,n$ be the Young functions of $K$, with $f_1$ defined on $\R_x$ and $f_2$ on $\R_y$. Let us consider the generalized Orlicz ball $K' \in \R^{n-1}$, with the Young functions $\Phi_i = f_{i+1}$ for $i > 1$ and $\Phi_1(t) = t$ — that is, we replace the first two functions with a single identity function. For any $x \in \R$ let $P_x$ denote the set $K' \cap (\{x\} \times \R^{n-2})$, and $|P_x| = \nu(P_x)$. As $K'$ is a convex set, from the Brunn-Minkowski inequality (see for instance [@ff]) the function $x \mapsto |P_x|$ is a log-concave function, which means that for any $t \in [0,1]$ we have $$|P_{tx + (1-t)y}| \geq |P_x|^t|P_y|^{1-t}.$$ In particular, for given real non-negative numbers $a,b,c$ we have $$|P_{a+c}| \geq |P_a|^{b \slash (b+c)} |P_{a+b+c}|^{c \slash (b+c)},$$ $$|P_{a+b}| \geq |P_a|^{c \slash (b+c)} |P_{a+b+c}|^{b \slash (b+c)},$$ and as a consequence when we multiply the two inequalities, $$\label{PABC} |P_{a+b}| \ |P_{a+c}| \geq |P_a| \ |P_{a+b+c}|.$$ Now let us take $a = f_1(x_1) + f_2(y_1)$, $b = f_1(x_2) - f_1(x_1)$ and $c = f_2(y_2) - f_2(y_1)$. As the Young functions are non-negative and increasing on $[0,\infty)$, the numbers $a,b,c$ are non-negative. From the definitions above we have: $$K_{x_1,y_1} = \{(z_3,\ldots,z_n) \in \R^{n-2} : f_1(x_1) + f_2(y_1) + \sum_{i=3}^n f_i(z_i) \leq 1\} = \{(z_i)_{i=3}^n : \Phi_1(a) + \sum_{i=3}^n \Phi_{i-1}(z_i) \leq 1\} = P_a.$$ Similarily we have $K_{x_2,y_1} = P_{a+b}$, $K_{x_1,y_2} = P_{a+c}$ and $K_{x_2,y_2} = P_{a+b+c}$. Substituting those values into inequality (\[PABC\]) we get the thesis. First we consider $K \subset \R^{n-1} \times \R_z$. Take any $z_2 > z_1 > 0$ and consider any c-set $B$ in $\R^{n-1}$. We define $\phi_1(x) = \1_K(x,z_1)$ and $\phi_2(x) = \1_K(x,z_2)$ for $x \in \R^{n-1}$. Let ${K_{+}}' = ({K_{+}})_{z = z_1}$. By Lemma \[sekcjajest\] ${K_{+}}'$ is a positive quadrant of some generalized Orlicz ball $K'$. \[metaderivative\] If $\bar{K}'$ is a derivative of $K'$, then there exists a generalized Orlicz ball $\bar{K}$ such that $\phi_j(x)$ on $\bar{K}'$ is equal to $\1_{\bar{K}}(x,z_j)$ for $j \in \{1,2\}$. We have a sequence $K' = K_0', K_1',\ldots,K_m' = \bar{K}'$ where $K_{i+1}'$ is some restriction of $K_i'$. We can, taking identical restrictions (that is, restrictions to hyperplanes defined by the same equations or to the same intervals with respect to the same variables), construct a sequence $K = K_0,K_1,\ldots,K_m = \bar{K}$ such that $K_i' = (K_i)_{z = z_1}$. As $z$ was not a variable of $\R^{n-1}$ of which $K'$ was a subset, on each step being a hyperplane restriction $z$ does not appear in the equation of the restriction hyperplane, thus we can speak of a $z$ variable in all $K_i$, and the isometric immersion $u : \bar{K} \hookrightarrow K$ maps $(\bar{K})_{z = z_j}$ into $K_{z = z_j}$. Thus $\1_{\bar{K}}(x,z_j) = \1_{K}(u(x,z_j))$, which (when, as always, we identify $\bar{K}$ with its image in $K$) gives the thesis. \[phithetat4\] For any generalized Orlicz ball $K \subset \R^{m-1} \times \R_z$, any $z_2 > z_1 > 0$, any coordinate-wise decomposition $\R^{m-1} = \R^k \times \R^{m-k-1}$ and any proper measure $\mi$ on $K' = K_{z=z_1}$ the function $$\theta^1_k(y) = \frac{\int_{\R^k} \1_K(x,y,z_2) d\mi_{|\R^k}(x)}{\int_{\R^k} \1_K(x,y,z_1) d\mi_{|\R^k}(x)}$$ is coordinate-wise decreasing on $\R^{m-k-1}$. Let $l = m-k-1$. Select any coordinate variable $y_i$ from $\R^l$ and fix all other variables $\mathbf{y}$ in $\R^l$ at some $\mathbf{y}_0$. For $y_1 \leq y_2$ we have to prove $$\frac{\int_{\R^k} \1_K(x,\mathbf{y}_0,y_1,z_2) d\mi_{|\R^k}(x)}{\int_{\R^k} \1_K(x,\mathbf{y}_0,y_1,z_1) d\mi_{|\R^k}(x)} \geq \frac{\int_{\R^k} \1_K(x,\mathbf{y}_0,y_2,z_2) d\mi_{|\R^k}(x)}{\int_{\R^k} \1_K(x,\mathbf{y}_0,y_2,z_1) d\mi_{|\R^k}(x)}.$$ The intersection $K_{\mathbf{y} = \mathbf{y}_0}$ is a generalized Orlicz ball from Lemma \[sekcjajest\] and the restriction of $\mi$ is a proper measure from Lemma \[sekcjami\]. Thus taking $K'' = K_{\mathbf{y} = \mathbf{y}_0}$ we have to prove $$\frac{\int_{\R^k} \1_{K''}(x,y_1,z_2) d\mi_{|\R^k}(x)}{\int_{\R^k} \1_{K''}(x,y_1,z_1) d\mi_{|\R^k}(x)} \geq \frac{\int_{\R^k} \1_{K''}(x,y_2,z_2) d\mi_{|\R^k}(x)}{\int_{\R^k} \1_{K''}(x,y_2,z_1) d\mi_{|\R^k}(x)}.$$ Note that even if the density of $\mi$ changes with $y$, it cancels out in both fractions, thus we can assume the density of $\mi$ changes only on $\R^k$. As a proper measure has a $1\slash m$-concave density, and thus a log-concave density, we can apply Lemma \[BMlemma\] to get the thesis. \[phitheta\] The functions $\phi_1$ and $\phi_2$ defined as above define a $\Theta$ function on $K'$. We have to check the four properties defining $\Theta$ functions. Property (\[t-1\]) is obvious, both $\phi_1$ and $\phi_2$ are bounded by one. Note that $K$ is a c-set, as it is convex and 1-symmetric, which immediately gives properties (\[t0\]) and (\[t1\]). Condition (\[t2\]) is a consequence of Lemma \[phithetat4\]. If $\bar{K}'$ is any derivative of $K'$, then from Lemma \[metaderivative\] we have some $\bar{K}$ such that $\phi_j$ restricted to $\bar{K}'$ are equal to $\1_{\bar{K}}(\cdot,z_j)$, and thus from Lemma \[phithetat4\] the appropriate ratio of integrals is coordinate-wise decreasing. \[phipthetas\] If $K$ is a proper generalized Orlicz ball, then $\phi_1$ and $\phi_2$ define a strict $\Theta$ function. The properties (\[s3\]) and (\[t4\]) are trivial. For property (\[s2\]) notice that as the Young functions are strictly increasing, ${{\rm Int}}K_{z = z_1} \supset K_{z=z_2}$. To check property (\[s4\]) we have to prove that $\int_{\R^k} \1_K(x,y,z_j) d\mi_{|\R^k}(x) = \mi_{|\R^k}(K_{y,z_j})$ is continuous in $y$ for $j = 1,2$ and $k > 0$. Let $\mi_k$ denote $\mi_{|\R^k}$. Let us take any sequence $y^i {{\rightarrow}}y^\infty$. First note that as the Young functions $f_l$ do not assume the value $+\infty$, they are continuous. Thus $\sum f_l(y_l^i) {{\rightarrow}}\sum f_l(y_l^i)$. Let $L_a = \{x \in \R^k : \sum f_i(x_i) \leq 1 - a\}$, let $a_l = \sum f_l(y_l^i) + f_z(z_j)$ and $a = \sum f_l(y_l) + f_z(z_j)$. We know $a_l {{\rightarrow}}a$, we want to prove $\mi_k (L_{a_l}) {{\rightarrow}}\mi_k(L_a)$. However, $$\begin{aligned} \lim_{l{{\rightarrow}}\infty} \mi_k(L_{a_l}) \leq \lim_{t {{\rightarrow}}0^+} \mi_k(L_{a + t}) = \mi_k(\bigcap_{t > 0} L_{a+t}) = \mi_k(L_a)\end{aligned}$$ as measure is continuous with respect to the set, and $$\begin{aligned} \lim_{l{{\rightarrow}}\infty} \mi_k(L_{a_l}) \geq \lim_{t {{\rightarrow}}0^-} \mi_k(L_{a + t}) = \mi_k(\bigcap_{t < 0} L_{a+t}) = \mi_k(L_a),\end{aligned}$$ where we use the fact that $\mi_k(\{x \in \R^k : \sum f_i(x_i) = 1 - a\}) = 0$, as $f_i$ are strictly increasing. Thus $\mi_k(K_{y_l,z_j}) {{\rightarrow}}\mi_k(L_{y,z_j})$, which proves property (\[s4\]). \[phinondeg\] For any generalized Orlicz ball $K$ the functions $\phi_1$ and $\phi_2$ define a non-degenerate $\Theta$ function. First we prove that $\phi_1$ and $\phi_2$ define a weakly non-degenerate $\Theta$ function. From Lemma \[approrlicz\] we can approximate $K$ with a proper generalized Orlicz ball $K'$ satisfying $K' \subset K$ and $\lambda(K \setminus K') < \eps \slash 2$. Additionally, from Corollary \[apprsection\] we may take $z_1'$ and $z_2'$ such that $K' \cap \{z = z_j'\}$ approximates $K \cap \{z = z_j\}$ up to a set of $\lambda$ measure $\eps$. We take $\phi_1'(x) = \1_{K'}(x,z_1')$ and $\phi_2'(x) = \1_{K'}(x,z_2')$. As the intersections of $K'$ at $z_j'$ were good approximations of intersections of $K$ at $z_i$, we have $\int |\phi_i - \phi_i'| d\lambda = \lambda(K_{z=z_1} \bigtriangleup K'_{z=z_1'}) \leq \eps$. From Lemma \[approrlicz\] we know $K'$ is a proper generalized Orlicz ball and $K' \subset K$. From Lemma \[phipthetas\] we know that $\phi_1'$ and $\phi_2'$ define a strict $\Theta$ function. Thus $\phi_1$ and $\phi_2$ define a weakly non-degenerate $\Theta$ function. As for the derivatives of the function defined by $\phi_1$ and $\phi_2$ by Lemma \[metaderivative\] they are constructed in the same manner on some derivative of $K$, and thus also define a weakly non-degenerate $\Theta$ function. Thus $\phi_1$ and $\phi_2$ define a non-degenerate $\Theta$ function. \[wniosekjedno\] For any generalized Orlicz ball $K \subset \R^n$ and any c-set $A \subset \R^{n-1}$ the function $$z \mapsto \frac{\int_{{\bar{A}}}\1_K(z,x) d\mi(x)}{\int_{\R^{n-1}} \1_K(z,x) d\mi(x)}$$ is a decreasing function of $z$ where defined. From Corollary \[phinondeg\] we can apply Theorem \[main\] to the $\Theta$ function defined by $\phi_1$, $\phi_2$ to get for any $0 \leq z_1 < z_2$: $$\label{pndg1} \frac{\int_A \1_K(x,z_2) d\mi(x)}{\int_A \1_K(x,z_1) d\mi(x)} \geq \frac{\int_{{\bar{A}}}\1_K(x,z_2) d\mi(x)}{\int_{{\bar{A}}}\1_K(x,z_1) d\mi(x)},$$ if both sides are defined. We can apply Fact \[obvi\] to make it $$\label{pndg2} \frac{\int_{\R^{n-1}} \1_K(x,z_2) d\mi(x)}{\int_{\R^{n-1}} \1_K(x,z_1) d\mi(x)} \geq \frac{\int_{{\bar{A}}}\1_K(x,z_2) d\mi(x)}{\int_{{\bar{A}}}\1_K(x,z_1) d\mi(x)}.$$ Switching the left numerator with the right denominator we get the thesis. If the right-hand side denominator in inequality (\[pndg1\]) is zero, the right-hand side numerator is also zero, as $z_1 < z_2$ and ${K_{+}}$ is a c-set. Thus both for $z_1$ and $z_2$ our function is either zero or undefined. If the left-hand side denominator is zero and the right-hand side is defined, again the left-hand side numerator is zero, thus in inequality (\[pndg2\]) we have an equality, which again gives the thesis. The general case — the $\psi$ function -------------------------------------- Let $\lambda_K$ denote the Lebesgue measure restricted to ${K_{+}}$. Recall that we set out to prove $$\lambda_K({{\bar{A}}}\times B ) \cdot \lambda_K(A \times {{\bar{B}}}) \geq \lambda_K(A \times B ) \cdot \lambda_K({{\bar{A}}}\times {{\bar{B}}})$$ for any c-sets $A \subset \R^k$ and $B \subset \R^{n-k}$. This is equivalent to $$\lambda_K(A \times {{\bar{B}}}) \cdot \lambda_K({{\bar{A}}}\times \R^{n-k}) \geq \lambda_K({{\bar{A}}}\times {{\bar{B}}}) \cdot \lambda_K(A \times \R^{n-k}).$$ If either $\lambda_K(A \times \R^{n-k})$ or $\lambda_K({{\bar{A}}}\times \R^{n-k})$ is zero, then respectively either $\lambda_K(A \times {{\bar{B}}})$ or $\lambda_K({{\bar{A}}}\times {{\bar{B}}})$ is zero and the thesis is satisfied. Thus it suffices to prove $$\frac{\lambda_K(A \times {{\bar{B}}})}{\lambda_K(A \times \R^{n-k})} = \frac{\int_A \int_{{\bar{B}}}\1_K(z,x) dz dx}{\int_A \int_{\R^{n-k}} \1_K(z,x) dz dx} \geq \frac{\int_{{\bar{A}}}\int_{{\bar{B}}}\1_K(z,x) dz dx}{\int_{{\bar{A}}}\int_{\R^{n-k}} \1_K(z,x) dz dx} = \frac{\lambda_K({{\bar{A}}}\times {{\bar{B}}})}{\lambda_K({{\bar{A}}}\times \R^{n-k})},$$ when both sides are defined, which means it is enough to prove $\psi_1(x) = \int_{\R^{n-k}} \1_K(z,x) dz$ and $\psi_2(x) = \int_{{\bar{B}}}\1_K(z,x) dz$ define a non-degenerate $\Theta$ function on $K' = K_{z = 0} \subset \R^k$ and apply Theorem \[main\]. \[dupaderivative\] If $\bar{K}'$ is a derivative of $K'$, then there exists a generalized Orlicz ball $\bar{K}$ such that $\psi_1(x)$ on $\bar{K}'$ is equal to $\int_{\R^{n-k}} \1_{\bar{K}}(z,x) dz$ and $\psi_2(x)$ is equal to $\int_{{\bar{B}}}\1_{\bar{K}}(z,x) dz$. The proof is identical to the proof of Lemma \[metaderivative\]. For any generalized Orlicz ball $K \subset \R^n$, any coordinate-wise decomposition $\R^n = \R^k \times \R^{n-k}$ and any c-set $B \subset \R^{n-k}$ the functions $\psi_1$ and $\psi_2$ define a $\Theta$ function on $K$. Property \[t-1\] follows from the fact that $K$ is bounded. Property \[t0\] follows from the fact ${K_{+}}$ is a c-set. Property \[t1\] follows from the fact that $B \subset \R^{n-k}$. As before, the tricky part is to prove property \[t2\]. Consider any coordinate-wise decomposition $\R^k = \R^{k_1} \times \R^{k_2}$. Choose any variable $v$ in $\R^{k_1}$ and fix all the others at some fixed $\mathbf{v}_0$. We have: $$\frac{\int_{\R^{k_2}} \psi_2(v,\mathbf{v}_0,y) d\mi(y)}{\int_{\R^{k_2}} \psi_1(v,\mathbf{v}_0,y) d\mi(y)}= \frac{\int_{\R^{k_2}} \int_{{\bar{B}}}\1_K(v,\mathbf{v}_0,y,z) d\mi(y) dz}{\int_{\R^{k_2}} \int_{\R^{n-k}} \1_K(v,\mathbf{v}_0,y,z) d\mi(y) dz} = \frac{\int_{\R^{k_2} \times {{\bar{B}}}} \1_K(v,\mathbf{v}_0,y,z) d\mi(y) dz}{\int_{\R^{l_2} \times \R^k} \1_K(v,\mathbf{v}_0,y,z) d\mi(y) dz}.$$ We have to prove this function is decreasing in $v$ where defined. Let us restrict ourselves to the generalized Orlicz ball $\hat{K} = K_{\mathbf{v} = \mathbf{v}_0}$. Notice that $\R^{l_2} \times A$ is a c-set in $\R^{l_2} \times \R^k$ and $\mi \otimes \lambda$ is a proper measure in $\R^{l_2} \times \R^k$. We have to prove $$\frac{\int_{\R^{l_2} \times {{\bar{A}}}} \1_{\hat{K}}(v,y,z) d(\mi\otimes \lambda)(y,z)}{\int_{\R^{l_2} \times \R^k} \1_{\hat{K}}(v,y,z) d(\mi \otimes \lambda)(y,z)}$$ is decreasing in $v$, but this is exactly the thesis of Corollary \[wniosekjedno\]. Again, as in Lemma \[phitheta\], due to Lemma \[dupaderivative\], the appropriate ratio is also decreasing for any derivative $\bar{K}$ of $K$. For any generalized Orlicz ball $K \subset \R^n$, any coordinate-wise decomposition $\R^n = \R^k \times \R^{n-k}$ and any c-set $B \subset \R^{n-k}$ the functions $\psi_1$ and $\psi_2$ define a non-degenerate $\Theta$ function on $K$. Again the derivatives of $\psi$ are again functions formed as in Lemma \[dupaderivative\], so it is enough to prove $\psi$ is weakly non-degenerate. Take any $\eps > 0$. From Lemma \[approrlicz\] we may take a proper generalized Orlicz ball $\hat{K} \subset K$ with $\lambda(K \setminus \hat{K}) < \eps \min\{\lambda(K),1\} \slash 2$ and $\lambda_k(K_{z=0} \setminus \hat{K}_{z=0}) < \eps \min\{\lambda_k(K_{z=0}),1\})$ from Lemma \[apprsection\]. Denote $\hat{K}_{z=0}$ by $\hat{K}'$. Let $z_1$ be any coordinate in $\R^{n-k}$, take $B' = B \cup (\{\mathbf{z}: z_1 < \delta\} \cap {K_{+}})$, where $\delta$ is so small that the addition is of $\lambda_{n-k}$ measure less than $\eps \slash 2$. $B'$ is a sum of two c-sets and thus a c-set. We define $\psi_1'(x) = \int_{\R^{n-k}} \1_{\hat{K}}(z,x) dz$ and $\psi_2'(x) = \int_{{{\bar{B}}}'} \1_{\hat{K}}(z,x) dz$. We have $\lambda_k(K' \setminus \hat{K}') < \eps \lambda_k(K')$ from the definition of $\hat{K}$. Also $\psi_1'$ and $\psi_2'$ are indeed good approximations of $\psi_1$ and $\psi_2$, as $$\int_{\R^k} |\psi_1(x) - \psi_1'(x)| dx \leq \int_{\R^k} \int_{\R^{n-k}} |\1_K(x,z) - \1_{\hat{K}}(x,z)| dz dx = \lambda (K \bigtriangleup \hat{K}) \leq \eps \slash 2,$$ and $$\begin{aligned} \int_{\R^k} |\psi_2(x) - \psi_2'(x)| dx &= \int_{\R^k} \Big|\int_{\R^{n-k}} \1_K(x,z) \1_{{\bar{B}}}(x,z) - \1_{\hat{K}}(x,z) \1_{{{\bar{B}}}'}(x,z) dz\Big|dx \leq \mi((K \cap {{\bar{B}}}) \bigtriangleup (\hat{K} \cap {{\bar{B}}}')) \\ &\leq \lambda(K \setminus \hat{K}) + \lambda_K ({{\bar{B}}}\bigtriangleup {{\bar{B}}}') = \lambda(K \setminus \hat{K}) + \lambda_K (B \bigtriangleup B') \leq \eps.\end{aligned}$$ Thus we only have to prove that $\psi_1'$ and $\psi_2'$ define a strict $\Theta$ function on $\hat{K}$. Property (\[s3\]) is true as $\hat{K}$ is proper — $\hat{K}'$ is defined by those Young functions of $\hat{K}$ which act on the variables of $\R^k$. Property (\[t4\]) is obvious from the definition of $\psi_1'$. The function $\psi_2'$ is 0 on the set $\sum f_i(x_i) > 1 - f_{z_1}(\delta)$ from the definition of $B'$ — any point in ${{\bar{B}}}'$ has $z_1 > \delta$, hence property (\[s2\]). Finally (\[s4\]) is checked exactly as in Lemma \[phipthetas\]. Thus $\psi_1$ and $\psi_2$ do define a non-degenerate $\Theta$ function, which ends the proof of Theorem \[orliczglownetw\]. [99]{} M. Anttila, K. Ball and I. Perissinaki, The central limit problem for convex bodies. Trans. Amer. Math. Soc., 355 (2003), pp. 4723–-4735. K. Ball and I. Perissinaki, The subindependence of coordinate slabs in $\ell_p^n$ balls, Israel J. Math., 107 (1998), pp. 289-299. F. Barthe, O. Gudeon, S. Mendelson and A. Naor, A Probabilistic Approach to the Geometry of the $\ell_p^N$-ball, Annals of Probability, 33 (2005), pp. 480–513. R. J. Gardner, The Brunn-Minkowski Inequality, Bull. Amer. Math. Soc. 39 (2002), pp. 355-405 S. G. Bobkov, F. L. Nazarov, On convex bodies and log-concave probability measures with unconditional basis. Geometric aspects of functional analysis, 53–69, Lecture Notes in Math., 1807, Springer, Berlin, 2003. C. Borell, Convex measures on locally convex spaces. Ark. Mat. 12 (1974), 239–252. B. Fleury, O. Guedon, G. Paouris, A stability result for mean width of $L_p$-centroid bodies. Preprint. Available at http://www.institut.math.jussieu.fr/$\tilde{\ }$guedon/Articles/06/FGP-Accepted.pdf A. A. Giannopoulos, Notes on isotropic convex bodies, Institute of Mathematics, Polish Academy of Sciences, Warsaw (2003), available at http://users.uoa.gr/$\tilde{ }$apgiannop/isotropic-bodies.ps. B. Klartag, A central limit theorem for convex sets, Invent. Math., Vol. 168, (2007), 91–131. B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal., Vol. 245, (2007), pp. 284–310. S. Kwapie[ń]{}, R. Lata[ł]{}a and K. Oleszkiewicz, Comparison of Moments of Sums of Independent Random Variables and Differential Inequalities. Journal of Functional Analysis, 136 (1996), pp. 258–268. E. Meckes and M. Meckes, The Central Limit Problem for Random Vectors with Symmetries. Preprint. Available at http://arxiv.org/abs/math.PR/0505618. V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space. Lecture Notes in Mathematics, 1376 (1989), pp. 64–104. V. Milmanc G. Schechtman, Asymptotic theory of finite-dimensional normed spaces. With an appendix by M. Gromov. Lecture Notes in Mathematics, 1200. Springer-Verlag, Berlin, 1986. C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y. L. Tong (ed.), Inequalities in Statistics and Probability, Hayward, CA, pp. 127–140. Qi-Man Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), 343-356. J. O. Wojtaszczyk, The square negative correlation property for generalized Orlicz balls. Preprint, to be published in GAFA. Available at http://www.mimuw.edu.pl/\~onufry/papers/Orlicz.pdf [^1]: Partially supported by MEiN Grant no 1 PO3A 012 29

ArXiv

--- abstract: 'We show that Pb and Bi adatoms and dimers have a large tunneling anisotropic magnetoresistance (TAMR) of up to 60% when adsorbed on a magnetic transition-metal surface due to strong spin-orbit coupling and the hybridization of $6p$ orbitals with $3d$ states of the magnetic layer. Using density functional theory, we have explored the TAMR effect of Pb and Bi adatoms and dimers adsorbed on a Mn monolayer on W(110). This surface exhibits a noncollinear cycloidal spin spiral ground state with an angle of 173$^\circ$ between neighboring spins which allows to rotate the spin quantization axis of an adatom or dimer quasi-continuously and is ideally suited to explore the angular dependence of TAMR using scanning tunneling microscopy (STM). We find that the induced magnetic moments of Pb and Bi adatoms and dimers are small, however, the spin-polarization of the local density of states (LDOS) is still very large. The TAMR obtained from the anisotropy of the vacuum LDOS is up to 50-60 % for adatoms. For dimers the TAMR depends sensitively on the dimer orientation with respect to the crystallographic directions of the surface due to the formation of bonds between the adatoms with the Mn surface atoms and the symmetry of the spin-orbit coupling induced mixing. Dimers oriented along the spin spiral direction of the Mn monolayer display the largest TAMR of 60 % which is due to hybrid $6p-3d$ states of the dimers and the Mn layer.' author: - Soumyajyoti Haldar - Mara Gutzeit - Stefan Heinze title: 'Tunneling anisotropic magnetoresistance of Pb and Bi adatoms and dimers on Mn/W(110): A first-principles study' --- Introduction ============ The tunneling magnetoresistance (TMR), in which the flow of current depends on the relative magnetization directions of two magnetic layers, has a significant impact on modern day applications ranging from spintronics to magnetic data storage. Using spin-polarized scanning tunneling microscopy (STM), it is even possible to detect the TMR effect for single magnetic adatoms on surfaces [@Yayon2007; @Meier82; @Tao2009; @Loth2010; @Ziegler2011; @Lazo2012; @Khajetoorians55]. The resistance can also depend on the magnetization direction relative to the current direction because of spin-orbit coupling (SOC), which is known as the tunneling anisotropic magnetoresistance (TAMR) [@Bode2002; @Gould2004]. The TAMR is driven by SOC which couples spin and orbital momentum degrees of freedom by the Hamiltonian $H_{SOC} = \xi \, \mathbf{L} \cdot \mathbf{S}$, where $\xi$, $\mathbf{L}$, and $\mathbf{S}$ are the SOC constant, orbital momentum operator and spin operator, respectively. SOC and magnetocrystalline anisotropy effects depend on the environment of an adatom and hence can be tuned by adatom adsorption which have been studied quite extensively  [@Gambardella1130; @Hirjibehedin1199; @Loth1628; @Khajetoorians2011; @Rau988]. The TAMR can be observed with only one ferromagnetic electrode and it does not require any coherent spin-dependent transport. Hence, the TAMR is very attractive for spintronics applications [@Fert2008; @Sinova2012]. The TAMR was first observed for a double layer of Fe on W(110) [@Bode2002]. Subsequently, the TAMR has been observed in various systems, [[e.g.,]{}]{} planar ferromagnetic surfaces [@Shick2006; @Chantis2007], tunnel junctions [@Gould2004; @Matos2009a; @Matos2009b; @Gao2007], mechanically controlled break junctions [@Viret2006; @Bolotin2006]. The observed values of TAMR in the above cases are $\approx$ 10%. Attempts have been made to increase the value of TAMR by using $3d$ or $5d$ elements, [[e.g.,]{}]{} using isolated adatoms  [@Neel2013; @Schoneberg2016], bimetallic alloys [@Shick2010] and with antiferromagnetic electrodes [@Park2011]. Recently, Hervé [[*et al.*]{}]{} have reported a TAMR of up to 30% for Co films on Ru(0001) mediated by surface states [@Herve2018b]. Another approach to tune SOC is to use single atoms and dimers of $6p$ elements. The strength of SOC scales with atomic number ($Z$), principal quantum number ($n$), and orbital quantum number ($l$) as $\xi \propto Z^{4}n^{-3}l^{-2}$. Hence, $6p$ elements such as Pb and Bi have a higher SOC strength as compared to that of the $3d$ or $5d$ elements studied before. Further tuning of SOC can be achieved by reducing the high rotational symmetry of single atom, [[i.e.,]{}]{} by using dimers of these elements. The effect of strong SOC on unsupported $6p$ dimers has been discussed recently [@Borisova2016a]. In an experimental and theoretical study, Schöneberg [[*et al.*]{}]{} [@Schoneberg2018] have achieved TAMR values of $\approx$ 20% by using suitably oriented Pb dimers on the Fe bilayer on W(110) substrate where magnetic domains with out-of-plane magnetization and domain walls with in-plane magnetization can be observed [@Bode2002]. In recent years noncollinear magnetic structures at transition-metal interfaces have gained popularity as promising candidates for spintronic applications due to their interesting dynamical and transport properties [@Fert2013; @Nagaosa2013]. A monolayer Mn grown on W(110) surface (Mn/W(110)) is a prominent example which exhibits a noncollinear magnetic structure with a cycloidal 173$^{\circ}$ spin-spiral ground state along the $[1\overline{1}0]$ direction [@Bode2007] that is driven by the Dzyaloshinskii-Moriya interaction. Using this magnetic surface with a noncollinear spin structure, it is possible to control the spin direction of adsorbed Co adatoms due to local exchange coupling which has been demonstrated in recent experiments using scanning tunneling microscopy (STM) by Serrate [[*et al.*]{}]{} [@Serrate2010; @Serrate2016]. The noncollinear spin state of the Mn monolayer is reflected due to hybridization even in the orbitals of the adsorbed Co adatom [@Haldar2018]. The possibility of controlling the magnetization direction of an adatom on this surface without the presence of external magnetic field makes this system very promising for TAMR studies. Compared to the domain walls of Fe/W(110) used in previous studies [@Schoneberg2018] the spin structure of this surface is known on the atomic scale and allows a quasi continuous rotation of the local spin quantization axis. Recently, Caffrey [[*et al.*]{}]{} have predicted TAMR values up to 50% for Ir adatoms, i.e. a $5d$ transition metal, on Mn/W(110) [@Caffrey2014a], however, experimental evidence is missing. Here we have explored Pb and Bi adatoms and dimers on Mn/W(110) in order to explore the magnitude of TAMR and its dependence on the $6p$ element and atomic arrangement on the surface. We have used first-principles density functional theory (DFT) calculations to investigate the adsorption of Pb and Bi adatoms and dimers on Mn/W(110) and studied their electronic and magnetic properties. The spin structure of Mn/W(110) is locally well approximated as a two-dimensional antiferromagnet [@Heinze2000]. We considered two limiting cases of spin directions which are possible due to the cycloidal nature and propagation direction of the spin spiral in the Mn layer: (i) a magnetization direction perpendicular to the surface (out-of-plane) and (ii) a magnetization direction pointing along the $[1\overline{1}0]$ direction (in-plane). Our results indicate that the adsorption of these adatoms facilitates local enhancement of SOC above the surface leading to very large values of the TAMR of 50% to 60% for adatoms. The orientation of Pb and Bi dimers is shown to be crucial in order to achieve even larger TAMR values. This can be understood based on the symmetry of the matrix elements of the SOC Hamiltonian as well as the hybridization of $6p$ adsorbate with $3d$ substrate states. This paper is organized as follows. First, we briefly discuss the computational methods used in our calculations. Then we proceed to discuss the structural, electronic, and magnetic properties, as well as the TAMR of adatoms and the same for dimers in different orientations. The TAMR effects are discussed focusing on the local density of states at the adsorbate atoms and the Mn layer and the vacuum density of states and interpreted based on a simplified model. We summarize our main conclusions in the final section. Computational details {#sec:compdet} ===================== In this work we used first-principles calculations using a plane wave based DFT code <span style="font-variant:small-caps;">vasp</span> [@vasp1; @vasp2] within the projector augmented wave method (PAW) [@blo; @blo1]. For the exchange-correlation, we have used the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) [@PBE; @PBEerr]. For SOC, we followed the methods described by Hobbs [[*et al.*]{}]{} [@Hobbs2000]. We used a 450 eV energy cutoff for the plane wave basis set convergence. Structural relaxations are performed using a $6\times6\times1$ k-point Monkhorst-Pack mesh [@Monkhorst1976]. The vacuum local density of states (LDOS) was calculated by placing an empty sphere at a specific height of 5.3 [Å]{} above the adatoms onto which the LDOS was projected. For the calculation of electronic properties, magnetic properties and LDOS, we have used $20\times20\times1$ k-point Monkhorst-Pack mesh. Structural details {#subsec:structure} ------------------ ![(a) Top view and (b) perspective view of a $c(4\times4)$ supercell used for the adatom on Mn/W(110) calculation. Gray spheres represent W atoms while Mn atoms are depicted as green (light blue) spheres with arrows showing the ferromagnetic (antiferromagnetic) magnetic moments with respect to the $6p$ adatom (dark blue sphere). ‘nn’ and ‘nnn’ are the nearest neighbor and next nearest neighbor Mn atoms to the adatom. $x$ and $y$ refer to the direction of coordinates for the supercell. (c) Top view of a $c(6\times6)$ supercell used for dimer adsorption on Mn/W(110) along with the three dimer orientations considered in our calculations. $m_\perp$ and $m_\parallel$ denote the direction of a perpendicular magnetization and a parallel magnetization with respect to the surface, respectively.[]{data-label="fig:geom"}](figure1.png) We modeled Mn/W(110) using a symmetric slab consisting of five atomic layers of W with a pseudomorphic Mn layer on each side. We have approximated the local magnetic order of the system as antiferromagnetic, [[i.e.,]{}]{} collinear due to the long periodicity of the spin spiral ground state [@Heinze2000; @Bode2007; @Serrate2010]. The effect of the noncollinearity of the spin structure on the electronic states of adatoms has been studied before [@Haldar2018]. We used a $c(4\times4)$ AFM surface unit cell, as shown in Fig. \[fig:geom\](a)-(b). The GGA calculated lattice constant of W, [[i.e.,]{}]{} 3.17 [Å]{} is used for our calculations as it is in good agreement with the experimental value of 3.165 [Å]{}. A thick vacuum layer of $\approx$ 25 [Å]{} is included in the $z$ direction normal to the surface to remove interactions between repeating slabs. We added Pb or Bi adatoms at the hollow-site position on each Mn monolayer. The $c(4\times4)$ unit cell is large enough to keep the interactions between the periodic images of the adatoms small. For the adsorption of dimers, we have used a larger $c(6\times6)$ AFM surface unit cell (see Fig. \[fig:geom\](c)) to keep the interactions coming from the periodic images of the dimers negligible. In the case of Pb or Bi dimers, we have considered three possible dimer orientations on the surface: (i) along the \[001\] direction, (ii) along the $[1\overline{1}0]$ direction, and (iii) along the $[1\overline{1}1]$ direction as shown in Fig. \[fig:geom\](c). The magnetization direction in calculations including SOC has been chosen normal to the surface, $\perp$, and along the $[1\bar{1}0]$ in-plane direction, $\parallel$, as enforced by the cycloidal nature of the underlying spin spiral structure of Mn/W(110) [@Bode2007]. The position of the adatoms, dimers and the Mn layers are relaxed with 0.01 eV/[Å]{} force tolerance. We have kept the coordinates of W atoms fixed in all our calculations. Tunneling anisotropic magnetoresistance {#subsec:tamr} --------------------------------------- Using the spectroscopic mode of an STM, the TAMR can be obtained by measuring the differential conductance (d$I$/d$V$) above an adatom or a dimer for two different magnetization directions. The TAMR is obtained from $$\begin{aligned} \mathrm{TAMR} &= \frac{[(\mathrm{d}I/\mathrm{d}V)_\perp-(\mathrm{d}I/\mathrm{d}V)_\parallel]}{(\mathrm{d}I/\mathrm{d}V)_\perp}\; ,\end{aligned}$$ where $\perp$ and $\parallel$ denote a perpendicular magnetization and a parallel magnetization with respect to the surface, respectively. Within the Tersoff-Hamann model [@Tersoff1983; @Tersoff1985], the d$I$/d$V$ signal is directly proportional to the local density of states (LDOS), $n(z, \epsilon)$, at the tip position in the vacuum, $z$, a few [Å]{}ngströms above the surface. Hence, the TAMR can be calculated theoretically from the anisotropy of the LDOS arising due to SOC [@Bode2002; @Neel2013]. Then the TAMR can be calculated as: $$\begin{aligned} \mathrm{TAMR} &= \frac{n_\perp(z, \epsilon)-n_\parallel(z, \epsilon)}{n_\perp(z, \epsilon)}\; . \label{eq:TAMR}\end{aligned}$$ Results and Discussion {#sec:results} ====================== Pb and Bi adatoms on Mn/W(110) {#sbsec:adatoms} ------------------------------ ### Structural and magnetic properties {#sub2sec:str_mag} ---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ ------------------------- $d_{\text{nn}}$ $d_{\text{nnn}}$ $\Delta z_{\text{nn}}$ $\Delta z_{\text{nnn}}$ $\Delta x_{\text{nn}}$ $\Delta y_{\text{nnn}}$ Pb 2.76 3.17 $-0.13$ +0.02 $-0.01$ +0.02 Bi 2.70 2.97 $-0.10$ +0.06 $-0.05$ +0.03 ---- ----------------- ------------------ ------------------------ ------------------------- ------------------------ ------------------------- : Relaxed distances (in [Å]{}) of Pb and Bi adatoms from the Mn atoms of the Mn/W(110) surface. $d_{\text{nn}}$ and $d_{\text{nnn}}$ denotes the nearest neighbor (nn) and the next-nearest neighbor (nnn) Mn atoms, respectively. $\Delta x$, $\Delta y$, and $\Delta z$ are the displacements with respect to the clean surface of Mn atoms after the adsorption of the adatoms. Positive (negative) values imply that the Mn atoms move towards (away from) the adatom.[]{data-label="tab:distance"} We begin our discussion with the local structural relaxations upon adsorption of Pb and Bi adatoms on Mn/W(110) which are tabulated in Table \[tab:distance\]. Our calculations indicate that the hollow site (see Fig. \[fig:geom\](a)) is the most stable adsorption site for both Pb and Bi adatoms. The other sites, [[e.g.,]{}]{} bridge and top sites are unstable for both adatoms in our calculations and collapse to the hollow site position. The adsorption of the adatoms creates a buckling in the underlying Mn layer in the vicinity of the adsorption sites (see Table \[tab:distance\]). Significant changes can be observed for the nearest neighbor (nn) Mn atoms, which move away from the adatoms, while the next nearest neighbor (nnn) Mn adatoms move slightly towards the adatoms. ---- -------- ------------------ ------------------- --------------------- Adatom Mn$_{\text{nn}}$ Mn$_{\text{nnn}}$ Mn$_{\text{clean}}$ Pb +0.00 +2.36 $-3.34$ $\pm$3.41 Bi +0.08 +2.60 $-3.35$ $\pm$3.41 ---- -------- ------------------ ------------------- --------------------- : Magnetic moments (in [[$\mu_{\text{B}}$]{}]{}) of the adsorbed Pb and Bi adatoms and the nearest neighbor (nn) and next nearest neighbor (nnn) Mn atoms of the Mn monolayer on W(110). For comparison the value of the clean Mn/W(110) surface is given.[]{data-label="tab:magmom"} The magnetic properties of these systems are affected by the hybridization between the $6p$ adatoms and underlying Mn atoms of Mn monolayer (see Table \[tab:magmom\]). The clean Mn surface of Mn/W(110) have magnetic moments $\pm$ 3.41 [[$\mu_{\text{B}}$]{}]{}. The $p_z$ orbitals of the adatoms mainly hybridize with the $d_{z^2}$ orbitals of nn Mn adatoms. The magnetic moments of the nn Mn adatoms drop quite significantly for both atom types. They are reduced by 1.05 [[$\mu_{\text{B}}$]{}]{} and 0.81 [[$\mu_{\text{B}}$]{}]{} for Pb and Bi adsorption, respectively. Due to the hybridization, the induced magnetic moment on Bi adatom is 0.08 [[$\mu_{\text{B}}$]{}]{}, whereas the Pb adatom is non-magnetic. The effect of hybridization is less prominent for the nnn Mn adatoms where a slight reduction of magnetic moment $\sim$ 0.06 [[$\mu_{\text{B}}$]{}]{} occurs for both adatoms. ### Electronic properties  Next, we discuss the electronic properties of the $6p$ adatoms adsorbed on the Mn/W(110) surface. Fig. \[fig:spinpol\_adatom\] shows the spin-resolved LDOS of the Pb and Bi adatom adsorbed on Mn/W(110), the LDOS of the neighboring Mn atoms, and $m_l$ decomposed $p$ states of Pb and Bi adatom. These calculations have been performed in the scalar relativistic approximation, [[i.e.,]{}]{} neglecting SOC. A possible hybridization can be observed by calculating and comparing the spin-resolved LDOS of the adatoms with the neighboring Mn states as shown in Fig. \[fig:spinpol\_adatom\]. This hybridization effect is clearly observed just below [[$E_{\text{F}}$]{}]{} where minority Mn peaks are located at the same position as $p_x$ and $p_z$ states of the adatoms. Further interactions are observed for Pb around [[$E_{\text{F}}$]{}]{}$-$0.50 eV, [[$E_{\text{F}}$]{}]{}$-$0.34 eV, where the states from the adatoms interact with the states from the nn Mn atoms. In this energy range one also sees reduced exchange splitting of the nn Mn states as compared to the nnn Mn states which affects the magnetic moment of the nn Mn atoms as mentioned in section \[sub2sec:str\_mag\]. The magnetic moment of nn Mn atoms drops to 2.36 [[$\mu_{\text{B}}$]{}]{} and 2.6 [[$\mu_{\text{B}}$]{}]{} upon adsorption of the Pb and Bi adatom, respectively. Despite the small spin splitting observed for both adatoms, the spin polarization of the adatoms is quite large. The spin polarization of the adatoms varies in-between $\pm 40$% which is mainly arising from the $p_z$ and $p_y$ states of the adatoms. ![(a, b) Spin-resolved partial charge density plots at 3 [Å]{} above the Pb adatom on Mn/W(110) in the energy range \[$E_\text{F}-0.065$, $E_\text{F}-0.045$ eV\]. (c, d) cross-sectional plots through the Pb adatom parallel to the \[001\] direction for the charge densities of the top panel. []{data-label="fig:chargedensity_Pb"}](figure3.png) Previously, it has been shown that the spin direction of adsorbed Co adatoms on Mn/W(110) can be detected in spin-polarized STM images at small bias voltages due to the different orbital symmetry of $d$ states in majority and minority spin channel [@Serrate2010]. We find a similar effect for the $p$ orbitals of Pb close to the Fermi energy, $E_F$. Figure \[fig:chargedensity\_Pb\] shows top and cross-sectional spin-resolved partial charge density plots in a small energy window \[[[$E_{\text{F}}$]{}]{}$-$0.065, [[$E_{\text{F}}$]{}]{}$-$0.045 eV\] for Pb adatom adsorbed on Mn/W(110). A strong interaction between the minority $p_x$ states of the adatom and the minority $d_{z^2}$ orbitals of the neighboring Mn atoms is clearly visible in the cross-sectional plot along the $[001]$ direction shown in Figs. \[fig:chargedensity\_Pb\](d). Here, the axes of the $d_{z^2}$ Mn orbitals are distorted pointing towards the Pb atom and a large part of the charge density is concentrated at the interface between adsorbate and substrate. However, such hybridization is less prominent in the majority channel which displays the rotationally symmetric shape of a $p_z$ orbital \[Fig. \[fig:chargedensity\_Pb\](c)\]. The partial charge density calculated at a height of 3 [Å]{} in the vacuum \[Fig. \[fig:chargedensity\_Pb\](a-b)\] shows that the both spin channels are clearly distinguishable from each other due to the shape of their orbitals. For the majority channel one can clearly observe the $p_z$ states of the adatom in the vacuum. In the minority channel, the double-lobed structure of the $p_x$ state protrudes rotationally symmetric states such as $s$, $p_z$ and $d_{z^2}$ orbitals which usually extend further into the vacuum. Similar behavior has been reported previously by Serrate [[*et al.*]{}]{} for different $d$-states of a Co adatom adsorbed on Mn/W(110) [@Serrate2010]. Hence, we can conclude that in an STM experiment with a magnetic tip it will be possible to identify the spin direction of the Pb adatom by means of the respective orbitals dominating near [[$E_{\text{F}}$]{}]{} yielding similar effects observed in spin-polarized STM [@Serrate2010; @Serrate2016; @Haldar2018]. However, for the Bi adatom the above mentioned feature is not present in the vicinity of $E_F$ which is accessible for STM. In this case, the majority $p_y$ states of Bi are completely covered by the rotationally symmetric orbitals in the vacuum (not shown). Therefore, the orbital shapes for majority and minority states for the charge densities calculated in the vicinity of [[$E_{\text{F}}$]{}]{} do not differ from one another. ### TAMR of Pb and Bi adatoms on Mn/W(110) {#sub2sec:TAMR_adatom} ![(a) Total (black lines) and spin-resolved (Majority: blue, Minority: red) vacuum LDOS including SOC above the Pb adatom on Mn/W(110) for out-of-plane ($\perp$, solid lines) and in-plane (parallel to the $[1\overline{1}0]$ direction) magnetizations ($\parallel$, dashed lines). (b) TAMR obtained from the spin-averaged vacuum LDOS according to Eq. (\[eq:TAMR\]). (c) Orbital decomposition of the LDOS of the Pb adatom in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:PbAdatom"}](figure4.pdf) ![(a) Total (black lines) and spin-resolved (Majority: blue, Minority: red) vacuum LDOS including SOC above the Bi adatom on Mn/W(110) for out-of-plane ($\perp$, solid lines) and in-plane (parallel to the $[1\overline{1}0]$ direction) magnetizations ($\parallel$, dashed lines). (b) TAMR obtained from the spin-averaged vacuum LDOS according to Eq. (\[eq:TAMR\]). (c) Orbital decomposition of the LDOS of the Bi adatom in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:BiAdatom"}](figure5.pdf) In this section we will focus on the description of the electronic structure of $6p$ adatoms adsorbed on the Mn monolayer of W(110). Especially the anisotropy of the LDOS due to SOC and the subsequent TAMR effect will be discussed in detail. Fig. \[fig:PbAdatom\](a) shows both the total (spin-averaged) and spin-resolved vacuum LDOS above the Pb adatom – in an energy range around $E_F$ typically accessible to STM – calculated for the two magnetization directions including SOC: (i) perpendicular to the surface (out-of-plane) denoted as $n_\perp(z, \epsilon)$ and parallel to the $[1\overline{1}0]$ direction (in-plane) denoted as $n_\parallel(z, \epsilon)$. Differences between both magnetization components are clearly discernible in the energy range below the Fermi level ([[$E_{\text{F}}$]{}]{}). The most significant feature is located at $-0.37$ eV in $n_\parallel(z, \epsilon)$ and corresponds to a peak of majority $p_z$ states being split and shifted towards lower energies as the magnetization rotates from the film plane ($\parallel$) to the perpendicular ($\perp$) direction of the surface. The same effect, although much less prominent, is also visible for the minority states. This behavior leads to a maximum value in the TAMR of $-49$% (see Fig. \[fig:PbAdatom\](b)). Around [[$E_{\text{F}}$]{}]{} this effect is considerably smaller and of opposite sign with TAMR values up to +22%. Similar observations can be seen in the vacuum LDOS of the Bi adatom on Mn/W(110) shown in Fig. \[fig:BiAdatom\](a). Here, the dominant peak of majority $p_z$ states which splits likewise upon rotation of the magnetization direction is shifted by 0.2 eV towards lower energies compared to Pb. Linked to this state, the value of the TAMR first takes a local maximum of +42% at $-0.66$ eV before dropping abruptly to a minimum of $-61$% at $-0.57$ eV below [[$E_{\text{F}}$]{}]{}. Similar to Pb, differences concerning $n_\perp$ and $n_\parallel$ for the minority channel are small in this energy range and the main part of the TAMR originates from majority states. In contrast, states with minority character are causing a modest TAMR of +20% just below [[$E_{\text{F}}$]{}]{}. For both adatoms the anisotropy of the vacuum LDOS shows only little magnetization-direction dependent differences in the unoccupied regions and giant values in the TAMR effect are restricted to areas below [[$E_{\text{F}}$]{}]{}. A closer look at the orbitally resolved LDOS of the adatoms in Fig. \[fig:PbAdatom\](c) and Fig. \[fig:BiAdatom\](c) reveals that the above-mentioned changes between both magnetization components in the vacuum can be attributed to $p_z$ states of the adatoms which are mostly below [[$E_{\text{F}}$]{}]{}. The curves in the vacuum almost coincide with the ones for states of this character calculated directly at the respective adatom. As the $p_z$ states are oriented along the surface normal, they preponderate in the vacuum compared to the $p_x$ and $p_y$ states. In contrast, the prominent peak of majority $p_y$ states dominating the LDOS in the vicinity of [[$E_{\text{F}}$]{}]{} of both Pb and Bi is not visible in the vacuum LDOS because they are aligned parallel to the film plane. The shift of this peak, from a position of 0.2 eV above [[$E_{\text{F}}$]{}]{} for Pb, towards occupied regions for Bi can be explained by the increasing number of electrons in the $p$ shell. On the other hand, the shift of the majority states with $p_z$ character which are identified to generate the large anisotropy of the vacuum LDOS and hence the shift of the position of the maximum TAMR can be ascribed to the different strength of the attractive potential acting between valence electrons and nucleus. Due to the larger nuclear charge these potentials lead to a stronger binding of the $p_z$ states to the nucleus for Bi. Further reasons for the majority $p_z$ states of the Bi adatom being shifted towards lower energies is the higher spin polarization compared to Pb as well as the smaller distance from its nearest neighbor Mn atom in the Mn monolayer. Hereby the orbital overlap increases resulting in a larger splitting of the states. ### Modeling of the TAMR In order to explain the large TAMR found for $6p$ adatoms adsorbed on Mn/W(110), we revert to the Hamiltonian of SOC mentioned in the introduction. As shown in Ref. [@Abate1965], the SOC operator can be written as a matrix in the following way: $$\mathcal{H}_{SOC}= \frac{\xi}{2} \begin{pmatrix} M&N\\ -N^{*}&M^{*} \end{pmatrix}\; .$$ Here, the diagonal matrices $M$ describe the coupling of two states with equal spin direction, whereas the secondary diagonal matrices $N$ denote the interaction of states with different spin character via SOC. Both can be calculated for an arbitrary orientation of the spin quantization axis by applying ladder operators of spin and angular momentum to linear combinations of complex spherical harmonics which represent both $p$ and $d$ orbitals. This approach yields the matrix element describing a spin-orbit induced hybridization between states with $p_z$ and $p_x$ symmetry in the same spin channel as [@Schoeneberg2016Diss]: $$\langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \uparrow \rangle= i\sin\theta\sin\phi \;, \label{eq:H_SOC_up_up}$$ and the element for coupling states of the same symmetry, but with opposite spin direction as $$\langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \downarrow \rangle=\cos\phi+i\sin\phi\cos\theta \;. \label{eq:H_SOC_up_down}$$ In the first case (cf. Eq. (\[eq:H\_SOC\_up\_up\])) the matrix element vanishes for the perpendicular magnetization direction ($\phi$=0$^{\circ}$, $\theta$=0$^{\circ}$) and becomes maximal for its magnetization pointing along the $[1\overline{1}0]$ direction ($\phi$=90$^{\circ}$, $\theta$=90$^{\circ}$), i.e. we expect a mixing of the two states only for a spin-quantization axis chosen along the film plane. The reverse is true if both states have opposite spin direction (cf. Eq. (\[eq:H\_SOC\_up\_down\])). Evaluating the matrix elements given in Ref. [@Schoeneberg2016Diss] for a potential hybridization mediated by SOC for states with $p_z$ and $p_y$ character shows that such interaction can not be realized on the Mn/W(110) surface for the two above mentioned magnetization directions, which are possible on the substrate due to the spin spiral ground state. For this reason the discussion concerning the anisotropy of the vacuum LDOS is restricted to $p_x$ and $p_z$ states for both $6p$ adatoms and dimers in this paper. Applying the above considerations first to the case of a Bi adatom on Mn/W(110) \[Fig. \[fig:BiAdatom\]\], one can explain the maximum value of the TAMR at $-0.57$ eV below [[$E_{\text{F}}$]{}]{} by a magnetization-direction dependent mixing of $p_x$ and $p_z$ orbitals of opposite spin channels. At this energy the prominent peak of majority $p_z$ states whose in-plane magnetization component resembles a single peak is split and shifted towards lower energies upon rotation of the spin-quantization axis (see Fig. \[fig:BiAdatom\](c)). According to the matrix elements, this behavior hints at a SOC-mediated hybridization with a minority $p_x$ state which can be found at $-0.82$ eV. The TAMR of the Pb adatom can also be understood based on the matrix elements of $H_{\rm SOC}$. E.g. the vacuum LDOS of the minority spin channel \[Fig. \[fig:PbAdatom\](a)\] just below $E_F$ is reduced upon rotating the magnetization direction from in-plane to out-of-plane. This is due to mixing by SOC in the minority spin channel \[Fig. \[fig:PbAdatom\](c)\] between a $p_z$ state located at $-0.12$ eV and a peak at $-0.05$ eV of $p_x$ orbital character. For an in-plane magnetization direction, which allows mixing within the same spin channel by SOC according to Eq. (\[eq:H\_SOC\_up\_up\]), the $p_x$ minority state peak at $-0.05$ eV splits into two peaks which coincide with the positions of two minority states $p_z$ peaks. This creates a large negative TAMR within the minority spin channel of $-56$% (not shown here). However, the TAMR is obtained from the total, spin-averaged LDOS. Just below the Fermi energy it is positive with a value of $+22$% due to a majority $p_z$ peak whose height is reduced due to SOC for an in-plane magnetization \[Fig. \[fig:PbAdatom\](c)\]. The maximum TAMR effect of the Pb adatom of $-49$% occurs at $0.37$ eV below [[$E_{\text{F}}$]{}]{}. It originates from the majority spin channel \[Fig. \[fig:PbAdatom\](a)\] and it is due to the splitting of a majority $p_z$ state as can be seen from the orbital decomposition at the Pb atom \[Fig. \[fig:PbAdatom\](c)\]. Since the mixing occurs for a magnetization direction perpendicular to the surface it can be explained by a SOC induced mixing with $p_x$ states of the opposite spin channel according to Eq. (\[eq:H\_SOC\_up\_down\]). While changes in the minority $p_x$ LDOS can be noted within the relevant energy interval it is not possible to unambiguously propose a single peak which is responsible for the mixing. As will be discussed in detail for the Pb dimers at the end of this manuscript, there is also an impact of the Mn $3d$ states which are also subject to SOC and with which the Pb $p$ states are hybridizing. Pb and Bi dimers on Mn/W(110) {#subsec:dimers} ----------------------------- ### Structural and magnetic properties {#sub2sec:str_mag_dimers} -------------------- ------ -------------------------- ------ -------------------------- $d$ [[$\mu_{\text{B}}$]{}]{} $d$ [[$\mu_{\text{B}}$]{}]{} $[001]$ 3.23 +0.08 3.22 +0.12 $[1\overline{1}0]$ 3.35 +0.18 3.93 +0.02 $[1\overline{1}1]$ 3.11 $\pm$0.02 3.12 $\pm$0.02 Free 2.96 0.67 2.68 0.0 -------------------- ------ -------------------------- ------ -------------------------- : The dimer bond lengths $d$ (in Å) and the individual magnetic moments (in [[$\mu_{\text{B}}$]{}]{}) of the adsorbed Pb and Bi dimers on the Mn monolayer on W(110). For comparison, the bond lengths and the magnetic moments of the free dimers (calculated with SOC) are given.[]{data-label="tab:dimer_d_m"} Since the Pb and Bi adatoms adsorb in the hollow-site position of the Mn layer, the dimers can be oriented along the $[001]$, $[1\overline{1}0]$, and $[1\overline{1}1]$ directions (see Fig. \[fig:geom\](c)). The relaxed dimer bond lengths and the magnetic moments for the three orientations along with the values for free dimers are given in Table \[tab:dimer\_d\_m\]. The dimer bond lengths increase after the adsorption due to structural relaxation from the bond length values of free dimers. For Pb dimers an increase of $\approx$ 10% in bond length can be observed. For Bi dimers, a larger increase of $\approx$ 20% in bond length has been observed except along the $[1\overline{1}0]$ orientation. In this case, we find an increase of bond length values by $\approx$ 45 %. In Pb dimers, the individual atoms carry small induced magnetic moments for all three orientations due to the hybridization with the Mn monolayer. Among the three orientations, the largest individual magnetic moment of +0.18 [[$\mu_{\text{B}}$]{}]{} is observed for the $[1\overline{1}0]$ orientation. These induced moments for Pb dimers are in contrast with the single atom adsorption where Pb remains nonmagnetic. Similar to the single adatom adsorption, Bi dimers also pick up small induced magnetic moment for all orientations with the largest value of +0.12 [[$\mu_{\text{B}}$]{}]{} along the $[001]$ direction. Similar to the single adatom adsorption, reduction of magnetic moments for both nn and nnn Mn adatoms have been observed here as well. ### Electronic properties  We proceed by describing and comparing the electronic structure of the dimers with those presented for the single adatoms before explaining the anisotropy of the LDOS. It should be pointed out here that the notation of the $p$ orbitals of the dimers refers to the global coordinate axes of the Mn/W(110) surface as shown in Fig. \[fig:geom\], i.e. no local system for the adsorbates rotated for different orientations has been used. Therefore, the $p_x$ and $p_y$ orbitals of both \[001\] and $[1\overline{1}0]$ dimers are aligned along the \[001\] and $[1\overline{1}0]$ direction, respectively. As a result, the orbitals responsible for the covalent bond are changing. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\] shows the $m_l$ decomposed $p$ states of both Pb and Bi dimers on a large energy scale around [[$E_{\text{F}}$]{}]{}. We will exemplify the differences in the LDOS compared to the single adatoms by means of the Bi dimers; similar observations can be made for the respective Pb adsorbates. Compared with the Bi adatom (cf. Fig. \[fig:spinpol\_adatom\](f)), the Bi dimer along the $[001]$ orientation exhibits the largest modifications in its $p_x$ orbitals which are responsible for the covalent bond in this case hereby forming $\sigma$ orbitals (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](b)). This becomes most evident in the minority channel just below [[$E_{\text{F}}$]{}]{} where the corresponding states of the single adatom are shifted by 1 eV to the left due to the orbital overlap of the two atoms of the dimer. In contrast, the $p_y$ and $p_z$ states only show minor differences compared to the Bi adatom; especially the large peak of majority $p_y$ states dominating close to [[$E_{\text{F}}$]{}]{} of the adatom is also found for the $[001]$ dimer. Owing to the large distance of 3.93 [Å]{} between the two Bi atoms of the $[1\overline{1}0]$ dimer the overlap of their orbitals is small resulting in similar features as for the adatom (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](d)). The main change in its $m_l$ resolved LDOS is the disappearance of the dominant majority $p_y$ peak at [[$E_{\text{F}}$]{}]{} which can be attributed to the fact that these orbitals are forming $\sigma$ bonds in this case. For the Bi $[1\overline{1}1]$ dimer, differences in the $m_l$ decomposed LDOS (cf. Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\](f)) are clearly discernible compared to the single adatom (cf. Fig. \[fig:spinpol\_adatom\](f)). This observation can partly be ascribed to the small bond length of 3.12 [Å]{} and hence a large orbital overlap and partly to the combination of $p_x$ and $p_y$ states forming $\sigma$ bonds. Both orbitals are tilted with respect to the global coordinate axes of the substrate leading to a covalent bond of a mixture of the two states with different symmetry. The $p_z$ states which are crucial for STM are relatively weakly affected for all dimers. ### TAMR effect of the Pb dimers on Mn/W(110) {#sub2sec:TAMR_dimers_Pb}  In the following section we study the TAMR of Pb dimers on Mn/W(110) for the three different dimer orientation discussed before. In Fig. \[fig:Pb\_dimer\_dos\](a) both the total and spin-resolved components of the vacuum LDOS of the Pb dimer oriented along the \[001\] direction are plotted for the two different magnetization directions which can occur due to the spin spiral states of the Mn monolayer on W(110). Note, that the in-plane magnetization direction is perpendicular to the dimer axis in this case. Compared to the respective adatom (cf. Fig. \[fig:PbAdatom\]), only small differences between $n_{\perp}$ and $n_{\parallel}$ can be observed in the occupied regions below [[$E_{\text{F}}$]{}]{}. The most striking features occur now at energies $-0.7$ eV, $-0.42$ eV, $-0.2$ eV, and $0.05$ eV leading to maximum values in the TAMR of $\pm 28$% (see Fig. \[fig:Pb\_dimer\_dos\](b)). Unlike the Pb adatom, the anisotropy of the vacuum LDOS takes another local maximum of +29% just above [[$E_{\text{F}}$]{}]{}. As one can see from the spin-resolved curves in Fig. \[fig:Pb\_dimer\_dos\](a), the TAMR at $-0.7$ eV stems from a modification of both spin channels upon rotation of the magnetization direction, whereas at [[$E_{\text{F}}$]{}]{} only majority states contribute whose perpendicular magnetization components are clearly enhanced compared to the parallel ones. Similar to the Pb adatom, differences between $n_{\perp}$ and $n_{\parallel}$ in the unoccupied regions are barely noticeable for both spin directions of the \[001\] dimer. The orbitally decomposed LDOS of this dimer plotted in Fig. \[fig:Pb\_dimer\_dos\](c), shows further similarities with the adatom. It is dominated by a prominent peak of majority $p_y$ states at 0.15 eV above [[$E_{\text{F}}$]{}]{} which is not reflected in the vacuum LDOS and exhibits discernible changes in the $p_z$ states with respect to both magnetization directions below [[$E_{\text{F}}$]{}]{}. The $p_z$ states predominate the vacuum LDOS due to their double-lobed orbitals pointing along the surface normal. However, they experience a small shift towards lower energies as well as a splitting which is a consequence of the interaction between both Pb atoms composing the dimer. Compared to the $[001]$ Pb dimer, the anisotropy of the vacuum LDOS is much larger for the dimer oriented along the $[1\overline{1}0]$ direction representing the propagation direction of the spin spiral on Mn/W(110) (see Fig. \[fig:Pb\_dimer\_dos\](d)-(e)). For this dimer orientation the in-plane magnetization direction is along the dimer axis (cf. Fig. \[fig:geom\]). As for the single adatom, the appearance of the LDOS in the vacuum below [[$E_{\text{F}}$]{}]{} is characterized by magnetization-direction dependent differences of the majority $p_z$ states where the main contribution comes from a dominant peak of the in-plane magnetized dimer at $-0.45$ eV below [[$E_{\text{F}}$]{}]{}. Being split multiple times upon reorientation of the spin-quantization axis, it creates a steep descent in the TAMR up to $-64$% thereby even exceeding the maximum value of the Pb adatom by 15%. Just above [[$E_{\text{F}}$]{}]{}, the anisotropy of the vacuum LDOS takes a local minimum of $-38$% which is due to the shift of a minority $p_z$ state as the magnetization rotates from the perpendicular direction to the film plane along the $[1\overline{1}0]$ direction. The substantial similarity of the LDOS of this dimer in the $p_z$ orbitals compared to the single adatom (cf. Fig. \[fig:PbAdatom\]) can be explained by means of the relatively large distance of both Pb atoms (see Table \[tab:dimer\_d\_m\]). If they are further apart, their interaction, i.e. the overlap of their orbitals, will be small thereby causing a similar behavior as for a single atom (cf. Fig. \[fig:PbAdatom\](c) and Fig. \[fig:Pb\_dimer\_dos\](f)). The $p_y$ orbitals, on the other hand, which are forming $\sigma$ bonds in the case of the $[1\overline{1}0]$ dimer are expected to show more remarkable differences in comparison with the adatom. This becomes mostly evident in the unoccupied regions where the prominent peak of majority $p_y$ states is completely absent as seen in Fig.  \[fig:Pb\_dimer\_dos\](f) with respect to the adatom (cf. Fig. \[fig:PbAdatom\](c)).  The electronic structure of the Pb dimer along $[1\overline{1}1]$ direction is shown in Fig. \[fig:Pb\_dimer\_dos\](g)-(i) for both magnetization directions. As for the \[001\] Pb dimer, only minor differences between $n_{\perp}$ and $n_{\parallel}$ are visible in the vacuum LDOS above the $[1\overline{1}1]$ dimer along the diagonal of the unit cell. At $-0.6$ eV below [[$E_{\text{F}}$]{}]{}, a peak of majority states for parallel magnetization direction is enhanced compared to the perpendicular magnetization direction resulting in a negative TAMR of $-25$%. The vacuum LDOS of the unoccupied spectrum on the other hand is mainly characterized by magnetization-direction dependent changes of the minority states causing maximum values of +28% in the TAMR at +0.62 eV. The magnitude of TAMR is much smaller for $[1\overline{1}1]$ dimer orientation as compared to the $[1\overline{1}0]$, which is oriented along the natural magnetization direction of the Mn/W(110) substrate. However, it is very similar as for the dimer oriented along $[001]$, which is perpendicular to the magnetization of the surface. We will discuss these behavior in more detail in Section \[TAMR\_origin\_Dimers\]. ### TAMR of Bi dimers on Mn/W(110) {#sub2sec:TAMR_dimers_Bi} Similar changes of the TAMR with the dimer orientation can be observed for Bi dimers on Mn/W(110). Considering first the electronic structure of the \[001\] Bi dimer which is shown in Fig. \[fig:Bi\_dimer\_dos\](a)-(c), one notices that as for the \[001\] Pb dimer the curves for both magnetization directions of the spin-averaged vacuum LDOS do not differ significantly from each other. The largest differences are now located in the energy range between $-0.9$ and $-0.3$ eV leading to a local maximum in the TAMR of $-37$% at $-0.75$ eV. The negative TAMR value is much smaller than for the Bi adatom (cf. Fig. \[fig:BiAdatom\](b)). Consistent with the single Bi adatom, the energy range just below [[$E_{\text{F}}$]{}]{} is dominated by a large peak of minority states with $p_z$ character (see Fig. \[fig:Bi\_dimer\_dos\](c)), whereas the prominent peak of majority $p_y$ orbitals is absent in the vacuum LDOS due to its orientation within the film plane. In contrast to the anisotropy of the vacuum LDOS of the corresponding Pb dimer, changes between $n_{\perp}$ and $n_{\parallel}$ vanish in the case of the \[001\] Bi dimer directly at [[$E_{\text{F}}$]{}]{}. A closer look at the orbitally resolved LDOS reveals that the $p_z$ states which dominate the LDOS above the surface are affected by the mutual interaction of both Bi atoms composing the dimer. In comparison with the single Bi adatom, they are shifted towards lower energies and experience a larger splitting which is mostly apparent in the majority channel. The dominant peak of majority $p_z$ states causing the large TAMR of $-61$% at $-0.57$ eV for the single Bi adatom (cf. Fig. \[fig:BiAdatom\](b)) is therefore located outside of the presented energy range for the dimers. However, owing to the interaction of the orbitals it is split as well and less pronounced than for the single Bi adatom (not shown). The hybridization of majority $p_z$ orbitals is less prominent in the case of a Bi dimer oriented along the $[1\overline{1}0]$ direction (Fig. \[fig:Bi\_dimer\_dos\](d)-(f)) of the Mn/W(110) surface due to the large distance of nearly 4 [Å]{} between both Bi atoms. As one can see from its orbital decomposition in Fig. \[fig:Bi\_dimer\_dos\](f), $p_z$ states move closer to [[$E_{\text{F}}$]{}]{} showing similar characteristics as in the case of the single Bi adatom (cf. Fig. \[fig:BiAdatom\]). At $E_{\rm F}-0.55$ eV a dominant peak of majority $p_z$ states for the in-plane magnetization orientation becomes visible which both decreases in height and shifts towards lower energies upon rotation of the spin-quantization axis. This is also the largest observable change between $n_{\perp}$ and $n_{\parallel}$ of the total (spin-averaged) vacuum LDOS for the $[1\overline{1}0]$ Bi dimer that is plotted in Fig. \[fig:Bi\_dimer\_dos\](d). The just mentioned magnetization-direction dependent changes in the majority $p_z$ states thereby correspond to a huge TAMR effect of $-64$% at $-0.55$ eV. Hence, the anisotropy of the LDOS is in the same order of magnitude as for the single Bi adatom and takes its largest value at the same energetic position as well. The TAMR is considerably smaller for the rest of the presented energy range, especially at [[$E_{\text{F}}$]{}]{} where differences for parallel and perpendicular magnetizations of both spin channels only create a modest TAMR of approximately 15%. As observed for the corresponding Pb dimer, only minor differences between both magnetization directions occur in the spin-averaged vacuum LDOS of the Bi dimer placed along the diagonal of the unit cell, i.e. the $[1\overline{1}1]$ direction. Whereas the spin-resolved curves are indeed clearly characterized by changes upon rotation of the magnetization in the occupied regions (see Fig. \[fig:Bi\_dimer\_dos\](g)), changes in the sum of both spin channels only lead to small values in the TAMR ranging from $-16$% at $-0.8$ eV up to $-10$% at [[$E_{\text{F}}$]{}]{}. As shown in the orbital decomposed LDOS of the dimer in Fig. \[fig:Bi\_dimer\_dos\](i), the main part of the anisotropy is created by the $p_z$ orbitals. Additionally, this dimer orientation exhibits the smallest TAMR of all studied Bi dimer geometries and consistent with the previously presented results for $[1\overline{1}1]$ Pb dimers on Mn/W(110). ### Origin of the TAMR for $6p$ dimers on Mn/W(110) {#TAMR_origin_Dimers} The variation of the TAMR magnitude depending on the orientation of both Pb and Bi dimers can partially be explained by means of a physical model considering two atomic states coupled via SOC proposed in Ref. . For a possible SOC induced hybridization of the dimer $p$ states, we refer to the matrix elements presented in section \[sub2sec:TAMR\_adatom\]. The orbitals are defined with respect to the global coordination axes of the unit cell. Within this simplified model, our expectations match quite well with the $6p$ dimers oriented along the $[1\overline{1}0]$ direction showing the largest TAMR of all studied configurations. However, the SOC induced hybridization is not so clearly visible between their $p_z$ and $p_x$ states forming $\pi_z$ and $\pi_x$ molecular orbitals, respectively. The reduction of the TAMR effect for the $[1\overline{1}1]$ $6p$ dimers can be understood as well using the simplified model of two atomic states with differing orbital symmetry. For the case of a dimer orientation along the diagonal of the supercell the magnetization direction of the substrate is rotated with respect to the bonding axis of the atoms leading to a reduction of the respective matrix elements. For instance the hybridization between the $p$ states, $ \langle \uparrow,p_z|\mathcal{H}_{SOC}|p_x, \uparrow \rangle \propto \sin\theta\sin\phi$, is reduced for an azimuth angle of 45$^{\circ}$. Since changes in the LDOS scale with the square of the matrix elements [@Bode2002], for both $6p$ dimers, the TAMR in $[1\overline{1}1]$ orientation is diminished by a factor of 4 compared to the $[1\overline{1}0]$ dimer ($\sim$ 15% vs. $\sim$ 60%; cf. Fig. \[fig:Pb\_dimer\_dos\](e), Fig. \[fig:Pb\_dimer\_dos\](h) and Fig. \[fig:Bi\_dimer\_dos\](e), Fig. \[fig:Bi\_dimer\_dos\](h)). The same behavior has recently been observed for Pb dimers on a Fe bilayer on W(110) [@Schoneberg2018]. In addition, the $p_x$ states are partially involved in the formation of molecular $\sigma$ bonds along the dimer axis and thereby not available for the mixing with the $p_z$ states which further reduces the possible value of the TAMR. The reduction of the LDOS of $p_x$ states in the shown energy range due to hybridization is even more apparent in the $[001]$ and $[1\bar{1}1]$ Bi dimers (cf. Fig. \[fig:Bi\_dimer\_dos\](c), Fig. \[fig:Bi\_dimer\_dos\](i)) while it is similar to that of the Bi adatom (cf. Fig. \[fig:BiAdatom\]) for the $[1\bar{1}0]$ dimer (cf. Fig. \[fig:Bi\_dimer\_dos\](f)). For the $[001]$ Pb dimer, within the simplified model, one can interpret the changes in the curves of the minority $p_x$ and $p_z$ states upon rotation of the spin-quantization axis at $-0.52$ eV due to hybridization mediated by SOC (see Fig. \[fig:Pb\_dimer\_dos\](c)). The same effect could already be observed for the single Pb adatom directly at [[$E_{\text{F}}$]{}]{} (cf. Fig. \[fig:PbAdatom\](c)). If the dimer orientation ($[001]$) is perpendicular to the magnetization direction ($[1\overline{1}0]$) of the substrate, one would expect SOC to mix molecular $\pi_z$ and antibonding $\sigma^{*}$ orbitals which are composed of $p_x$ states here [@Schoneberg2018]. Molecular orbitals of this symmetry are located further apart in the energy spectrum than $\pi_z$ and $\pi_x$ molecular orbitals that can easily hybridize via SOC for a dimer axis along the magnetization of the Mn/W(110) surface, [[i.e.,]{}]{} the $[1\overline{1}0]$ direction. Hence, within this simple model, the $[001]$ $6p$ dimers were expected to exhibit a much smaller variation of their electronic structure under the influence of SOC. However, our DFT calculations show that the anisotropy of the LDOS actually takes an unexpected high value of $-28$% and $-37$% for the case of the $[001]$ Pb and Bi dimer adsorbed on Mn/W(110), respectively. Hence, this model based on only two atomic/molecular states is not sufficient to quantitatively understand the TAMR for dimers along the $[001]$ axis. ![(a) and (b): top view (cross section) of the spin-resolved partial charge density plots of the Pb \[001\] dimer on Mn/W(110) in the energy range \[[[$E_{\text{F}}$]{}]{}$-0.51$, [[$E_{\text{F}}$]{}]{}$-0.49$ eV\]. (c) and (d): cross-sectional plots through the Pb dimer parallel to the \[001\] direction for the charge densities shown in (a) and (b).[]{data-label="fig:pb_001_chg"}](figure9.png) ![(a) and (b): top view (cross section) of the spin-resolved partial charge density plots of the Pb $[1\overline{1}0]$ dimer on Mn/W(110) in the energy range \[[[$E_{\text{F}}$]{}]{}$-0.46$, [[$E_{\text{F}}$]{}]{}$-0.44$ eV\]. (c) and (d): cross-sectional plots through the Pb dimer parallel to the $[1\overline{1}0]$ direction for the charge densities shown in (a) and (b). []{data-label="fig:pb_110_chg"}](figure10.png) In order to achieve a deeper understanding of the effect of $p-d$ hybridization on TAMR for the Pb and Bi dimers on Mn/W(110), we have calculated the partial charge densities within the scalar-relativistic approximation, [[i.e.,]{}]{} neglecting SOC, for a small energy range where the TAMR appears most prominent. The inclusion of SOC will not affect the hybridization as evident from the LDOS in the scalar relativistic approximation \[Fig. \[fig:Pb\_Bi\_dimer\_dos\_full\]\] and including SOC \[Figs. \[fig:Pb\_dimer\_dos\], \[fig:Bi\_dimer\_dos\]\]. In the following we exemplify the influence of the substrate by means of both $[001]$ and $[1\overline{1}0]$ Pb dimers only as we observe similar characteristic behaviors for the Bi dimers. The spin-resolved partial charge density of the $[001]$ Pb dimer at approximately [[$E_{\text{F}}$]{}]{}$-0.5$ eV at which the large TAMR of $\sim$ 28% occurs (cf. Fig. \[fig:Pb\_dimer\_dos\](b)) is shown in Fig. \[fig:pb\_001\_chg\]. From the top view \[Fig. \[fig:pb\_001\_chg\](a,b)\] which represents a cross section through the dimer one can clearly see the molecular $\pi_z$ and $\sigma_x$ character of the adsorbate for both spin channels, whereas from the side view \[Fig. \[fig:pb\_001\_chg\](c,d)\] a strong hybridization with the $d$ orbitals of the Mn atoms of the surface becomes visible. In the majority spin channel the axes of the $p_z$ orbitals at the Pb dimer deviate from the $z$ direction of the unit cell and the $d$ orbitals of the Mn atoms are twisted towards the adsorbate \[Fig. \[fig:pb\_001\_chg\](c)\]. This leads to a tilt of the upper lobes of the $p_z$ orbitals towards the center of the dimer and an overlap of the lower lobes with the Mn $d$ states in the case of the majority channel. For the minority channel on the other hand \[Fig. \[fig:pb\_001\_chg\](d)\] a clear differentiation between the Pb and Mn states is not possible anymore due to the strong hybridization which becomes manifest in an accumulation of the charge density at the interface. These observations already indicate that for the explanation of the TAMR effect of the $6p$ dimers on Mn/W(110) more than two atomic states have to be taken into account. Hybrid Pb-Mn interface states are also present in the majority channel of the $[1\overline{1}0]$ Pb dimer at the position of the maximum TAMR around 0.45 eV below [[$E_{\text{F}}$]{}]{} (see Fig. \[fig:pb\_110\_chg\]). A closer look at the calculated charge density reveals that its majority $p_z$ orbitals only interact with the Mn atom below the dimer axis, but not with the other atoms of the Mn monolayer \[Fig. \[fig:pb\_110\_chg\](c)\]. Exactly the same behavior can also be observed for the corresponding $[1\overline{1}0]$ Bi dimer on Mn/W(110) (not shown). The reason for this hybridization can be explained by means of the different distances of the $6p$ atoms and their neighboring Mn atoms. While the central Mn atom and one atom of the Pb dimer are separated by just 2.84 [Å]{}, the respective distance towards the next Mn atoms is 3.15 [Å]{} and hence significantly larger. However, the interaction with the central Mn atom described above cannot be realized for the $[1\overline{1}1]$ dimer since the respective atom of the substrate is missing below a bonding axis along the diagonal (see Fig. \[fig:geom\](c)). ![Orbital decomposed LDOS including SOC of the central Mn atoms below the $[001]$ and $[1\overline{1}0]$ Pb dimer in terms of the majority (up) and minority (down) states. Solid (dashed) lines correspond to the magnetization direction perpendicular (parallel) to the surface plane. The orange up and down arrow indicates majority and minority spin channels, respectively.[]{data-label="fig:Mn_Soc_Dos"}](figure11.pdf) Keeping in mind the studies of the partial charge densities, we did a further investigation of the LDOS of the central Mn atoms below the Pb dimer axes for the case of the $[001]$ and $[1\overline{1}0]$ direction. Fig. \[fig:Mn\_Soc\_Dos\] shows their orbitally decomposed $d$ states in an energy interval of $\pm$1 eV around [[$E_{\text{F}}$]{}]{} along with the $p$ states of the adsorbates for the two different magnetization directions discussed before. It is evident that the Mn atoms below both Pb dimers are likewise affected by the rotation of the magnetization direction and bear resemblance to the changes in the states of the Pb atoms at same energetic positions. For the $[001]$ direction this becomes mostly apparent at 0.15 eV where a large peak of majority $d_{yz}$ orbitals of the central Mn atom shows the same behavior upon a change of the spin-quantization axis as the dominant $p_y$ state of the adsorbed dimer (cf. Fig. \[fig:Pb\_dimer\_dos\](c)). Moreover one can observe an enhancement of the parallel magnetization component of majority states with $d_{z^2}$ and $d_{x^2-y^2}$ character at $-0.41$ eV which corresponds with a small peak of $n_{\parallel}$ for the majority $p_z$ states at the same energy. Further resemblance regarding magnetization-direction dependent differences in the LDOS are found between $-0.70$ eV and $-0.50$ eV for the minority $d_{xz}$ orbitals of Mn and $p_z$ and $p_x$ orbitals of the $[001]$ Pb dimer. For the $[1\overline{1}0]$ Pb dimer this SOC-dependent hybridization is most prominent for the $d_{yz}$ states both at [[$E_{\text{F}}$]{}]{} in the minority channel and at [[$E_{\text{F}}$]{}]{}$-0.45$ eV in the majority channel. Especially at the last-mentioned position, for the in-plane magnetization, it becomes clear that the majority Mn $d_{yz}$ states and the majority Pb $p_z$ states interacts quite strongly \[Fig. \[fig:Mn\_Soc\_Dos\]\] and produce a large TAMR value (see Fig. \[fig:Pb\_dimer\_dos\](e)). Conclusion ========== In conclusion, we have presented a detailed study of the spin-resolved electronic structure of single Pb and Bi adatoms and dimers adsorbed on the Mn monolayer on W(110) including the effect of spin-orbit coupling. Using density functional theory, we calculated the tunneling anisotropic magnetoresistance effect from two magnetization directions, imposed due to the cycloidal spin spiral ground state in the Mn layer, for the respective $6p$ adsorbate: perpendicular to the surface (out-of-plane) and parallel to the $[1\overline{1}0]$ direction representing the propagation direction of the spin spiral ground state (in-plane). Our calculations for the $6p$ adatoms which are characterized by large spin-orbit coupling constants predict an enhancement of the TAMR up to 49% for Pb and 61% for Bi adatoms. In both cases it can mainly be attributed to magnetization-direction dependent changes of majority $p_z$ states of the adatom. The anisotropy of the LDOS of both adatoms can generally be explained by means of a simplified physical model which considers the coupling of two atomic states with different orbital symmetry ($p_z$ and $p_x$ in the present case) via spin-orbit coupling. Although Pb and Bi adatoms carry almost no magnetic moment, they exhibit a large spin polarization directly at the surface and also in the vacuum due to the hybridization with the substrate. The spin polarization becomes maximal with values up to 60% around [[$E_{\text{F}}$]{}]{} for the Pb adatom. We have also investigated the TAMR for three different dimer orientations adsorbed on the Mn/W(110) surface. Consistent with the expectations both Pb and Bi dimers with their bonding axis along the magnetization direction of the substrate, i.e. the $[1\bar{1}0]$ direction, show the maximum anisotropy of the vacuum LDOS with values of 64% in the occupied regions. The origin of this large effect is a molecular $\pi_z$ orbital with majority spin character which strongly interacts with the central Mn atom below the dimer axis. Similar interactions are also found for a dimer orientation perpendicular to the magnetization direction of Mn/W(110), but with much smaller TAMR values of 37% for Bi and 28% for Pb, respectively. The TAMR becomes minimal for $6p$ dimers along the diagonal $[1\overline{1}1]$ direction (16% in the case of Bi, 27% for Pb) due to reduced SOC induced mixing of the $p$ states on the one hand and due to missing Mn atoms for hybridization below their bonding axes on the other hand. A further exploration of the central Mn atoms below the $[001]$ and $[1\overline{1}0]$ dimers has shown that their $d$ orbitals are likewise affected by changes upon rotation of the magnetization direction which has to be taken into account for the comprehension of the TAMR effect apart from the simple model of only two atomic states interacting by SOC. Acknowledgments {#acknowledgments .unnumbered} =============== We acknowledge the DFG via SFB677 for financial support. We gratefully acknowledge the computing time at the supercomputer of the North-German Supercomputing Alliance (HLRN). We thank N. M. Caffrey for valuable discussions. [50]{} ifxundefined \[1\][ ifx[\#1]{} ]{} ifnum \[1\][ \#1firstoftwo secondoftwo ]{} ifx \[1\][ \#1firstoftwo secondoftwo ]{} ““\#1”” @noop \[0\][secondoftwo]{} sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{} @startlink\[1\] @endlink\[0\] @bib@innerbibempty [****,  ()](https://doi.org/10.1103/PhysRevLett.99.067202) [****,  ()](https://doi.org/10.1126/science.1154415) [****,  ()](https://doi.org/10.1103/PhysRevLett.103.057202) [****, ()](https://doi.org/10.1038/nphys1616) [****,  ()](http://stacks.iop.org/1367-2630/13/i=8/a=085011) [****,  ()](https://doi.org/10.1103/PhysRevB.86.180406) [****,  ()](https://doi.org/10.1126/science.1228519) [****, ()](https://doi.org/10.1103/PhysRevLett.89.237205) [****, ()](https://doi.org/10.1103/PhysRevLett.93.117203) [****,  ()](https://doi.org/10.1126/science.1082857) [****,  ()](https://doi.org/10.1126/science.1146110) [****,  ()](https://doi.org/10.1126/science.1191688) [****, ()](https://doi.org/10.1103/PhysRevLett.106.037205) [****,  ()](https://doi.org/10.1126/science.1252841) [****,  ()](https://doi.org/https://doi.org/10.1016/j.tsf.2008.08.172) [****,  ()](https://doi.org/10.1038/nmat3304) [****,  ()](https://doi.org/10.1103/PhysRevB.73.024418) [****, ()](https://doi.org/10.1103/PhysRevLett.98.046601) [****,  ()](https://doi.org/10.1103/PhysRevB.79.155303) [****, ()](https://doi.org/10.1103/PhysRevB.80.045312) [****, ()](https://doi.org/10.1103/PhysRevLett.99.226602) [****,  ()](https://doi.org/10.1140/epjb/e2006-00201-3) [****, ()](https://doi.org/10.1103/PhysRevLett.97.127202) [****, ()](https://doi.org/10.1103/PhysRevLett.110.037202) [****, ()](https://doi.org/10.1021/acs.nanolett.5b05071) [****, ()](https://doi.org/10.1103/PhysRevB.81.212409) [****, ()](https://doi.org/10.1038/nmat2983) [****,  ()](https://doi.org/10.1103/PhysRevB.97.220406) [****,  ()](https://doi.org/10.1134/S0021364016070043) [****,  ()](https://doi.org/10.1103/PhysRevB.97.041114) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****, ()]{} [****,  ()](https://doi.org/10.1103/PhysRevB.98.220401) [****,  ()](https://doi.org/10.1088/0953-8984/26/39/394010) [****,  ()](https://doi.org/10.1126/science.288.5472.1805) [****,  ()](https://doi.org/10.1103/PhysRevB.54.11169) @noop [****,  ()](https://doi.org/10.1103/PhysRevB.50.17953) [****,  ()](https://doi.org/10.1103/PhysRevB.59.1758) [****, ()](https://doi.org/10.1103/PhysRevLett.77.3865) [****,  ()](https://doi.org/10.1103/PhysRevLett.78.1396) [****,  ()](https://doi.org/10.1103/PhysRevB.62.11556) [****,  ()](https://doi.org/10.1103/PhysRevB.13.5188) [****,  ()](https://doi.org/10.1103/PhysRevLett.50.1998) [****,  ()](https://doi.org/10.1103/PhysRevB.31.805) @noop [****,  ()]{} **, @noop [Ph.D. thesis]{},  ()

ArXiv

--- abstract: 'Dipolar Bose and Fermi gases, which are currently being studied extensively experimentally and theoretically, interact through anisotropic, long-range potentials. Here, we replace the long-range potential by a zero-range pseudo-potential that simplifies the theoretical treatment of two dipolar particles in a harmonic trap. Our zero-range pseudo-potential description reproduces the energy spectrum of two dipoles interacting through a shape-dependent potential under external confinement very well, provided that sufficiently many partial waves are included, and readily leads to a classification scheme of the energy spectrum in terms of approximate angular momentum quantum numbers. The results may be directly relevant to the physics of dipolar gases loaded into optical lattices.' author: - 'K. Kanjilal' - 'John L. Bohn' - 'D. Blume' title: 'Pseudo-potential treatment of two aligned dipoles under external harmonic confinement' --- Introduction ============ Many-body systems with dipolar interactions have attracted a lot of attention recently. Unlike the properties of ultracold atomic alkali vapors, which can be described to a very good approximation by a single scattering quantity (the $s$-wave scattering length), those of dipolar gases additionally depend on the dipole moment. This dipole moment can be magnetic, as in the case of atomic Cr [@grie05; @stuh05], or electric, as in the case of heteronuclear molecules such as OH [@meer05; @boch04], KRb [@wang04a] or RbCs [@kerm04]. Furthermore, dipolar interactions are long-ranged and anisotropic, giving rise to a host of novel many-body effects in confined dipolar gases such as roton-like features [@dell03; @sant03; @rone06a] and rich stability diagrams [@sant00; @yi00; @gora00; @mart01; @yi01; @gora02; @rone06; @bort06]. The physics of dipolar gases loaded into optical lattices promises to be particularly rich. For example, this setup constitutes the starting point for a range of quantum computing schemes [@bren99; @jaks00; @demi02; @bren02]. Additionally, a variety of novel quantum phases have already been predicted to arise [@gora02a; @dams03; @barn06; @mich06]. Currently, a number of experimental groups are working towards loading dipolar gases into optical lattices. This paper investigates the physics of doubly-occupied optical lattice sites in the regime where the tunneling between neighboring sites and the interactions with dipoles located in other lattice sites can be neglected. In this case, the problem reduces to treating the interactions between two dipoles in a single lattice site. Assuming that the lattice potential can be approximated by a harmonic potential, the center of mass motion separates and the problem reduces to solving the Schrödinger equation for the relative distance vector $\vec{r}$ between the two dipoles. The interaction between the two aligned dipoles is angle-dependent and falls off as $1/r^3$ at large interparticle distances. In this work, we replace the shape-dependent interaction potential by an angle-dependent zero-range pseudo-potential, which is designed to reproduce the scattering properties of the full shape-dependent interaction potential, and derive an implicit eigenequation for two interacting identical bosonic dipoles and two interacting identical fermionic dipoles analytically. Replacing the full interaction potential or a shape-dependent pseudo-potential by a zero-range pseudo-potential [@ferm34; @huan57; @busc98; @blum02; @bold02; @kanj04; @stoc04] often allows for an analytical description of ultracold two-body systems in terms of a few key physical quantities. Here we show that the eigenequation for appropriately chosen zero-range pseudo-potentials reproduces the energy spectrum of two dipoles under harmonic confinement interacting through a shape-dependent model potential; that the applied zero-range treatment readily leads to an approximate classification scheme of the energy spectrum in terms of angular momentum quantum numbers; and that the proposed pseudo-potential treatment breaks down when the characteristic length of the dipolar interaction becomes comparable to the characteristic length of the external confinement. The detailed understanding of two interacting dipoles obtained in this paper will guide optical lattice experiments and the search for novel many-body effects. Section \[sec\_pp\] introduces the Hamiltonian under study and discusses the anisotropic zero-range pseudo-potential that is used to describe the scattering between two interacting dipoles. In Sec. \[sec\_ho\], we derive an implicit eigen equation for two dipoles under external spherical harmonic confinement interacting through the zero-range pseudo-potential and show that the resulting eigenenergies agree well with those obtained for a shape-dependent model potential. Finally, Sec. \[sec\_conclusion\] concludes. System under study and anisotropic pseudo-potential {#sec_pp} =================================================== Within the mean-field Gross-Pitaevskii formalism, the interaction between two identical bosonic dipoles, aligned along the space-fixed $\hat{z}$-axis by an external field, has been successfully modeled by the pseudo-potential $V_{pp}(\vec{r})$ [@yi00], $$\begin{aligned} \label{eq_dipole} V_{pp}(\vec{r})= \frac{2 \pi \hbar^2}{\mu} a_{00} \delta(\vec{r})+ d^2 \frac{1-3 \cos^2 \theta}{r^3}.\end{aligned}$$ Here, $\mu$ denotes the reduced mass of the two-dipole system, $d$ the dipole moment, and $\theta$ the angle between $\hat{z}$ and the relative distance vector $\vec{r}$. The $s$-wave scattering length $a_{00}$ depends on both the short- and long-range parts of the true interaction potential. The second term on the right hand side of Eq. (\[eq\_dipole\]) couples angular momentum states with $l=l'$ ($l >0$) and $|l - l'| = 2$ (any $l,l'$). For identical fermions, $s$-wave scattering is absent and the interaction is described, assuming the long-range dipole-dipole interaction is dominant, by the second term on the right hand side of Eq. (\[eq\_dipole\]). Our goal in this paper is to determine the eigenequation of two identical bosonic dipoles and two identical fermionic dipoles under external spherically harmonic confinement with angular trapping frequency $\omega$ analytically. The Schrödinger equation for the relative position vector $\vec{r}$ reads $$\begin{aligned} \label{eq_se} [H_0 + V_{int}(\vec{r}) ] \psi(\vec{r}) = E \psi (\vec{r}),\end{aligned}$$ where the Hamiltonian $H_0$ of the non-interacting harmonic oscillator is given by $$\begin{aligned} \label{eq_ham} H_0 = -\frac{\hbar^2}{2 \mu} \nabla^2 _{\vec{r}} +\frac{1}{2} \mu \omega^2 r^2.\end{aligned}$$ In Eq. (\[eq\_se\]), $V_{int}(\vec{r})$ denotes the interaction potential. The pseudo-potential $V_{pp}(\vec{r})$ cannot be used directly in Eq. (\[eq\_se\]) since both parts of the pseudo-potential lead to divergencies. The divergence of the $\delta$-function potential arises from the singular $1/r$ behavior at small $r$ of the spherical Neumann function $n_0(r)$, and can be cured by introducing the regularization operator $\frac{\partial}{\partial r} r$ [@huan57]. Curing the divergence of the long-ranged $1/r^3$ term of $V_{pp}$ is more involved, since it couples an infinite number of angular momentum states, each of which gives rise to a singularity in the $r \rightarrow 0$ limit. The nature of each of these singularities depends on the quantum numbers $l$ and $l'$ coupled by the pseudo-potential, and hence has to be cured separately for each $l$ and $l'$ combination. In this work, we follow Derevianko [@dere03; @dere05] and cure the divergencies by replacing $V_{pp}(\vec{r})$ with a regularized zero-range potential $V_{pp,reg}(\vec{r})$, which contains [*[infinitely]{}*]{} many terms, $$\begin{aligned} \label{eq_ppreg} V_{pp,reg}(\vec{r}) = \sum_{ll'} V_{ll'}(\vec{r}).\end{aligned}$$ The sum in Eq. (\[eq\_ppreg\]) runs over $l$ and $l'$ even for identical bosons, and over $l$ and $l'$ odd for identical fermions. For $l \ne l'$, $V_{ll'}$ and $V_{l'l}$ are different and both terms have to be included in the sum. In Sec. \[sec\_ho\], we apply the pseudo-potential to systems under spherically symmetric external confinement. For these systems, the projection quantum number $m$ is a good quantum number, i.e., the energy spectrum for two interacting dipoles under spherically symmetric confinement can be solved separately for each allowed $m$ value. Consequently, a separate pseudo-potential can be constructed for each $m$ value. In the following, we restrict ourselves to systems with vanishing projection quantum number $m$; the generalization of the pseudo-potential to general $m$ is discussed at the end of this section. The $V_{ll'}$ are defined through their action on an arbitrary $\vec{r}$-dependent function $\Phi(\vec{r})$ [@dere03; @dere05], $$\begin{aligned} \label{eq_llprime} V_{ll'}(\vec{r}) \Phi(\vec{r}) = g_{ll'} \frac{\delta(r)}{r^{l'+2}} Y_{l'0}(\theta,\phi) \times \nonumber \\ \left[ \frac{\partial^{2l+1}}{\partial r^{2l+1}} r^{l+1} \int Y_{l0}(\theta,\phi) \Phi(\vec{r}) d \Omega \right]_{r \rightarrow 0}\end{aligned}$$ with $$\begin{aligned} \label{eq_ppregstrength} g_{ll'} = \frac{\hbar^2}{2 \mu} \frac{a_{ll'}}{k^{l+l'}} \frac{(2l+1)!! (2l' +1)!!}{(2l+1)!},\end{aligned}$$ where $k$ denotes the relative wave vector, $k=\sqrt{2 \mu E/\hbar^2}$, and the $a_{ll'}$ generalized scattering lengths. Since we are restricting ourselves to $m=0$, the $V_{ll'}$ are written in terms of the spherical harmonics $Y_{lm}$ with $m=0$. When applying the above pseudo-potential we treat a large number of terms in Eq. (\[eq\_ppreg\]), and do not terminate the sum after the first three terms as done in Refs. [@dere03; @dere05; @yi04]. We note that the non-Hermiticity of $V_{pp,reg}$ does not lead to problems when determining the energy spectrum; however, great care has to be taken when calculating, e.g., structural expectation values [@reic06]. To understand the functional form of the zero-range pseudo-potential defined in Eqs. (\[eq\_ppreg\]) through (\[eq\_ppregstrength\]), let us first consider the piece of Eq. (\[eq\_llprime\]) in square brackets. If we decompose the incoming wave $\Phi(\vec{r})$ into partial waves, $$\begin{aligned} \Phi(\vec{r}) = \sum_{n_il_im_i} c_{n_il_im_i} Q_{n_il_i}(r) Y_{l_im_i}(\theta,\phi),\end{aligned}$$ where the $c_{n_il_im_i}$ denote expansion coefficients and the $Q_{n_il_i}$ radial basis functions, the spherical harmonic $Y_{l0}$ in the integrand of $V_{ll'}$ acts as a projector or filter. After the integration over the angles, only those components of $\Phi(\vec{r})$ that have $l_i=l$ and $m_i=0$ survive. The operator $\frac{\partial^{2l+1}}{\partial r^{2l+1}} r^{l+1}$ in Eq. (\[eq\_llprime\]) is designed to then first cure the $r^{-l-1}$ divergencies of the $Q_{n_il}$, which arise in the $r \rightarrow 0$ limit, and to then second “extract” the coefficients of the regular part of the $Q_{n_il}(r)$ that go as $r^{l}$ [@huan57]. Alltogether, this shows that the square bracket in Eq. (\[eq\_llprime\]) reduces to a constant when the $r \rightarrow 0$ limit is taken. To understand the remaining pieces of the pseudo-potential, we multiply Eq. (\[eq\_llprime\]) from the left with $Q_{n_ol_o}^* Y^*_{l_om_o}$ and integrate over all space. The spherical harmonic $Y_{l'0}$ in Eq. (\[eq\_llprime\]) then ensures that the integral is only non-zero when $l'=l_o$ and $m_o=0$. When performing the radial integration, the $\delta(r)/r^{l'}$ term ensures that the coefficients of the regular part of the $Q_{n_ol_o}$ that go as $r^{l_o}$ are being extracted (note that the remaining $1/r^2$ term cancels the $r^2$ in the volume element). Alltogether, the analysis outlined in the previous paragraph shows that the functional form of $V_{ll'}$ ensures that the divergencies of the radial parts of the incoming and outgoing wave is cured in the $r \rightarrow 0$ limit and that the $l$th component of the incoming wave is scattered into the $l'$th partial wave. The sum over all $l$ and $l'$ values in Eq. (\[eq\_ppreg\]) guarantees that any state with quantum number $l$ can be coupled to any state with quantum number $l'$, provided the corresponding generalized scattering length $a_{ll'}$ is non-zero. We note that the regularized pseudo-potential given by Eqs. (\[eq\_ppreg\]) through (\[eq\_ppregstrength\]) is only appropriate if the external confining potential in Eq. (\[eq\_ham\]) has spherical symmetry [@idzi06]. Generalizations of the above zero-range pseudo-potential, aimed at treating interacting dipoles under elongated confinement, require the regularization scheme to be modified to additionally cure divergencies of cylindrically symmetric wave functions. These extensions will be subject of future studies. We now discuss the generalized scattering lengths $a_{ll'}$, which determine the scattering strengths of the $V_{ll'}$. The $a_{ll'}$ have units of length and are defined through the K-matrix elements $K_{lm}^{l'm'}$ [@newt], $$\begin{aligned} \label{eq_scatt} a_{ll'} = \lim_{k \rightarrow 0} \frac{-K_{l0}^{l'0}(k)}{k}\end{aligned}$$ for $m=0$. The scattering lengths $a_{ll'}$ and $a_{l'l}$ are identical because the K-matrix is symmetric. In general, the scattering lengths $a_{ll'}$ have to be determined from the K-matrix elements for the “true” interaction potential, which contains the long-range dipolar and a short-ranged repulsive part, of two interacting dipoles. As discussed further in Sec. \[sec\_ho\], an approach along these lines is used to obtain the squares shown in Fig. \[fig3\]. Alternatively, it has been shown that the K-matrix elements (except for $K_{00}^{00}$, see below) for realistic potentials, such as for the Rb-Rb potential in a strong electric field [@yi00] or an OH-OH model potential [@rone06], are approximated with high accuracy by the K-matrix elements for the dipolar potential only, calculated in the first Born approximation. Applying the Born approximation to the second term on the right hand side of Eq. (\[eq\_dipole\]), we find for $m=0$ and $l=l'$ ($l \ge 1$) $$\begin{aligned} \label{eq_born1} a_{ll}= -\frac{2D_*}{(2l-1)(2l+3)},\end{aligned}$$ and for $m=0$ and $l=l' + 2$ $$\begin{aligned} \label{eq_born2} a_{l,l-2} = -\frac{D_*}{(2l-1) \sqrt{(2l+1)(2l-3)}}.\end{aligned}$$ For $l'=2$ and $l=0$, e.g., Eq. (\[eq\_born2\]) reduces to $a_{20}=-D_*/(3 \sqrt{5})$, in agreement with Ref. [@dere03]. The scattering lengths $a_{l-2,l}$ are equal to $a_{l,l-2}$, and all other generalized scattering lengths are zero. In Eqs. (\[eq\_born1\]) and (\[eq\_born2\]), $D_*$ denotes the dipole length, $D_* = \mu d^2/\hbar^2$. All non-zero scattering lengths $a_{ll'}$ are negative, depend on $l$ and $l'$, and are directly proportional to $d^2$. Furthermore, for fixed $D_*$, the absolute value of the non-zero $a_{ll'}$ decreases with increasing angular momentum quantum number $l$, indicating that the coupling between different angular momentum channels decreases with increasing $l$. However, this decrease is quite slow and, in general, an accurate description of the two-dipole system requires that the convergence with increasing $l_{max}$ be assessed carefully. One can now show readily that the K-matrix elements $K_{l0}^{l'0}$ of $V_{pp,reg}$, calculated in the first Born approximation, with $a_{ll'}$ given by Eqs. (\[eq\_born1\]) and (\[eq\_born2\]) coincide with the K-matrix elements $K_{l0}^{l'0}$ of $V_{pp}$. This provides a simple check of the zero-range pseudo-potential construction and proofs that the prefactors of $V_{ll'}$ are correct. In turn, this suggests that the applicability regimes of $V_{pp}$ and $V_{pp,reg}$ are comparable, if the generalized scattering lengths $a_{ll'}$ used to quantify the scattering strengths of $V_{ll'}$ are approximated by Eqs. (\[eq\_born1\]) and (\[eq\_born2\]). The applicability regime of $V_{pp,reg}$ may, however, be larger than that of $V_{pp}$ if the full energy-dependent K-matrix of a realistic potential is used instead. To generalize the zero-range pseudo-potential defined in Eqs. (\[eq\_ppreg\]) through (\[eq\_ppregstrength\]) for projection quantum numbers $m=0$ to any $m$, only a few changes have to be made. In Eq. (\[eq\_llprime\]), the spherical harmonics $Y_{l0}$ have to be replaced by $Y_{lm}$, and the generalized scattering lengths have to be defined through $\lim_{k \rightarrow 0} -K_{lm}^{l'm'}/k$. Correspondingly, Eqs. (\[eq\_born1\]) and (\[eq\_born2\]) become $m$-dependent. Two dipoles under external confinement {#sec_ho} ====================================== Section \[sec\_hoA\] derives the implicit eigenequation for two dipoles interacting through the pseudo-potential under external harmonic confinement and Section \[sec\_hoB\] analyzes the resulting eigen spectrum. Derivation of the eigenequation {#sec_hoA} ------------------------------- To determine the eigen energies of two aligned dipoles with $m=0$ under spherical harmonic confinement interacting through the zero-range potential $V_{pp,reg}$, we expand the eigenfunctions $\Psi(\vec{r})$ in terms of the orthonormal harmonic oscillator eigen functions $R_{n_il_i}Y_{l_i0}$, $$\begin{aligned} \label{eq_expansion} \Psi(\vec{r}) = \sum_{n_il_i} c_{n_il_i} R_{n_il_i}(r) Y_{l_i0}(\theta,\phi).\end{aligned}$$ The pseudo-potential $V_{pp,reg}$ enforces the proper boundary condition of $\Psi(\vec{r})$ at $r=0$, and thus determines the expansion coefficients $c_{n_il_i}$. To introduce the key ideas we first consider $s$-wave interacting particles [@busc98], for which the pseudo-potential reduces to a single term, and then consider the general case, in which the pseudo-potential contains infinitely many terms. Including only the term with $l$ and $l'=0$ in Eq. (\[eq\_ppreg\]), the Schrödinger equation becomes, $$\begin{aligned} \label{eq_swave} \sum_{n_il_i} c_{n_il_i} (E_{n_il_i} - E + V_{00}) R_{n_il_i}(r) Y_{l_i0}(\theta,\phi) =0,\end{aligned}$$ where the $E_{n_il_i}$ denote the eigenenergies of the non-interacting harmonic oscillator, $$\begin{aligned} \label{eq_hoen} E_{n_il_i}=\left( 2 n_i + l_i + \frac{3}{2} \right) \hbar \omega.\end{aligned}$$ In what follows, it is convenient to express the energy $E$ of the interacting system in terms of a non-integer quantum number $\nu$, $$\begin{aligned} \label{eq_nu} E = \left( 2 \nu + \frac{3}{2} \right) \hbar \omega.\end{aligned}$$ Multiplying Eq. (\[eq\_swave\]) from the left with $R^*_{n_ol_o}Y^*_{l_o0}$ with $l_o>0$ and integrating over all space, we find that the $c_{n_il_i}$ with $l_i>0$ vanish. This can be understood readily by realizing that the $s$-wave pseudo-potential $V_{00}$, as discussed in detail in Sec. \[sec\_pp\], only couples states with $l=l'=0$. To determine the expansion coefficients $c_{n_i0}$, we multiply Eq. (\[eq\_swave\]) from the left with $R^*_{n_o0}Y^*_{00}$ and integrate over all space. This results in $$\begin{aligned} \label{eq_swave1} c_{n_o0} (2 n_o - 2 \nu) \hbar \omega + R_{n_o0}^*(0) g_{00} B_0 =0,\end{aligned}$$ where $B_0$ denotes the result of the square bracket in Eq. (\[eq\_llprime\]), $$\begin{aligned} \label{eq_swave2} B_0 = \left[ \frac{\partial}{\partial r} \left( r \sum_{n_i=0}^{\infty} c_{n_i0} R_{n_i0}(r) \right) \right]_{r \rightarrow 0}.\end{aligned}$$ Note that $B_0$ is constant and independent of $n_i$. In Eq. (\[eq\_swave1\]), the $r$-independent term $R_{n_o0}^*(0)$ arises from the radial integration over the $\delta$-function of the pseudo-potential. If we solve Eq. (\[eq\_swave1\]) for $c_{n_o0}$ and plug the result into Eq. (\[eq\_swave2\]), the unknown constant $B_0$ cancels and we obtain an implicit eigenequation for $\nu$, $$\begin{aligned} 1 = g_{00} \left[ \frac{\partial}{\partial r} \left( r \sum_{n_i=0}^{\infty} \frac{R_{n_i0}^*(0) R_{n_i0}(r)} {(2 \nu - 2 n_i) \hbar \omega} \right) \right]_{r \rightarrow 0}.\end{aligned}$$ Using Eqs. (\[eq\_app1\]) and (\[eq\_app6\]) from the Appendix to simplify the term in square brackets, we obtain the well-known implicit eigenequation for two particles interacting through the $s$-wave pseudo-potential under spherical harmonic confinement [@busc98], $$\begin{aligned} \label{eq_swavefinal} \frac{\Gamma \left( \frac{-E}{2 \hbar \omega}+\frac{1}{4} \right) } {2 \Gamma \left( \frac{-E}{2 \hbar \omega} + \frac{3}{4} \right)} - \frac{a_{00}}{a_{ho}} =0.\end{aligned}$$ Here, $a_{ho}$ denotes the harmonic oscillator length, $a_{ho}=\sqrt{\hbar/(\mu \omega)}$. The derivation of the implicit eigenequation for two dipoles under external harmonic confinement interacting through the pseudo-potential with infinitely many terms proceeds analogously to that outlined above for the $s$-wave system. The key difference is that each $V_{ll'}$ term in Eq. (\[eq\_llprime\]) with $l \ne l'$ couples states with different angular momenta, resulting in a set of coupled equations for the expansion coefficients $c_{n_il_i}$. However, since $V_{pp,reg}$ for dipolar systems couples only angular momentum states with $|l-l'| \le 2$ \[see, e.g., the discussion at the beginning of Sec. \[sec\_pp\] and around Eqs. (\[eq\_born1\]) and (\[eq\_born2\])\], the coupled equations can, as we outline in the following, be solved analytically by including successively more terms in $V_{pp,reg}$. To start with, we plug the expansion given in Eq. (\[eq\_expansion\]) into Eq. (\[eq\_se\]), where the interaction potential $V_{int}$ is now taken to be the pseudo-potential $V_{pp,reg}$ with infinitely many terms. To obtain the general equation for the expansion coefficients $c_{n_il_i}$, we multiply as before from the left with $R^*_{n_ol_o}Y^*_{l_o0}$ and integrate over all space, $$\begin{aligned} \label{eq_general1} c_{n_ol_o} (2 n_o + l_o - 2\nu) \hbar \omega + \left [\frac{R^*_{n_ol_o}(r)}{r^{l_o}} \right]_{r \rightarrow 0} \times \nonumber \\ \left[ g_{l_o-2,l_o} B_{l_o-2} + g_{l_ol_o} B_{l_o}+ g_{l_o+2,l_o} B_{l_o+2} \right]=0.\end{aligned}$$ Here, the $B_{l_o-2}$, $B_{l_o}$ and $B_{l_o+2}$ denote constants that are independent of $n_i$, $$\begin{aligned} \label{eq_general2} B_{l_o} = \left[ \frac{\partial^{2l_o+1}}{\partial r^{2l_o+1}} \left\{ r^{l_o+1} \left( \sum_{n_i=0}^{\infty} c_{n_il_o} R_{n_il_o}(r) \right) \right\} \right]_{r \rightarrow 0}.\end{aligned}$$ The three terms in the square bracket in the second line of Eq. (\[eq\_general1\]) arise because the $V_{l'-2,l'}$, $V_{l'l'}$ and $V_{l'+2,l'}$ terms in the pseudo-potential $V_{pp,reg}$ couple the state $R^*_{n_ol_o}Y^*_{l_o0}$, for $l'=l_o$, with three components of the expansion for $\Psi$, Eq. (\[eq\_expansion\]). Importantly, the constants $B_{l_o-2}$, $B_{l_o}$ and $B_{l_o+2}$, defined in Eq. (\[eq\_general2\]), depend on the quantum numbers $l_o-2$, $l_o$ and $l_o+2$, respectively, which implies that Eq. (\[eq\_general1\]) defines a set of infinitely many coupled equations that determine, together with Eq. (\[eq\_general2\]), the expansion coefficients $c_{n_il_i}$. Notice that Eqs. (\[eq\_general1\]) and (\[eq\_general2\]) coincide with Eqs. (\[eq\_swave1\]) and (\[eq\_swave2\]) if we set $l_o=0$ and $g_{ll'}=0$ if $l$ or $l' > 0$. We now illustrate how Eqs. (\[eq\_general1\]) and (\[eq\_general2\]) can be solved for identical bosons, i.e., in the case where $l$ and $l'$ are even (the derivation for identical fermions proceeds analogously). Our strategy is to solve these equations by including successively more terms in the coupled equations, or equivalently, in the pseudo-potential. As discussed above, if $a_{00}$ is the only non-zero scattering length, the eigenenergies are given by Eq. (\[eq\_swavefinal\]). Next, we also allow for non-zero $a_{20}$, $a_{02}$ and $a_{22}$, i.e., we consider $l$ and $l' \le 2$ in Eq. (\[eq\_ppreg\]). In this case, the coefficients $c_{n_i0}$ and $c_{n_i2}$ are non-zero and coupled, but all $c_{n_il_i}$ with $l_i>2$ are zero. Using the expressions for $B_0$ and $B_2$ given in Eq. (\[eq\_general2\]), we decouple the equations. Finally, using Eqs. (\[eq\_app1\]) and (\[eq\_app6\]) from the Appendix, the eigenequation can be compactly written as $$\begin{aligned} \label{eq_uptotwo} t_0+ \frac{q_2}{t_2}=0,\end{aligned}$$ where $$\begin{aligned} \label{eq_tl} t_l = \frac{\Gamma( \frac{-E}{2 \hbar \omega} + \frac{1}{4} - \frac{l}{2})} {2^{2l+1} \Gamma(\frac{-E}{2 \hbar \omega} +\frac{3}{4} + \frac{l}{2} )} - (-1)^l \frac{a_{ll}}{k^{2l} a_{ho}^{2l+1}},\end{aligned}$$ and $$\begin{aligned} \label{eq_ql} q_l = -\frac{a_{l-2,l}^2}{k^{4l-4}a_{ho}^{4l-2}}.\end{aligned}$$ Equation (\[eq\_uptotwo\]) can be understood as follows. If only $a_{00}$ is non-zero, it reduces to $t_0=0$, in agreement with Eq. (\[eq\_swavefinal\]). If only $a_{00}$, $a_{02}$ and $a_{20}$ are non-zero, Eq. (\[eq\_uptotwo\]) remains valid if $a_{22}$ in $t_2$ is set to zero. This shows that the term $q_2$ and the first term on the right hand side of $t_2$ arise due to the coupling between states with angular momenta $0$ and $2$. The second term of $t_2$, in contrast, arises due to a non-zero $a_{22}$. Finally, for non-zero $a_{00}$ and $a_{22}$ but vanishing $a_{20}$ and $a_{02}$, Eq. (\[eq\_uptotwo\]) reduces to $t_0t_2=0$. In this case, we recover the eigenequations $t_0=0$ for $s$-wave interacting particles [@busc98] and $t_2=0$ for $d$-wave interacting particles [@stoc04]. We now consider $l$ and $l'$ values with up to $l_{max}=4$ in Eq. (\[eq\_ppreg\]), i.e., we additionally allow for non-zero $a_{24}$, $a_{42}$ and $a_{44}$, and discuss how the solution changes compared to the $l_{max}=2$ case. The equation for the expansion coefficients $c_{n_i0}$ remains unchanged while that for $c_{n_i2}$ is modified. Furthermore, the expansion coefficients $c_{n_i4}$ are no longer zero. Consequently, we have three coupled equations, which can be decoupled, resulting in the following implicit eigenequation, $t_0+q_2/(t_2+q_4/t_4)=0$. In analogy to the $l_{max}=2$ case, the $q_4$ term and the first part on the right hand side of the $t_4$ term arise due to the “off-diagonal” scattering lengths $a_{24}$ and $a_{42}$, and the second term of $t_4$ arises due to the “diagonal” scattering length $a_{44}$. Next, let us assume that we have found the implicit eigenequation for the case where we include terms in Eq. (\[eq\_ppreg\]) with $l$ and $l'$ up to $l_{max}-2$. If we now include terms with $l$ and $l'$ up to $l_{max}$, only the equations for the expansion coefficients $c_{n_o l_o}$ with $l_o = l_{max}-2$ and $l_{max}$ change; those for the expansion coefficients $c_{n_ol_o}$ with $l_o \le l_{max}-4$ remain unchanged. This allows the $l_{max}/2+1$ coupled equations for the expansion coefficients to be decoupled analytically using the results already determined for the case where $l$ and $l'$ go up to $l_{max}-2$. Following this procedure, we find the following implicit eigenequation $$\begin{aligned} \label{eq_eigen} T_{l_{max}}=0,\end{aligned}$$ where $T_{l_{max}}$ itself can be written as a continued fraction. For identical bosons we find, $$\begin{aligned} \label{eq_tlmax} T_{l_{max}} = t_0 + \frac{q_2}{t_2 + \frac{q_4}{t_4 + \cdots +\frac{q_{l_{max}}}{t_{l_{max}}}}}.\end{aligned}$$ Taking $l_{max} \rightarrow \infty$ gives the eigenequation for two identical bosons under spherical harmonic confinement interacting through $V_{pp,reg}$ with infinitely many terms. For two identical fermions, Eqs. (\[eq\_tl\]) through (\[eq\_tlmax\]) remain valid if the subscripts $0,2,\cdots$ in Eq. (\[eq\_tlmax\]) are replaced by $1,3,\cdots$. The derived eigenequation reproduces the eigenenergies in the known limits. For the non-interacting case (all $a_{ll'}=0$), the eigenenergies coincide with the eigenenergies of the harmonic oscillator, i.e., $E_{nl}= ( 2n + l + 3/2 ) \hbar \omega$, where $n=0,1,2,\cdots$ and $l=0,2,4,\cdots$ (in the case of identical bosons) and $l=1,3,\cdots$ (in the case of identical fermions). The $k$th levels, with energy $(2k+ 3/2) \hbar \omega$ for bosons and $(2k+5/2) \hbar \omega$ for fermions, has a degeneracy of $k+1$, $k=0,1,\cdots$. Non-vanishing $a_{ll'}$ lead to a splitting of degenerate energy levels but leave the number of energy levels unchanged. If $a_{ll}$ is the only non-zero scattering length, the eigenequation reduces to that obtained for spherically symmetric pseudo-potentials with partial wave $l$ [@stoc04]. Analysis of the energy spectrum {#sec_hoB} ------------------------------- This section analyses the implicit eigenequation, Eq. (\[eq\_eigen\]), derived in the previous section for the zero-range pseudo-potential for $m=0$ and compares the resulting energy spectrum with that obtained for a shape-dependent model potential. The implicit eigenequation, Eq. (\[eq\_eigen\]), can be solved readily numerically by finding its roots in different energy regions. The solutions of the Schrödinger equation for the shape-dependent model potential are otained by expanding the eigenfunctions on a B-spline basis. Lines in Figs. \[fig1\](a) and (b) ![ Relative eigenenergies $E$ for (a) two identical bosonic dipoles and (b) two identical fermionic dipoles interacting through $V_{pp,reg}$ \[using $a_{00}=0$ in (a)\] under spherical harmonic confinement as a function of $D_*/a_{ho}$. The line style indicates the predominant character of the corresponding eigenstates. In (a), a solid line refers to $l \approx 0$, a dashed line to $l \approx 2$, and a dotted line to $l \approx 4$; in (b), a solid line refers to $l \approx 1$, a dashed line to $l \approx 3$, and a dotted line to $l \approx 5$. []{data-label="fig1"}](fig1.ps){width="8cm"} show the eigenenergies obtained by solving Eq. (\[eq\_eigen\]) for two identical bosons and two identical fermions, respectively, interacting through $V_{ps,reg}(\vec{r})$ under external spherically symmetric harmonic confinement as a function of the dipole length $D_*$. In both panels, we assume that the interaction between the two dipoles is purely dipolar, i.e., in Fig. \[fig1\](a) we set $a_{00}=0$. The other scattering lengths $a_{ll'}$ are approximated by Eqs. (\[eq\_born1\]) and (\[eq\_born2\]). Interestingly, for identical bosons, the lowest gas-like level, which starts at $E=1.5 \hbar \omega$ for $D_*=0$, increases with increasing $D_*$. For identical fermions, in contrast, the lowest gas-like state decreases with increasing $D_*$. In addition to obtaining the eigenenergies themselves, the pseudo-potential treatment allows the spectrum to be classified in terms of angular momentum quantum numbers. To this end, we solve the implicit eigenequation, Eq. (\[eq\_eigen\]), for increasing $l_{max}$, and monitor how the energy levels shift as additional angular momenta are included in $V_{pp,reg}$. Since a level with approximate quantum number $l$ changes only little as larger angular momentum values are included in the pseudo-potential, this analysis reveals the predominant character of each energy level. In Fig. \[fig1\](a), the eigenfunctions of energies shown by solid, dashed and dotted lines have predominantly $l=0$, 2 and 4 character, respectively. In Fig. \[fig1\](b), the eigenfunctions of energies shown by solid, dashed and dotted lines have predominantly $l=1$, 3 and 5 character, respectively. We find that the lowest excitation frequency between states with predominantly $l=0$ \[$l=1$\] character, increases \[decreases\] for identical bosons \[fermions\] with increasing $D_*$. These predictions can be verified directly experimentally. To assess the accuracy of the developed zero-range pseudo-potential treatment, we consider two interacting bosons with non-vanishing $s$-wave scattering length $a_{00}$. We imagine that the dipole moment of two identical polarized bosonic polar molecules is tuned by an external electric field. As the dipole moment $d$ is tuned, the $s$-wave scattering length $a_{00}$, which depends on the short-range and the long-range physics of the “true” interaction potential, changes. To model this situation, we solve the two-body Schrödinger equation, Eq. (\[eq\_se\]), numerically for a shape-dependent model potential with hardcore radius $b$ and long-range dipolar tail. In this case, $V_{int}$ is given by $$\begin{aligned} \label{eq_model} V_{model}(\vec{r}) = \left\{ \begin{array}{ll} d^2\frac{1 - 3 \cos^2 \theta} {r^3} & \mbox{if $r \ge b$}\\ \infty & \mbox{if $r < b$} \end{array} \right. .\end{aligned}$$ For $d=0$, the $s$-wave scattering length $a_{00}$ for $V_{model}$ is given by $b$. As the dipole length $D_*$ increases, $a_{00}$ goes through zero, and becomes negative. Just when the two-body potential supports a new bound state, $a_{00}$ goes through a resonance and becomes large and positive. As $D_*$ increases further, $a_{00}$ decreases. This resonance structure repeats itself with increasing $D_*$ (see Fig. 1 of Ref. [@bort06]; note, however, that the lengths $a_{ho}$ and $D_*$ defined throughout the present work differ from those defined in Ref. [@bort06]). For the model potential $V_{model}$, $a_{00}$ depends on the ratio between the short-range and long-range length scales, i.e., on $b/D_*$. To compare the pseudo-potential energies and the energies for the model potential, we fix $b$ and calculate $a_{00}$ for each $D_*$ considered. The dipole-dependent $s$-wave scattering length is then used in the zero-range pseudo-potential $V_{pp,reg}$. The other scattering lengths are, as before, approximated by the expressions given in Eqs. (\[eq\_born1\]) and (\[eq\_born2\]). Solid lines in Fig. \[fig2\](a) and (b) show the eigenenergies ![ Panel (a) shows the relative energies $E$ for two aligned identical bosonic dipoles under external spherical harmonic confinement as a function of $D_*/a_{ho}$. Solid lines show the numerically determined energies obtained using $V_{model}$ with $b=0.0097 a_{ho}$. Crosses show the energies obtained using $V_{pp,reg}$ with essentially infinitely many terms, and $a_{00}$ calculated for $V_{model}$. Panel (b) shows a blow-up of the energy region around $E \approx 5.5 \hbar \omega$. Note that the horizontal axis in (a) and (b) are identical. []{data-label="fig2"}](fig2.ps){width="8cm"} obtained for $V_{model}$ as a function of $D_*$. Crosses show the eigenenergies obtained for $V_{pp,reg}$ using a value of $l_{max}$ that results in converged eigenenergies. The overview spectrum shown in Fig. \[fig2\](a) shows that one of the energy levels dives down to negative energies close to that $D_*$ value at which the two-body potential $V_{model}$ supports a new bound state. The blow-up, Fig. \[fig2\](b), around $E \approx 5.5 \hbar \omega$ shows excellent agreement between the energies obtained using $V_{pp,reg}$ (crosses) and those obtained using $V_{model}$ (solid lines); the maximum deviation for the energy range shown is 0.05 %. As before, we can assign approximate quantum numbers to each energy level. At $D_* \ll a_{ho}$, the three energy levels around $E \approx 5.5 \hbar \omega$ have, from bottom to top, approximate quantum numbers $l=2$, 4 and 0. After two closely spaced avoided crossings around $D_* \approx 0.025 a_{ho}$, the assignment changes to $l=0$, 2 and 4 (again, from bottom to top). If the maximum angular momentum $l_{max}$ of the pseudo-potential is set to 2, the energy level with approximate quantum number $l=4$ would be absent entirely. This illustrates that a complete and accurate description of the energy spectrum requires the use of a zero-range pseudo-potential with infinitely many terms. The energy of a state with approximate quantum number $l$ requires $l_{max}$ to be at least $l$ for the correct degeneracy be obtained and at least $l+2$ for a quantitative description. The sequence of avoided crossings at $D_* \approx 0.025 a_{ho}$ suggests an interesting experiment. Assume that the system is initially, at small electric field (i.e., small $D_*/a_{ho}$), prepared in the excited state with angular momentum $l \approx 0$ and $E \approx 5.52 \hbar \omega$. The electric field is then slowly swept across the first broad avoided crossing at $D_* \approx 0.019 a_{ho}$ to transfer the population from the state with $l \approx 0$ to the state with $l \approx 2$. We then suggest to sweep quickly across the second narrower avoided crossing at $D_* \approx 0.028 a_{ho}$ (the ramp speed must be chosen so minimize population transfer from the state with $l \approx 2$ to the state with $l \approx 4$). As in the case of $s$-wave scattering only [@dunn04], the time-dependent field sequence has to be optimized to obtain maximal population transfer. The proposed scheme promises to provide an efficient means for the transfer of population between states with different angular momenta and for quantum state engineering. Figure \[fig2\] illustrates that the pseudo-potential treatment reproduces the eigenenergies of the shape-dependent model potential $V_{model}$. To further assess the validity of the pseudo-potential treatment, we now consider two interacting bosonic dipoles for which the dipolar interaction is dominant, i.e., we consider $a_{00}=0$. For $V_{model}$ with $b = 0.0031 a_{ho}$, we determine a set of $D_*$ values at which $a_{00}=0$. Note that the number of bound states with predominantly $s$-wave character increases by one for each successively larger $D_*$. Crosses in Figs. \[fig3\](a)-(c) show the eigenenergies ![ Crosses show the relative eigenenergies $E$ as a function of $D_*/a_{ho}$ for two identical bosons with $a_{00}=0$ interacting through $V_{model}$ with $b = 0.0031 a_{ho}$ in three different energy regions. Lines show $E$ for two identical bosons with $a_{00}=0$ interacting through $V_{ps,reg}$ with $a_{ll'}$ given by Eq. (\[eq\_born1\]) and (\[eq\_born2\]). As in Fig. \[fig1\](a) solid, dashed and dotted lines show the energies of levels characterized by approximate quantum numbers $l \approx 0$, 2 and 4. The agreement between the crosses and the lines is good at small $D_*/a_{ho}$ but less good at larger $D_*/a_{ho}$. Squares show the eigenenergies obtained for the energy-dependent pseudo-potential at $D_*=0.242a_{ho}$; the agreement between the squares and the crosses is excellent, illustrating that usage of the energy-dependent K-matrix greatly enhances the applicability regime of $V_{pp,reg}$. []{data-label="fig3"}](fig3.ps){width="8cm"} for $V_{model}$ with $a_{00}=0$ as a function of $D_*$ in the energy ranges around $1.5$, $3.5$ and $5.5 \hbar \omega$. For comparison, lines show the eigenenergies obtained for the regularized pseudo-potential with $a_{00}=0$. As in Fig. \[fig1\], the linestyle indicates the predominant character of the energy levels (solid line: $l \approx 0$; dashed line: $l \approx 2$; and dotted line: $l \approx 4$). The agreement between the energies obtained for the pseudo-potential with $a_{ll'}$ given by Eqs. (\[eq\_born1\]) and (\[eq\_born2\]) and for the model potential for small $D_*$ is very good, thus validating the applicability of the pseudo-potential treatment. The agreement becomes less good, however, as $D_*$ increases. This can be explained readily by realizing that the dipole length $D_*$ approaches the harmonic oscillator length $a_{ho}$. In general, the description of confined particles interacting through zero-range pseudo-potentials is justified if the characteristic lengths of the two-body potential are smaller than the characteristic length of the confining potential. For example, in the case of $s$-wave interactions only, the van der Waals length has to be smaller than the oscillator length [@blum02; @bold02]. The model potential $V_{model}$ is characterized by a short-range length scale, the hardcore radius $b$, and the dipole length $D_*$; in Fig. \[fig3\], it is the relatively large value of $D_*/a_{ho}$ that leads, eventually, to a break-down of the pseudo-potential treatment. As in the case of spherical interactions, the break-down can be pushed to larger $D_*$ values by introducing energy-dependent generalized scattering lengths $a_{ll'}(k)$, defined through $-K_{l0}^{l'0}(k)/k$ for $m=0$, and by then solving the eigenequation, Eq. (\[eq\_eigen\]), self-consistently [@blum02; @bold02]. Figure \[fig4\] shows three selected scattering lengths $a_{ll'}(k)$ for the model potential $V_{model}$ with $D_*=78.9b$ as a function of energy. This two-body potential supports eight bound states with projection quantum number $m=0$, which have predominantly $s$-wave character. Both energy and length in Fig. \[fig4\] are expressed in oscillator units to allow for direct comparison with the data shown in Fig. \[fig3\]. The scattering length $a_{00}(k)$, shown by a solid line in Fig. \[fig4\], is zero at zero energy and increases with increasing energy. Both $a_{20}(k)$ (dashed line) and $a_{22}(k)$ (dash-dotted line) are negative. Their zero-energy values coincide with those calculated in the Born approximation (horizontal dotted lines). Using these energy-dependent $a_{ll'}(k)$ to parametrize the strengths of the pseudo-potential and solving the eigenequation, Eq. (\[eq\_eigen\]), self-consistently, we obtain the squares in Fig. \[fig3\]. The energies for $V_{pp,reg}$ with [*[energy-dependent]{}*]{} $a_{ll'}$ (squares) are in much better agreement with the energies obtained for the model potential (crosses) than the energies obtained using the [*[energy-independent]{}*]{} $a_{ll'}$ to parametrize the pseudo-potential (lines). ![Energy-dependent scattering lengths $a_{00}(k)$ (solid line), $a_{20}(k)$ (dashed line) and $a_{22}(k)$ (dash-dotted line) for the model potential $V_{model}$ with $D_*=78.9b$ as a function of the relative energy $E$. In oscillator units, $V_{model}$ is characterized by $b=0.0031 a_{ho}$ and $D_*=0.242 a_{ho}$. For comparison, horizontal dotted lines show the energy-independent scattering lengths $a_{22}$, Eq. (\[eq\_born1\]), and $a_{20}$, Eq. (\[eq\_born2\]), calculated in the first Born approximation. []{data-label="fig4"}](fig4.ps){width="8cm"} This suggests that the applicability regime of the regularized zero-range pseudo-potential can be extended significantly by introducing energy-dependent scattering lengths. Since the proper treatment of resonant interactions within the regularized zero-range pseudo-potential requires that the energy-dependence of the generalized scattering lengths be included, future work will address this issue in more depth. Summary {#sec_conclusion} ======= This paper applies a zero-range pseudo-potential treatment to describe two interacting dipoles under external spherically harmonic confinement. Section \[sec\_pp\] introduces the regularized zero-range pseudo-potential $V_{pp,reg}$ used in this work, which was first proposed by Derevianko [@dere03; @dere05]. Particular emphasis is put on developing a simple interpretation of the individual pieces of the pseudo-potential. Furthermore, we clearly establish the connection between $V_{pp,reg}$ and the pseudo-potential $V_{pp}$, which is typically employed within a mean-field framework. We argue that the applicability regime of these two pseudo-potentials is comparable if the scattering strengths of $V_{pp,reg}$, calculated in the first Born approximation, are chosen so as to reproduce those of $V_{pp}$. We then use the regularized zero-range pseudo-potential to derive an implicit eigen equation for two dipoles under external confinement, a system which can be realized experimentally with the aid of optical lattices. In deriving the implicit eigenequation, we again put emphasis on a detailed understanding of how the solution arises, thus developing a greater understanding of the underlying physics. The implicit eigenequation can be solved straightforwardly, and allows for a direct classification scheme of the resulting eigenspectrum. By additionally calculating the eigen energies for two dipoles interacting through a finite range model potential numerically, we assess the applicability of the developed zero-range pseudo-potential treatment. We find good agreement between the two sets of eigenenergies for small $D_*$ and quantify the deviations as $D_*$ increases. Finally, we show that the validity regime of $V_{pp,reg}$ can be extended by parametrizing the scattering strengths of $V_{pp,reg}$ in terms of the energy-dependent K-matrix calculated for a realistic model potential. This may prove useful also when describing resonantly interacting dipoles. At first sight it may seem counterintuitive to replace the [*[long-range]{}*]{} dipolar interaction by a [*[zero-range]{}*]{} pseudo-potential. However, if the length scales of the interaction potential, i.e., the van der Waals length scale characterizing the short-range part and the dipole length characterizing the long-range part of the potential, are smaller than the characteristic length of the trap $a_{ho}$, this approach is justified since the zero-range pseudo-potential is designed to reproduce the K-matrix elements of the “true” interaction potential. This is particularly true if the pseudo-potential is taken to contain infinitely many terms, as done in this work. In summary, this paper determines the eigenenergies of two interacting dipoles with projection quantum number $m=0$. The applied zero-range pseudo-potential treatment is validated by comparing the resulting eigenenergies with those obtained numerically for a shape-dependent model potential. The analysis presented sheds further light on the intricate properties of angle-dependent scattering processes and their description through a regularized zero-range pseudo-potential with infinitely many terms. The calculated energy spectrum may aid on-going experiments on dipolar Bose and Fermi gases. Acknowledgements: KK and DB acknowledge support by the NSF through grant PHY-0555316 and JLB by the DOE. Appendix ======== In this Appendix, we evaluate the following infinite sum, $$\begin{aligned} \label{eq_app1} C_l = \left[ \frac{\partial^{2l+1}}{\partial r^{2l+1}} \left\{ r^{l+1} \sum_{n=0}^{\infty} \frac{ \left[\frac{R^*_{nl}(r)}{r^l}\right]_{r\rightarrow0} R_{nl}(r)} {2 \left(\nu-n-\frac{l}{2} \right) \hbar \omega} \right\} \right]_{r \rightarrow 0}.\end{aligned}$$ Writing the radial harmonic oscillator functions $R_{nl}(r)$ in terms of the Laguerre polynomials $L_n^{(l+1/2)}$, $$\begin{aligned} \label{eq_app2} R_{nl}(r)= \sqrt{\frac{2^{l+2}}{(2l+1)!!\pi^{1/2}L_n^{(l+1/2)}(0) a_{ho}^{3}}} \times \nonumber \\ \exp \left(-\frac{r^2}{2a_{ho}^2} \right) \left( \frac{r}{a_{ho}} \right)^l L_n^{(l+1/2)}(r^2/a_{ho}^2),\end{aligned}$$ we find $$\begin{aligned} \label{eq_app3} \left[\frac{R_{nl}(r)}{r^l}\right]_{r\rightarrow0}= \sqrt{\frac{2^{l+2}L_n^{(l+1/2)}(0)}{(2l+1)!!\pi^{1/2} a_{ho}^{2l+3}}}.\end{aligned}$$ Using Eqs. (\[eq\_app2\]) and (\[eq\_app3\]), the $C_l$ can be rewritten as $$\begin{aligned} \label{eq_app4} C_l= \frac{2^{l+1}}{(2l+1)!!\pi^{1/2} a_{ho}^{2l+3}} \times \nonumber \\ \left[\frac{\partial^{2l+1}}{\partial r^{2l+1}} \left( \exp\left(\frac{-r^2}{2a_{ho}^2}\right) r^{2l+1}\sum_{n=0}^{\infty}{\frac{L_n^{(l+1/2)}\left((\frac{r}{a_{ho}})^2 \right)} {\left(\nu -n-\frac{l}{2} \right) \hbar \omega}} \right) \right]_{r\rightarrow0}.\end{aligned}$$ We evaluate the infinite sum in Eq. (\[eq\_app4\]) using the properties of the generating function [@abranote1], $$\begin{aligned} \label{eq_app5} \sum_{n=0}^{\infty}{\frac{L_n^{(l+1/2)}((r/a_{ho})^2)}{\nu -n-\frac{l}{2}}}= \nonumber \\ -\Gamma(-\nu + l/2)U(-\nu + l/2, l+3/2, (r/a_{ho})^2).\end{aligned}$$ Using Eq. (\[eq\_app5\]) together with the small $r$ behavior of the hypergeometric function $U$ [@abranote2], the expression for the $C_l$ reduces to $$\begin{aligned} \label{eq_app6} C_l = \frac{(-1)^l 2^{2l+2}(2l)!!}{(2l+1)!!} \frac{\Gamma \left(-\nu + \frac{l}{2} \right)} {\Gamma \left(-\nu-\frac{l+1}{2} \right)} \frac{1}{\hbar \omega a_{ho}^{2l+3}}.\end{aligned}$$ [10]{} A. 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ArXiv

--- abstract: 'We introduce a new family of shift spaces — the subordinate shifts. Using subordinate shifts we prove in an elementary way that for every nonnegative real number $t$ there is a shift space with entropy $t$.' author: - 'Marcin Kulczycki, Dominik Kwietniak, and Jian Li' title: Entropy of subordinate shift spaces --- Introduction. ============= Let ${\mathscr{A}}$ be a finite collection of *symbols* which we call an *alphabet*. Typical choices are ${\mathscr{A}}=\{0,1,\ldots,r-1\}$ for some integer $r$ or a set of Roman letters, for example ${\mathscr{A}}=\{a,b,c,d\}$. The *full shift* over ${\mathscr{A}}$ is the set ${\mathscr{A}}^{\mathbb{N}}$ of all infinite sequences of symbols. A *block* is any finite sequence of symbols. A *shift space* over ${\mathscr{A}}$ is a subset of ${\mathscr{A}}^{\mathbb{N}}$ defined by establishing some constraints on blocks, which are allowed to appear as subsequences. For example, the set of all infinite sequences of $0$’s and $1$’s that do not contain the block $11$ is a shift space over $\{0,1\}$. A shift space becomes a dynamical system when we equip it with the *shift map*, which discards the first element of a sequence and shifts the remaining elements by one place to the left. The branch of mathematics concerning the study of shift spaces is known as *symbolic dynamics*. Shift spaces are mathematical models for digitized information, arising frequently as a result of the discretization of dynamical processes. For example, imagine a point moving along a trajectory in space. Partition the space into finitely many pieces and assign a symbol to each piece. Now write down the sequence of symbols labelling the successive partition elements visited by the point as it follows the trajectory. We have encoded information about the trajectory in a sequence from the shift space over the set of labels of the partition cells. As is often the case in mathematics, there is a notion of isomorphism for shift spaces known as *conjugacy* showing us when two seemingly different shift spaces are practically the same. In other words, conjugate shift spaces can be treated as two instances of the same underlying object (for more details we refer the reader to [@LM95 Def. 1.5.9]). For example, there is neither loss nor gain of information when the full $\{0,1\}$-shift is transformed into the full $\{4,7\}$-shift by switching every $0$ to $4$ and every $1$ to $7$. A common problem in symbolic dynamics is to decide whether two shift spaces are conjugate or not. Various invariants are devised to help us distinguish shift spaces which are *not* conjugate. A property of a shift space is a *conjugacy invariant* if whenever a shift space $X$ possesses that property, then every shift space conjugate to $X$ also possesses that property. Among the most important invariants is *entropy*, which is a measure of the complexity of a shift space. Entropy is a nonnegative real number equal to the asymptotic growth rate of the number of blocks that occur in a shift space. Entropy is a conjugacy invariant [@LM95 Cor. 4.1.10]. Therefore, if two shift spaces have different entropy, then they have different structure — they are not conjugate. For example, the full shift over $\{0,1\}$ is not conjugate to the full shift over $\{a,b,c\}$, as they have different entropies. There is one problem with this useful tool: computing the entropy of an arbitrary shift space is often a difficult or even a hopeless task. For example, it is known that for every nonnegative real number $t$ there is a shift space with entropy $t$, but this result is a part of nontrivial theory (see for example [@Walters pp. 178–9]). We will show, however, that this can be proven in an elementary way. To this end, we will define a new class of shift spaces — the subordinate shifts — for some of which the calculation of entropy is straightforward and requires only basic combinatorics. Notation and definitions. ------------------------- Throughout this paper, the symbol ${\mathbb{N}}$ denotes the set of *positive* integers. We denote the number of elements of a finite set $A$ by $|A|$. Given any real number $x$, by $\lfloor x\rfloor$ we mean the largest integer not greater than $x$. Recall that a sequence of real numbers $\{a_n\}_{n=1}^\infty$ is *subadditive* if $a_{m+n}\leq a_m+a_n$ for all $m,n\in{\mathbb{N}}$. Let $\{a_n\}_{n=1}^\infty$ be a subadditive sequence of nonnegative real numbers. Then the sequence $\{a_n/n\}_{n=1}^\infty$ converges to a limit equal to the infimum of the terms of this sequence, that is $$\lim_{n\to\infty}\frac{a_n}{n} = \inf_{n\in{\mathbb{N}}}\frac{a_n}{n}.$$ We follow the notation of Lind and Marcus [@LM95] as close as possible. One notable exception is that we consider only one-sided shifts, while Lind and Marcus consider two-sided (invertible) shifts throughout most of their book. Let ${\mathscr{A}}$ be a finite set, which we call the *alphabet*. We refer to the elements of ${\mathscr{A}}$ as *symbols*. The *full ${\mathscr{A}}$-shift* is the collection of all infinite sequences of symbols from ${\mathscr{A}}$. The full ${\mathscr{A}}$-shift is denoted by $${\mathscr{A}}^{\mathbb{N}}=\{x=(x_i)_{i=1}^\infty : x_i\in{\mathscr{A}}\text{ for all } i\in{\mathbb{N}}\}.$$ We usually write an element of ${\mathscr{A}}^{\mathbb{N}}$ as $x=(x_i)_{i=1}^\infty=x_1x_2x_3\ldots$. Often we identify a finite set ${\mathscr{A}}$ such that $|{\mathscr{A}}|=r$ with $\{0,1,\ldots,r-1\}$. A *full $r$-shift* is then the full shift over the alphabet $\{0,1,\ldots,r-1\}$ and a *full binary shift* is the full $2$-shift. A *block over ${\mathscr{A}}$* is a finite sequence of symbols from ${\mathscr{A}}$. We write blocks without separating their symbols, so a block over ${\mathscr{A}}=\{0,1,2\}$ might look like $01220120$. The *length of a block $u$* is the number of symbols it contains. An *$n$-block* stands for a block of length $n$. We identify a symbol with the $1$-block consisting of this symbol. An *empty block* is the unique block with no symbols and length zero that we denote ${\varepsilon}$. The set of all blocks over ${\mathscr{A}}$ (including ${\varepsilon}$) is denoted by ${\mathscr{A}}^*$. A *concatenation* of two blocks $u=a_1\ldots a_k$ and $v=b_1\ldots b_l$ is the block $uv$ obtained by writing $u$ first and then $v$, that is, $uv=a_1\ldots a_k b_1\ldots b_l$. The concatenation is an associative operation, because $(uv)w=u(vw)$ for any blocks $u,v,w\in{\mathscr{A}}^*$. For this reason we may write $uvw$, or indeed concatenate any sequence of blocks (finite or not) without ambiguity. If $n\ge 1$, then $u^n$ stands for the concatenation of $n$ copies of $u$. Given a nonempty block $u\in{\mathscr{A}}^*$ we denote by $u^\infty$ the sequence $uuu\ldots\in{\mathscr{A}}^{\mathbb{N}}$. Let $x=(x_i)_{i=1}^\infty\in{\mathscr{A}}^{\mathbb{N}}$ and let $1\le i \le j$ be integers. We write $x_{[i,j]}=x_ix_{i+1}\ldots x_j$ for the block of symbols in $x$ starting from the $i$-th and ending at the $j$-th position. We say that a block $w\in A^*$ *occurs in $x$* and $x$ *contains* $w$ if $w=x_{[i,j]}$ for some integers $1\le i \le j$. Note that ${\varepsilon}$ occurs in every sequence from ${\mathscr{A}}^{\mathbb{N}}$. Similarly, given an $n$-block $w=w_1\ldots w_n \in{\mathscr{A}}^*$ we define $w_{[i,j]}=w_iw_{i+1}\ldots w_j\in{\mathscr{A}}^*$ for each $1\le i\le j\le n$. A *prefix* of a block $z\in{\mathscr{A}}^*$ is a block $u$ such that $z=uv$ for some $v\in{\mathscr{A}}^*$. For every ${\mathscr{A}}$ we define the *shift map* $\sigma\colon{\mathscr{A}}^{\mathbb{N}}\rightarrow{\mathscr{A}}^{\mathbb{N}}$. It maps a sequence $x=(x_i)_{i=1}^\infty$ to the sequence $\sigma(x)=(x_{i+1})_{i=1}^\infty$. Equivalently, $\sigma(x)$ is the sequence obtained by dropping the first symbol of $x$ and moving the remaining symbols by one position to the left. Given any collection ${\mathscr{F}}$ of blocks over ${\mathscr{A}}$ (i.e., a subset of ${\mathscr{A}}^*$) we define a *shift space specified by ${\mathscr{F}}$*, denoted by $X_{\mathscr{F}}$, as the set of all sequences from ${\mathscr{A}}^{\mathbb{N}}$ which do not contain any blocks from ${\mathscr{F}}$. We say that ${\mathscr{F}}$ is a collection of *forbidden blocks for $X_{\mathscr{F}}$* as blocks from ${\mathscr{F}}$ are forbidden to occur in $X_{\mathscr{F}}$. A *shift space* is a set $X\subset{\mathscr{A}}^{\mathbb{N}}$ such that $X=X_{\mathscr{F}}$ for some ${\mathscr{F}}\subset{\mathscr{A}}^*$. A *binary shift space* is a shift space over the alphabet $\{0,1\}$. Show that for every shift space $X$ we have $\sigma(X)\subset X$. Find a shift space $X$ for which $\sigma(X)\neq X$. With each shift space $X$ over ${\mathscr{A}}$, we may associate a set of blocks over ${\mathscr{A}}$ which occur in some sequence $x\in X$. We call this set the *language of $X$* and denote it by ${\mathscr{B}}(X)$. We write ${\mathscr{B}}_n(X)$ for the set of all $n$-blocks contained in ${\mathscr{B}}(X)$. Show that if $\mathcal{L}$ is a language of some shift space over ${\mathscr{A}}$, then $\mathcal{L}$ is: 1. *factorial*, meaning that if $u\in\mathcal{L}$ and $u=vw$ for some blocks $v,w\in {\mathscr{A}}^*$, then both $v$ and $w$ also belong to $\mathcal{L}$, 2. *prolongable*, meaning that for every block $u$ in $\mathcal{L}$ there is a symbol $a\in {\mathscr{A}}$ such that $ua$ also belongs to $\mathcal{L}$. Actually, the converse is also true. Given a factorial and prolongable subset $\mathcal{L}\subset{\mathscr{A}}^*$ there is a shift space $X$ such that $\mathcal{L}$ is the language of $X$. A collection of forbidden blocks defining $X$ is ${\mathscr{F}}={\mathscr{A}}^*\setminus\mathcal{L}$. We can also characterize points in a shift space $X$. \[lem:char\] Let $\mathcal{L}\subset{\mathscr{A}}^*$ be factorial and prolongable. Let $X$ be a shift space such that $\mathcal{L}={\mathscr{B}}(X)$. Then a point $x\in {\mathscr{A}}^{\mathbb{N}}$ is in $X$ if and only if $x_{[i,j]}\in\mathcal{L}$ for all $i,j\in{\mathbb{N}}$ with $ i < j$. Shift spaces are determined by their language. In other words, two shift spaces are equal if and only if they have the same language [@LM95 Proposition 1.3.4]. Hence there is a one-to-one correspondence between shift spaces over ${\mathscr{A}}$ and factorial, prolongable subsets of ${\mathscr{A}}^*$. To define a shift space, one can either specify its set of forbidden blocks or its language. Let $x\in{\mathscr{A}}^{\mathbb{N}}$. Let ${\mathscr{B}}_x$ be the collection of all blocks occurring in $x$. Since ${\mathscr{B}}_x$ is a factorial and prolongable language, it defines a shift space, which we denote by $\Sigma_x$. Let $S\subset{\mathbb{N}}\cup\{0\}$. We define $${\mathscr{F}}_S=\{1\underbrace{0\ldots 0}_p1:p\notin S\}.$$ The binary shift defined by forbidding blocks from ${\mathscr{F}}_S$ is called the *$S$-gap shift* and is denoted by $X_S$. In particular, we call $X_{\mathbb{N}}$ the *golden mean shift* (a sequence belongs to it if and only if it does not contain the block $11$). The following fact merely states that we may construct inductively an *infinite* sequence starting from an infinite collection of *finite* sequences such that each sequence in it coincides with any shorter one as long as both are defined. \[lem:growing-words\] Let $\{w^{(n)}\}_{n=1}^\infty$ be a sequence in ${\mathscr{A}}^*$ and for each $n\in{\mathbb{N}}$ let $l(n)$ be the length of $w^{(n)}$. If for each $k\in{\mathbb{N}}$ the block $w^{(k)}$ is a prefix of $w^{(k+1)}$, then there is a point $x\in{\mathscr{A}}^{\mathbb{N}}$ such that for each $n\in{\mathbb{N}}$ we have $x_{[1,l(n)]}=w^{(n)}$. Moreover, if $\lim_{n\rightarrow\infty} l(n)=\infty$, then $x$ is unique. We use superscripts in brackets as above to denote indices for sequences of blocks. That is, we write $\{w^{(n)}\}_{n=1}^\infty$ to denote a sequence of blocks. This way we may reserve subscripts for enumerating symbols within the block: $w^{(n)}=w^{(n)}_1w^{(n)}_2\ldots w^{(n)}_k$. Finally, we are ready to move to the entropy itself. In full generality, this concept was defined by Adler, Konheim and McAndrew [@AKM] for an arbitrary compact topological space $X$ and a continuous map $f\colon X\to X$. The definition below applies only to shift spaces but the resulting number is equal to the Adler, Konheim and McAndrew entropy of the shift treated as a dynamical system (see [@LM95 Exercise 6.3.8]). Let $\log$ denote the logarithm to base $2$ (choosing a different base would also yield a valid definition; it would change the value of entropy only by a multiplicative constant). Let $X\subset{\mathscr{A}}^{\mathbb{N}}$ be a nonempty shift space and let $m,n\in{\mathbb{N}}$. Observe that every block $w\in{\mathscr{B}}_{m+n}(X)$ can be written in a unique way as a concatenation $w=uv$, where $u\in {\mathscr{B}}_m(X)$ and $v\in {\mathscr{B}}_n(X)$. Therefore $|{\mathscr{B}}_{m+n}(X)|\le|{\mathscr{B}}_{m}(X)|\cdot |{\mathscr{B}}_{n}(X)|$, and hence $$\log |{\mathscr{B}}_{m+n}(X)|\le \log|{\mathscr{B}}_{m}(X)|+\log |{\mathscr{B}}_{n}(X)|.$$ By applying Fekete’s Lemma to the nonnegative sequence $\log|{\mathscr{B}}_n(X)|$ we may now define the *entropy of $X$*, denoted by $h(X)$, as $$h(X)=\lim_{n\to\infty} \frac{1}{n}\log |{\mathscr{B}}_n(X)|=\inf_{n\ge 1} \frac{1}{n}\log |{\mathscr{B}}_n(X) |.$$ Roughly speaking, the entropy measures the complexity of a shift space $X$ in terms of the asymptotic growth rate of the number of $n$-blocks that appear in the language $X$. In other words, the number of $n$-blocks in a shift space of entropy $h\ge 0$ roughly equals $2^{nh}$. Every finite shift space has entropy zero. The full ${\mathscr{A}}$-shift has entropy $\log|{\mathscr{A}}|$. In particular, the full $r$-shift has entropy $\log r$. Observe that if $X,Y\subset{\mathscr{A}}^{\mathbb{N}}$ and $X\subset Y$, then $h(X)\leq h(Y)$. Since a shift space over ${\mathscr{A}}$ is a subset of the full ${\mathscr{A}}$-shift, we may conclude that the entropy of any shift space over ${\mathscr{A}}$ is a nonnegative real number bounded above by $\log|{\mathscr{A}}|$ (for systems that are not shifts it may well be infinite — see [@ALM Example 4.2.6]). As mentioned in the introduction, computing the entropy of a shift space is a hard problem — try, for example to verify straight from the definition that the entropy of the golden mean shift is $\log((1+\sqrt{5})/2)$. There are (relatively rare) families of shift spaces (e.g. shifts of finite type, see [@LM95]) for which we can actually provide a (theoretically) computable formula for entropy. Even in these special cases, one needs to apply some non-trivial tools. For the sake of illustration we recall some results from [@LM95 Exercise 4.3.7]. Let $S\subset{\mathbb{N}}\cup\{0\}$ and let $X_S$ be the associated $S$-gap shift. Then $h(X_S)=\log\lambda$, where $\lambda$ is the unique positive solution of the equation $$\sum_{j\in S} x^{-j-1}=1.$$ For every $t\in[0,1]$ there is a set $S\subset{\mathbb{N}}\cup\{0\}$ ($S$ depends on $t$) such that $h(X_S)=t$. Subordinate shifts. =================== The goal of this section is to introduce a family of shift spaces with easily calculable entropy; a family rich enough to contain a shift space with every possible nonnegative entropy. For the remainder of the paper, we fix ${\mathscr{A}}=\{0,1,\ldots, r-1 \}$. We say that a block $w=w_1\ldots w_k\in{\mathscr{A}}^*$ *dominates* a block $v=v_1\ldots v_k\in{\mathscr{A}}^*$ if $v_i\le w_i$ for $i=1,\ldots,k$. In an analogous way we define when one sequence from ${\mathscr{A}}^{\mathbb{N}}$ dominates another. A *subordinate of $\mathcal{L}\subset {\mathscr{A}}^*$* is the set $\mathcal{L}^\le$ of all blocks over ${\mathscr{A}}$ that are dominated by some block in $\mathcal{L}$. Observe that if $\mathcal{L}$ is factorial and prolongable, then the same holds for $\mathcal{L}^\le$. In particular, given a point $x\in{\mathscr{A}}^{\mathbb{N}}$, we may define a *subordinate shift of $x$*, denoted by $X^{\le x}$, as a shift space given by the language ${\mathscr{B}}^\le_x$, where ${\mathscr{B}}_x$ is the language of blocks occurring in $x$. All binary blocks of length $3$ are dominated by $111$. The blocks $0000$, $0001$, $0100$, and $0101$ are the only blocks dominated by $0101$. Subordinate shifts are *hereditary* (this is a notion introduced in [@KerrLi] and examined in [@Kwietniak]). It can be shown that a hereditary shift is subordinate if and only if it is irreducible in the sense of [@LM95 Definition 1.3.6]. It turns out that a shift space that has been recently extensively studied is an example of a subordinate shift. Recall that a positive integer $n$ is *square-free* if there is no prime number $p$ such that $p^2$ divides $n$. Let $\eta$ be a point in $\{0,1\}^{\mathbb{N}}$ given by $$\eta_n=\begin{cases} 1&\text{ if $n$ is square-free,}\\ 0& \text{ otherwise.} \end{cases}$$ In other words, $\eta_n=(\mu(n))^2$, where $\mu\colon{\mathbb{N}}\to{\mathbb{N}}$ is the famous Möbius function. It can be shown that $S=X^{\le \eta}$ is the *square-free flow*; that is a shift space, whose structure is strongly tied to the statistical properties of square-free numbers. For more details see [@Peckner; @Sarnak]. The study of the square-free flow has been recently extended to the more general context of $\mathcal{B}$-free integers; that is to say integers with no factor in a given family $\mathcal{B}$ of pairwise relatively prime integers, the sum of whose reciprocals is finite, see [@B-free-dynamical; @B-free-measures]. We aim to show that if given $t\in [0,1]$, then we are able to choose a point $x(t)$ from the full binary shift such that $h(X^{\le x(t)})=t$. First we tackle rational entropies. \[lem:rational-case\] If $w\in\{0,1\}^*$ is a block of length $q$ with $p$ occurrences of the symbol $1$ and $x=w^\infty$, then $h(X^{\le x})=p/q$. Since replacing in $w$ any subset of $1$’s with $0$’s leads to a block dominated by $w$ which is in ${\mathscr{B}}(X^{\le x})$, we know that there are at least $2^p$ blocks in ${\mathscr{B}}_q(X^{\le x})$. It follows that $h(X^{\le x})\geq p/q$. On the other hand, the periodicity of $x=w^\infty$ implies that for each $j\in{\mathbb{N}}$ there are at most $q$ different blocks of length $qj$ in ${\mathscr{B}}_{x}$. Each such block dominates exactly $2^{pj}$ blocks from $\{0,1\}^*$. Therefore there are at most $q\cdot 2^{pj}$ blocks in ${\mathscr{B}}_{qj}(X^{\le x})$. Consequently, $$\begin{aligned} h(X^{\le x}) & = \lim_{n\to\infty} \frac{1}{n}\log|{\mathscr{B}}_n(X^{\le x})| = \lim_{j\to\infty} \frac{1}{qj}\log|{\mathscr{B}}_{qj}(X^{\le x})|\\ & \le \lim_{j\to\infty} \frac{1}{qj}\log (q\cdot 2^{pj})=p/q.\qedhere\end{aligned}$$ We now show how to construct a shift space with entropy $\pi/8$. We hope that this example will make the general construction that comes after it much clearer. Let $t=\pi/8=0.3926990816\ldots$. We consider the following sequence of rational approximations of $t$ from the above: $$0.4,\, 0.4,\, 0.393,\, 0.3927,\,0.3927,\,0.3927,\,0.3926991,\,0.39269909,\,\ldots .$$ We obtain our $n$-th approximation by rounding up $t$ to the nearest number with no more than $n$ digits after the decimal point. Let $p_n$ be this $n$-th approximation times $10^n$. We now inductively build a sequence of blocks $\{w^{(n)}\}_{n=1}^\infty$ from $\{0,1\}^*$ such that for every $n\in{\mathbb{N}}$: 1. $w^{(n)}$ has length $10^n$, 2. the symbol $1$ appears exactly $p_n$ times in $w^{(n)}$, 3. $w^{(n)}$ is a prefix of $w^{(n+1)}$, 4. $w^{(n+1)}$ is dominated by $(w^{(n)})^{10}$. We can visualize the construction of $w^{(n+1)}$ as a process with two steps. In the first step we concatenate ten copies of $w^{(n)}$. In the second step we keep the first $p_{n+1}$ occurrences of the symbol $1$ in $w^{(n+1)}$ and replace the rest by $0$’s. In our example, we first put $w^{(1)}=1111000000$. We want the symbol $1$ to appear $p_2=40$ times in $w^{(2)}$, so we define $w^{(2)}=(w^{(1)})^{10}$ — there is no need to remove any $1$’s. Then in $w^{(3)}$ we need to see $393$ appearances of $1$, so we define $w^{(3)}$ as a concatenation of nine copies of $w^{(2)}$ and a block $v$ of length $100$ which agrees with $w^{(2)}$ except that the last seven $1$’s appearing in $w^{(2)}$ are replaced by $0$’s in $v$. The construction continues on inductively. Applying Lemma \[lem:growing-words\] to the sequence of binary blocks $\{w^{(n)}\}_{n=1}^\infty$, we obtain a point $x\in\{0,1\}^{\mathbb{N}}$. The entropy of the subordinate shift $X^{\le x}$ is $t=\pi/8$. The proof of this fact is contained in the general result below. We are now equipped to tackle the main theorem of this paper. \[thm:binary-case\] For every $t\in[0,1]$ there is a binary subordinate shift with entropy $t$. For every $n\in{\mathbb{N}}$ let $s_n$ be the rational approximation of $t$ obtained by rounding up the decimal expansion of $t$ to the nearest number with no more than $n$ digits after the decimal point. Let $p_n=s_n\cdot 10^n$. We now inductively build a sequence of blocks $\{w^{(n)}\}_{n=1}^\infty$ from $\{0,1\}^*$ such that for every $n\in{\mathbb{N}}$: 1. $w^{(n)}$ has length $10^n$, 2. the symbol $1$ appears exactly $p_n$ times in $w^{(n)}$, 3. $w^{(n)}$ is a prefix of $w^{(n+1)}$, 4. $w^{(n+1)}$ is dominated by $(w^{(n)})^{10}$. We start with $$w^{(1)}=\underbrace{1\ldots1}_{p_1}\underbrace{0\ldots 0}_{10-p_1}.$$ Assume now that we have defined $w^{(1)},\ldots,w^{(n)}$ so that the four conditions above are satisfied to the extent to which they apply to $w^{(1)},\ldots,w^{(n)}$. We now concatenate ten copies of $w^{(n)}$. We keep the first $p_{n+1}$ occurrences of the symbol $1$ unaltered in the sequence, replace the rest by $0$, and call the result $w^{(n+1)}$. It is easy to verify that the conditions above now hold to the extent to which they apply to $w^{(1)},\ldots,w^{(n+1)}$, so the inductive construction is complete. Applying Lemma \[lem:growing-words\] to the sequence $\{w^{(n)}\}_{n=1}^\infty$ we obtain a point $x\in\{0,1\}^{\mathbb{N}}$. Observe that for every $n\in{\mathbb{N}}$ the point $x$ is dominated by $(w^{(n)})^\infty$. It follows that $X^{\leq x}\subset X^{\le (w^{(n)})^\infty}$, and so $h(X^{\leq x})\leq h(X^{\le (w^{(n)})^\infty})$. By Lemma \[lem:rational-case\] we have $h(X^{\le (w^{(n)})^\infty})=s_n$, and therefore $$h(X^{\leq x})\leq\lim_{n\to\infty}h(X^{\le (w^{(n)})^\infty})=\lim_{n\to\infty}s_n=t.$$ On the other hand, observe that for every $n\in{\mathbb{N}}$ we have $w^{(n)}\in{\mathscr{B}}_{10^n}(X^{\leq x})$, so all $2^{p_n}$ blocks dominated by $w^{(n)}$ are also in ${\mathscr{B}}_{10^n}(X^{\leq x})$. Therefore $\log |{\mathscr{B}}_{10^n}(X^{\leq x})|\ge \log 2^{p_n} = p_n$, and so $$t=\lim_{n\to\infty}s_n=\lim_{n\to\infty}\frac{p_n}{10^n}\le \lim_{n\to\infty} \frac{1}{10^n}\log |{\mathscr{B}}_{10^n}(X^{\leq x})|=h(X^{\leq x}),$$ which completes the proof of $h(X^{\leq x})=t$. Prove that the shift space $X^{\leq x}$ constructed in the proof of Theorem \[thm:binary-case\] is *irreducible* (see [@LM95 Definition 1.3.6]), meaning that given any pair of blocks $u,v\in{\mathscr{B}}(X^{\leq x})$ there is a block $y\in{\mathscr{B}}(X^{\leq x})$ such that $uyv\in{\mathscr{B}}(X^{\leq x})$. It remains to show that the conclusion of Theorem \[thm:binary-case\] is true for every $t> 1$. For every $t\in (1,\infty)$ there is a shift space with entropy $t$. Let $t\in (1,\infty)$ be given. Pick $k\in{\mathbb{N}}$ and $s\in (0,1)$ such that $ks=t$. Use Theorem \[thm:binary-case\] to obtain a binary subordinate shift $X$ with entropy $s$. Using $X$ we now construct a shift space $Y\subset\{0,1,\ldots,2^k-1\}^{\mathbb{N}}$. We start with $Y=\emptyset$. For every point $w\in X$ we perform the following procedure: 1. express $w$ as a concatenation $w_1w_2\ldots$, where each block $w_n$ has length $k$, 2. for every $w_n$ we determine the number $a_n\in\{0,1,\ldots,2^k-1\}$ such that $w_n$ is the binary notation for $a_n$ (for example, for $k=2$, we get $00\mapsto 0$, $01\mapsto 1$, $10\mapsto 2$, and $11\mapsto 3$), 3. we add the sequence $a_1a_2\ldots$ to $Y$ (note that it is the $a_n$’s that are symbols here, not their digits). It is elementary to check that $Y$ is a shift space. It suffices to analyze how the language of $Y$ is created from the language of $X$. Observe that for every $n\in{\mathbb{N}}$ we have $|{\mathscr{B}}_n(Y)|=|{\mathscr{B}}_{kn}(X)|$. Therefore $$h(Y)=\lim_{n\to\infty}\frac{1}{n}\log |{\mathscr{B}}_n(Y)|=k\cdot\lim_{n\to\infty}\frac{1}{kn}\log |{\mathscr{B}}_{kn}(X)|=k\cdot h(X)=t.\qedhere$$ Acknowledgment. {#acknowledgment. .unnumbered} =============== The authors would like to thank the referees for their thorough and careful work. Preparing this article we asked our students and colleagues to comment on it. We are grateful to: Jakub Byszewski, Vaughn Climenhaga, Jakub Konieczny, Marcin Lara, Simon Lunn, Martha [Ł]{}[a]{}cka, Dariusz Matlak, Samuel Roth, and Maciej Ulas for their remarks and suggestions. The research of Dominik Kwietniak was supported by the National Science Centre (NCN) under grant Maestro 2013/08/A/ST1/00275. The research of Jian Li was supported by Scientific Research Fund of Shantou University (YR13001). [10]{} R. L. Adler, A. G. Konheim, M. H. McAndrew, Topological entropy, [*Trans. Amer. Math. Soc.*]{} [**114**]{} (1965) 309–319. L. Alsedá, J. Llibre, M. Misiurewicz, [*Combinatorial Dynamics and Entropy in Dimension One.*]{} Second edition. World Scientific, River Edge, NJ, 2000. E.-H. El Abdalaoui, M. Lemańczyk, T. De La Rue, A dynamical point of view on the set of $\mathscr{B}$-free integers (2013), available at <http://arxiv.org/abs/1311.3752>. D. Kerr, H. Li, Independence in topological and $C\sp *$-dynamics, [*Math. Ann.*]{} [**338**]{} (2007) 869–926. J. Ku[ł]{}aga-Przymus, M. Lemańczyk, B. Weiss, On invariant measures for $\mathscr{B}$-free systems (2014), available at <http://arxiv.org/abs/1406.3745>. D. Kwietniak, Topological entropy and distributional chaos in hereditary shifts with applications to spacing shifts and beta shifts, [*Discrete Contin. Dyn. Syst.*]{} [**33**]{} (2013) 2451–2467. D. Lind, B. Marcus, [*An Introduction to Symbolic Dynamics and Coding.*]{} Cambridge University Press, Cambridge, 1995. R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow (2014), available at <http://arxiv.org/abs/1205.2905>. P. Sarnak, Three lectures on the Möbius function, randomness, and dynamics (2011), available at <http://publications.ias.edu/sarnak/paper/512>. P. Walters, [*An Introduction to Ergodic Theory.*]{} Graduate Texts in Mathematics, Vol. 79, Springer-Verlag, New York-Berlin, 1982.

ArXiv

--- abstract: 'We consider non-selfadjoint operator algebras ${{\mathfrak{L}(G,\lambda)}}$ generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs $G$. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with broad conditions that ensure the double commutant property. It is also shown that the double commutant property may fail for ${{\mathfrak{L}(G,\lambda)}}$ in the case of the single vertex graph with two edges and a suitable choice of left weight function $\lambda$.' address: - 'Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1' - 'School of Mathematics & Statistics, University College Dublin, Belfield, Dublin 4, Ireland' - 'Department of Mathematics & Statistics, Lancaster University, Lancaster, U.K., LA1 4YF' author: - 'David W. Kribs' - 'Rupert H. Levene' - 'Stephen C. Power' bibliography: - 'myrefs.bib' title: Commutants of Weighted Shift Directed Graph Operator Algebras --- Introduction ============ For over two decades, operator algebras associated with directed graphs and their generalizations have received intense interest in the operator algebra and mathematics community. This class of algebras includes many interesting examples, often with connections to different areas, such as dynamical systems, and at the same time is sufficiently broad that results for them have given insights to the general theory of operator algebras. The most fundamental non-selfadjoint algebras in this class are the tensor algebras [@muhlysolel; @popescu] and free semigroupoid algebras of directed graphs [@katsouliskribs; @kribspower; @kribssolel], including free semigroup algebras [@davidsonpitts; @kennedy]. Each of these is generated by creation operators on a Fock-type Hilbert space defined by the graph, and there is now an extensive body of work for these algebras. In this paper we consider weighted creation operator generalizations, in the weak operator topology (WOT) closed setting, and we investigate their algebraic structure. The resulting [*weighted shift directed graph algebras*]{} ${{\mathfrak{L}(G,\lambda)}}$ may be viewed as the minimal generalization of two different classes of non-selfadjoint algebras: the free semigroupoid algebras of directed graphs on the one hand, and on the other, the classical unilateral weighted shift algebras associated with single variable weighted Bergman spaces. The paper is organized as follows. In the next section we introduce the notation $\lambda$, $\rho$, for certain left and right weight functions for the path semigroupoid of a directed graph $G$, and define their associated weighted creation operators (which need not be bounded) and their respective operator algebras, ${{\mathfrak{L}(G,\lambda)}}$ and ${{\mathfrak{R}(G,\rho)}}$, on the Fock space $\H_G$. In the subsequent section we investigate the structure of the commutant algebra ${{\mathfrak{L}(G,\lambda)}}'$ and obtain its characterization under the natural condition (left-boundedness of $\lambda$) that all the weighted left creation operators are bounded. In the proof we identify a simple commuting square condition that relates the left weight $\lambda$ to a particular right weight $\rho$ which is relevant to the commutant, and we exploit this to show that ${{\mathfrak{L}(G,\lambda)}}' ={{\mathfrak{R}(G,\rho)}}$ for this right weight. In the fourth section we investigate the double commutant ${{\mathfrak{L}(G,\lambda)}}''$ and obtain broad conditions which ensure the double commutant property ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$. A range of illuminating examples is also given. In particular, for the single vertex graph with two edges it is shown that there exist left-bounded weights $\lambda$ for which ${{\mathfrak{L}(G,\lambda)}} '' = \B(\H_G)$. On the other hand, for the directed $2$-cycle graph, with two vertices and two edges, necessary and sufficient conditions are obtained for the double commutant property. Our focus here is on the analysis of generalized weighted shifts and the non-selfadjoint operator algebras they generate, in a setting that embraces both commutative and non-commutative versions, and is built upon the contemporary directed graph operator algebra framework. In fact the first foray in this direction for single vertex directed graphs gave sufficient conditions for the determination of the commutant and for reflexivity [@kribs04], the basic general goals being to extend results from the single variable commutative case and to expose new phenomena in the non-commutative directed graph setting. Our concern in the present paper is to characterize commutants for the left and right algebras by identifying explicit conditions at the level of weighted graphs. It would be interesting to connect this double commutant investigation with the recent work [@marcouxmastnak] on a general double commutant theorem for non-selfadjoint algebras, and with recent work on weighted Hardy algebras of correspondences [@dor-on; @muhlysolel-weighted]. We leave the natural problems of invariant subspace structure and reflexivity for these algebras for investigation elsewhere. It should be possible to identify a large class of these algebras as being reflexive, and in doing so, extend results from the case of weighted Bergman spaces [@Shields74] and partial results from the weighted free semigroupoid algebra case [@kribs04]. Additionally, non-reflexive examples have not yet been constructed in the non-commutative case. This should also be possible with extended notions of strictly cyclic weighted shifts to our setting. Weighted Shift Directed Graph Algebras ====================================== Let $G$ be a countable directed graph with edge set $E(G)=\{e,f,\ldots\}$ and vertex set $V(G)=\{x,y,\ldots\}$. We will write $G^+=\{ u,v,w,\ldots \}$ for the set of finite paths in $G$, including the vertices regarded as paths of length $0$. Note that if $G$ is finite (by which we mean that both $V(G)$ and $E(G)$ are finite), then the set $\{w\in G^+\colon |w|< k\}$ is finite for each $k\ge1$, where $|w|$ denotes the length of a path $w$. We write $s(w)$ and $r(w)$ for the source and range vertices of a path $w$; in particular, if $x\in V(G)$, then $r(x)=x=s(x)$. We will also take a right to left orientation for path products, so that $w = r(w) w s(w)$ for all $w\in G^+$, and for $v,w\in G^+$ we have $wv\in G^+$ if and only if $s(w)=r(v)$. To each such graph $G$ we associate the Hilbert space $\H_G=\ell^2(G^+)$, called the Fock space of $G$, with canonical orthonormal basis $\{ \xi_v : v\in G^+ \}$. The vectors $\xi_x$ for $x\in V(G)$ are called vacuum vectors. The left creation operators on $\H_G$ are the partial isometries defined as follows: $L_w \xi_v = \xi_{wv}$ whenever $wv\in G^+$, and $L_w \xi_v = 0$ otherwise. (These operators may also be viewed as generated by the left regular representation of the path semigroupoid of the graph.) Pictorially, as an accompaniment to the directed graph, one can view the actions of the generators $L_e$ as tracing out downward tree structures that lay out the basis vectors for $\H_G$. One tree is present for each vertex $x$ in $G$, with the top tree vertex in each tree corresponding to a vacuum vector $\xi_x$, and the tree edges corresponding to the basis pairs $(\xi_w, \xi_{ew})$. We call a function $\lambda\colon G^+\times G^+\to [0,\infty)$ a *left weight* on $G$ if 1. $\lambda(v,w)>0\iff wv\in G^+$; and 2. $\lambda$ satisfies the *(left) cocycle condition*: $$\lambda(v,w_2w_1) = \lambda(w_1v,w_2)\lambda(v,w_1)$$ for all $v,w_1,w_2\in G^+$ with $w_2w_1v\in G^+$. Note that if $v\in G^+$, then $r(v)\in V(G)$ satisfies $r(v)=r(v)^2$, hence $\lambda(v,r(v))=\lambda(v,r(v))^2$ and so $\lambda(v,r(v))=1$. In particular, for $x\in V(G)$ we have $x=s(x)=r(x)$ and so $\lambda(s(x),x)=\lambda(x,r(x))=1$. Note also that the edge weights $\lambda(v,e)$ (where $ev\in G^+$ and $e\in E(G)$) determine the entire function $\lambda$ through the left cocycle condition. Indeed, if we attach the weight $\lambda(v,e)$ to the edge in the Fock space tree corresponding to the move $\xi_v\mapsto \xi_{ev}$ defined by $L_e\xi_v = \xi_{ev}$, then we can view $\lambda(v,w)$ (when non-zero) as the product of the individual weights one crosses when moving from $\xi_v$ to $\xi_{wv}$ in that tree. See subsequent sections for more discussion on this “forest” perspective. Given such a left weight $\lambda$, we define (by a mild abuse of notation) $$\lambda(w)=\sup_{v\in G^+} \lambda(v,w)\in [0,\infty]$$ for each $w\in G^+$. We say that $\lambda$ is *left-bounded at $w$* if $\lambda(w) < \infty$, and that $\lambda$ is *left-bounded* if this condition holds for all $w\in G^+$. The cocycle condition gives $\lambda(w_2w_1)\leq \lambda(w_2)\lambda(w_1)$ whenever $w_2w_1\in G^+$, so $\lambda$ is left-bounded if and only if $\lambda$ is left-bounded at every edge $e\in E(G)$. If $\lambda$ is left-bounded at $w$, then we define the weighted left shift operator $L_{\lambda,w}\in \B(\H_G)$ to be the continuous linear extension of $$L_{\lambda,w} \, \xi_v = \begin{cases} \lambda(v,w) \,\xi_{wv} &\text{if $wv\in G^+$}\\ 0 &\text{otherwise.} \end{cases}$$ Since $\lambda(w)<\infty$, it is easy to see that this gives a well-defined operator with $\|L_{\lambda,w}\|=\lambda(w)$. Moreover, if $\lambda$ is left-bounded, then by the cocycle condition we see that $w\mapsto L_{\lambda,w}$ is a semigroupoid homomorphism: $$L_{\lambda,w_2w_1}=L_{\lambda,w_2}L_{\lambda,w_1}\quad \text{whenever $w_2w_1\in G^+$}.$$ We remark that one could also consider complex-valued left weight functions rather than weights taking non-negative values only. However, the corresponding weighted left shift operators would be jointly unitarily equivalent to weighted shift operators defined by a non-negative weight function. To see this, consider a complex-valued left weight $\mu\colon G^+\times G^+\to \bC$, by which we mean that $\mu(v,w)\ne0\iff wv\in G^+$ and $\mu$ satisfies the left cocycle condition. Define corresponding weighted left shifts $L_{\mu,w}$ exactly as above, let $\lambda\colon G^+\times G^+\to [0,\infty)$ be the non-negative left weight $\lambda(v,w)=|\mu(v,w)|$ and consider $\beta\colon G^+\times G^+\to \bT$, $\beta(v,w)=\tfrac{\lambda(v,w)}{\mu(v,w)}$ when $wv\in G^+$, and $\beta(v,w)=1$ otherwise. Note that $\beta$ then satisfies the left cocycle condition, so in particular we have $\beta(s(v),wv)=\beta(v,w)\beta(s(v),v)$ whenever $wv\in G^+$. The unitary operator $U_\beta$ mapping $\xi_v$ to $\beta(s(v),v)\xi_v$ satisfies $$U_\beta L_{\mu,w}\xi_v = \mu(v,w)\beta(s(v),wv)\xi_{wv}=\beta(s(v),v)\lambda(v,w)\xi_{wv} = L_{\lambda,w}U_\beta\xi_v$$ whenever $wv\in G^+$, hence $U_\beta L_{\mu,w}=L_{\lambda,w}U_\beta$, i.e., $L_{\mu,w}=U_\beta^*L_{\lambda,w}U_\beta$. Observe also that the requirement that $\lambda(v,w) \ne 0$ when $wv\in G^+$ is equivalent to requiring $L_{\lambda,w}$ to be injective on the set $\{\xi_v : wv\in G^+\}$, and is thus an assumption we build into the weights to avoid degeneracies in the analysis. Finally note that each operator $L_{\lambda,w}$ factors as a product of $L_w$ and a diagonal (with respect to the standard basis) weight operator, just as in the single variable case of [@Shields74] which is recovered when $G$ consists of a single vertex with a single loop edge. We now define the algebras ${{\mathfrak{L}(G,\lambda)}}$ that we shall consider in the paper. In the case of the single vertex, single loop edge graph, these algebras include classical unilateral weighted shift algebras such as those associated with weighted Bergman spaces; see the survey article [@Shields74] for an entrance point into the literature. The case of a single vertex graph and multiple loop edges was first considered along with some reflexivity type problems in [@kribs04]. If ${\lambda}$ is a left weight on a directed graph $G$, then we write ${{\mathfrak{L}(G,\lambda)}}$ for the WOT-closed unital operator algebra generated by the family of weighted left shift operators $\{ L_{\lambda,w} : w \in G^+,\ \lambda(w)<\infty\}$. If (as is often the case below) $\lambda$ is left-bounded, then the set $$\{L_{\lambda,w}\colon w\in V(G)\cup E(G)\}$$ also generates ${{\mathfrak{L}(G,\lambda)}}$ as a WOT-closed unital operator algebra. Let us call a strictly positive function $\alpha\colon G^+\to (0,\infty)$ with $\alpha(x)=1$ for all $x\in V(G)$ a *path weight* on $G$. For any such $\alpha$, there is a corresponding left weight $\lambda_\alpha$ on $G$ given by $$\lambda_\alpha(v,w)= \begin{cases} \frac{\alpha(wv)}{\alpha(v)}&\text{if $wv\in G^+$}\\ 0&\text{otherwise.} \end{cases}$$ Conversely, from any left weight $\lambda$, we obtain a corresponding path weight $\alpha_\lambda\colon v\mapsto \lambda(s(v),v)$, and these correspondences are inverses of one another. This observation allows us to easily construct examples of left weights. The left-handed notions above have right-handed counterparts which will play an important role in describing commutants. A *right weight* on $G$ is a function $\rho\colon G^+\times G^+\to [0,\infty)$ satisfying $\rho(v,u)>0\iff vu\in G^+$ and the *(right) cocycle condition* $$\rho(v,u_1u_2)=\rho(vu_1,u_2)\rho(v,u_1)$$ for all $v,u_1,u_2\in G^+$ with $vu_1u_2\in G^+$. We then have $\rho(v,s(v))=1$ for all $v\in G^+$. We write $\rho(u)=\sup_v\rho(v,u)$, and say $\rho$ is *right-bounded at $u$* if $\rho(u)<\infty$. We may then consider the weighted right shift operator $R_{\rho,u}\in \B(\H_G)$ (with $\|R_{\rho,u}\|=\rho(u)$) which satisfies the defining equation $$R_{\rho,u}\xi_v= \begin{cases} \rho(v,u) \,\xi_{vu} &\text{if $vu\in G^+$}\\ 0 &\text{otherwise.} \end{cases}$$ We have $\rho(u_1u_2)\leq \rho(u_2)\rho(u_1)$, and $R_{\rho,u_1u_2}=R_{\rho,u_2}R_{\rho,u_1}$ whenever $u_1u_2\in G^+$ and $\rho$ is right-bounded at $u_1$ and at $u_2$. We write ${{\mathfrak{R}(G,\rho)}}$ for the WOT-closed unital operator algebra generated by $ \{R_{\rho,u}\colon u\in G^+,\ \rho(u)<\infty\} $. A right weight is *right-bounded* if it is right-bounded at every $u\in G^+$. Finally, we observe that $$\rho_\alpha(v,u)= \begin{cases} \frac{\alpha(vu)}{\alpha(v)}&\text{if $vu\in G^+$}\\ 0&\text{otherwise} \end{cases}$$ defines a one-to-one correspondence between path weights $\alpha$ and right weights $\rho=\rho_\alpha$. Each of these right-handed definitions may be derived by applying the corresponding left-handed definition to the opposite graph of $G$, and making appropriate identifications. Note that the suprema defining $\lambda(u)$ and $\rho(u)$ are taken over the first argument, in $\lambda(\cdot,u)$ and $\rho(\cdot,u)$, and so in particular the notion of right-boundedness for a left weight function does not arise. A path weight $\alpha$, on the other hand, may be said to be left (resp. right) bounded if the associated map $\lambda_\alpha$ (resp. $\rho_\alpha$) is left-bounded (resp. right-bounded). The weighted shift creation operators $L_{\lambda, e}$ for edges of $e$, and also sums of these operators, are in fact special cases of a wide class of weighted shift operators defined on general countable trees, rather than our graph generated trees. The single operator theory for these general shifts, such as conditions for hyponormality and $p$-hyponormality, is developed in the recent book of Jablo´nski, Jung, and Stochel [@jjs-12]. Muhly and Solel have recently defined weighted shift versions of the Hardy algebras $H^\infty(E)$ [@muhlysolel-weighted] that can be associated with a correspondence $E$ (a self-dual right Hilbert $C^*$-module) over a $W^*$-algebra $M$. The Hardy algebras $\A=H^\infty(E)$ in fact provide generalizations of the free semigroupoid graph algebras in which the self-adjoint (diagonal) subalgebra $\A \cap \A^*$ is no longer commutative. At the expense of a much higher level of technicality, the weighted shift versions of these Hardy algebras similarly extend the weighted shift directed graph algebras ${{\mathfrak{L}(G,\lambda)}}$. Commutant Structure =================== Let $\lambda$ be a left weight on $G$, and let $\rho$ be a right weight on $G$. We say that the pair $(\lambda,\rho)$ satisfies the *commuting square condition* at $(w,u)\in G^+\times G^+$ if $$\rho(wv,u)\lambda(v,w)=\lambda(vu,w)\rho(v,u)$$ for every $v\in G^+$ with $wvu\in G^+$. If $\lambda$ is left-bounded at $w$ and $\rho$ is right-bounded at $u$, then a simple computation shows that this condition holds if and only if $$R_{\rho,u}L_{\lambda,w}=L_{\lambda,w}R_{\rho,u}.$$ Recall that associated with the left weight $\lambda$ is a forest graph whose vertices are labelled by the elements of $G^+$ and whose edges $(v,ev)$ are labelled by the individual weights $\lambda(v,e)$ for $e\in E(G)$. We may now augment this ${\lambda}$-labelled forest by additional ${\rho}$-edges $(v,ve)$, which are labelled by the individual nonzero weights ${\rho}(v,e)$. The resulting labelled graph is the union of two labelled edge-disjoint forests which share the same vertex set. The commuting square condition can be viewed as a commuting square within this labelled graph, for the weights indicated in Figure \[f:weights\]. ![The commuting square condition. Solid lines are paths made of edges labelled by $\lambda$-weights, and dashed lines are paths of edges labelled by $\rho$-weights.[]{data-label="f:weights"}](diagram) Let $\lambda$ be a left weight on $G$. A right weight $\rho$ on $G$ is a *right companion* to $\lambda$ if $(\lambda,\rho)$ satisfies the commuting square condition at every $(w,u)\in G^+\times G^+$. We call a right companion $\rho$ to $\lambda$ *canonical* if $\rho(r(e),e)=\lambda(s(e),e)$ for all $e\in E(G)$. \[prop:canonical\] For any left weight $\lambda$ on $G$, there is a unique canonical right companion $\rho$ to $\lambda$, namely $\rho=\rho_\alpha$ where $\alpha$ is the path weight with $\lambda=\lambda_\alpha$. Moreover, if $\rho_1$ and $\rho_2$ are both right companions to $\lambda$, then ${{\mathfrak{R}(G,\rho_1)}}={{\mathfrak{R}(G,\rho_2)}}$. Let $\alpha=\alpha_\lambda$ be the path weight given by $\alpha(v)=\lambda(s(v),v)$. Then $\lambda=\lambda_\alpha$ and by an easy calculation, the right weight $\rho_\alpha$ (defined in the previous section) is a canonical right companion to $\lambda$. If $\rho_1$ and $\rho_2$ are both right companions to $\lambda$, then applying the commuting square condition for $\rho_1$ and $\rho_2$ with $v=r(u)=s(w)$ shows that $q(u):=\frac{\rho_2(r(u),u)}{\rho_1(r(u),u)}>0$ satisfies $\rho_2(w,u)=q(u)\rho_1(w,u)$ for any $w,u$ with $wu\in G^+$. So $\rho_1$ is right-bounded at $u$ if and only if $\rho_2$ is right-bounded at $u$, and in this case $R_{\rho_1,u}=q(u)R_{\rho_2,u}$; hence ${{\mathfrak{R}(G,\rho_1)}}={{\mathfrak{R}(G,\rho_2)}}$. If $\rho_1$ and $\rho_2$ are both canonical right companions to $\lambda$, then $q(e)=1$ for all $e\in E(G)$. Applying the cocycle condition for $\rho_1$ and $\rho_2$ to the relation $\rho_2=q\cdot \rho_1$ shows that $q(u_1u_2)=q(u_2)q(u_1)$ whenever $u_1u_2\in G^+$; hence $q(u)=1$ for all $u\in G^+$, so $\rho_1=\rho_2$. For $k\geq0$, let $Q_k$ be the orthogonal projection of $\H_G$ onto the closed linear span of $\{ \xi_v : |v|=k \}$. For $j\in\bZ$, define a complete contraction $\Phi_j\colon \B(\H_G)\to \B(\H_G)$ by $$\Phi_j(X) = \sum_{m\geq \max\{ 0,-j\}} Q_m X Q_{m+j}.$$ Also for $k\in\bN$, define $\Sigma_k\colon \B(\H_G)\to \B(\H_G)$ via the Cesaro-type sums $$\Sigma_k(X) = \sum_{|j|<k} \Big( 1 - \frac{|j|}{k} \Big) \Phi_j(X).$$ The following lemma is well-known. For completeness, we include a short proof. \[lem:Sigma-k\] For $k\ge0$ and $X\in \B(\H_G)$, we have $\|\Sigma_k(X)\|\leq \|X\|$, and $\Sigma_k(X)$ converges to $X$ in the strong operator topology as $k\to \infty$. Let $z\in \bT$ and let $U_z$ be the diagonal unitary operator on $\H_G$ for which $U_z\xi_v = z^{|v|}\xi_v$ for each $v \in G^+$. Then $U_zQ_sXQ_tU_z^* = z^{s-t}Q_sXQ_t$ for any $s,t\ge0$. Since $\sum_{\ell\ge0}Q_\ell=I$, it follows that for $j\in \bZ$, we have $$\Phi_j(X) = \int _{|z|=1}z^{j}U_zXU_z^*dz.$$ Writing $F_k(z)=\sum_{j=-k}^k(1-\frac{|j|}{k+1})z^j$ for the usual Fejér kernel, we see that$$\Sigma_k(X) = \int_{|z|=1}F_{k-1}(z)U_zXU_z^*dz.$$ Considering the scalars $\langle \Sigma_k(X)\xi, \zeta \rangle$, for $\xi, \zeta \in \H_G$, and the fact that $\|F_{k-1}\|_{L^1(\bT)}=1$, it follows that $\|\Sigma_k(X)\| \leq \|X\|$ for all $k$. Let $\xi \in \H_G$. Then $$\|(X-\Sigma_k(X))\xi\| \leq \int_{|z|=1}F_{k-1}(z)\|(X - U_zXU_z^*)\xi\|dz.$$ The operators $U_z, U_z^*$ converge to the identity operator in the strong operator topology, as $z$ tends to $1$, and $F_k$ tends weak star to the unit point mass measure at $z=1$ as $k\to \infty$. It follows that $\Sigma_k(X)\xi \to X\xi$ as $k\to \infty$, and so $\Sigma_k(X){\stackrel{\textsc{sot}}{\to}}X$. \[lem:Xf\] Let $\rho$ be a right weight on $G$ and suppose that $f\colon G^+\to \bC$ has the property that if $f(u)\ne0$, then $\rho(u)<\infty$. Let $\H_0$ be the dense subspace of $\H_G$ spanned by $\{\xi_v\colon v\in G^+\}$, and consider the sesquilinear form $A_f\colon \H_0\times\H_0\to \bC$ with $$A_f(\xi_v,\xi_w)= \begin{cases} f(u)\rho(v,u)&\text{if $w=vu$ for some $u\in G^+$}\\ 0&\text{otherwise}. \end{cases}$$ If $A_f$ is bounded on $\H_0\times \H_0$, then the operator $X_f\in \B(\H_G)$ implementing the continuous extension of $A_f$ to $\H_G\times\H_G$ satisfies $X_f\in {{\mathfrak{R}(G,\rho)}}$. Let $(G_1,G_2,\dots)$ be a sequence of finite subgraphs of $G$ which increases to $G$; that is, $V(G_n)$ and $E(G_n)$ are finite sets for each $n$, and $V(G_n)\uparrow V(G)$ and $E(G_n)\uparrow E(G)$. For $n\ge1$, let $P_n$ be the projection onto the closure of the subspace of $\H_G$ spanned by $\{\xi_v\colon v\in G_n^+\}$. For $v,w\in G^+$, a calculation shows that $\langle P_n\Phi_j(X_f)P_n\xi_v,\xi_w\rangle=0$ unless $v,w\in G_n^+$ with $w=vu$ for some $u\in G_n^+$ with $|u|=-j$ and $\rho(u)<\infty$; and that in the latter case, $$\langle P_n\Phi_j(X_f)P_n\xi_v,\xi_w\rangle=A_f(\xi_v,\xi_w)=f(u)\rho(v,u).$$ It follows that $P_n\Phi_j(X_f)P_n= P_nF_{j,n} P_n$ where $$F_{j,n}= \displaystyle\sum_{\substack{u\in G_n^+,\,|u|=-j,\\\rho(u)<\infty}} f(u)R_{\rho,u}.$$ (We have $F_{j,n}=0$ if $j>0$.) Since $V(G_n)$ and $E(G_n)$ are finite, $F_{j,n}$ is a finite linear combination of operators $R_{\rho,u}$, so $F_{j,n}\in {{\mathfrak{R}(G,\rho)}}$. Now $P_n{\stackrel{\textsc{sot}}{\to}}I$, so $$\Phi_j(X_f)=\operatorname*{{\mbox{\scshape sot}}-lim}_{n\to \infty} P_n F_{j,n} P_n.$$ In fact, we will shortly see that $$\Phi_j(X_f)=\operatorname*{{\mbox{\scshape wot}}-lim}_{n\to \infty} F_{j,n}.$$ From this, it follows that $\Phi_j(X_f)\in {{\mathfrak{R}(G,\rho)}}$, so $\Sigma_k(X_f)\in {{\mathfrak{R}(G,\rho)}}$ for all $k\ge1$, allowing us to conclude, by Lemma \[lem:Sigma-k\], that $X_f=\operatorname*{{\mbox{\scshape sot}}-lim}_{k\to\infty}\Sigma_k(X_f)\in {{\mathfrak{R}(G,\rho)}}$ as desired. To see this, we will first show that $\{\|F_{j,n}\|\colon n\ge 1\}$ is bounded. Note that for any $v\in G^+$, we have the norm-convergent sums $$X_f\xi_v=\sum_{w\in G^+} \langle X_f\xi_v,\xi_w\rangle \xi_w = \sum_{w\in G^+}A_f(\xi_v,\xi_w)\xi_w = \sum_{u\in G^+}f(u)\rho(v,u)\xi_{vu}.$$ Moreover, $$F_{j,n}\xi_v=\sum_{\substack{u\in G_n^+,\,|u|=-j,\\\rho(u)<\infty}}f(u)\rho(v,u)\xi_{vu}$$ so $\|F_{j,n}\xi_v\|\leq \|X_f\|$. For $i=1,2$, if $v_iu_i\in G^+$ and $|u_1|=|u_2|=-j$, then $v_1\ne v_2\implies v_1u_1\ne v_2u_2$. It follows that $\{ F_{j,n}\xi_v\colon v\in G^+\}$ is a pairwise orthogonal family of vectors for each $n\ge1$, hence $F_{j,n}=\sum^{\oplus}_{v\in G^+} F_{j,n}\xi_v\xi_v^*$ and so $$\|F_{j,n}\|=\sup_{v\in G^+} \|F_{j,n} \xi_v\|\leq \|X_f\|.$$ Now $P_n^\perp:=I-P_n{\stackrel{\textsc{sot}}{\to}}0$ as $n\to \infty$, so $P_nF_{j,n}P_n^\perp{\stackrel{\textsc{sot}}{\to}}0$ and $P_n^\perp F_{j,n}{\stackrel{\textsc{wot}}{\to}}0$ as $n\to \infty$. Hence $$F_{j,n}=P_nF_{j,n}P_n+P_nF_{j,n}P_n^\perp+P_n^\perp F_{j,n}{\stackrel{\textsc{wot}}{\to}}\Phi_j(X_f) \quad\text{as $n\to \infty$},$$ which completes the proof. It is not difficult to see that if the function $f$ in Lemma \[lem:Xf\] has finite support, then $$X_f= \sum_{\substack{u\in G^+,\\\rho(u)<\infty}}f(u)R_{\rho,u}.$$ Heuristically, it is useful to think of $X_f$ as the formal series given by this formula even when the support of $f$ is infinite. \[lem:ker\] Let $\lambda$ be a left-bounded left weight on $G$. If $K\in {{\mathfrak{L}(G,\lambda)}}'$ and $K\xi_x=0$ for all $x\in V(G)$, then $K=0$. Given $w\in G^+$, consider $x=s(w)$. We have $$K\xi_w=\lambda(x,w)^{-1}KL_{\lambda,w}\xi_x=\lambda(x,w)^{-1}L_{\lambda,x}K\xi_x=0,$$ so $K=0$. We are now ready to prove our main result. \[commutant\_thm\] If $\lambda$ is a left-bounded left weight on $G$ and $\rho$ is its canonical right companion, then the commutant of ${{\mathfrak{L}(G,\lambda)}}$ coincides with ${{\mathfrak{R}(G,\rho)}}$. The observations at the start of this section show that $L_{\lambda,w}$ commutes with $R_{\rho,u}$ whenever $w,u\in G^+$ and $\rho(u)<\infty$. Hence ${{\mathfrak{L}(G,\lambda)}}^\prime$ contains ${{\mathfrak{R}(G,\rho)}}$. To prove the other inclusion, begin by fixing $S \in {{\mathfrak{L}(G,\lambda)}}^\prime$. For $u\in G^+$, consider the coefficients $a_u\in \bC$ defined by $$a_u=\langle S\xi_{r(u)},\xi_u\rangle.$$ Observe that for any $x\in V(G)$, the operator $L_{\lambda,x}=L_x$ is a projection with range spanned by $\{\xi_u\colon u\in r^{-1}(x)\}$, and $L_{x} S\xi_x=SL_{x}\xi_x=S\xi_x$. Hence $$S\xi_x=\sum_{u\in G^+} \langle S\xi_x,\xi_u\rangle \xi_u=\sum_{u\in r^{-1}(x)} a_u\xi_u$$ with convergence in norm. If $v,w\in G^+$, then $\xi_v=\lambda(s(v),v)^{-1}L_{\lambda,v} \xi_{s(v)}$ and $[S,L_{\lambda,v}]=0$, and $L_{\lambda,v}^*\xi_w=0$ unless $w=vu$ for some $u\in G^+$, and $L_{\lambda,v}^*\xi_{vu}=\lambda(u,v)\xi_u$. Now $\frac{\lambda(u,v)}{\lambda(s(v),v)}=\frac{\rho(v,u)}{\rho(r(u),u)}$ by the commuting square condition, and it follows that $$\begin{aligned} \langle S\xi_v,\xi_w\rangle &= \begin{cases} \frac{\rho(v,u)}{\rho(r(u),u)}a_u &\text{if $w=vu$ for some $u\in G^+$}\\0&\text{otherwise}. \end{cases}\end{aligned}$$ In particular, if $a_u\ne 0$, then $\rho(u)<\infty$ since $$\|S\|\ge\sup_{\{v\in G^+\colon vu\in G^+\}}|\langle S\xi_v,\xi_{vu}\rangle | = \sup_{v\in G^+}\frac{\rho(v,u)}{\rho(r(u),u)}|a_u|=\frac{\rho(u)}{\rho(r(u),u)}|a_u|.$$ In view of this, if we define $f\colon G^+\to \bC$ by $f(u)=a_u \rho(r(u),u)^{-1}$, then we may legitimately consider the bilinear form $A_{f}\colon \H_0\times \H_0\to \bC$, defined as in Lemma \[lem:Xf\]. Consider the operators $\Sigma_{k}(S)$. If $v,w\in G^+$ and $\big||w|-|v|\big|<k$, then $$\begin{aligned} \langle \Sigma_{k}(S)\xi_v,\xi_w\rangle &= \sum_{|j|<k}\left(1-\frac{|j|}k\right)\sum_{m\geq \max\{0,-j\}}\langle SQ_{m+j}\xi_v,Q_m\xi_w\rangle\\ &= \left(1-\frac{\big||w|-|v|\big|}k\right)\langle S\xi_v,\xi_w\rangle\\ &= \begin{cases} \left(1-\frac{|w|-|v|}k\right)f(u)\rho(v,u)&\text{if $w=vu$ for some $u\in G^+$}\\0&\text{otherwise} \end{cases}\\ &= \left(1-\frac{|w|-|v|}k\right)A_f(\xi_v,\xi_w).\end{aligned}$$ Hence for any $\xi,\eta\in \H_0$, we have $A_f(\xi,\eta)=\lim_{k\to \infty}\langle \Sigma_k(S)\xi,\eta\rangle$, so by Lemma \[lem:Sigma-k\], $$|A_f(\xi,\eta)|\leq \sup_{k\ge1}\|\Sigma_k(S)\|\,\|\xi\|\,\|\eta\|\leq \|S\|\,\|\xi\|\,\|\eta\|.$$ Thus $A_f$ is bounded on $\H_0\times \H_0$. By Lemma \[lem:Xf\], the bounded linear operator $X=X_f$ implementing $A_f$ is in ${{\mathfrak{R}(G,\rho)}}$. So $X\in {{\mathfrak{L}(G,\lambda)}}'$, and (as above) we conclude that for $x\in V(G)$, the vector $X\xi_x$ is in the closed subspace spanned by $\{\xi_u\colon u\in r^{-1}(x)\}$. Moreover, for any $u\in r^{-1}(x)$ we have $$\langle X\xi_x,\xi_u\rangle = A_f(\xi_x,\xi_u)=f(u)\rho(x,u)=a_u=\langle S\xi_x,\xi_u\rangle.$$ So $X\xi_x=S\xi_x$ for all $x\in V(G)$. Since $X,S\in {{\mathfrak{L}(G,\lambda)}}'$, we have $K=X-S\in {{\mathfrak{L}(G,\lambda)}}'$ and $K\xi_x=0$ for all $x\in V(G)$. Hence $S=X$ by Lemma \[lem:ker\], and so $S\in {{\mathfrak{R}(G,\rho)}}$, which completes the proof. This result generalizes and improves on a few previous results. The case of the single vertex and single edge graph yields classical single variable weighted shift operators, and there, the notion of right-boundedness simply corresponds to the weight sequence being bounded below. Hence this result generalizes the fundamental commutant theorem for weighted Bergman spaces $H^\infty(\beta)$ [@Shields74]. In the case of a single vertex graph with $n$ edges and unit weights this result captures the commutant theorem for free semigroup algebras $\L_n$ [@davidsonpitts2], and it improves on the commutant result of [@kribs04], which established a special case of the theorem in the single vertex multi-edged weighted shift case. Finally, this result generalizes the commutant theorem for free semigroupoid algebras [@kribspower] which are determined by general unweighted directed graphs. Double Commutant Theorems ========================= When $\rho$ is right-bounded, we obtain the following mirror image of Theorem \[commutant\_thm\] which may be established with a flipped version of the preceding proof. For brevity, we will instead pass to the opposite graph ${{{G}^t}}$ of $G$, which is essentially “$G$ with the edges reversed”. More formally, we set $V({{{G}^t}})=V(G)$, $E({{{G}^t}})=E(G)$ and $({{{G}^t}})^+=\{{{{v}^t}}\colon v\in G^+\}$, where ${{{(wv)}^t}}={{{v}^t}}{{{w}^t}}$ for $wv\in G^+$, and ${{{u}^t}}=u$ for $u\in V(G)\cup E(G)$; the source and range maps for ${{{G}^t}}$ are given by $ {{{s}^t}}({{{v}^t}})=r(v)$ and ${{{r}^t}}({{{v}^t}})=s(v)$. If $\lambda$ is a left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then the commutant of ${{\mathfrak{R}(G,\rho)}}$ coincides with ${{\mathfrak{L}(G,\lambda)}}$. Let ${{{G}^t}}$ be the opposite graph of $G$ and let ${{{\rho}^t}}({{{v}^t}},{{{u}^t}})={\rho}(v,u)$ for $v,u\in G^+$; since $\rho$ is a right-bounded right weight on $G$, it follows that ${{{\rho}^t}}$ is a left-bounded left weight on $G^t$. A calculation using the path weight associated with $\lambda$ and $\rho$ shows that the canonical right companion to ${{{\rho}^t}}$ is the right weight ${{{\lambda}^t}}$ on $G^t$ given by ${{{\lambda}^t}}(v^t,w^t)=\lambda(v,w)$. Let $U\colon \H_{{{{G}^t}}}\to \H_G$ be the unitary with $U\xi_{{{{v}^t}}}=\xi_v$. For $u,w\in G^+$ with $\lambda(w)<\infty$, by checking values on basis vectors we see that $$UL_{\rho^t,{{{u}^t}}}U^*=R_{\rho,u} {{\quad\text{and}\quad}}UR_{\lambda^t,{{{w}^t}}}U^*=L_{\lambda,w},$$ so $$U{{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}} U^*={{\mathfrak{R}(G,\rho)}} {{\quad\text{and}\quad}}U {{\mathfrak{R}({{{G}^t}},{{{{\lambda}}^t}})}}U^*={{\mathfrak{L}(G,\lambda)}}.$$ By Theorem \[commutant\_thm\], ${{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}}'={{\mathfrak{R}({{{G}^t}},{{{\lambda}^t}})}}$, hence $$\begin{aligned} {{\mathfrak{R}(G,\rho)}}'=(U{{\mathfrak{L}({{{G}^t}},{{{{\rho}}^t}})}}U^*)'&=U{{\mathfrak{L}({{{G}^t}},{{{\rho}^t}})}}'U^*\\&=U{{\mathfrak{R}({{{G}^t}},{{{{\lambda}}^t}})}}U^*={{\mathfrak{L}(G,\lambda)}}.\qedhere \end{aligned}$$ Combining the previous two results leads us to the following double commutant theorem. \[preconj\] If $\lambda$ is a left-bounded left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then ${{\mathfrak{L}(G,\lambda)}}$ coincides with its double commutant: $${{\mathfrak{L}(G,\lambda)}}'' = {{\mathfrak{L}(G,\lambda)}}.$$ If $\alpha\colon G^+\to (0,\infty)$ is any path weight with $\sup_v\alpha(v)<\infty$ and $\inf_v\alpha(v)>0$, then plainly $\lambda_\alpha$ is left-bounded and its canonical right companion $\rho_\alpha$ is right-bounded, giving a large class of weights satisfying the hypotheses of this result. In particular, if $|G^+|<\infty$ (i.e., if $G$ is a finite *acyclic* directed graph), then for any left weight $\lambda$ on $G$, we see that ${{\mathfrak{L}(G,\lambda)}}$ is an algebra of $n\times n$ matrices with ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$, where $n=|G^+|$. On the other hand, there are many weights which satisfy the hypotheses of Theorem \[preconj\] but violate these boundedness conditions for the path weight $\alpha$. For example, if $G^+$ contains paths of arbitrary length and $\alpha(v)=f(|v|)$ where $f\colon \bN_0\to (0,\infty)$ is any decreasing function with $f(0)=1$ and $f(k)\to 0$ as $k\to \infty$, then $\lambda_\alpha$ is left-bounded and $\rho_\alpha$ is right-bounded, but $\inf_v\alpha(v)=0$. We now show one way to weaken the hypotheses in Theorem \[preconj\], at least if $G$ is a finite directed graph. We first require a technical lemma. \[lem:phi\] Let $\rho$ be a right weight on $G$ and let $u\in G^+$ with $\rho(u)<\infty$. Let $k\ge0$ and let $X\in \B(\H_G)$. If $[X,R_{\rho,u}]=0$, then $[\Sigma_{k}(X),R_{\rho,u}]=0$. By calculating values on canonical basis vectors, we observe that $$Q_mR_{\rho,u}= \begin{cases} R_{\rho,u}Q_{m-|w|}&\text{if $m\ge|u|$}\\ 0&\text{if $0\leq m<|u|$.} \end{cases}$$ So if $[X,R_{\rho,u}]=0$, then $$\Phi_j(X)R_{\rho,u}=\sum_{m\geq \max\{|u|,|u|-j\}}R_{\rho,u}Q_{m-|u|}XQ_{m+j-|u|}=R_{\rho,u}\Phi_j(X).$$ Since $\Sigma_{k}(X)$ is a linear combination of the operators $\Phi_j(X)$, which all commute with $R_{\rho,u}$, we see that $\Sigma_{k}(X)$ commutes with $R_{\rho,u}$. For any right weight ${\rho}$ on $G$, let us write $$G^+_{\rho}=\{u\in G^+\colon {\rho}(u)<\infty\}.$$ Since $\rho(x)=1$ for $x\in V(G)$, we have $V(G)\subseteq G^+_\rho$. Moreover, since $\rho(vw)\leq \rho(v)\rho(w)$ for any $vw\in G^+$, we see that $G^+_\rho$ is a subsemigroupoid in $G^+$. \[thm:tails\] Let ${\lambda}$ be a left-bounded left weight on a finite directed graph $G$, with canonical right companion $\rho$. If $$\label{eq:tails} \forall\,v\in G^+\ \exists\,u_v\in G^+_\rho\colon vu_v\in G^+_\rho,$$ then ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$. It suffices to show that ${{\mathfrak{L}(G,\lambda)}}''\subseteq {{\mathfrak{L}(G,\lambda)}}$; equivalently (by Theorem \[commutant\_thm\]) that ${{\mathfrak{R}(G,\rho)}}'\subseteq {{\mathfrak{L}(G,\lambda)}}$. Suppose that $T\in{{\mathfrak{R}(G,\rho)}}'$. If $v\in G^+$, then $R_{\rho,s(v)}=R_{s(v)}$ is a projection in ${{\mathfrak{R}(G,\rho)}}$ with $\xi_v=R_{s(v)}\xi_v$ and $[T,R_{s(v)}]=0$, from which it follows that $T\xi_v\in R_{s(v)}\H_G$. Moreover, if $u\in G^+_\rho$, then the restriction of $R_{\rho,u}^*R_{\rho,u}$ to $R_{r(u)}\H_G$ is an injective diagonal operator since it maps $\xi_v$ to $\rho(v,u)^2\xi_v$ if $s(v)=r(u)$. Now suppose $K\in {{\mathfrak{R}(G,\rho)}}'$ and $K\xi_x=0$ for all $x\in V(G)$; we claim that we necessarily have $K=0$. To see this, let $v\in G^+$, let $u_v$ be as in Eq. (\[eq:tails\]) and note that $r(u_v)=s(v)$, since $vu_v\in G^+$. Now $$\begin{aligned} R_{\rho,u_v}^*R_{\rho,u_v}K\xi_v&= R_{\rho,u_v}^*KR_{\rho,u_v}\xi_v =\rho(v,u_v)R^*_{\rho,u_v}K\xi_{vu_v} \\&= \rho(v,u_v)\rho(r(v),vu_v)^{-1}R_{\rho,u_v}^*R_{\rho,vu_v}K\xi_{r(v)}=0, \end{aligned}$$ so $K\xi_v=0$ by the observations of the previous paragraph, establishing the claim. Now let $T\in {{\mathfrak{R}(G,\rho)}}'$ be arbitrary. Since $G$ is finite, for each $k\in \bN$ the set $\{w\in G^+\colon |w|<k\}$ is finite and we may consider the operator $$p_k(T)=\sum_{\{w\in G^+\colon |w|<k\}} \left(1-\frac{|w|}k\right) a_w \lambda(s(w),w)^{-1}L_{\lambda,w}$$ where $a_w=\langle T\xi_{s(w)},\xi_w\rangle$ for $w\in G^+$. Clearly, $p_k(T)\in {{\mathfrak{L}(G,\lambda)}}$. As observed above, we have $T\xi_x\in R_x\H_G$, so $$T\xi_x=\sum_{w\in s^{-1}(x)} a_w\xi_w.$$ By Lemma \[lem:phi\], the operators $\Sigma_k(T)$ are in ${{\mathfrak{R}(G,\rho)}}'$. Hence $K=\Sigma_k(T)-p_k(T)\in {{\mathfrak{R}(G,\rho)}}'$, and a calculation gives $K\xi_x=0$ for all $x\in V(G)$. Hence $\Sigma_k(T)=p_k(T)\in {{\mathfrak{L}(G,\lambda)}}$. Since ${{\mathfrak{L}(G,\lambda)}}$ is strongly closed and $\Sigma_k(T)\to T$ strongly, we obtain $T\in {{\mathfrak{L}(G,\lambda)}}$. Hence ${{\mathfrak{R}(G,\rho)}}'\subseteq{{\mathfrak{L}(G,\lambda)}}$ which completes the proof. We now give some examples illustrating this result. If $u\in G^+$, it will be useful to write $$G^+u=\{vu\colon v\in G^+ \text{ and } vu\in G^+\}.$$ ![The path weight $\alpha$ considered in Example \[exa:rho(v)\][]{data-label="f:eg47a"}](eg45-alpha) ![The path weight $\alpha$ considered in Example \[exa:rho(v)\][]{data-label="f:eg47a"}](eg47-alpha) \[exa:rho(v)1\] Let $G$ be the directed graph with a single vertex $\phi$ and two loop edges, $e$ and $f$, so that $G^+=\{\phi,e,f,ee=e^2,ef,fe,f^2,eee=e^3,\dots\}$ and $s(w)=r(w)=\phi$ for every $w\in G^+$ (see Figure \[f:eg45\]). As indicated in Figure \[f:eg45a\], we consider the path weight $\alpha\colon G^+\to (0,\infty)$ given by $$\alpha(v)= \begin{cases} 2^{-|v|}&\text{if $v\in G^+e$}\\ 1&\text{otherwise}, \end{cases}$$ and let $\lambda=\lambda_\alpha$. It is easy to check that the left weight $\lambda$ is left-bounded (in fact $\lambda(w)\leq 1$ for all $w\in G^+$); the canonical right companion of $\lambda$ is $\rho=\rho_\alpha$ by Proposition \[prop:canonical\]. We claim that while $\rho$ is not right-bounded, we have $$G^+_{\rho}=\{\phi\}\cup G^+e = G^+\setminus G^+f,$$ so that Eq. (\[eq:tails\]) holds with $u_v=e$ for all $v\in G^+$, hence ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$. Let us check that $G^+_{\rho}$ is indeed of this form. We have ${\rho}(v,u)=\frac{\alpha(vu)}{\alpha(v)}$ for any $ v,u\in G^+$. In particular, if $u\in G^+f$, then $\alpha(vu)=1$ and so $\rho(v,u)=2^{|v|}$ for $v\in G^+e$, so ${\rho}(u)=\infty$. On the other hand, if $u\in G^+\setminus G^+f$, then $\alpha(vu)=2^{-|vu|}\leq \alpha(v)$ for all $v\in G^+$, so $\rho(u)\leq 1$. \[exa:twovertex\] Let $G$ be the directed $2$-cycle, so that $V(G)=\{x,y\}$ and $E(G)=\{e,f\}$ where $s(e)=r(f)=x$ and $s(f)=r(e)=y$ (see Figure \[f:eg46\]). For this particular graph $G$, we will show that if $\lambda$ is any left-bounded left weight on $G$ with right companion $\rho$, then ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$ if and only if $G_\rho^+$ satisfies Eq. (\[eq:tails\]). Note that the edges in any path in $G^+$ must alternate: $$G^+=\{x,y,e,f,ef,fe,efe,fef,efef,fefe,\dots\}.$$ We first show that $ef\in G^+_\rho$. Let $v\in G^+\setminus V(G)$ with $vef\in G^+$. Either $v=(ef)^{k}$ or $v=f(ef)^{k-1}=(fe)^{k-1}f$ for some $k\ge1$. Let $\alpha\colon G^+\to (0,\infty)$ be the path weight with $\lambda=\lambda_\alpha$ and $\rho=\rho_\alpha$. Since $(ef)^kef=(ef)^{k+1}=ef(ef)^k$, we have $$\rho((ef)^k,ef)=\frac{\alpha((ef)^{k+1})}{\alpha((ef)^k)} = \lambda((ef)^k,ef)\leq \lambda(ef)<\infty$$ and since $f(ef)^{k-1}ef=f(ef)^k=(fe)^kf$, we have $$\rho(f(ef)^{k-1},ef)=\frac{\alpha((fe)^{k}f)}{\alpha((fe)^{k-1}f)}=\lambda((fe)^{k-1}f,fe)\leq \lambda(fe)<\infty,$$ so $\rho(ef)<\infty$, i.e., $ef\in G^+_\rho$. Similarly, $fe\in G^+_\rho$. Since $G_\rho^+$ is a semigroupoid, we have $\langle ef,fe\rangle \subseteq G^+_\rho$ where $\langle ef,fe\rangle := V(G)\cup \{ (ef)^k,(fe)^k\colon k\ge1\}$. If $G_\rho^+\supsetneq\langle ef,fe\rangle$, then $G_\rho^+$ contains an element of odd length. By symmetry, we may assume this is of the form $e(fe)^n$ for some $n\ge0$. We then also have $e(fe)^m=e(fe)^n(fe)^{m-n}\in G_\rho^+$ for any $m> n$, so if we define $u_v$ for $v\in G^+$ by $$u_v= \begin{cases} s(v)&\text{if $v\in \langle ef,fe\rangle$}\\ (fe)^n&\text{if $v=e(fe)^k$ for some $k\ge0$}\\ e(fe)^n&\text{if $v=f(ef)^k$ for some $k\ge0$,} \end{cases}$$ then $u_v\in G^+_\rho$ and $vu_v\in G^+_\rho$ for all $v\in G^+$, so Eq. (\[eq:tails\]) holds and so ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{L}(G,\lambda)}}$ by Theorem \[thm:tails\]. On the other hand, if $G^+_\rho=\langle ef,fe\rangle$, then Eq. (\[eq:tails\]) does not hold, since if $|v|$ is odd and $u\in G^+_\rho$ with $vu\in G^+$, then $|vu|$ is also odd, so $vu\not\in G_\rho^+$. In this case, it is not difficult to see that the orthogonal projection $P$ onto the closed linear span of $\{\xi_v\colon \text{$v=x$ or $v=(fe)^k$, $k\ge1$}\}$ commutes with $R_{\rho,u}$ for $u\in \{x,y,ef,fe\}$, hence $P\in {{\mathfrak{R}(G,\rho)}}'={{\mathfrak{L}(G,\lambda)}}''$. If $T\in {{\mathfrak{L}(G,\lambda)}}$, then $\langle T\xi_x,\xi_x\rangle = \langle T\xi_f,\xi_f\rangle$. Since $P\xi_x=\xi_x$ and $P\xi_f=0$, we have $P\not\in {{\mathfrak{L}(G,\lambda)}}$. So ${{\mathfrak{L}(G,\lambda)}}''\ne {{\mathfrak{L}(G,\lambda)}}$. We note that it is indeed possible for Eq. (\[eq:tails\]) to fail for this graph $G$. For example, let $\alpha\colon G^+\to (0,\infty)$ with $\alpha(v)=1$ for all $v\in s^{-1}(x)$, and $\alpha(v)=2^{f(|v|)}$ for $v\in s^{-1}(y)$ where $f\colon \bN_0\to \bZ$ is a function with $f(0)=0$, and $f(n+1)\in \{f(n)-1,f(n)+1\}$ for all $n\in \bN_0$ and with $\sup_n f(n)=\infty$ and $\inf_n f(n)=-\infty$. One may then check that the left weight $\lambda_\alpha$ is left-bounded, and that its canonical right companion $\rho$ satisfies $G_\rho^+=\langle ef,fe\rangle$, so Eq. (\[eq:tails\]) fails. \[exa:rho(v)\] For a general left-bounded weight $\lambda$, the double commutant property for ${{\mathfrak{L}(G,\lambda)}}$ can fail very badly. For example, let $G$ again be the directed graph with a single vertex $\phi$ and two loop edges $e$ and $f$, and let us now define a path weight $\alpha\colon G^+\to (0,\infty)$ recursively by setting $\alpha(\phi)=\alpha(e)=\alpha(f)=1$, and $$\begin{aligned} \alpha(ewe)&=\tfrac12 \alpha(we),&\alpha(fwf)&=\tfrac 12 \alpha(wf),\\ \alpha(ewf)&=\alpha(wf), &\alpha(fwe)&=\alpha(we). \end{aligned}$$ This is illustrated in Figures \[f:eg45\] and \[f:eg47a\]. Take $\lambda=\lambda_\alpha$ and $\rho=\rho_\alpha$. Observe that ${\lambda}$ is a left-bounded left weight since $\alpha(wv)\leq \alpha(v)$ for all $w,v\in G^+$. For any $k\in \bN$ and $w\in G^+$, $${\rho}(we)\geq {\rho}(f^k,we)=\frac{\alpha(f^kwe)}{\alpha(f^k)}=2^{k-1} \alpha(we)\to \infty\text{ as~$k\to \infty$,}$$ hence ${\rho}(we)=\infty$; by symmetry, ${\rho}(wf)=\infty$. Hence $ G^+_{\rho}=\{\phi\}$. Since $R_{\rho,\phi}=R_\phi=I$, we have ${{\mathfrak{R}(G,\rho)}}=\bC I$ and so ${{\mathfrak{L}(G,\lambda)}}''={{\mathfrak{R}(G,\rho)}}'=\B(\H_G)\ne {{\mathfrak{L}(G,\lambda)}}$. The commutant result yields other structural results on the algebras, such as the following. Let ${\lambda}$ be a left-bounded left weight on $G$ with canonical right companion $\rho$. If either $\rho$ is right-bounded, or $G$ is finite and $G^+_{\rho}$ satisfies Eq. (\[eq:tails\]), then ${{\mathfrak{L}(G,\lambda)}}$ is inverse closed. This is a well-known property of commutants: if $A\in{{\mathfrak{L}(G,\lambda)}} = {{\mathfrak{R}(G,\rho)}}'$ is invertible in $\B(\H_G)$, then for all $R\in{{\mathfrak{R}(G,\rho)}}$, $A^{-1}R = A^{-1} RAA^{-1} = RA^{-1}$, and hence $A^{-1}\in{{\mathfrak{R}(G,\rho)}}' = {{\mathfrak{L}(G,\lambda)}}$. If ${\lambda}$ is a left-bounded left weight on $G$ whose canonical right companion $\rho$ is right-bounded, then every normal element of ${{\mathfrak{L}(G,\lambda)}}$ lies in the SOT-closure of the linear span of the projections $L_{\lambda,x}$ for $x\in V(G)$. Let $N$ be a normal element of ${{\mathfrak{L}(G,\lambda)}}$. Set $a_x = \langle N\xi_x,\xi_x\rangle$ for $x\in V(G)$ and let $M$ be the SOT-convergent sum $M=\sum_{x\in V(G)}a_xL_{\lambda,x}$. Observe that each $\xi_x$ is an eigenvector for ${{\mathfrak{L}(G,\lambda)}}^*$, as for all $u\in G^+\setminus\{x\}$ and $A\in {{\mathfrak{L}(G,\lambda)}}$, we have $$\langle A^*\xi_x, \xi_u\rangle =\tfrac1{{\rho}(r(u),u)} \langle\xi_x, A R_{\rho,u} \xi_{r(u)}\rangle= \tfrac1{{\rho}(r(u),u)}\langle R_{\rho,u}^*\xi_x,A\xi_{r(u)}\rangle=0.$$ Thus $N^* \xi_x = \overline{a_x} \xi_x$, and by normality $N\xi_x = a_x \xi_x$. Hence for all $u\in G^+$, $$N\xi_u = \tfrac1{{\rho}(r(u),u)}N R_{\rho,u} \xi_{r(u)} = \tfrac1{{\rho}(r(u),u)}R_{\rho,u} N\xi_{r(u)} = a_{r(u)} \xi_u=M\xi_u,$$ so $N=M$. Building on Theorems \[preconj\] and \[thm:tails\], a natural open problem is to determine weighted graph conditions that fully characterize when the algebra ${{\mathfrak{L}(G,\lambda)}}$ and its double commutant ${{\mathfrak{L}(G,\lambda)}}''$ coincide. This work was partly supported by a London Mathematical Society travel grant. The first named author was partly supported by NSERC Discovery Grant 400160 and a University Research Chair at Guelph. We are grateful to the referee for helpful suggestions.

ArXiv

--- abstract: 'The potential weakness of the Y-00 direct encryption protocol when the encryption box ENC is not chosen properly is demonstrated in a fast correlation attack by S. Donnet et al in Phys. Lett. A 356 (2006) 406-410. In this paper, we show how this weakness can be eliminated with a proper design of ENC. In particular, we present a Y-00 configuration that is more secure than AES under known-plaintext attack. It is also shown that under any ciphertext-only attack, full information-theoretic security on the Y-00 seed key is obtained for any ENC when proper deliberate signal randomization is employed.' author: - | Horace P. Yuen [^1] and Ranjith Nair\ Center for Photonic Communication and Computing\ Department of Electrical and Computer Engineering\ Department of Physics and Astronomy\ Northwestern University, Evanston, IL 60208 title: 'On the Security of Y-00 under Fast Correlation and Other Attacks on the Key' --- Introduction ============ The quantum-noise based direct encryption protocol Y-00, called $\alpha\eta$ in our earlier papers \[1-6\], was repeatedly misrepresented in previous criticisms, but that situation has apparently changed with our recent papers \[7-9\]. For the first time, a meaningful attack on Y-00 type protocols beyond exhaustive search has been developed in [@donnet]. A fast correlation attack (FCA) was presented that was shown to succeed by simulations for moderate signal levels when the ENC box in Y-00 is a LFSR (linear feedback shift register) of a few taps and length up to 32. Even though such Y-00 is already insecure against what we call assisted brute-force search [@nair06] due to the small seed key size $|K| \leq 32$, such FCA is of interest as it brings forth the whole issue of Y-00 seed key security against similar and other attacks. The attack in [@donnet] is geared toward only the experiment reported in [@optlett03]. We have emphasized all along [@yuen04; @ptl05; @spie05] that the use of LFSR in the reported experiments was just for proof of principle demonstration, that the ENC box must be chosen appropriately in a final design, and that other techniques need to be deployed for proper security. To quote from [@spie05], “Similar to encryption based on nonlinearly combining the LFSR’s, Eve can launch a correlation attack using the following strategy: $\ldots$ many of the LFSR’s could be trivially attacked.” Thus, we were aware of the possible weakness of some ENC and in particular of FCA type attacks. Indeed, Hirota and Kurosawa [@hirota06] have already desribed a counter-measure to FCA via a “keyed mapper”, the incorporation of which in ASK-signal Y-00 [@hirota05] has been developed and is being tested. Generally speaking, it is important to study LFSR-based Y-00 despite its possible weakness, because LFSR is a practically convenient choice in various applications similar to the situation in standard cryptography. In this paper, we first briefly describe general attacks on the Y-00 seed key as a problem of decoding in real noise – a viewpoint which includes all FCA’s. For both ciphertext-only attacks (CTA) and known-plaintext attacks (KPA), we show that Y-00 may be considered as a classical stream cipher, the ENC box, with real physical noise added on top. We comment on the possible defenses involving just a properly chosen LFSR, or an added keyed mapper, or with a keyed connection polynomial for the LFSR. We describe an AES-based Y-00 that is more secure against KPA than AES (Advanced Encryption Standard) alone, in the sense that if it is broken then AES is also broken but not the other way around. The practical security advantage of such AES-based Y-00 will be indicated. Finally, for CTA, we show that Deliberate Signal Randomization (DSR) introduced in [@yuen04] provides full information-theoretic security on the Y-00 seed key for any ENC. We hope that these results would establish beyond doubt that Y-00 is an important cryptosystem to consider in theory and in practice. Attacks on Y-00 seed key ======================== Consider the original quantum-noise randomized cipher Y-00 [@prl; @yuen04] as depicted in Fig. 1. Alice encodes each data bit into a $2M-$ary phase-shifted coherent state in a qumode of energy $\alpha^2_0$. A seed key $K$ of bit length $|K|$ is used to drive a conventional stream cipher ENC to produce a running key $K'$ that is used to determine, for each qumode carrying the bit, which pair of antipodal coherent states, referred to as a basis, is to be used as a binary phase-shift keying (BPSK) signal set for Bob. With a synchronous ENC at the receiver, Bob discriminates the BPSK signals for each qumode by an appropriate receiver. With a differential (DPSK) implementation [@prl; @yuen04; @optlett03; @ptl05; @pra05; @spie05], there is no need to phase lock between Alice and Bob as is true in ordinary communications. \[htbp\] The optimum quantum receiver performance for both Bob and Eve is the same as in the non-differential case in principle, the differential implementation being a practical convenience. Even with a full copy of the quantum state granted to Eve in our KCQ approach of performance analysis [@yuen04; @pla05; @yuen05qph; @nair06], security on the data is nearly perfect when the seed key induced correlation is neglected [@prl]. Generally, it is a horrendous problem with yet no solution for meaningfully quantifying the data security of a symmetric-key cipher. In current practice, it is assumed that CTA on the data is not a problem if $|K|$ is “large”, and attention is focused on KPA on the key. For conventional or standard [@yuen05qph; @nair06] ciphers, the key is usually completely protected from CTA for uniformly random data. This is, however, not the case for the bare Y-00 [@yuen04; @pla05; @yuen05qph; @nair06]. In this paper, we address both CTA and KPA on the Y-00 seed key, the (classical) ciphertext being obtained from some quantum measurement on the qumodes assumed to be in Eve’s possession. It is seen from Fig. 1 that a CTA or KPA on the Y-00 seed key is equivalent to the corresponding attack on the standard stream cipher ENC with its output stream observed in noise resulting from the coherent state randomization of the signal phase. Thus, it is equivalent to a CTA or KPA on the ENC alone as a stream cipher but with noise on top. The connection between the running key bits $K'$ and the basis, called the “*mapper*” [@pra05; @spie05; @hirota06], a crucial component of Y-00, and the noise effect on $K'$ are described in [@spie05; @donnet]. In a FCA on a conventional stream cipher composed of, say, a nonlinear combination of the outputs of a bank of $m$ LFSR’s, one focuses on one LFSR $L_i$ at a time and looks for correlation between the final stream cipher output $K'$ and the output $k'_i$ of $L_i$. Thus, even though the complete cipher is nonrandom, $K'$ constitutes a noisy observation of $k'_i$ from which a good estimate of $k'_i$ may perhaps be obtained. Such a divide-and-conquer strategy can be repeated to yield all the keys $k_i$ for each $L_i$. For Y-00, there is real noise from the coherent states, but a similar FCA can be launched if there is a significant correlation between $K'$ and the observed $2M$-ary signal, as obtained, say, by heterodyning. In general, attack on the Y-00 seed key is *exactly* a decoding problem on a memoryless channel for both CTA and KPA. This can be seen by regarding the seed key as information bits and the observed sequence of $2M$-ary signals translated by the mapper to $K'$ as the codeword, with independent coherent-state noise for each qumode so that the memoryless channel alphabet has size $\log_2 2M$ in a CTA and $\log_2 M$ in a KPA. Note that this code from ENC, as in the case of AES, could be nonlinear with no useful linear approximation, making linear decoding not a viable attack. It is not known whether information-theoretic security may be obtained in Y-00 for a properly designed ENC, i.e., whether a (decoding) algorithm may be found that would succeed in determining the seed key with some nonvanishing probability [@yuen05qph; @nair06]. And there is the further question, if such an algorithm exists, of its complexity as the general syndrome decoding of even a linear code is exponential. In contrast, for KPA on standard “nondegenerate” nonrandom ciphers, the key is actually uniquely determined at a bit length $n_1=n_d$, the nondegeneracy distance [@yuen05qph; @nair06] which is often not very long. Thus, such cipher has no information-theoretic security against KPA, although there is still the problem of attack complexity in finding $K$ that may allow complexity-based security which can be practically as good as information-theoretic security [@yuen05qph]. The key point in this connection is that randomization introduces real noise that is otherwise absent in a nonrandom cipher, signifying its role in adding security to KPA. For standard stream ciphers built upon LFSR’s, the class of FCA described above is powerful enough to break them for sufficiently long observed length $N$ of the output. However, the complexity of all known FCA algorithms is exponential in either the memory needed or the number $t$ of tap coefficients in the LFSR \[13\]. Thus, practically there are LFSR-based stream ciphers that are not broken by any known attack when the LFSR length $|K|$ and $t$ are sufficiently large. Shorter LFSR’s or ones with long $|K|$ and with few taps are more convenient and cheaper to use in practice, but are vulnerable to computationally intensive but feasible attacks. If such LFSR is used in the ENC in Y-00, the cipher becomes vulnerable even for moderate signal level if long enough $N$ is employed when that does not lead to an undue increase in memory required. For the $|K|=32$ single LFSR case reported in [@donnet], only $N=1500$ is needed in a CTA to undermine the system at the signal level $\alpha^2_0 \sim 1.5\times 10^4$, roughly the numbers used in [@optlett03]. The convolutional-code based algorithm chosen in [@donnet] is not suited to attacking long $|K|$ LFSR with a few taps, and thus would not be able to break the $|K|=4400$ and $t=3$ LFSR used in our system in [@pra05; @corndorf04]. However, a different FCA would no doubt be able to break that system, such as those designed for small $t$. Defenses Against Fast Correlation Attacks ========================================= We have already observed that one may use practical LFSR that resists known FCA in the ENC of Y-00. There are many other ways to defeat such and even more general attacks on the Y-00 seed key, as we will discuss in the rest of this paper. First, a properly designed deterministic mapper that determines the $2M-$ary signal from the running key $K'$ would spread the noise into the different bit positions of $K'(m)$, increasing the minimum complexity of attacks. The mapper may also be keyed, e.g., the mapper function may be chosen for each qumode from the running key $K_m'$ from another ${ENC}_m$ with another seed key $K_m$. This results in a product cipher of ENC in noise and $ENC_m$, for which no obvious modification of the FCA can be made that does not involve exponential search over $K_m$. In particular, one cannot plot Fig. 3 in [@donnet] which is the basic starting point of their attack. This defense has already been proposed [@hirota06], although there is “correlation immunity” for such ciphers only under an approximation. Secondly, the connection polynomial in the LFSR can be keyed, i.e., chosen randomly from an exponential number of possibilities. The known FCA’s on LFSR all require knowledge of the LFSR connection polynomial. In a future paper, we will present information-theoretic analysis of the effect of a keyed connection polynomial. Such ciphers can clearly be implemented in software, and to a considerable extent in hardware with field programmable logic, thus retaining much of the convenience of LFSR in practical applications. We do not believe information-theoretic security can be obtained this way, but it may greatly increase the complexity of at least FCA type attacks, thus providing useful practical security in some situations. Thirdly, we now give an ENC design for Y-00 that leads to exponential complexity for CTA according to current knowledge, and more security that AES for KPA generally. Consider the ENC of Fig. 2 where a bank of $m$ parallel AES in a stream cipher mode is used to provide the $m=\log_2 M$ bits running key segment $K'(m)$ which determines, through the mapper, the basis of a qumode. Typically in our previous experimental demonstrations, $m \sim 10$ and $|K|$ is in the thousands. Thus each $K_i$ may be readily chosen to be of 256 bits. Under heterodyne or any other quantum measurement by Eve, the result is a noisy version of $K'(m)$ with independent coherent-state induced randomization for each qumode. According to the present state of knowledge, no KPA on AES is better than exhaustive (exponential) search [@stinson]. Even in a divide-and-conquer type attack as in FCA, so that a single AES is to be considered, one needs to deal with the KPA problem of artifical noise from such strategy with the addition of *real* coherent-state noise, in a CTA on the Y-00 seed key. Let $N_1$ be the length of the qumode sequence used for the attack, so that Eve may parallelize $N/{N_1}$ attacks simultaneously from the total length $N$. It is clear that even without noise, the attack complexity remains exponential for any realistic $N \leq 2^{80}$ and any $N_1$. In a KPA, the comparison is to be made with the same $N_1$ for no parallelization. Thus, Y-00 is equivalent to AES in a stream cipher mode with output observed in noise, thus harder than AES alone which does not have the decoding in noise problem. In particular, it is easily seen that if the Y-00 in the configuration of Fig. 2 can be broken, then each $AES_i$ itself can be broken. \[htbp\] [ {width="4.5in" height="2in"}]{} The question arises as to what constitutes a fair comparison between a given stream cipher ENC versus Y-00 on top of ENC. A different configuration was given for ENC in [@nair06], where a single classical stream cipher (say AES) is used without parallelization but is adjusted to give the same clock rate for encrypting each data bit. The present scheme appears simpler in principle and more secure in practice when AES is used in ENC, because the functionality of multiple AES in parallel cannot be replaced by a single AES. However, with such parallelization for maintaining the same clock rate as AES (or ENC alone), the question arises as to whether the added security from Y-00 can be obtained from, say, nonlinearly combining the parallel AES’s. This question cannot be answered until security is precisely defined and quantified. However, it may be observed in this connection that there is no known attack developed for AES observed *in noise*, and the intrinsic nonlinearity of AES renders all known decoding attacks inapplicable. The major qualitative advantage of Y-00 [@yuen05qph; @nair06] compared to a standard nonrandom cipher is that the quantum noise automatically provides high speed true randomization not available otherwise, thus giving it a different kind of protection from nonrandom ciphers. Furthermore, one has to attack such physics-based cryptosystem at the communication line with physical (measurement) equipment, which is not available to everyone at every place, whereas one only needs to sit at a computer terminal to attack conventional ciphers. In this connection, it may be mentioned that the high rate heterodyne attack needed on Y-00 is currently not quite technologically feasible, though it may be in the not-too-far future. Y-00 can be employed to realize these benefits not available otherwise. However, if it is intrinsically less secure than conventional ciphers, its utility would be in serious doubt. The configuration of Fig. 2 shows this is not the case – it can in fact be more secure that ENC or $AES_i$ by itself. There is also no known attack applicable to AES in noise. Deliberate Signal Randomization =============================== In contrast to a nondegenerate nonrandom classical cipher for which the key is completely protected in the information-theoretic sense against CTA when the data is uniformly random [@yuen05qph; @nair06], there is little distinction between CTA and KPA for the bare Y-00. Only a factor of 2 in the per qumode alphabet size is obtained in KPA versus CTA as indicated above, and expounded in [@nair06]. The question arises as to whether full information-theoretic security against CTA can be restored by modifying the bare Y-00. The authors of [@donnet] appear to be pessimistic on the possibility of achieving this. To quote: “While randomization methods might increase the security level, it remains to be seen if they will provide perfect secrecy.” In the following, we show how this is possible with Deliberate Signal Randomization (DSR) independently of the mechanism of running key generation. The reason why the seed key cannot be attacked in CTA is clear for an additive stream cipher with uniformly random data. The “channel” between the seed key and the output observation has zero capacity due to the data which acts as random noise. In particular, it is clear that no FCA can be launched. The coherent-state noise in Y-00 is not big enough for high signal level to produce a similar effect. However, further randomization may in principle be produced to achieve this end, both classically and quantum mechanically. Since the coherent-state noise in Y-00 can in principle be replaced, in an equivalent classical system, by deliberate randomization of the classical signal from Alice as we have repeatedly emphasized [@yuen04; @pla05; @yuen05qph; @nair06], we first consider this classical situation. Let $\theta_s$ be the signal point on the circle of Fig. 1, $x$ the data bit, $k'$ the running key segment that determines the basis. Before deliberate or noise randomization, $\theta_s(x,k')$ is uniquely determined by $x$ and $k'$. From $\theta_s$ one randomizes it to $\theta_r$ according to a probability density $p(\theta_r|\theta_s)$. We use continuous $\theta$’s here but the argument is identical for discrete $\theta$’s. More generally, let $\theta$ be Eve’s observed signal point, so that $\theta=\theta_r$ in a classical noiseless system with deliberate randomization. Then, $$\label{pdf} p(\theta|x,k')=\int p(\theta|\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d} \theta_r.$$ In the classical noiseless case with just signal randomization, $p(\theta|\theta_r)=\delta(\theta-\theta_r)$, the BPSK signal may be correctly discriminated when the observed $\theta$ falls within the half-circle centred around $\theta_s$. Thus we pick $p(\theta_r|\theta_s)$ to be the uniform distribution on the half-circle with midpoint $\theta_s$. If $x$ is uniformly random, then from (1) $$\label{pdf2} p(\theta|k')=\frac{1}{2} \sum_{x=0,1} p(\theta|x,k')$$ is the uniform distribution on the full circle independent of $k'$. This proves the observation of $\theta$ to Eve yields no information at all on $k'$. In other words, Eve’s channel on $k'$ has zero capacity from DSR and uniformly random data which acts as added noise unknown to her, similar to a nondegenerate nonrandom stream cipher. For coherent-state noise described in the wedge approximation [@pla05; @nair06], whereupon a heterodyne or phase measurement the observed $\theta$ is taken to be uniformly distributed within a standard deviation around $\theta_r$ and zero outside, the same $k'-$independence for $p(\theta|k')$ obtains when $\theta_r$ is chosen in a discrete number of positions for given $\theta_s$ so that $p(\theta_r|\theta_s)$ fills out a uniform half-circle again. We have assumed an integral number of wedges would do this, which can be guaranteed by choice of the signal level $\alpha_0$. Going beyond the wedge approximation, one needs to determine the function $p(\theta_r|\theta_s)$ in (1) for a coherent state/fixed measurement $p(\theta|\theta_r)$ so that $p(\theta|x,k')$ is uniformly distributed in a half-circle, where $p(\theta|\theta_r)$ is obtained from Eve’s optimal individual qumode quantum measurement. In this case, there is the problem that the resulting error probability for Bob may be higher than the designed level even with knowledge of the seed key $K$. In principle, this problem can be handled in one of two different ways without affecting the data security as measured by the Shannon limit [@yuen05qph; @nair06]. First, one may increase $S$ and correspondingly $M$ while maintaining the same Y-00 random cipher characteristic $\Gamma = M/{\pi\sqrt{S}}$ defined in [@nair06]. Doing so will make the tail of the probability distribution that causes Bob’s error arbitrarily small. Indeed, in the classical limit $S \rightarrow \infty, M \rightarrow \infty, M/{\sqrt{S}} \rightarrow \pi\Gamma$, a constant, the error vanishes. A second way is to employ an error correcting code for Bob and randomize the entire codeword of $n$-bits in a correlated fashion in the signal space $\mathcal{C}^n$, where $\mathcal{C}$ is the coherent-state circle in $\mathbb{R}^2$. This is done by moving the $n$-bit codewords within mutually exclusive but jointly exhaustive regions that fill the entire signal space $\mathcal{C}^n$, similar to the filling of the circle $\mathcal{C}$ in the one-bit case. Detailed quantitative treatment of these will appear elsewhere. Note that Y-00 is only a random cipher for a given quantum measurement, it is not a random quantum cipher. See [@nair06]. A convenient way to make it a quantum random cipher is to randomize the parameter $\theta_s$ to $\theta_r$ that determines the quantum state $\rho(\theta_r)$ to be transmitted. The resulting output state is then, analogous to (1), $$\label{state} \rho(x,k')=\int \rho(\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d}\theta_r.$$ It may be seen from (3) that by uniformly randomizing $\theta_s$ as above, for any state modulation $\rho(\theta_r)$, the output quantum state itself is independent of $k'$ upon averaging over $x$ as before. Thus, such quantum DSR would protect the key against CTA with the most general joint (quantum measurement) attack. In this case, there is generally a larger probability of error that Bob would decide on $x$ incorrectly as compared to no DSR, similar to the specific coherent state case under heterodyne attack. One of the above two approaches in the fixed measurement case can be similarly employed to bring the error down to any desired level. It may be noted that the deployment of full DSR just described above is practically difficult at present if only because high speed random numbers are needed. On the other hand, it may be possible to delve into the qumode sequence to take advantage of the randomization inherent in such sequence for selected deliberate randomization while providing essentially the same overall result. Detailed treatment of concrete DSR on Y-00 will be given elsewhere. Conclusion ========== We have shown that Y-00 can be designed to be secure against fast correlation attacks including that of ref. [@donnet], and that it can be configured to be more secure than AES while retaining the same high speed and its advantage as a physics-based cipher. We also prove the full information-theoretic security of Y-00 with proper deliberate signal randomization against ciphertext-only attacks. Quantitative security against known-plaintext attacks, as in the case of conventional ciphers, is a difficult, open, and important area of research. Acknowledgements ================ We would like to thank E. Corndorf, G. Kanter, P. Kumar, and C. Liang for useful discussions. This work has been supported by DARPA under grant F30602-01-2-0528 and AFOSR grant FA9550-06-1-0452. [10]{} url \#1[`#1`]{}urlprefix G. Barbosa, E. Corndorf, P. Kumar, H.P. Yuen, “Secure communication using mesoscopic coherent states”, Phys. Rev. Lett. 90 (2003) 227901. H.P. Yuen, “<span style="font-variant:small-caps;">KCQ</span>: A new approach to quantum cryptography <span style="font-variant:small-caps;">I</span>. <span style="font-variant:small-caps;">G</span>eneral principles and qumode key generation”, quant-ph/0311061. E. Corndorf, G. Barbosa, C. Liang, H. Yuen, P. Kumar, “High-speed data encryption over 25km of fiber by two-mode coherent-state quantum cryptography”, Opt. Lett. 28, 2040-2042, 2003. C. Liang, G.S. Kanter, E. Corndorf, and P. Kumar, “Quantum noise protected data encryption in a WDM network”, Photonics Tech. Lett. 17, pp. 1573-1575, 2005. E. Corndorf, C. Liang, G.S. Kanter, P. Kumar, and H.P. Yuen, “Quantum-noise–protected data encryption for WDM fiber-optic networks”, Phys. Rev. A 71 (2005) p. 062326. G.S. Kanter, E. Corndorf, C. Liang, V.S. Grigoryan, and P. Kumar, in *Fluctuation and Noise in Photonics and Quantum Optics III*, ed. P.R. Hemmer etc., Proc. of SPIE vol. 58 42 (SPIE, Bellingham, WA, 2005), pp. 74-86. H.P. Yuen, P. Kumar, E. Corndorf, R. Nair, “Comment on ‘How much security does Y-00 protocol provide us?’”, Phys. Lett. A, 346 (2005) 1-6; quant-ph/0407067. H.P. Yuen, R. Nair, E. Corndorf, G.S. Kanter, P. Kumar, To appear in *Quantum Information & Computation* Vol. 6 No. 7 (Nov 2006) 561-582; quant-ph/0509091 v. 3. R. Nair, H.P. Yuen, E. Corndorf, T. Eguchi, P. Kumar, “Quantum Noise Randomized Ciphers”, quant-ph/0603263 v. 5; To appear in Phys. Rev. A. S. Donnet, A. Thangaraj, M. Bloch, J. Cussey, J-M. Merolla, L. Larger, Phys. Lett. A, 356 (2006) 406-410. O. Hirota, K. Kurosawa, quant-ph/0604036; to appear in *Quant. Info. Proc.*. O. Hirota, M. Sohma, M. Fuse, and K. Kato, ‘Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme’’, Phys. Rev. A. 72 (2005) 022335; quant-ph/0507043. F. Jonsson, Ph.D. Thesis, Lund University, Sweden, 2002; Available online at [www.pcc.lth.se/PrimePub/primepub.asp?AemneID=572&SpraakID=1&RotID=1](www.pcc.lth.se/PrimePub/primepub.asp?AemneID=572&SpraakID=1&RotID=1) E. Corndorf, G. Kanter, C. Liang, and P. Kumar, ‘Quantum-noise protected data encryption for WDM networks,’ in 2004 Conference on Lasers Electro Optics (CLEO’04), San Francisco, CA, Postdeadline CPDD8. D.R. Stinson, *Cryptography: Theory and Practice*, Chapman and Hall/CRC, 3rd ed, 2006. [^1]: [email protected]

ArXiv

--- abstract: 'Protein-stabilised emulsions can be seen as mixtures of unadsorbed proteins and of protein-stabilised droplets. To identify the contributions of these two components to the overall viscosity of sodium caseinate o/w emulsions, the rheological behaviour of pure suspensions of proteins and droplets were characterised, and their properties used to model the behaviour of their mixtures. These materials are conveniently studied in the framework developed for soft colloids. Here, the use of viscosity models for the two types of pure suspensions facilitates the development of a semi-empirical model that relates the viscosity of protein-stabilised emulsions to their composition.' author: - Marion Roullet - 'Paul S. Clegg' - 'William J. Frith' bibliography: - 'References.bib' title: 'Viscosity of protein-stabilised emulsions: contributions of components and development of a semi-predictive model' --- Introduction ============ Despite their complexity, food products can be conveniently studied from the perspective of colloid science [@mezzenga:2005]. In the last three decades, research in the field of food colloids has led to major advances in understanding their structure over a wide range of lengthscales [@dickinson:2011], which has proved key to developing a good control of their flavour and texture properties [@vilgis:2015]. Many food products such as mayonnaise, ice cream, and yogurt involve protein-stabilised emulsions either during their fabrication or as the final product. Proteins have particularly favourable properties as emulsifiers because of their ability to strongly adsorb at oil/water interfaces and to stabilise oil droplets by steric and electrostatic repulsion. However, proteins do not completely adsorb at the interface, leaving a residual fraction of protein suspended in the continuous phase after emulsification [@srinivasan:1996; @srinivasan:1999]. Protein-stabilised emulsions are thus mixtures of protein-stabilised droplets and suspended proteins, as illustrated in Figure \[Fig:CartoonProtDrop\]. Understanding the contributions of these two components to the properties of the final emulsion remains a challenge. ![Illustration of a protein assembly, protein-stabilised droplet, and protein-stabilised emulsion seen as a mixture of droplets and un-adsorbed proteins. []{data-label="Fig:CartoonProtDrop"}](Fig1){width="80.00000%"} When considered separately, the droplets in protein-stabilised emulsions can be considered as colloidal particles with some degree of softness. It is thus possible to compare the rheological properties of protein-stabilised emulsions to other types of soft particle suspension and to model their behaviour. From a theoretical point of view, particles, colloidal or not, can be described as soft if they have the ability to change size and shape at high concentration [@vlassopoulos:2014]. Such a definition covers a striking variety of systems, including gel microparticles [@adams:2004; @shewan:2015], microgels [@cloitre:2003; @tan:2005], star polymers [@roovers:1994; @winkler:2014] and block co-polymer micelles [@lyklema:2005:4]. These systems have been the focus of many studies in the last two decades, however one major challenge to comparing the behaviour of such diverse materials is the availability of a well defined volume fraction $\phi$ for the suspensions. To overcome the challenge of defining the volume fraction of soft colloids, a common approach is to use an effective volume fraction $\phi_{eff}$ proportional to the concentration $c$, $\phi_{eff}=k_0\times c$, where $k_0$ is a constant indicating the voluminosity of the soft particle of interest, usually determined in the dilute or semi-dilute regime. Such a definition for $\phi_{eff}$ does not take into account the deformation or shrinking of the particle at high concentrations, high values of ($\phi_{eff}>1$) can thus be reached. $k_0$ can be estimated using osmometry [@farrer:1999], light scattering [@vlassopoulos:2001] or viscosimetry [@tan:2005; @roovers:1994; @boulet:1998]. In this study, $k_0$ was estimated, for each individual component of the emulsions, by modelling the relative zero-shear viscosity $\eta_0/\eta_s$ behaviour of the pure suspensions in the semi-dilute regime with Batchelor equation for hard spheres [@batchelor:1977]: $$\label{Eq:Batchelor} \frac{\eta_0}{\eta_s} = 1+2.5 \phi_{eff}+6.2 \phi_{eff}^2$$ Sodium caseinate is used here to stabilise emulsions as a case-study, because of its outstanding properties as a surface-active agent and stabiliser, and because it is widely used in industry. Sodium caseinate is produced by replacing the calcium in native milk casein micelles, with sodium, to increase its solubility [@dalgleish:1988], a process which also leads to the disruption of the micelles. It has been established that sodium caseinate is not present as a monomer in suspension, but rather in the form of small aggregates [@lucey:2000]. The exact nature of the interactions in play in the formation of these aggregates is not well-known but they have been characterised as elongated and their size estimated to be around $\SI{20}{\nano\metre}$ [@farrer:1999; @lucey:2000; @huppertz:2017]. Some larger aggregates can also form in presence of residual traces of calcium or oil from the original milk, however these only represent a small fraction of the protein [@dalgleish:1988]. The viscosity behaviour of sodium caseinate as a function of concentration shows similarities with hard-sphere suspensions at relatively small concentrations, but at higher concentrations, over $c>\SI{130}{\gram\per\liter}$, the viscosity continues to increase with a power-law rather than diverging [@farrer:1999; @pitowski:2008] as would be expected for a hard sphere suspension [@faroughi:2014]. In this study, the rheology of protein-stabilised emulsions is examined within the framework of soft colloidal particles. Modeling proteins in this way ignores protein-specific elements, such as surface hydration, conformation changes, association, and surface charge distribution [@sarangapani:2013; @sarangapani:2015], but it provides a convenient theoretical framework to separate and discuss the contributions of both sodium caseinate and the droplets to the viscosity of emulsions. Similarly, protein-stabilised droplets can be seen as comprising an oil core and a soft protein shell [@bressy:2003], allowing for a unifying approach for both components of the emulsions. The aim of this study is to present a predictive model of the viscosity of protein-stabilised emulsions, that takes into account the presence and behaviour of both the protein stabilised droplets and the unadsorbed protein. A first step is to characterise separately the flow behaviour and viscosity of suspensions of purified protein-stabilised droplets, and of protein suspensions over a wide range of concentrations. This also allows a critical assessment of the soft colloidal approach. These components are then combined to form mixtures of well-characterised composition and their viscosity is compared to a semi-empirical model. Because they are well dispersed, most of the suspensions and emulsions display a Newtonian behavior at low shear, with shear thinning at higher shear-rates. In this context, we model the concentration dependence of zero-shear viscosity and the shear thinning behaviour separately to confirm the apparent colloidal nature of the components of the emulsions and protein suspensions. Materials & Methods =================== Preparation of protein suspensions ---------------------------------- Because of its excellent ability to stabilise emulsions, sodium caseinate (Excellion S grade, spray-dried, kindly provided by DMV, Friesland Campina, Netherlands), was used in this study. It was further purified by first suspending it in deionised water, at $5-9 \%$ (w/w), and then by mixing thoroughly with a magnetic stirrer for $\SI{16}{\hour}$. After complete dispersion, a turbid suspension was obtained, which was centrifuged at $\num{40000}\times$g (Evolution RC, Sorvall with rotor SA 600, Sorvall and clear $\SI{50}{\milli\liter}$ tubes, Beckmann) for $\SI{4}{\hour}$ at $\SI{21}{\degreeCelsius}$. Subsequently, the supernatant, made of residual fat contamination, and the sediment were separated from the suspension, that was now clearer. The solution was then filtered using a $\SI{50}{\milli\liter}$ stirred ultra-filtration cell (Micon, Millipore) with a $\SI{0.45}{\micro\meter}$ membrane (Sartolon Polyamid, Sartorius). In order to avoid spoilage of the protein solution 0.05% of ProClin 50 (Sigma Aldrich) was added. The suspension at $5\%$ (w/w) was then diluted to the required concentration. Concentrated suspensions of sodium caseinate were prepared by evaporating a stock solution of sodium caseinate at $\SI{5}{\percent}$(w/w), prepared following the previous protocol, using a rotary evaporator (Rotavapor R-210, Buchi). Mild conditions were used to avoid changing the structure of the proteins: the water bath was set at $\SI{40}{\degreeCelsius}$ and a vacuum of $\SI{45}{\milli\bar}$ was used to evaporate water. The concentration of all the suspensions after purification was estimated by refractometry, using a refractometer RM 50 (Mettler Toledo), LED at $\SI{589.3}{\nano\metre}$ and a refractive index increment of $dn/dc =\SI[separate-uncertainty=true]{0.1888(00033)}{\milli\liter\per\gram}$ [@zhao:2011]. ![Size distributions of sodium caseinate after the purification protocol. The sample was fractionated by Asymetric Flow Field Flow Fractionation (kindly performed by PostNova Analytics Ltd), and the sizes were measured online by Dynamic Light Scattering (dot line, red) and by Multi Angle Light Scattering (dash dot line, orange). The relative percentage of each class is weighted by the intensity of the scattered light. The inset is a zoom of the small fraction of proteins that are present as larger aggregates []{data-label="Fig:SizeProtFFF"}](Fig2){width="90.00000%"} Size analysis by Flow Field Fractionation (kindly performed by PostNova Analytics Ltd) showed that the resulting suspensions of sodium caseinate were composed of small aggregates of a hydrodynamic radius of $\SI{11}{\nano\metre}$ at $\SI{96}{\percent}$, while the remaining $\SI{4}{\percent}$ formed larger aggregates with a wide range of sizes (hydrodynamic radii from $\SI{40}{\nano\metre}$ to $\SI{120}{\nano\metre}$) as shown in Figure \[Fig:SizeProtFFF\]. Preparation of emulsions ------------------------ Nano-sized caseinate-stabilised droplets were prepared in two steps. First, the pre-emulsion was produced by mixing $\SI{45}{\milli\gram\per\milli\liter}$ sodium caseinate solution (prepared as detailed previously) with glyceryl trioctanoate ($\rho=\SI{0.956}{\gram\per\milli\liter}$, Sigma Aldrich) at a weight ratio 4:1 using a rotor stator system (L4R, Silverson). This pre-emulsion was then stored at $\SI{4}{\degreeCelsius}$ for $\SI{4}{\hour}$ to reduce the amount of foam. It was then passed through a high-pressure homogeniser (Microfluidizer, Microfluidics) with an input pressure of $\SI{5}{\bar}$, equivalent to a pressure of $\approx\SI{1000}{\bar}$ in the micro-chamber, three times consecutively. After 3 passes, a stationary regime was reached where the size of droplets could not be reduced any further. This protocol for emulsification produced droplets of radius around $\SI{110}{\nano\metre}$ as measured by Dynamic Light Scattering (Zetasizer, Malvern) and $\SI{65}{\nano\metre}$ by Static Light Scattering (Mastersizer, Malvern). Because not all the protein content was adsorbed at the interface, an additional centrifugation step was required to separate the droplets from the continuous phase of protein suspension. This separation was performed by spinning the emulsion at $\num{235000}\times$g with an ultra-centrifuge (Discovery SE, Sorvall, with fixed-angle rotor $45$Ti, Beckmann Coulter) for $\SI{16}{\hour}$ at $\SI{21}{\degreeCelsius}$. The concentrated droplets then formed a solid layer at the top of the subnatant that could be carefully removed with a spatula. The subnatant containing proteins and some residual droplets was discarded. The drying of a small fraction of the concentrated droplet layer and the weighing of its dry content yielded a concentration of the droplet paste of $\SI[separate-uncertainty=true]{0.519(0008)}{\gram\per\milli\liter}$, so the concentration in droplets of all the suspensions were derived from the dilution parameters. Only one centrifugation step was employed to separate the droplets from the proteins, as it was felt that further steps may lead to protein desorption and coalescence. The pure nano-sized droplets were then re-dispersed at the required concentration, in the range $\num{0.008}$ to $\SI{0.39}{\gram\per\milli\liter}$ in deionised water for $1$ to $\SI{30}{\minute}$ with a magnetic stirrer. Preparation of mixtures ----------------------- To prepare emulsions with a controlled concentration of proteins in suspension, the concentrated droplets were re-suspended in a protein suspension at the desired concentration using a magnetic stirrer and a stirring plate for $\SI{5}{\minute}$ to $\SI{2}{\hour}$. Viscosity measurements ---------------------- Rotational rheology measurements were performed using a stress-controlled MCR 502 rheometer (Anton Paar) and a Couette geometry (smooth bob and smooth cup, $\SI{17}{\milli\liter}$ radius) at $\SI{25}{\degreeCelsius}$. For each sample, three measurements are performed and averaged to obtain the flow curve. The values of viscosity on the plateau at low shear are averaged to determine the zero-shear viscosity. Viscosity measurements were performed at different concentrations for protein suspensions, protein-stabilised droplet suspensions, and mixtures. Results & Discussion {#S:1} ==================== In order to study the rheological behaviour of protein-stabilised emulsions, the approach used here is to separate the original emulsion into its two components, namely un-adsorbed protein assemblies and protein-coated droplets, and to characterise the suspensions of each of these components. Despite their intrinsic complexity due to their biological natures, random coil proteins such as sodium caseinate can conveniently be considered as colloidal suspensions, as we demonstrate in the discussion below. Viscosity of suspensions in the semi-dilute regime: determination of volume fraction ------------------------------------------------------------------------------------ The weight concentration (in $\SI{}{\gram\per\milli\liter}$) is a sufficient parameter to describe the composition in the case of one suspension, but only the use of the volume fraction of the suspended particles allows meaningful comparisons between protein assemblies and droplets. In the framework of soft colloids, the effective volume fraction $\phi_{eff}$ of a colloidal suspension can be determined by modelling the viscosity in the semi-dilute regime with a hard-sphere model. ![Relative viscosity of sodium caseinate suspensions ($\square$, navy blue) and sodium caseinate-stabilised droplets ($\bigcirc$, cyan) as a function of the concentration of dispersed material. The lines denote Batchelor model for hard spheres in the dilute regime, Equation\[Eq:Batchelor\].[]{data-label="Fig:DiluteRegimeConc"}](Fig3){width="90.00000%"} The relative zero-shear viscosities of semi-dilute samples are displayed in Figure \[Fig:DiluteRegimeConc\] as a function of the mass concentration of protein or droplets (viscosity data at the full range of concentration can be found in Figure S2 in the supplementary material). As can be seen, protein suspensions reach a higher viscosity at a lower weight fraction than droplet suspensions. This is because the protein is highly hydrated and swollen, and so occupies a greater volume per unit mass than do the droplets, where the main contributor to the occupied volume is the oil core. The viscosity behaviour of each type of suspension in the semi-dilute regime can be described by a theoretical model such as Batchelor’s equation [@batchelor:1977], Equation \[Eq:Batchelor\] as a function of the volume fraction $\phi$. This involves assuming that the particles in the suspension of interest do not have specific interparticle interactions or liquid interfaces in this regime, and can be accurately described as hard spheres. In addition, as a first approximation, the effective volume fraction $\phi_{eff}$ of soft particles in suspension is assumed to be proportional to the weight concentration $c$: $$\label{Eq:EffPhi_proportional} \phi_{eff}=k_0 \times c$$ where $k_0$ is a constant expressed in $\SI{}{\milli\liter\per\gram}$. This equation is combined with Equation \[Eq:Batchelor\] in order to obtain an expression for the viscosity as a function of the concentration. When applied to experimental viscosity values for suspensions of protein or droplets at concentrations in the semi-dilute regime, such an expression allows estimation of $k_0$. The effective volume fraction $\phi_{eff}$ of the suspensions can then be calculated using Equation \[Eq:EffPhi\_proportional\]. When fitted to the viscosity data for pure sodium caseinate and pure droplets, as described above, Equation \[Eq:Batchelor\] gives satisfactory fits as shown in Figure \[Fig:DiluteRegimeConc\]. The resulting values for $k_0$ are, for protein suspensions, $k_{0,prot}=\SI[separate-uncertainty=true]{8.53(23)}{\milli\liter\per\gram}$, and for droplet suspensions, $k_{0,drop}=\SI[separate-uncertainty=true]{2.16(13)}{\milli\liter\per\gram}$. The protein result is in reasonable agreement with previous results, where determinations of the volume fraction using the intrinsic viscosity gave $\phi_{eff,prot} = \num[separate-uncertainty=true]{6.4}~c$ [@pitowski:2008] and $\phi_{eff,prot} = \num[separate-uncertainty=true]{6.5(5)}~c$ [@huppertz:2017], while osmometry measurements (at a higher temperature) gave $\phi_{eff,prot}=\num{4.47}~c$ [@farrer:1999]. For droplet suspensions, $k_{0,drop}$ corresponds to the voluminosity of the whole droplets. If these were purely made of a hard oil core, their voluminosity would be $1/\rho_{oil}=\SI{1.05}{\milli\liter\per\gram}$. The higher value observed can be attributed to the layer of adsorbed proteins at the surface of the droplets. This is an indication that the nano-sized droplets can be modelled as core-shell particles. These results make it possible to calculate the effective volume fractions $\phi_{eff}$ of both types of suspensions, which is a necessary step to allowing their comparison. It is however important to keep in mind that $\phi_{eff}$ is an estimate of the volume fraction using the hard sphere-assumption, which is likely to break down as the concentration is increased, where deswelling, deformation and interpenetration of the particles may occur [@vlassopoulos:2014]. Modelling the viscosity behaviours of colloidal suspensions ----------------------------------------------------------- In order to identify the contributions of the components to the viscosity of the mixture, it is important to characterise the viscosity behaviours of the pure suspensions of caseinate-stabilised nano-sized droplets and of sodium caseinate. This is achieved by modelling the volume fraction dependence of the viscosity with equations for hard and soft colloidal particles. ### Suspensions of protein-stabilised droplets {#Sec:ViscoDropModel} The viscosity of protein-stabilised droplet suspensions is displayed in Figure \[Fig:ViscoDropQuemada\]. A sharp divergence is observed at high volume fraction and this behaviour is typical of hard-sphere suspensions [@dekruif:1985]. It is thus appropriate to use one of the relationships derived for such systems to model the viscosity behaviour of droplet suspensions. ![Relative viscosity of sodium caseinate-stabilised droplets ($\circ$, cyan) as a function of the effective volume fraction. The red dashed line denotes Quemada equation for hard spheres, Equation \[Eq:QuemadaHS\] with $\phi_m =\num{0.79}$[]{data-label="Fig:ViscoDropQuemada"}](Fig4){width="80.00000%"} Amongst the multiple models for the viscosity of hard-sphere suspensions that have been proposed over time, the theoretical model developed by Quemada [@quemada:1977] is used in this work: $$\label{Eq:QuemadaHS} \frac{\eta_0}{\eta_s} = \left(1-\frac{\phi}{\phi_m}\right)^{-2}$$ Where the parameter $\phi_m$ is the maximum volume fraction at which the viscosity of the suspension diverges: $$\label{Eq:DivergenceVisco} \lim_{\phi\to\phi_m} \frac{\eta_0}{\eta_s}=\infty$$ The Quemada model fits remarkably well to the experimental data of the relative viscosity $\frac{\eta_0}{\eta_s}$ of suspensions of droplets. The value for the maximum volume fraction is found to be $\phi_m =\num[separate-uncertainty=true]{0.79(2)}$. Despite the similarity in viscosity behaviour between the droplet suspensions and hard-sphere suspensions, the maximum volume fraction found here is considerably higher than the theoretical value of $\phi_m=\phi_{rcp}=\num{0.64}$ for randomly close-packed hard spheres. A possible explanation for this discrepancy is the polydispersity of the droplet suspension. Indeed, random close-packing is highly affected by the size distribution of the particles, as smaller particles can occupy the gaps between larger particles [@farris:1968]. In a recent study, Shewan and Stokes modelled the viscosity of hard sphere suspensions using a maximum volume fraction predicted by a numerical model developed by Farr and Groot [@shewan:2015b; @farr:2009], which allows the maximum volume fraction of multiple hard-sphere suspensions to be predicted from their size distribution. Here, the same approach is used with the size distributions of the protein-stabilised droplets obtained from both the Mastersizer and the Zetasizer. The numerically estimated random close-packing volume fraction $\phi_{rcp}$ is close for both size distributions, and its value is $\phi_{rcp}=\num{0.68}$. Although this is a higher maximum volume fraction than for a monodisperse hard-sphere suspension, it is still considerably lower than the experimental value, $\phi_m=\num{0.79}$. Such a high random close-packing fraction can be achieved numerically only if a fraction of much smaller droplets is added to the distribution obtained by light scattering. The hypothesis of the presence of small droplets, undetectable by light scattering without previous fractionation is supported by the observation of such droplets upon fractionation of a very similar emulsion in a previous study [@dalgleish:1997]. It is also possible that other mechanisms than the polydispersity come into play at high volume fractions of droplets. Although it would be hard to quantify, it is likely that the soft layer of adsorbed proteins may undergo some changes at high volume fraction, such as deswelling or interpenetration. ### Protein suspensions Sodium caseinate is known to aggregate in solution to form clusters or micelles [@farrer:1999; @pitowski:2008; @huppertz:2017]. These differ from protein-stabilised droplets because of their swollen structure, and likely dynamic nature. The viscosity behaviour of the suspensions they form is displayed in Figure \[Fig:ViscoProtSoftQuemada\]. ![Relative viscosity of sodium caseinate suspensions ($\square$, navy) as a function of the effective volume fraction. The red dashed line denotes the modified Quemada equation, Equation \[Eq:ModifQuemadaSoft\], the values for $n$ and $\phi_m$ are listed in table \[Tab:ModifQuemadaProtParameters\].[]{data-label="Fig:ViscoProtSoftQuemada"}](Fig5){width="80.00000%"} At high concentrations, the viscosity does not diverge as quickly as for the suspensions of droplets. This result is in agreement with previous studies on sodium caseinate, in which suspensions at higher concentrations were studied [@farrer:1999; @pitowski:2008; @loveday:2010]. In these works, it was shown that the viscosity does not diverge but follows a power law $\eta_0/\eta_s\propto(\phi_{eff,prot})^{12}$ . The behaviour displayed by sodium caseinate resembles that of core-shell microgels [@tan:2005] and soft spherical brushes[@vlassopoulos:2001], hence a soft colloid framework (as reviewed e.g. in Ref. [@vlassopoulos:2014]) seems suitable for the study of these suspensions. A general feature of the viscosity behaviour of soft colloidal suspensions is the oblique asymptote at high concentrations. This behaviour is beleived to arise because, as the concentration increases, the effective volume occupied by each particle decreases, by de-swelling or interpenetration. Thus, the strong viscosity divergence of hard-sphere suspensions is absent for soft colloids. To describe the behaviour of such suspensions, a model is thus required, that takes into account this distinctive limit at high concentrations while retaining the hard sphere behaviour at lower concentrations. A semi-empirical modification that fulfills the above criteria is the substitution of the maximum volume fraction $\phi_m$ by a $\phi$-dependent parameter $\phi_m^*$ that takes the form: $\phi_m^* = \left({\phi_m}^n+\phi^n\right)^{1/n}$. As a result, a modified version of Equation \[Eq:QuemadaHS\] can be derived, that takes into account the softness of the particles via a concentration-dependent maximum volume fraction $\phi_m^*$. This semi-empirical viscosity model is expressed: $$\label{Eq:ModifQuemadaSoft} \frac{\eta_0}{\eta_s} = \left(1-\frac{\phi}{\phi_m^*}\right)^{-2}$$ where: $$\phi_m^* = \phi_m\left(1+\left(\frac{\phi}{\phi_m}\right)^n\right)^{1/n}$$ The addition of the exponent $n$ as a parameter expresses the discrepancy from the hard-sphere model. The smaller $n$, the lower the volume fraction $\phi$ at which $\phi^*_m$ diverges from $\phi_m$, and the less sharp the divergence in viscosity. The model in Equation \[Eq:ModifQuemadaSoft\] was applied to fit the experimental data displayed in Figure \[Fig:ViscoProtSoftQuemada\], and the resulting fitting parameters are listed in Table \[Tab:ModifQuemadaProtParameters\]. Parameter Value Standard Error ----------- ------- ---------------- $\phi_m$ $n$ : Parameters for modified Quemada model for soft colloids, Equation \[Eq:ModifQuemadaSoft\], applied to sodium caseinate suspensions \[Tab:ModifQuemadaProtParameters\] The use of this approach gives a good fit of the viscosity behaviour of sodium caseinate in the range of concentrations used here. In addition, this semi-empirical model also satisfactorily describes the viscosity of sodium caseinate suspensions at higher concentration from Ref. [@farrer:1999; @pitowski:2008]. It is worth noting that the inflection of viscosity is slightly sharper for the model than for the experimental data. The power-law towards which the relative viscosity $\eta_0/\eta_s$ described by Equation  \[Eq:ModifQuemadaSoft\] tends at high concentration (ie $\phi>\phi_m$) can be calculated by developing $\phi_m^*$. Indeed, at high concentration $\phi_m^*$ converges towards $\phi\times\left(1+\frac{1}{n} \times \left(\frac{\phi_m}{\phi}\right)^n\right)\propto(\phi_{eff})^{n}$, so $\eta_0/\eta_s$ converges towards $\left(1+n\left(\frac{(\phi_{eff})}{\phi_m}\right)^{n}\right)^{2}$ (detailed calculations are provided in the supplementary material). Using the value in Table \[Tab:ModifQuemadaProtParameters\], the relative viscosity of sodium caseinate suspensions is found to follow the power law $\eta_0/\eta_s\propto(\phi_{eff,prot})^{\num[separate-uncertainty=true]{12.2(8)}}$. This value is in good agreement with the literature where $\eta_0/\eta_s\propto(\phi_{eff,prot})^{12}$ in the concentrated regime [@farrer:1999; @pitowski:2008; @loveday:2010]. It is interesting to note that Equation \[Eq:ModifQuemadaSoft\] provides a good model for the behaviour of particle suspensions and emulsions whose particles have a wide range of softness, as will be detailed elsewhere. Within this context the concentration behaviour of sodium caseinate suspensions seems to indicate that they can also be regarded as suspensions of soft particles. This interpretation of the behaviour can be further tested by considering the shear-rate dependent response of both the emulsions and sodium caseinate suspensions. Shear thinning behaviour of protein and droplet suspensions ----------------------------------------------------------- Over most of the concentration range studied here, the protein suspensions display Newtonian behaviour. However, at high concentration of protein, shear thinning is observed at high shear-rates (flow curves in supplementary material). By comparison, the droplet suspensions display shear thinning at a much broader range of concentrations. This behaviour is common in colloidal suspensions [@dekruif:1985; @helgeson:2007], as well as polymer and surfactant solutions and arises from a variety of mechanisms [@cross:1965; @cross:1970]. In non-aggregated suspensions of Brownian particles shear-thinning arises from the competition between Brownian motion (which increases the effective diameter of the particles) and the hydrodynamic forces arising from shear. Shear thinning then occurs over a range where the two types of forces balance, as characterised by the dimensionless reduced shear stress ($\sigma_{r}$) being of order unity. $\sigma_{r}$ is given by $$\sigma_{r} = \frac{\sigma R^3}{k T} \label{Eq:SigmaRC}$$ where $R$ is the radius of the colloidal particle, $k$ is the Boltzmann constant and $T$ is the temperature of the suspension (here $T = \SI{298}{\kelvin}$). In such suspensions, the flow curve can be described using the following equation for the viscosity as a function of shear stress [@woods:1970; @krieger:1972; @frith:1987]: $$\frac{\eta}{\eta_s} = \eta_\infty + \frac{\eta_0-\eta_\infty}{1+\left(\sigma_r/\sigma_{r,c}\right)^m} \label{Eq:ShearThin}$$ Where $\eta_0$ is the zero-shear viscosity, $\eta_\infty$ is the high-shear limit of the viscosity, $m$ is an exponent that describes the sharpness of the change in regime between $\eta_0$ and $\eta_\infty$, and $\sigma_{r,c}$ is the reduced critical shear stress. Because shear thinning arises from the competition between Brownian motion and the applied external flow, the use of a dimensionless stress that takes into account the size of the colloidal particles allows meaningful comparisons between the different suspensions[@woods:1970; @frith:1987]. Here, we use this approach to compare the flow behaviour of the protein and droplet suspensions, and to test further the hypothesis that the protein suspensions can be considered to behave as though they are suspensions of soft particles. ![Shear thinning behaviour of concentrated suspensions of sodium caseinate ($\square$, navy), and sodium-caseinate stabilised droplets ($\circ$, cyan) as characterised by the critical shear stress for shear-thinning. (a) Critical shear stress $\sigma_c$ as a function of the zero-shear relative viscosity $\eta_0/\eta_s$ for several concentrated suspensions. $\sigma_c$ and $\eta_0$ were estimated by fitting the flow curves (Figure S1 in supplementary material) with Equation \[Eq:ShearThin\]. (b) Reduced critical shear stress $\sigma_{r,c}$ \[Eq:SigmaRC\] as a function of the zero-shear relative viscosity $\eta_0/\eta_s$. The error bars indicate the uncertainty of the fitting parameters (more details are provided in the supplemetary material), and the lines are indicated as guide for the eye.[]{data-label="Fig:SigmaCShearThin"}](Fig6){width="90.00000%"} Fitting the flow curves with Equation \[Eq:ShearThin\] allows for the extraction of the critical stress $\sigma_c$. The behaviour of this parameter as a function of the zero shear relative viscosity (as a proxy for concentration) is shown in Figure \[Fig:SigmaCShearThin\](a). The corresponding values of $\sigma_{r,c}$ are calculated using $R_{drop} \equiv R_{h,drop} = \SI{110}{\nano\metre}$ and $R_{prot}\equiv R_{h,prot} = \SI{11}{\nano\metre}$, are displayed in Figure \[Fig:SigmaCShearThin\] (b). As can be observed, the protein suspensions require a much higher stress to produce a decrease in viscosity than do the droplet suspensions, as $\sigma_c$ is more than two orders of magnitudes higher. However, this difference is largely absent when the reduced critical shear-stress is used, indicating that the main difference between both systems is the size of the particles and that there are no differences in interparticle interactions at high concentrations, notably no further extensive aggregation of sodium caseinate. Shear thinning is thus another aspect of the rheology of sodium caseinate that shows an apparent colloidal behaviour rather than polymeric behaviour . This result reinforces the relevance of the soft colloidal framework as an approach for studying the viscosity of sodium caseinate and sodium caseinate-stabilised droplets. Viscosity of mixtures ===================== After having studied separately the components of protein-stabilised emulsions, the next logical step is to investigate mixtures of both with well-characterised compositions by combining purified droplets and protein suspensions. In addition, the soft colloidal framework developed above provides a basis for the development of a predictive approach to the viscosity of mixtures of proteins and droplets, as formed upon emulsification of oil in a sodium caseinate suspension. These topics are the subject of the current section. ![Composition of suspensions of sodium caseinate ($\square$, navy), sodium-caseinate stabilised droplets ($\circ$, cyan), and of mixtures ($\triangle$, colour-coded as a function of $\chi_{prot}$ defined in Equation \[Eq:ChiProt\]). []{data-label="Fig:SamplesVisco"}](Fig7){width="70.00000%"} These mixtures are composed of water and of two types of colloidal particles (droplets and protein aggregates), hence they are conveniently represented as a ternary mixture, as displayed on Figure \[Fig:SamplesVisco\]. This representation is limited by the high volume fractions reached by proteins in suspension, hence some data points lie outside of the diagram. The two-dimensional space of composition for the mixtures can be described by the total effective volume fraction $\phi_{eff,tot} = \phi_{eff,prot}+\phi_{eff,drop}$ and the ratio of their different components $\chi_{prot}$: $$\label{Eq:ChiProt} \chi_{prot}=\frac{\phi_{eff,prot}}{\phi_{eff,prot}+\phi_{eff,drop}}$$ $\chi_{prot}$ describes the relative percentage of protein in the emulsion compared to the droplets: $\chi_{prot}=1$ for samples containing only proteins, $\chi_{prot}=0$ for samples containing only protein-stabilised droplets, and $\chi_{prot}=0.5$ for mixtures containing an equal volume fraction of proteins and protein-stabilised droplets. The viscosity of the mixtures containing both proteins and protein-stabilised droplets was measured as for the pure suspensions. The values can be compared with the pure suspensions using the total volume fraction for the mixtures $\phi_{eff,tot}$, and are displayed in Figure \[Fig:RawViscoMix\]. ![Relative viscosities $\eta_0/\eta_s$ of suspensions as a function of the effective volume fraction $\phi_{eff}$: sodium caseinate suspensions ($\square$, navy), sodium-caseinate stabilised droplets suspensions ($\circ$, cyan), and suspensions of mixtures ($\triangle$, colour-coded as a function of $\chi_{prot}$ defined in Equation \[Eq:ChiProt\]).[]{data-label="Fig:RawViscoMix"}](Fig8){width="90.00000%"} The mixtures all display viscosities between those of the pure droplets and of the pure proteins at a given volume fraction, their exact value depending on their compositional index $\chi_{prot}$. Notably, no phase separation is observed in the emulsion samples on the timescale of the experiments. This is an unusual result as sodium caseinate-stabilised emulsions are notoriously prone to depletion induced-flocculation caused by the presence of unadsorbed sodium caseinate [@bressy:2003; @srinivasan:1996; @dickinson:1997; @dickinson:2010; @dickinson:1999]. Presumably, this unusual behaviour is due to the small size of the droplets, which are only one order of magnitude larger than the naturally-occurring caseinate structures. The knowledge and models introduced for the suspensions of proteins and droplets in the previous sections can be used to develop a semi-empirical model to describe the viscosity of mixtures. Semi-empirical predictive model ------------------------------- Models have been developed previously to predict the viscosity of suspensions of multi-modal particles, for example in references [@mendoza:2017] or [@mwasame:2016a], the latter was then extended for mixtures of components of different viscosity behaviours in [@mwasame:2016b]. However these models are mathematically complex and do not describe accurately our experimental results. Instead, a simple and useful approach is to consider that each component of the mixture is independent from the other, as in the early model for multi-modal suspensions described in [@farris:1968]. In this case, the protein suspension acts as a viscous suspending medium for the droplets, whose viscosity behaviour was previously characterised and modelled by Equation \[Eq:QuemadaHS\]. Because the viscosity behaviour of the protein suspension is also known, it can be combined with the droplet behaviour to determine the viscosity of the mixture. This approach is illustrated on Figure \[Fig:MixViscoSchema\]. ![Development of a semi-empirical model to predict the viscosity of emulsions. The contribution of the proteins in suspension to the viscosity of the emulsion is modelled by an increase of viscosity of the continuous medium.[]{data-label="Fig:MixViscoSchema"}](Fig9){width="90.00000%"} ### Development of the model Considering the suspending medium alone first, it is useful to consider the protein content of the aqueous phase residing in the interstices between the droplets, $\phi_{prot}^i$: $$\label{Eq:PhiProtInter} \phi_{prot}^i=\frac{V_{prot}}{V_{prot}+V_{water}}=\frac{\phi_{prot}}{\phi_{prot}+\phi_{water}}=\frac{\phi_{prot}}{1-\phi_{droplet}}$$ Where it is assumed that $\phi_{prot}\simeq \phi_{eff,prot} = k_{0,prot}\times c_{prot}$ and $\phi_{droplet} \simeq \phi_{eff,drop} = k_{0,drop}\times c_{drop}$ according to Eq. \[Eq:EffPhi\_proportional\], with $k_{0,prot}$ and $k_{0,drop}$ determined previously using the Batchelor equation fitted to the viscosities of semi-dilute suspensions of pure proteins and pure droplets. The study of the pure suspensions of protein-stabilised droplets and of proteins makes it possible to model the viscosity behaviour of both suspensions: - The relative viscosity of a suspension of protein-stabilised droplets $\eta_{r,drop}(\phi)$ is described by Equation \[Eq:QuemadaHS\] with the parameter $\phi_{m}=\num[separate-uncertainty=true]{0.79(2)}$ (Quemada model for hard spheres [@quemada:1977]) - The relative viscosity of a suspension of sodium caseinate $\eta_{r,prot}(\phi)$ is described by Equation \[Eq:ModifQuemadaSoft\] with the parameters listed in Table \[Tab:ModifQuemadaProtParameters\] (modified Quemada model) and using $\phi_{prot}^i$ as described above. These elements are then combined to predict the relative viscosity of the mixture $\eta_{r,mix}^p$, in the absence of specific interactions between the droplets and the proteins, thus: $$\label{Eq:ViscoMixModel} \eta_{r,mix}^p(\phi_{eff,prot}, \phi_{eff,drop}) = \eta_{r,prot}\left(\phi_{prot}^i\right) \times \eta_{r,drop}\left(\phi_{eff,drop}\right)$$ ### Application of the model The values of the relative viscosity calculated for each mixture using Equation \[Eq:ViscoMixModel\] are compared to the experimentally measured relative viscosity $\eta_{r,mix}^m$, in Figure \[Fig:RawViscoMix\]. Details of the estimated viscosity of the continuous phase of the mixture can be found in the supplementary material (Figure S3). ![Predicted relative viscosity of mixture suspensions $\eta_{r,mix}^p$, calculated with Equation \[Eq:ViscoMixModel\], as a function of the measured viscosity $\eta_{r,mix}^m$ from Figure \[Fig:RawViscoMix\]. Each point is a mixture of different composition , and its colour indicates the value of the compositional index $\chi_{prot}$ defined by Equation \[Eq:ChiProt\]. The straight line represents y=x. The error bars indicate the uncertainty arising from the calculations (more details are provided in the supplemetary material).](Fig10){width="80.00000%"} Despite the simplicity of this model, it provides a reasonably accurate prediction of the viscosity of protein-stabilised emulsions. This result seems to indicate that there are no specific interactions between the proteins and the droplets, neither at a molecular scale between un-adsorbed and adsorbed proteins, nor at a larger length scale where depletion interactions could occur. This is likely to be related to the small size of the droplets in this specific system, and increasing the droplet size may result in a decreased accuracy of this simple model. The small inaccuracies in the predicted viscosities probably lie in the slightly imperfect fit of Equations  \[Eq:QuemadaHS\] and  \[Eq:ModifQuemadaSoft\]. First, at moderate viscosity ($\eta_r<10$), the slight discrepancy between predicted and measured viscosity of the samples with a high $\chi_{prot}$ is probably a reflection of the modest underestimation of the viscosity of protein suspensions for $2<\phi_{eff}<10$ by Equation \[Eq:ModifQuemadaSoft\]. At higher concentrations, the effective volume fraction approximation may break down. Indeed, as observed previously for pure suspensions, $\phi_{eff}$ can reach high values and may not correspond exactly to the volume fraction actually occupied by the particles, especially in the case of $\phi_{eff,prot}$ . A natural consequence is that the relationship $\phi_{eff,prot} + \phi_{eff,drop} + \phi_{eff,water} = 1$ may not be verified, leading to an overestimation of $\phi_{prot}^i$ when calculated by Equation \[Eq:PhiProtInter\]. It should be noted that the lack of unifying definition of the volume fraction for soft colloids is a particularly relevant challenge when dealing with mixtures. An approach to address this problem could be to take the viscosity behaviour of one of the two components as a reference, and map the volume fraction of the other component to follow this reference viscosity [@mwasame:2016b], but it would considerably increase the complexity of the model. Finally, another possible source of discrepancy is the assumption that the proteins in the interstices will reach the same random close packing fraction as for proteins in bulk $\phi_{rcp,prot}$. However, at high droplet volume fraction, there are geometrical arguments to support the hypothesis of a different random close packing volume fraction due to excluded volume effects. Therefore, this assumption may lead to a decreased accuracy of the model at high concentrations. To summarize, in this section we have shown that the preliminary study of the individual components of a mixture allows the subsequent prediction of the viscosity of mixtures of these components with reasonable accuracy, providing that the composition of the mixtures is known. ### Reversal of the model: estimation of the composition of emulsions A common challenge when formulating protein-stabilised emulsions is to estimate the amount of protein adsorbed at the interface as opposed to the protein suspended in the aqueous phase. Here we suggest that reversing the semi-empirical model developed in the previous section allows estimation of the amount of proteins in suspension after emulsification with a simple viscosity measurement, which can be performed on-line in advanced industrial processing lines. The calculation process is illustrated in Figure \[Fig:PredictViscoReverse\]. ![Reversal of semi-predictive model for the viscosity of protein-stabilised emulsions. The measurement of the emulsion viscosity $\eta_{r,mix}$ makes possible the calculation of the volume fraction of un-adsorbed proteins $\phi_{eff,prot}$, given that the volume fraction of droplets $\phi_{eff,drop}$ is known from the preparation protocol.[]{data-label="Fig:PredictViscoReverse"}](Fig11){width="90.00000%"} To assess the accuracy of the suggested method, a case in point is the emulsion used to prepare the sodium caseinate droplets in this study after microfluidisation. It is composed of $\num{20}\%$(wt) oil and $\num{4.0}\%$(wt) sodium caseinate, and its relative viscosity was measured to be $\eta_{r,mix}^m=\num{10}$. The first step is to calculate the contribution of the oil droplets to the viscosity of the mixture, in order to isolate the protein contribution. A $\num{20}$(wt)$\%$ content in oil corresponds to $\phi_{eff,drop}=\num{0.40}$, so $\eta_{r,drop}=\left(1-\phi_{eff,drop}/\phi_m\right)^{-2}=\num{4.1}$. It is then possible, using the Equation \[Eq:ViscoMixModel\], to calculate the viscosity of the continuous phase $\eta_{r,prot}\left(\phi_{prot}^i\right)=\eta_{r,mix}^m/\eta_{r,drop}=2.4$, assumed to arise from the presence of un-adsorbed proteins. In order to estimate the volume fraction of proteins in the interstices $\phi_{prot}^i$, the equation below has to be solved: $$\label{Eq:MixReverse} \left(1+\left(\frac{\phi_{eff,prot,m}}{\phi_{prot}^i}\right)^n\right)^{-1/n}=1-\frac{1}{\sqrt{\eta_{r,prot}}}$$ Finally, numerically solving Equation \[Eq:MixReverse\] with the values for $n$ and $\phi_m$ from Table \[Tab:ModifQuemadaProtParameters\] gives $\phi_{prot}^i=\num{0.33}$. This result corresponds to a volume fraction of un-adsorbed proteins in the overall emulsion $\phi_{eff,prot} = \phi_{prot}^i (1-\phi_{eff,drop})=\num{0.20}$, or expressed as a concentration in the emulsion: $c=\SI{23}{\milli\gram\per\milli\liter}$. This has to be compared with the initial concentration of $\SI{45}{\milli\gram\per\milli\liter}$ in proteins before emulsification. Thus, only half of the amount of proteins adsorb at the interface, while the other half is still in suspension. This result can be converted into a surface coverage to be compared with studies on sodium caseinate-stabilised emulsions using micron-sized droplets. It is estimated that $\SI{1}{\liter}$ of emulsion containing $\num{20}$(wt)$\%$ of oil, and with a droplet size of $R_{opt,c}=\SI{65}{\nano\metre}$ presents a surface area of oil of $\SI{920}{\metre\squared}$, and from the viscosity $\SI{22}{\gram}$ of sodium caseinate is adsorbed at the interfaces. Thus, the surface coverage is around $\SI{24}{\milli\gram\per\meter\squared}$. This result is in good correspondence with studies on similar emulsions at larger droplet sizes [@srinivasan:1996; @srinivasan:1999], and thus provides a validation for the use of the measurement of the viscosity as a tool to estimate the amount of unadsorbed proteins present in emulsions. The semi-empirical model for the viscosity of emulsions developed in this study, once calibrated, can thus be used not only as a predictive tool for mixtures of droplets and proteins of known composition, but also as a method to estimate the amount of adsorbed proteins without the need for further separation of the components. Conclusion ========== Previous studies have attempted to compare the rheological properties of sodium caseinate to those of a suspension of hard spheres, and found that agreement at high concentrations is poor[@farrer:1999; @pitowski:2008]. As a result it was concluded that a colloidal model is inadequate to describe the observed behaviour. Here we argue that this is mainly due to the choice of hard spheres as colloidal reference. We have shown that using the framework developed for soft colloidal particles, such as microgels and block co-polymer micelles [@vlassopoulos:2014], helps toward a better description of the viscosity behaviour of the protein dispersions. Although this approach neglects the additional layer of complexity due to the biological nature of the sodium caseinate, such as inhomogeneous charge distribution and dynamic aggregation [@sarangapani:2013; @sarangapani:2015], it gives a satisfactory model that can be used to build a better description of protein-stabilised emulsions. Interestingly, the soft colloidal approach can also be successfully applied to the rheology of non-colloidal food particles, such as fruit purees [@leverrier:2017]. In addition, a protocol was developed for preparing pure suspensions of protein-stabilised droplets rather than emulsions containing unadsorbed proteins. The viscosity behaviour of the nano-sized droplets appeared to be very similar to the hard sphere model. The main discrepancy is the high effective volume fraction at which the viscosity diverges, which may be due to the size distribution of droplets or arise from the softness of the layer of adsorbed proteins. Finally, examining protein-stabilised emulsions as ternary mixtures of water, unadsorbed proteins and droplets has allowed us to develop a semi-empirical model for their viscosity. The contributions of each component to the overall viscosity of the emulsions being quantified by the analysis of the properties of the pure suspensions of droplets or proteins. The model can also be reversed to estimate the composition, after emulsification, of a protein-stabilised emulsion given its viscosity. It should be noted, however, that the droplet size is likely to be critical to the success of the model, as it is expected that flocculation of droplets will occur for larger droplets [@bressy:2003; @srinivasan:1996; @dickinson:1997; @dickinson:2010; @dickinson:1999]. This is due to the depletion interaction generated by the proteins in the mixture, which is not taken into account in the present model. For this reason, it would be interesting to explore further the influence of the droplet size on the viscosity behaviour of emulsions. In addition, increasing the droplet size would change the hardness of the droplets by decreasing the internal pressure as well as the influence of the soft layer of proteins, adding further complexity to the system. Supplementary material {#supplementary-material .unnumbered} ====================== The Supplementary material contains information on the calculation of the error bars, the viscosity as a function of the concentration, calculations of the asymptotic behaviour of Equation \[Eq:ModifQuemadaSoft\], flow curves of the shear-thinning samples and the contributions to the viscosity of mixtures by the dispersed and continuous phases. This project forms part of the Marie Curie European Training Network COLLDENSE that has received funding from the European Union’s Horizon 2020 research and innovation programme Marie Skłodowska-Curie Actions under the grant agreement No. 642774. The authors wish to acknowledge DMV for graciously providing the sodium caseinate sample used in this study, and PostNova Analytics Ltd for graciously performing the field flow fractionation measurement of sodium caseinate.

ArXiv

--- abstract: 'We use a database of direct numerical simulations to derive parameterizations for energy dissipation rate in stably stratified flows. We show that shear-based formulations are more appropriate for stable boundary layers than commonly used buoyancy-based formulations. As part of the derivations, we explore several length scales of turbulence and investigate their dependence on local stability.' author: - Sukanta Basu - Ping He - 'Adam W. DeMarco' bibliography: - 'EDR.bib' title: Parameterizing the Energy Dissipation Rate in Stably Stratified Flows --- \[sec:level1\]Introduction ========================== Energy dissipation rate is a key variable for characterizing turbulence [@vassilicos15]. It is a sink term in the prognostic equation of turbulent kinetic energy (TKE; $\overline{e}$): $$\frac{\partial \overline{e}}{\partial t} + ADV = BNC + SHR + TRP + PRC - \overline{\varepsilon}, \label{TKE}$$ where, $\overline{\varepsilon}$ is the mean energy dissipation rate. The terms $ADV$, $BNC$, $SHR$, $TRP$, and $PRC$ refer to advection, buoyancy production (or destruction), shear production, transport, and pressure correlation terms, respectively. Energy dissipation rate also appears in the celebrated “-5/3 law” of Kolmogorov [@kolmogorov41a] and Obukhov [@obukhov41a; @obukhov41b]: $$E(\kappa) \approx \overline{\varepsilon}^{2/3} \kappa^{-5/3}, \label{K41}$$ where, $E(\kappa)$ and $\kappa$ denote the energy spectrum and wavenumber, respectively. In field campaigns or laboratory experiments, direct estimation of $\overline{\varepsilon}$ has always been a challenging task as it involves measurements of nine components of the strain rate tensor. Thus, several approximations (e.g., isotropy, Taylor’s hypothesis) have been utilized and a number of indirect measurement techniques (e.g., scintillometers, lidars) have been developed over the years. In parallel, a significant effort has been made to correlate $\overline{\varepsilon}$ with easily measurable meteorological variables. For example, several flux-based and gradient-based similarity hypotheses have been proposed [e.g., @wyngaard71a; @wyngaard71b; @thiermann92; @hartogensis05]. In addition, a handful of papers also attempted to establish relationships between $\overline{\varepsilon}$ and either the vertical velocity variance ($\sigma_w^2$) or TKE ($\overline{e}$). One of the first relationships was proposed by Chen [@chen74]. By utilizing the Kolmogorov-Obukhov spectrum (i.e., Eq. \[K41\]) with certain assumptions, he derived: $$\overline{\varepsilon} \approx \sigma_w^3. \label{C74}$$ Since this derivation is only valid in the inertial-range of turbulence, a band-pass filtering of vertical velocity measurements was recommended prior to computing $\sigma_w$. A few years later, Weinstock [@weinstock81] revisited the work of [@chen74] and again made use of Eq. \[K41\], albeit with different assumptions (see Appendix 2 for details). He arrived at the following equation: $$\overline{\varepsilon} \approx \sigma_w^2 N, \label{W81}$$ where, $N$ is the so-called Brunt-Väisäla frequency. Using observational data from stratosphere, Weinstock [@weinstock81] demonstrated the superiority of Eq. \[W81\] over Eq. \[C74\]. In a recent empirical study, by analyzing measurements from the CASES-99 field campaign, Bocquet et al. [@bocquet11] proposed to use $\overline{\varepsilon}$ as a proxy for $\sigma_w^2$. In the present work, we quantify the relationship between $\overline{\varepsilon}$ and $\overline{e}$ (as well as between $\overline{\varepsilon}$ and $\sigma_w$) by using turbulence data generated by direct numerical simulation (DNS). To this end, we first compute several well-known “outer” length scales (e.g., buoyancy length scale and Ozmidov scale), normalize them appropriately, and explore their dependence on local stability. Next, we investigate the inter-relationships of certain (normalized) outer length scales (OLS) which portray qualitatively similar stability-dependence. By analytically expanding these relationships, we arrive at two $\overline{\varepsilon}$–$\overline{e}$ and two $\overline{\varepsilon}$–$\sigma_w$ formulations; only the shear-based formulations portray quasi-universal scaling. The organization of this paper is as follows. In Sect. 2, we describe our DNS runs and subsequent data analyses. Simulated results pertaining to various length scales are included in Sect. 3. The $\overline{\varepsilon}$–$\overline{e}$ and $\overline{\varepsilon}$–$\sigma_w$ formulations are derived in Sect. 4. A few concluding remarks, including implications of our results for atmospheric modeling, are made in Sect. 5. In order to enhance the readability of the paper, either a heuristic or an analytical derivation of all the length scales is provided in Appendix 1. Given the importance of Eq. \[W81\], its derivation is also summarized in Appendix 2. Last, in Appendix 3, we elaborate on the normalization of various variables which are essential for the post-processing of DNS-generated data. Direct Numerical Simulation =========================== Over the past decade, due to the increasing abundance of high-performance computing resources, several studies probed different types of stratified flows by using DNS [e.g., @flores11; @garcia11; @brethouwer12; @chung12; @ansorge14; @shah14; @he15; @he16b]. These studies provided valuable insights into the dynamical and statistical properties of these flows (e.g., intermittency, structure parameters). In the present study, we use a DNS database which was previously generated by using a massively parallel DNS code, called HERCULES [@he16a], for the parameterization of optical turbulence [@he16c]. The computational domain size for all the DNS runs was $L_x \times L_y \times L_z = 18 h \times 10 h \times h$, where $h$ is the height of the open channel. The domain was discretized by $2304 \times 2048 \times 288$ grid points in streamwise, spanwise, and wall-normal directions, respectively. The bulk Reynolds number, $Re_b = \frac{U_b h}{\nu}$, for all the simulations was fixed at 20000; where, $U_b$ and $\nu$ denote the bulk (averaged) velocity in the channel and kinematic viscosity, respectively. The bulk Richardson number was calculated as: $Ri_b = \frac{\left(\Theta_{top}-\Theta_{bot}\right)g h}{U_b^2 \Theta_{top}}$; where, $\Theta_{top}$ and $\Theta_{bot}$ represent potential temperature at the top and the bottom of the channel, respectively. The gravitational acceleration is denoted by $g$. A total of five simulations were performed with gradual decrease in the temperature of the bottom wall (effectively by increasing $Ri_b$) to mimic the nighttime cooling of the land-surface. The normalized cooling rates ($CR$), $Ri_b/T_n$, ranged from $1\times10^{-3}$ to $5\times10^{-3}$; where, $T_n$ is a non-dimensional time ($=tU_b/h$). Since we were considering stably stratified flows in the atmosphere, the Prandtl number, $Pr = \nu/k$ was assumed to be equal to 0.7 with $k$ being the thermal diffusivity. All the simulations used fully developed neutrally stratified flows as initial conditions and evolved for upto $T_n = 100$. The simulation results were output every 10 non-dimensional time. To avoid spin-up issues, in the present study, we only use data for the last five output files (i.e., $60 \le T_n \le 100$). Furthermore, we only consider data from the region $0.1 h\le z \le 0.5 h$ to discard any blocking effect of the surface or avoid any laminarization in the upper part of the open channel. The turbulent kinetic energy and its mean dissipation are computed as follows (using Einstein’s summation notation): $$\overline{e} = \frac{1}{2} \overline{u_i' u_i'}$$ $$\overline{\varepsilon} = \nu \overline{\left(\frac{\partial u_i'}{\partial x_j} \frac{\partial u_i'}{\partial x_j}\right)}$$ In these equations and in the rest of the paper, the “overbar” notation is used to denote mean quantities. Horizontal (planar) averaging operation is performed for all the cases. The “prime” symbol is used to represent the fluctuation of a variable with respect to its planar averaged value. Length Scales ============= In this section, we discuss various length scales of turbulence. To enhance the readability of the paper, we do not elaborate on their derivations or physical interpretations here; for such details, the readers are requested to peruse Appendix 1. From the DNS-generated data, we first calculate the integral length scale ($\mathcal{L}$) and Kolmogorov length scale ($\eta$). They are defined as [@tennekes72; @pope00]: $$\mathcal{L} \equiv \frac{\overline{e}^{3/2}}{\overline{\varepsilon}},$$ $$\eta \equiv \left(\frac{\nu^3}{\overline{\varepsilon}}\right)^{1/4}.$$ In Fig. \[fig1\], normalized values of $\mathcal{L}$ and $\eta$ are plotted against the gradient Richardson number ($Ri_g = N^2/S^2$); where, $S$ is the magnitude of wind shear. It is evident that the simulations with larger cooling rates result in smaller $\mathcal{L}$ as would be physically expected. In contrast, $\eta$ marginally increases with higher stability due to lower $\overline{\varepsilon}$. The ratio of $\mathcal{L}$ to $\eta$ decreases from about 100 to 20 as stability is increased from weakly stable condition to strongly stable condition. {width="49.00000%"} {width="49.00000%"} Next, we compute four outer length scales: Ozmidov ($L_{OZ}$), Corrsin ($L_C$), buoyancy ($L_b$), and Hunt ($L_H$). They are defined as [@corrsin58; @dougherty61; @ozmidov65; @brost78; @hunt88; @hunt89; @sorbjan08; @wyngaard10]: $$L_{OZ} \equiv \left(\frac{\overline{\varepsilon}}{N^3}\right)^{1/2},$$ $$L_C \equiv \left(\frac{\overline{\varepsilon}}{S^3}\right)^{1/2}, \label{LC}$$ $$L_b \equiv \frac{\overline{e}^{1/2}}{N},$$ $$L_H \equiv \frac{\overline{e}^{1/2}}{S},$$ \[OLS\] Please note that, in the literature, $L_b$ and $L_H$ have also been defined as $\sigma_w/N$ and $\sigma_w/S$, respectively. Both $L_{OZ}$ and $L_C$ are functions of $\overline{\varepsilon}$, a microscale variable. In contrast, $L_b$ and $L_H$ only depend on macroscale variables. Both shear and buoyancy prefer to deform the larger eddies compared to the smaller ones [@itsweire93; @smyth00; @chung12; @mater13]. Eddies which are smaller than $L_C$ or $L_H$ are not affected by shear. Similarly, buoyancy does not influence the eddies of size less than $L_{OZ}$ or $L_b$. In other words, the eddies can be assumed to be isotropic if they are smaller than all these OLSs. {width="49.00000%"} {width="49.00000%"}\ {width="49.00000%"} {width="49.00000%"} Since $\mathcal{L}$ changes across the simulations, all the OLS values are normalized by corresponding $\mathcal{L}$ values and plotted as functions of $Ri_g$ in Fig. \[fig2\]. The collapse of the data from different runs, on to seemingly universal curves, is remarkable for all the cases except for $Ri_g > 0.2$. We would like to mention that similar scaling behavior was not found if other normalization factors (e.g., $h$) are used. Both normalized $L_{OZ}$ and $L_b$ decrease monotonically with $Ri_g$; however, the slopes are quite different. The length scales $L_C$ and $L_H$ barely exhibit any sensitivity to $Ri_g$ (except for $Ri_g > 0.1$). Even for weakly-stable condition, these length scales are less than 25% of $\mathcal{L}$. Based on the expressions of the OLSs (i.e., Eqs. \[OLS\]) and the definition of the gradient Richardson number, we can write: $$\frac{L_C}{L_{OZ}} = \left(\frac{N}{S}\right)^{3/2} = Ri_g^{3/4},$$ $$\frac{L_H}{L_{b}} = \left(\frac{N}{S}\right) = Ri_g^{1/2}.$$ Thus, for $Ri_g < 1$, one expects $L_C < L_{OZ}$ and $L_H < L_b$. Such relationships are fully supported by Fig. \[fig2\]. In comparison to the buoyancy effects, the shear effects are felt at smaller length scales for the entire stability range considered in the present study. {width="49.00000%"} {width="49.00000%"} Owing to their similar scaling behaviors, $L_b/\mathcal{L}$ against $L_{OZ}/\mathcal{L}$ are plotted in Fig. \[fig3\] (left panel). Once again, all the simulated data collapse nicely in a quasi-universal (nonlinear) curve. Since in a double-logarithmic representation (not shown) this curve is linear, we can write: $$\frac{L_b}{\mathcal{L}} \equiv \left(\frac{L_{OZ}}{\mathcal{L}}\right)^m, \label{Lb_vs_LOZ}$$ where, $m$ is an unknown power-law exponent. By using $L_b \equiv \overline{e}^{1/2}/N$ and the definitions of $L_{OZ}$ and $\mathcal{L}$, we arrive at: $$\frac{\overline{e}^{1/2}}{N} = \left(\frac{\overline{\varepsilon}}{N^3} \right)^{m/2} \left(\frac{\overline{e}^{3/2}}{\overline{\varepsilon}} \right)^{1-m}.$$ Further simplification leads to: $\overline{\varepsilon} = \overline{e} N$; please note that the exponent $m$ cancels out in the process. Instead of $\overline{e}^{1/2}$, if we utilize $\sigma_w$ in the definitions of $L_b$ and $\mathcal{L}$, we get: $\overline{\varepsilon} = \sigma_w^2 N$. This equation is identical to Eq. \[W81\] which was derived by Weinstock [@weinstock81]. His derivation, based on inertial-range scaling, is summarized in Appendix 2. In the right panel of Fig. \[fig3\], we plot $L_H/\mathcal{L}$ versus $L_{C}/\mathcal{L}$. Both these normalized length scales have limited ranges; nonetheless, they are proportional to one another. Like Eq. \[Lb\_vs\_LOZ\], we can write in this case: $$\frac{L_H}{\mathcal{L}} \equiv \left(\frac{L_{C}}{\mathcal{L}}\right)^n, \label{LH_vs_LC}$$ where, $n$ is an unknown power-law exponent. The expansion of this equation leads to either $\overline{\varepsilon} = \overline{e} S$ or $\overline{\varepsilon} = \sigma_w^2 S$, depending on the definition of $L_H$ and $\mathcal{L}$. Parameterizing the Energy Dissipation Rate ========================================== Earlier in Fig. \[fig3\], we have plotted normalized OLS values against one another. It is plausible that the apparent data collapse is simply due to self-correlation as same variables (i.e., $\mathcal{L}$, $N$, and $S$) appear in both abscissa and ordinate. To further probe into this problematic issue, we produce Fig. \[fig4\]. Here, we basically plot normalized $\overline{\varepsilon}$ as functions of normalized $\overline{e} N$, $\overline{e} S$, $\sigma_w^2 N$, and $\sigma_w^2 S$, respectively. These plots have completely independent abscissa and ordinate terms and do not suffer from self-correlation. Please note that the appearance of $Re_b$ and $Ri_b$ in these figures are due to the normalization of variables in DNS; Appendix 3 provides further details. Throughout the paper, the subscript “$n$” is used to denote a normalized variable. {width="49.00000%"} {width="49.00000%"}\ {width="49.00000%"} {width="49.00000%"} It is clear that the plots in the left panel of Fig. \[fig4\], which involve $N$, do not show any universal scaling. For low $CR$ values, normalized $\overline{\varepsilon}$ values do not go to zero; this behavior is physically realistic. One cannot expect $\overline{\varepsilon}$ to go to zero for neutral condition (i.e., $N \to 0$). With increasing cooling rates, the curves seem to converge to an asymptotic curve which passes through the origin. As $\overline{e}$ or $\sigma_w$ continually reduces with increasing stability, one does expect $\overline{\varepsilon}$ to approach zero. In a seminal paper, Deardorff [@deardorff80] proposed a parameterization for $\overline{\varepsilon}$ which for strongly stratified condition approaches to $0.25 \overline{e} N$. In Fig. \[fig4\] (top-left panel), we overlaid this line $\overline{\varepsilon} = 0.25 \overline{e} N$ on the DNS-generated data. Clearly, it only overlaps with the simulated data at the strongly stratified region. If $\overline{\varepsilon} = 0.25 \overline{e} N$ is used in the definition of $L_{OZ}$, after simplification, one gets $L_{OZ} = L_b/2$. The line $L_b = 2 L_{OZ}$ is drawn in Fig. \[fig3\]. As would be anticipated, it only overlaps with the simulated data when the OLS values are the smallest (signifying strongly stable condition). Compared to the left panels, the right panels of Fig. \[fig4\] portray very different scaling characteristics. All the data collapse on quasi-universal curves remarkably; especially, for the $\overline{\varepsilon} \approx \overline{e} S$ case. The slopes of the regression lines, estimated via conventional least-squares approach and bootstrapping [@efron82; @mooney93], are shown on these plots. Essentially, we have found: $$\overline{\varepsilon} = 0.23 \overline{e} S, \label{EDR1}$$ $$\overline{\varepsilon} = 0.63 \sigma_w^2 S. \label{EDR2}$$ We note that our estimated coefficient 0.63 is within the range of values reported by Schumann and Gerz [@schumann95] from laboratory experiments and large-eddy simulations (please refer to their Fig. 1). In summary, neither $\overline{\varepsilon} = \overline{e} N$ nor $\overline{\varepsilon} = \sigma_w^2 N$ are appropriate parameterizations for weakly or moderately stratified conditions; they may provide reasonable predictions for very stable conditions. In contrast, the shear-based parameterizations should be applicable from a wide range of stability conditions, from near-neutral to at least $Ri_g \approx 0.2$. Since within the stable boundary layer, $Ri_g$ rarely exceeds 0.2 [see @nieuwstadt84], we believe Eq. \[EDR1\] or Eq. \[EDR2\] will suffice for most practical boundary layer applications. However, for free atmosphere, where $Ri_g$ can exceed O(1), Deardorff’s parameterization (i.e., $\overline{\varepsilon} = 0.25 \overline{e} N$) might be a more viable option. Unfortunately, we cannot verify this speculation using our existing DNS dataset. Concluding Remarks ================== The boundary-layer community almost always utilizes buoyancy-based energy dissipation rate parameterizations for numerical modeling studies. Only a few studies [e.g., @grisogono08; @rodier17] have explored the possibilities of combining buoyancy-based and shear-based parameterizations. In this paper, we demonstrated that shear-based parameterizations are much more appropriate for stable boundary layer flows as long as $Ri_g$ does not exceed 0.2. Our DNS-based results are in complete agreement with the theoretical work (supported by numerical results) of Hunt et al. [@hunt88]. They concluded: > “...when the Richardson number is less than half, it is the mean shear ... (rather than the buoyancy forces) which is the dominant factor that determines the spatial velocity correlation functions and hence the length scales which determine the energy dissipation or rate of energy transfer from large to small scales.” We sincerely hope that our community will begin to appreciate this incredible insight of Hunt and his co-workers in future modeling studies (including large-eddy simulations). Before closing, we would like to emphasize that the proposed shear-based parameterizations are only applicable away from the surface. Near the surface, due to the blocking effect [see @hunt88; @hunt89], $L_C$ or $L_H$ cannot be a representative length scale. They should be properly combined with an explicit parameterization involving height above ground (e.g., the harmonic mean of $0.4z$ and $L_H$). Data and Code Availability {#data-and-code-availability .unnumbered} ========================== The DNS code (HERCULES) is available from: <https://github.com/friedenhe/HERCULES>. Upon acceptance of the manuscript, all the analysis codes and processed data will be made publicly available via [zenodo.org](zenodo.org). Given the sheer size of the raw DNS dataset, it will not be uploaded on to any repository; however, it will be available upon request from the authors. The first author thanks Bert Holtslag for thought-provoking discussions on this topic. The authors acknowledge computational resources obtained from the Department of Defense Supercomputing Resource Center (DSRC) for the direct numerical simulations. The views expressed in this paper do not reflect official policy or position by the U.S Air Force or the U.S. Government. Appendix 1: Derivation of Length Scales {#appendix-1-derivation-of-length-scales .unnumbered} ======================================= #### Integral Length Scale: Based on the original ideas of Taylor![@taylor35], both Tennekes and Lumley [@tennekes72] and Pope [@pope00] provided a heuristic derivation of the integral length scale. Given TKE ($\overline{e}$) and mean energy dissipation rate ($\overline{\varepsilon}$), an associated integral time scale can be approximated as $\overline{e}/\varepsilon$. One can further assume $\sqrt{\overline{e}}$ to be the corresponding velocity scale. Thus, an integral length scale ($\mathcal{L}$) can be approximated as $\overline{e}^{3/2}/\varepsilon$. In the literature, the autocorrelation function of the longitudinal velocity series is commonly used to derive an estimate of the integral length scale ($L_{11}$). The relationship between $\mathcal{L}$ and $L_{11}$ is discussed by Pope [@pope00]. #### Kolmogorov Length Scale: Pope [@pope00] paraphrased the first similarity hypothesis of Kolmogorov [@kolmogorov41a] as (the mathematical notations were changed by us for consistency): > “In every turbulent flow at sufficiently high Reynolds number, the statistics of the small-scale motions ($l \ll \mathcal{L}$) have a universal form that is uniquely determined by $\nu$ and $\overline{\varepsilon}$.” Based on $\nu$ and $\overline{\varepsilon}$, the following length scale can be formulated using dimensional analysis: $\eta \equiv \left(\frac{\nu^3}{\overline{\varepsilon}}\right)^{1/4}$. At this scale, TKE is converted into heat by the action of molecular viscosity. #### Ozmidov Length Scale: Dougherty [@dougherty61] and Ozmidov [@ozmidov65] independently proposed this length scale. Here, we briefly summarize the derivation of Ozmidov [@ozmidov65]. Based on Kolmogorov [@kolmogorov41a], the first-order moment of the velocity increment ($\Delta u$) in the vertical direction ($z$) can be written as: $$\overline{u\left(z+\Delta z\right) - u(z)} = \overline{\Delta u} = \Delta \overline{u} \approx \overline{\varepsilon}^{1/3} \Delta z^{1/3},$$ where, the overlines denote ensemble averaging. Using this equation, the vertical gradient of longitudinal velocity component can be approximated as: $$\frac{\partial \overline{u}}{\partial z} \approx \frac{\Delta \overline{u}}{\Delta z} \approx \overline{\varepsilon}^{1/3} \Delta z^{-2/3}.$$ Similar equation can be written for the vertical gradient of the lateral velocity component ($\frac{\partial \overline{v}}{\partial z}$). Thus, the magnitude of wind shear ($S$) can be written as: $$S \approx \overline{\varepsilon}^{1/3} \Delta z^{-2/3},$$ By definition, $Ri_g = N^2/S^2$. Thus, $$Ri_g \approx \frac{N^2}{\overline{\varepsilon}^{2/3} \Delta z^{-4/3}} \label{Rig_OZ}$$ Ozmidov [@ozmidov65] assumed that for a certain critical $Ri_g$ (which is assumed to be an unknown constant), $\Delta z$ becomes the representative outer length scale ($L_{OZ}$). Thus, Eq. \[Rig\_OZ\] can be re-written as: $$L_{OZ} \equiv \left(\frac{\overline{\varepsilon}}{N^3}\right)^{1/2}.$$ The unknown proportionality constant is a function of the critical $Ri_g$ and is assumed to be on the order of one. #### Buoyancy Length Scale: The following heuristic derivation is based on Brost and Wyngaard [@brost78] and Wyngaard [@wyngaard10]. In an order-of-magnitude analysis, the inertia term of the Navier-Stokes equations, can be written as: $$\frac{\partial u_i}{\partial t} \sim \frac{U_s}{T_s} \sim \frac{U_s}{L_s/U_s} \sim \frac{U_s^2}{L_s}, \label{LB1}$$ where, $L_s$, $T_s$, and $U_s$ represent certain length, time, and velocity scales, respectively. In a similar manner, the buoyancy term can be approximated as: $$\left(\frac{g}{\Theta_0}\right)\left(\theta'\right) \sim \left(\frac{g}{\Theta_0}\right)\left(\frac{\partial \overline{\theta}}{\partial z}\right) \left(L_s\right) \sim N^2 L_s, \label{LB2}$$ where, $\Theta_0$ and $\theta'$ denote a reference temperature and temperature fluctuations, respectively. Equating the inertia and the buoyancy terms, we get: $$L_s^2 = \frac{U_s^2}{N^2}.$$ For stably stratified flows, either $\overline{e}^{1/2}$ or $\sigma_w$ can be used as an appropriate velocity scale. Accordingly, the length scale ($L_s$) can be approximated as $\frac{\overline{e}^{1/2}}{N}$ or $\frac{\sigma_w}{N}$. In the literature, this length scale is commonly known as the buoyancy length scale ($L_b$). #### Corrsin Length Scale: The derivation of Corrsin [@corrsin58] leverages on a characteristic spectral time-scale, $T_s(\kappa)$, which is representative of the inertial-range. Based on dimensional argument, Onsager [@onsager49] proposed: $$T_s(\kappa) \equiv \frac{1}{\sqrt{\kappa^2E(\kappa)}}.$$ In order to guarantee local isotropy in the inertial-range, Corrsin [@corrsin58] hypothesized that $T_s(\kappa)$ must be much smaller than the time-scale associated with mean shear ($S$). In other words, $$\frac{1}{\sqrt{\kappa^2E(\kappa)}} \ll \frac{1}{S}.$$ Using the “-5/3 law” of Kolmogorov [@kolmogorov41a] and Obukhov [@obukhov41a; @obukhov41b], this equation can be re-written as: $$\frac{1}{\sqrt{\kappa^{4/3} \overline{\varepsilon}^{2/3}}} \ll \frac{1}{S}. \label{LC1}$$ If we assume that for a specific wavenumber $\kappa = 1/L_C$, the equality holds in Eq. \[LC1\], then we get: $$L_C^{2/3} = \frac{\overline{\varepsilon}^{1/3}}{S}.$$ From this equation, we can estimate $L_C$ as defined earlier in Eq. \[LC\]. #### Hunt Length Scale: Hunt [@hunt88] hypothesized that in stratified shear flows, $\overline{\varepsilon}$ is controlled by mean shear ($S$) and $\sigma_w$. From dimensional analysis, it follows that: $$\overline{\varepsilon} \equiv \sigma_w^2 S.$$ The associated length scale, $L_H$, is assumed to be on the order of $\sigma_w/S$. Appendix 2: Derivation of $\overline{\varepsilon} = \sigma_w^2 N$ by Weinstock {#appendix-2-derivation-of-overlinevarepsilon-sigma_w2-n-by-weinstock .unnumbered} ============================================================================== The starting point of Weinstock’s derivation was “-5/3 law” of Kolmogorov [@kolmogorov41a] and Obukhov [@obukhov41a; @obukhov41b]. He integrated this equation in the wavenumber space and set the upper integration limit to infinity. The lower integration limit was fixed at the buoyancy wavenumber ($\kappa_b$). Furthermore, he assumed that the eddies are isotropic for wavenumbers larger than $\kappa_b$ (i.e., in the inertial and viscous ranges). His derivation can be summarized as: $$\begin{split} \frac{3}{2} \sigma_w^2 & = \int_{\kappa_b}^{\kappa_2} \alpha \overline{\varepsilon}^{2/3} \kappa^{-5/3} d\kappa \\ & = \alpha \overline{\varepsilon}^{2/3} \int_{\kappa_b}^{\kappa_2} \kappa^{-5/3} d\kappa \\ & = \frac{3 \alpha}{2} \overline{\varepsilon}^{2/3} \left(\kappa_b^{-2/3} - \kappa_2^{-2/3} \right) \\ & \approx \frac{3 \alpha}{2} \overline{\varepsilon}^{2/3} \kappa_b^{-2/3}. \end{split} \label{W81x}$$ Weinstock [@weinstock81] assumed that $\kappa_b$ can be parameterized by $\frac{N}{\sigma_w}$ (basically, the inverse of the buoyancy length scale $L_b$). By plugging this parameterization into Eq. \[W81x\] and simplifying, we get: $$\begin{split} \overline{\varepsilon} & \approx \sigma_w^3 \kappa_b \\ & \approx \sigma_w^2 N. \end{split}$$ Appendix 3: Normalization of DNS Variables {#appendix-3-normalization-of-dns-variables .unnumbered} ========================================== In DNS, the relevant variables are normalized as follows: $$z_n = \frac{z}{h},$$ $$u_n = \frac{u}{U_b},$$ $$v_n = \frac{v}{U_b},$$ $$w_n = \frac{w}{U_b},$$ $$\theta_n = \frac{\theta-\Theta_{top}}{\Theta_{top}-\Theta_{bot}}.$$ After differentiation, we get: $$\frac{\partial u}{\partial z} = \frac{\partial u}{\partial z_n} \frac{\partial z_n}{\partial z} = \frac{\partial u}{\partial u_n} \frac{\partial u_n}{\partial z_n}\frac{\partial z_n}{\partial z} = \frac{U_b}{h} \frac{\partial u_n}{\partial z_n},$$ $$\frac{\partial v}{\partial z} = \frac{\partial v}{\partial z_n} \frac{\partial z_n}{\partial z} = \frac{\partial v}{\partial v_n} \frac{\partial v_n}{\partial z_n}\frac{\partial z_n}{\partial z} = \frac{U_b}{h} \frac{\partial v_n}{\partial z_n},$$ $$S = \sqrt{\left(\frac{\partial \overline{u}}{\partial z}\right)^2 + \left(\frac{\partial \overline{v}}{\partial z} \right)^2} = \frac{U_b}{h} S_n,$$ $$\frac{\partial \theta}{\partial z} = \frac{\partial \theta}{\partial z_n} \frac{\partial z_n}{\partial z} = \frac{\partial \theta}{\partial \theta_n} \frac{\partial \theta_n}{\partial z_n}\frac{\partial z_n}{\partial z} = \left(\frac{\Theta_{top}-\Theta_{bot}}{h}\right) \frac{\partial \theta_n}{\partial z_n}.$$ The gradient Richardson number can be expanded as: $$Ri_g = \frac{N^2}{S^2} = \frac{\left(\frac{g}{\Theta_0}\right)\left(\frac{\partial \overline{\theta}}{\partial z}\right)}{S^2} = \left(\frac{g}{\Theta_{top}}\right) \left(\frac{\Theta_{top}-\Theta_{bot}}{h}\right)\left(\frac{h}{U_b} \right)^2 \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$ Using the definition of $Ri_b$ (see Sect. 2), we re-write $Ri_g$ as follows: $$Ri_g = Ri_b \frac{\left(\frac{\partial \overline{\theta}_n}{\partial z_n} \right)}{S_n^2}.$$ Similarly, $N^2$ can be written as: $$N^2 = Ri_b \left(\frac{U_b^2}{h^2}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right). \label{EqN2}$$ The velocity variances and TKE can be normalized as: $$\sigma_{u_n}^2 = \frac{\sigma_u^2}{U_b^2},$$ $$\sigma_{v_n}^2 = \frac{\sigma_v^2}{U_b^2},$$ $$\sigma_{w_n}^2 = \frac{\sigma_w^2}{U_b^2}, \label{EqSigw}$$ $$\overline{e}_n = \frac{\overline{e}}{U_b^2}. \label{Eqe}$$ Following the above normalization approach, we can also derive the following relationship for the energy dissipation rate: $$\overline{\varepsilon} = \nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n. \label{EqEDR}$$ In order to expand $\overline{\varepsilon} = \overline{e} N$, we use Eq. \[EqN2\], Eq. \[Eqe\], and Eq. \[EqEDR\] as follows: $$\nu \left(\frac{U_b}{h}\right)^2 \overline{\varepsilon}_n = U_b^2 \overline{e}_n Ri_b^{1/2} \left(\frac{U_b}{h}\right) \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ This equation can be simplified to: $$\overline{\varepsilon}_n = Re_b Ri_b^{1/2} \overline{e}_n \left(\frac{\partial\overline{\theta}_n}{\partial z_n}\right)^{1/2}.$$ In a similar manner, $\overline{\varepsilon} = \overline{e} S$ can be re-written as: $$\overline{\varepsilon}_n = Re_b \overline{e}_n S_n.$$

ArXiv

--- abstract: 'In this short article we develop recent proposals to relate Yang-Baxter sigma-models and non-abelian T-duality. We demonstrate explicitly that the holographic space-times associated to both (multi-parameter)-$\beta$-deformations and non-commutative deformations of ${\cal N}=4$ super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.' --- [**Marginal and non-commutative deformations\ via non-abelian T-duality**]{} [Ben Hoare$^{a}$ and Daniel C. Thompson$^{b}$]{} [*$^{a}$ Institut für Theoretische Physik, ETH Zürich,\ Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland.*]{} [*$^{b}$ Theoretische Natuurkunde, Vrije Universiteit Brussel & The International Solvay Institutes,\ Pleinlaan 2, B-1050 Brussels, Belgium.*]{} [*E-mail:  *]{} [<[email protected]>, <[email protected]>]{} Introduction {#sec:intro} ============ There is a rich interplay between the three ideas of T-duality, integrability and holography. Perhaps the most well studied example of this is the use of the TsT transformation to ascertain the gravitational dual space-times to certain marginal deformations of ${\cal N}=4$ super Yang-Mills gauge theory [@Lunin:2005jy]. Whilst this employs familiar T-dualities of $U(1)$ isometries in space-time, T-duality can be extended to both non-abelian isometry groups and to fermionic directions in superspace. Such generalised T-dualities also have applications to holography. Fermionic T-duality [@Berkovits:2008ic; @Beisert:2008iq] was critical in understanding the scattering amplitude/Wilson loop duality at strong coupling. T-duality of non-abelian isometries has been employed as a solution generating technique in Type II supergravity [@Sfetsos:2010uq], relating for instance $AdS_5\times S^5$ to (a limit[^1] of) the space-times corresponding to ${\cal N}=2$ non-Lagrangian gauge theories. Developing the recent results of [@Hoare:2016wsk; @Borsato:2016pas] this note will investigate further the role generalised notions of T-duality can play in holography. A new perspective on deformations of the $AdS_5 \times S^5$ superstring has come from the study of Yang-Baxter deformations of string $\sigma$-models [@Klimcik:2002zj; @Klimcik:2008eq; @Klimcik:2014bta; @Delduc:2013fga; @Delduc:2013qra]. These are integrable algebraic constructions which deform the target space of the $\sigma$-model through the specification of an antisymmetric $r$-matrix solving the (modified) classical Yang-Baxter equation ((m)cYBE). If the $r$-matrix solves the mcYBE then, applied to the supercoset formulation of strings in $AdS_5\times S^5$ [@Metsaev:1998it; @Berkovits:1999zq], these give rise to $\eta$-deformed space-times which are conjectured to encode a quantum group $q$-deformation of ${\cal N}=4$ super Yang-Mills with a deformation parameter $q \in \mathbb{R}$ [@Delduc:2014kha; @Arutyunov:2013ega; @Arutyunov:2015qva]. However the $\eta$-deformed worldsheet theory appears to be only globally scale invariant [@Hoare:2015gda; @Hoare:2015wia], the target space-time does not solve exactly the Type II supergravity equations [@Arutyunov:2015qva] but rather a generalisation thereof [@Arutyunov:2015mqj]. Classically $\eta$-deformations are related via a generalised Poisson-Lie T-duality [@Vicedo:2015pna; @Hoare:2015gda; @Sfetsos:2015nya; @Klimcik:2015gba; @Klimcik:2016rov; @Delduc:2016ihq] to a class of integrable deformation of (gauged) WZW models known as $\lambda$-deformations [@Sfetsos:2013wia; @Hollowood:2014rla; @Hollowood:2014qma], which do however have target space-times solving the usual supergravity equations of motion [@Sfetsos:2014cea; @Demulder:2015lva; @Borsato:2016zcf; @Chervonyi:2016ajp]. There is also evidence that the latter class corresponds to a quantum group deformation of the gauge theory, but with $q$ a root of unity [@Hollowood:2015dpa]. If instead the $r$-matrix solves the unmodified cYBE (a homogeneous $r$-matrix), first considered in [@Kawaguchi:2014qwa], the YB $\sigma$-models have been demonstrated to give a wide variety of integrable target space-times including those generated by TsT transformations [@Matsumoto:2014nra; @Matsumoto:2015uja; @Matsumoto:2014gwa; @Matsumoto:2015jja; @vanTongeren:2015soa; @Kyono:2016jqy; @Osten:2016dvf]. For these models the corresponding dual theory can be understood in terms of a non-commutative $\mathcal{N} = 4$ super Yang-Mills with the non-commutativity governed by the $r$-matrix and the corresponding Drinfel’d twist [@vanTongeren:2015uha; @vanTongeren:2016eeb]. Recently it has been shown that such YB $\sigma$-models can be also be understood in terms of non-abelian T-duality: given an $r$-matrix one can specify a (potentially non-abelian) group of isometries of the target space with respect to which one should T-dualise [@Hoare:2016wsk]. The deformation parameter appears by first centrally extending this isometry group and then T-dualising. Following a Buscher-type procedure, the Lagrange multiplier corresponding to the central extension is non-dynamical. In particular it is frozen to a constant value and thereby plays the role of the deformation parameter. This conjecture was proven in the NS sector in [@Borsato:2016pas], where a slightly different perspective was also given. If one integrates out only the central extension, the procedure above can be seen to be equivalent to adding a total derivative $B$-field constructed from a 2-cocycle on the isometry group with respect to which we dualise and then dualising. In this note we develop this line of reasoning. We begin by outlining the essential features of Yang-Baxter $\sigma$-models and the technology of non-abelian T-duality in Type II supergravity. After demonstrating that a centrally-extended T-duality can be reinterpreted as as non-abelian T-duality of a coset based on the Heisenberg algebra, we show how the machinery of non-abelian T-duality developed for Type II backgrounds can be readily applied to the construction of [@Hoare:2016wsk; @Borsato:2016pas]. We confirm that the centrally-extended non-abelian T-duals produce the full Type II supergravity backgrounds corresponding to $\beta$-deformations (when the duality takes place in the $S^5$ factor of $AdS_5\times S^5$), non-commutative deformations (when performed in the Poincaré patch of $AdS_5$) and dipole deformations (when performed in both the $S^{5}$ and $AdS^{5}$ simultaneously). In appendices \[app:sugra\] and \[app:algconv\] we outline our conventions for supergravity and certain relevant algebras respectively. As a third appendix \[app:furtherexamples\] we include some additional worked examples including one for which the non-abelian T-duality is anomalous and the target space solves the generalised supergravity equations. The supergravity backgrounds in this note have appeared in the literature in the past but the derivation and technique presented here is both novel, simple and, we hope may have utility in the construction of more general supergravity backgrounds. Yang Baxter sigma-models {#sec:yangbaxter} ======================== Given a semi-simple Lie algebra $\mathfrak{f}$ (and corresponding group $F$) we define an antisymmetric operator $R$ obeying $$\label{eq:cybe} [R X , R Y] - R\left([R X, Y]+ [X,RY] \right) = c [ X, Y] \ , \quad X,Y \in \mathfrak{f} \ ,$$ where the cases $c=\pm 1$ and $c=0$ are known as the classical and modified classical Yang Baxter equations (cYBE and mcYBE) respectively. We adopt some notation $X\wedge Y = X\otimes Y - Y \otimes X$ and define e.g. $$r= T_1 \wedge T_2 + T_3 \wedge T_4 +\dots \ , \quad RX = {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes X)) \ .$$ We define an inner product by the matrix trace of generators, ${\operatorname{Tr}}(T_{A} T_{B})$, and lower and raise indices with this inner product and its inverse. In this way the $r$-matrix acts as $$\label{eq:rmatdef} R(T_{A}) \equiv R_{A}{}^{B}T_{B} \ , \quad R_{A}{}^{B} = {\operatorname{Tr}}\left( {\operatorname{Tr}}_2 ( r (\mathbb{I}\otimes T_{A})) T^{B} \right) \ .$$ Suppose we have a $\mathbb{Z}_{2}$ grading $\mathfrak{f} = \mathfrak{g}\oplus \mathfrak{k}$ for a subgroup $\mathfrak{g}$. Let $T_{A}$ be generators for $\mathfrak{f}$, $T_{\alpha}$ those of $\mathfrak{g}$ and $T_{i}$ the remaining orthogonal generators of $\mathfrak{k}$. We introduce a projector to the coset defined by $P(T_{\alpha})= 0$ and $P(T_{i})= T_{i}$ or, in matrix form, $$P(T_{A}) \equiv P_{A}{}^{B}T_{B} \ , \quad P_{A}{}^{B} = {\operatorname{Tr}}\left( P( T_{A}) T^{B} \right) \ .$$ We also define the adjoint action for $g\in F$ by $${\operatorname{Ad}}_{g} T_{A} \equiv gT_{A} g^{-1} \equiv D_{A}{}^{B}(g) T_{B} \ , \quad D_{AB} = {\operatorname{Tr}}(g T_{A} g^{-1} T_{B} ) \ .$$ Let the two-dimensional worldsheet field $g$ be a coset representative for $F/G$ with which we define the currents $$J_\pm = J_{\pm}^{A}T_{A} = g^{-1} \partial_{\pm} g \ , \quad J_{\pm}^{A}= {\operatorname{Tr}}(g^{-1} \partial_{\pm} g T^{A}) \ ,$$ where we use light-cone coordinates $\partial_\pm = \partial_0 \pm \partial_1$. The standard (bosonic) $\sigma$-model whose target is the coset space $F/G$ is $$\label{eq:cosetPCM} {\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ .$$ To define the Yang-Baxter model first we let $$R_{g} = {\operatorname{Ad}}_{g^{-1}} R {\operatorname{Ad}}_{g} \ , \quad (R_{g})_{A}{}^{B} = D(g)_{A}{}^{C}R_{C}{}^{D}D(g^{-1})_{D}{}^{B} \ ,$$ and define the operator $${\cal O} = \mathbb{I} - \eta R_{g}P \ , \quad {\cal O} _{A}{}^{B} = \delta_{A}{}^{B} - \eta P_{A}{}^{C} (R_{g})_{C}{}^{B} \ ,$$ in which we have explicitly introduced the deformation parameter $\eta$. Later we will restrict to the case $c=0$ in , in which case the parameter $\eta$ can be absorbed into the definition of $R$. The Yang-Baxter $\sigma$-model on a coset is given by [@Delduc:2014kha; @Matsumoto:2014nra] $${\cal L } = {\operatorname{Tr}}(J_+ P( {\cal O}^{-1}J_-) ) = J_{+}^{A} E_{AB} J_{-}^{B} \ , \quad E_{AB} = {\cal O}^{-1}{}_{B}{}^{C} P_{CA} \ .$$ Non-abelian T-duality Technology {#sec:natd} ================================ In this section we will mainly follow the approach of [@Sfetsos:2010uq; @Lozano:2011kb; @Itsios:2013wd] including the transformation of RR fluxes. Some subtleties are caused by the dualisation in a coset space and the approach here is slightly different to the one in [@Lozano:2011kb]. Let us consider the standard (bosonic) $\sigma$-model whose target is the coset space $F/G$ whose Lagrangian is given in eq. , and perform the non-abelian T-dual with respect to a subgroup $H \subset F$ (which need not, and in our applications mostly will not be, either semi-simple or a subgroup of $G$). Let $H_a$ be the generators of $\mathfrak{h}$ and $\tilde{H}^a$ generators of a dual algebra $\mathfrak{h}^\star$ normalised such that ${\operatorname{Tr}}(H_a \tilde{H}^b)= \delta^b_a$. Let us we parametrise the coset representative as $g= h \hat{g}$. We define $\hat{J}= \hat{g}^{-1} d \hat{g}$ and $L = L^a H_a = h^{-1}dh$ such that $$J= \hat{J}+ L^a H_a^{\hat{g} }\ , \quad H_a^{\hat{g} } ={\operatorname{Ad}}_{\hat{g}^{-1}} H_a \ .$$ We also define $$\begin{aligned} G_{ab} &= {\operatorname{Tr}}( H^{\hat{g}}_{a} P( H^{\hat{g}}_{b} ) ) \ , \quad Q_{a} &= {\operatorname{Tr}}( \hat{J} P( H^{\hat{g}}_{a} ) ) \ . \end{aligned}$$ In this notation the $H$ isometry of the target space is manifest since the metric corresponding to eq. is $$ds^2 = {\operatorname{Tr}}( \hat{J} P(\hat{J} )) + 2 Q^T L + L^T G L = {\operatorname{Tr}}( \hat{J} P(\hat{J} )) - Q^TG^{-1} Q + e^T e \ ,$$ where we introduce the frame fields $$G= \kappa^T \kappa \ , \quad e= \kappa \left( L + G^{-1} Q \right) .$$ We perform the dualisation by introducing a $\mathfrak{h}$-valued connection with components $A_{\pm} = A_{\pm}^{a}H_{a}$ and a $\mathfrak{h}^\star$-valued Lagrange multiplier $V= v_{a} \tilde{H}^{a}$. We covariantise currents $$J^{\nabla}_{\pm} = g^{-1} d g + g^{-1} A_{\pm } g \ ,$$ such that we are gauging a left action of some $\tilde{h} \in H$ $$g \rightarrow \tilde{h} g \ , \quad A \rightarrow \tilde{h} A \tilde{h} ^{-1} - d \tilde{h} \tilde{h}^{-1} \ ,$$ and consider $${\cal L }^{ \nabla} = {\operatorname{Tr}}(J^{\nabla}_+ P(J^{\nabla}_-) ) + {\operatorname{Tr}}(V F_{+ -} ) \ ,$$ where the field strength is $F_{+-} = \partial_{+} A_{-} - \partial_{-} A_{+} + [A_{+} , A_{-}]$. We continue by gauge fixing on the group element $g = \hat{g}$ i.e. $h=1$.[^2] Integrating the Lagrange multipliers enforces a flat connection and one recovers the starting model since $$\label{eq:puregauge} A_\pm = h^{-1}\partial_\pm h = L_\pm \ ,$$ and upon substituting back into the action one recovers the starting $\sigma$-model. On the other hand, integrating by parts the derivative terms of the gauge fields yields $${\cal L }^{ \nabla} = {\operatorname{Tr}}(\hat{J}_{+} P(\hat{J}_-) ) + A_{+}^{a}A_{-}^{b} M_{ab} + A_{+}^{a}( \partial_{-} v_{a}+ Q_{-a} ) - A_{-}^{a}( \partial_{+} v_{a} - Q_{+a} ) \ ,$$ in which we have pulled back the one-forms $Q$ and $\hat{J}$ to the worldsheet and defined $$\begin{aligned} F_{ab} &= {\operatorname{Tr}}([H_{a} ,H_{b}]V) = f_{ab}{}^{c} v_{c} \ , \quad M_{ab} =G_{ab} + F_{ab} \ . \end{aligned}$$ The gauge field equations of motion now read $$\label{eq:gauge} A_{-} = - M^{-1} ( \partial_{-} v + Q_{-} ) \ , \quad A_{+} = M^{-T} ( \partial_{+} v - Q_{+ } ) \ .$$ Combining these equations of motion for the gauge field in eqs.  and sets up the canonical transformation between T-dual theories. Substitution of the gauge field equation of motion into the action yields the T-dual model given by $$\begin{split}\label{eq:dualmodellag} {\cal L }_{dual} & = {\operatorname{Tr}}(\hat J_+ P (\hat J_-)) - A_+^T M A_- \\ & = {\operatorname{Tr}}(\hat{J}_{+} P(\hat{J}_-) ) + ( \partial_{+} v_{a} - Q_{+a} )(M^{-1})^{ab} ( \partial_{-} v_{b} + Q_{-b} ) \ , \end{split}$$ where in the first line $A_\pm$ are evaluated on the gauge field equation of motion eq. . The NS fields can be read directly from this $\sigma$-model and in particular the dual metric is given as $$\widehat{ds}^2 = {\operatorname{Tr}}( \hat{J} P(\hat{J} )) - Q^TG^{-1} Q + \widehat{e}_\pm^T \widehat{e}^{\vphantom{T}}_\pm \ ,$$ with $\widehat{e}_\pm$ given by the push forwards to target space of $$\label{eq:dualframe} \widehat{e}_\pm = \kappa \left( A_\pm + G^{-1} Q_\pm \right) \ ,$$ evaluated on the gauge field equation of motion . On the worldsheet left and right moving fermionic sectors couple to the frame fields $\widehat{e}_{+}$ and $\widehat{e}_-$ respectively. Since they define the same metric they are related by a local Lorentz rotation $$\label{eq:Lorentztrans} \Lambda \widehat{e}_{-} = \widehat{e}_{+} \ , \quad \Lambda = -\kappa M^{-T}M \kappa^{-1}$$ This Lorentz rotation lifts to spinors via $$\label{eq:Spinortrans} \Omega^{-1}\Gamma^a \Omega= ( \Lambda\cdot\Gamma)^a \ .$$ Using the Clifford isomorphism we convert the poly-form sum of RR fluxes $$\label{eq:Polyform} {\cal P}= e^{\Phi} ( F_{1 } + F_{3} + F_{5} - \star F_3 + \star F_1 ) \ ,$$ to a bi-spinor matrix. The T-duality rule is then given by $$\widehat{{\cal P}} ={\cal P} \cdot \Omega^{-1} \ .$$ The relationship between the local Lorentz rotations and RR field transformation in the case of abelian T-duality in curved space was made explicit in the work of Hassan [@Hassan:1999bv] and developed in the present context in [@Sfetsos:2010uq]. Note that although we have “bootstrapped” the transformation rule for the RR sector from knowledge of the NS sector it seems rather likely that the same conclusion can be reached in e.g. the pure spinor superstring by a straightforward extension of the arguments presented for abelian [@Benichou:2008it] and fermionic T-duality [@Sfetsos:2010xa].[^3] Finally let us turn to the transformation of the dilaton field under non-abelian T-duality. For the non-abelian duality to preserve conformality the dualisation procedure must avoid the introduction of a mixed gravitational-gauge anomaly [@Alvarez:1994np; @Elitzur:1994ri] and the structure constants of the algebra in which we dualise should satisfy $$n_a \equiv f_{ab}{}^b = 0 \ .$$ When this is the case the dual dilaton comes from the Gaussian integration in the path integral [@Buscher:1987qj] $$\label{eq:diltrans} \widehat{\Phi} = \Phi - \frac{1}{2}\log \det M \ .$$ On the other hand if $n_a \neq 0$ the dual model is not expected to be conformal, however it will be globally scale invariant. In this case we still define the dual “dilaton” to be given by . The global scale invariance then implies that, for example, the one-loop metric and $B$-field beta-functions (defined in of appendix \[app:sugra\]) only vanish up to diffeomorphisms and gauge transformations. This is in contrast to the conformal case, for which the beta-functions of the metric, $B$-field and dilaton vanish identically, while the RR fluxes solve the first order equations in eq. of appendix \[app:sugra\]. It transpires that the globally scale invariant models that arise from dualising with $n_a \neq 0$ satisfy a stronger set of equations than those of global scale invariance [@Hoare:2016wsk]. These are a modification of the Type II supergravity equations [@Arutyunov:2015mqj; @Wulff:2016tju; @Sakatani:2016fvh] that depend on a particular Killing vector $I$ of the background such that when $I = 0$ standard Type II supergravity is recovered. These equations are given in eqs. of appendix \[app:sugra\]. As mentioned above we take the dual “dilaton” field in these equations to still be defined in terms of the original dilaton via the transformation , while the one-forms $X$, $Z$ and $W$ are defined in terms of $\Phi$ and the Killing vector $I$ as in eq. of appendix \[app:sugra\]. To show that the dual background solves the modified supergravity equations we follow the derivation in [@Hoare:2016wsk]. After splitting the Lagrange multiplier as $v_a = u_a + y n_a$, it transpires that shifting $y$ is a symmetry of the dual background and T-dualising $y \to \tilde y$ gives a conformal $\sigma$-model with a dilaton linear in $\tilde y$. From the results of [@Arutyunov:2015mqj] this then implies that, in our conventions, the dual model solves the modified supergravity equations with $I^y = - 1$. The classical bosonic string Lagrangian in conformal gauge, $$\label{eq:cbsacg} \mathcal{L} = \partial_+ x^m (G_{mn} + B_{mn}) \partial_- x^n \ ,$$ has the property that when we replace $\partial_- x^m \to I^m$ it equals $W_n \partial_+ x^n$ where the one-form $W$, defined in eq. , is given by $$W_n = I^m (G_{mn} - B_{mn}) \ .$$ Following this procedure in the dual model with $I^y = - 1$ and the remaining components vanishing, we find that $$\label{eq:Ampush} W_n \partial_+ x^n = - A_+^a n_a \ ,$$ with $A_+$ evaluated on the gauge field equation of motion . To summarise; if the T-duality is anomalous then the background solves the modified supergravity equations with the one-form $W$, which can be used to define the modification, given by the push forward of the $A_+$ component of the gauge field evaluated on its equations of motion. Centrally-extended duality {#sec:centralext} ========================== Let us now consider non-abelian T-dualities with respect to centrally-extended algebras. In particular we consider the setup considered in [@Hoare:2016wsk; @Borsato:2016pas] in which case the dualities are equivalent to Yang-Baxter deformations for homogeneous $r$-matrices. The aim of this section is to extend this to the RR fluxes using the technology outlined in section \[sec:natd\]. We start by recalling that for a homogeneous $r$-matrix for a Lie algebra $\mathfrak{f}$ $$\label{eq:rmatans} r= \sum_{j} \eta_j \, \big( \sum_{i=1}^{n(j)} a_{ij} \, X_{ij} \wedge Y_{ij} \big) \ ,$$ the generators $\{X_{ij},Y_{ij}\}$ (for each fixed $j$) form a basis for a subalgebra $\mathfrak{h}$, which admits a central extension. In eq. $\eta_{j}$ are free parameters, while $a_{ij}$ are fixed real coefficients. For each free parameter we introduce a central extension, such that the centrally-extended algebra has a basis $\mathfrak{h}^{ext} = \{X_{ij}, Y_{ij}\} \oplus \{Z_j\}$, with commutation relations $[X_{ij},Y_{ij}]^{ext} = [X_{ij},Y_{ij}] + a_{ij}^{-1} Z_j$ (for fixed $i$ and $j$), and $[X_{ij},Z_j]^{ext} = [Y_{ij},Z_j]^{ext} = 0$. This is the centrally-extended algebra with respect to which we dualise. The precise relation between the centrally-extended non-abelian T-dual and the Yang-Baxter deformation was made in the NS sector in [@Borsato:2016pas]. The $R$-operator (see eq. ) governing a certain Yang-Baxter deformation defines an invertible map from $\mathfrak{h}^\star$ to $\mathfrak{h}$. Recalling our parametrisation of the $F/G$ coset representative $g= h \hat{g}$ with $h \in H$, we may write $h = \exp(R(X))$ for $X\in \mathfrak{h}^\star$. If $\mathfrak{h}$ is abelian then the relation between the Lagrange multipliers parametrising the T-dual model and the YB deformed model is simple: $V= \eta^{-1} R(X)$. When $\mathfrak{h}$ is non-abelian the relation is more involved [@Borsato:2016pas]. One can formally set up the non-abelian T-dual of the central extension by considering the coset of the centrally-extended algebra by the central generators. To see this precisely let us consider the Heisenberg algebra, i.e. the central extension of $U(1)^2$ $$[X, Y ]= Z \ , \quad [X , Z] = [ Y,Z ] = 0 \ .$$ We let $T_{1}= X, T_{2}= Y$ and $T_{3} = Z$ and hence the only non-vanishing structure constant is $f_{12}{}^{3}=1$. We introduce the matrix generators $$T_{1} = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \ , \quad T_{2} = \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right) \ , \quad T_{3 }= \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \ ,$$ and the group element $$g= \exp \left[ x_{1} T_{1} + x_{2}T_{2} + (x_{3}-\frac{1}{2} x_{1}x_{2} ) T_{3}\right] = \left( \begin{array}{ccc} 1& x_{1} &x_{3} \\ 0 &1 & x_{2}\\ 0 & 0 & 1 \\ \end{array} \right) \ .$$ The left-invariant one-forms $g^{-1}dg= L^{i} T_{i}$ are $$L^{1} = dx_{1} \ , \quad L^{2 } = dx_{2} \ , \quad L^{3} = dx_{3} - x_{1} dx_{2} \ .$$ We consider a $\sigma$-model based on this algebra $$\label{eq:Heisenberg} {\cal L} = E_{ab} L_+^a L_-^b = f_1 L^{1}_+ L^{1}_- + f_2 L^{2}_+ L^{2}_- + \lambda L^{3}_+ L^{3}_- \ ,$$ i.e. $E = \operatorname{diag} (f_1,f_2, \lambda)$, where we allow $f_{1,2}$ to be functions of any spectator coordinates. In the limit $\lambda \rightarrow 0$ the theory develops a gauge invariance (the coordinate $x_3$ drops out of the action all together) and reduces to the $\sigma$-model whose target space is simply $ds^2 = f_1 dx_1^2 + f_2 dx_2^2$. This Rube Goldberg construction allows us to now go head and perform a non-abelian T-duality on the coset following the techniques of [@Lozano:2011kb]. The resulting dual $\sigma$-model is given by $$\mathcal{L}_{dual} = \partial_+ v_a (M^{-1})^{ab} \partial_- v^b$$ in which $$\begin{split} M_{ab} = E_{ab} + f_{ab}{}^c v_c & = \left( \begin{array}{ccc} f_1 & v_3 & 0 \\ -v_3 & f_2 & 0 \\ 0 & 0 & \lambda \end{array} \right) \ , \\ (M^{-1})^{ab} & = \left( \begin{array}{ccc} h f_2 & - h v_3 & 0 \\ h v_3 & h f_1 & 0 \\ 0 & 0 & \frac{1}{\lambda} \end{array} \right) \ , \quad h= \frac{1}{f_1 f_2 + v_3^2}\ . \end{split}$$ The matrix $M^{-1}$ diverges in the limit of interest $\lambda \to 0$. In particular, the coefficient of the kinetic term for $v_3$ becomes infinite in the limit and this can be understood as freezing $v_3$ to a constant value. To see this let us rewrite the dual $\sigma$-model as $$\mathcal{L}_{dual} = \partial_+ v_\alpha (M^{-1})^{\alpha\beta} \partial_- v^\beta + \lambda a_+ a_- + a_+ \partial_-v_3 - a_- \partial_+ v_3 \ , \quad \alpha,\beta = 1,2 \ ,$$ where we integrate over $a_\pm$. Now taking $\lambda \to 0$ and then integrating out $a_\pm$ we find $\partial_\pm v_3 = 0$ and indeed $v_3$ is frozen to a constant value. This final step is analogous to the Buscher procedure considered in [@Hoare:2016wsk]. The true target space of the dual model is then spanned by the coordinates $v_1 \equiv y_2$ and $v_2 \equiv y_1$, while $v_3 \equiv \nu$ is a constant parameter. The dual metric, B-field and dilaton shift are easily ascertained: $$\label{eq:eq1} \widehat{ds}^2 = h ( f_1 dy_1^2 + f_2 dy_2^2 ) \ , \quad \widehat{B} = \nu h dy_1 \wedge dy_2 \ , \quad \widehat{\Phi} = \Phi + \frac{1}{2}\log h \ .$$ Frame fields for the dual geometry as seen by left and right movers [@Lozano:2011kb] are given by $$\label{eq:eq2} \widehat{e}_{+}^{\, i} = (\kappa\cdot M^{-1})^{a i} d v_a \, \quad \widehat{e}_{-}^{\,i} = - (\kappa \cdot M^{-1})^{ i a } dv_a \ , \quad i=1,2 \ , \quad a = 1,2,3 \ .$$ where $\frac{1}{2} (E+ E^T)= \kappa^T \kappa$. Explicitly we have $$\label{eq:eq3} \widehat{e}_+ = \left(\begin{array}{c} h \sqrt{f_1} (f_2 dy_2 + \nu dy_1) \\ h \sqrt{f_2}(f_1 dy_1 - \nu d y_2) \end{array}\right) \ , \quad \widehat{e}_- = \left(\begin{array}{c} h \sqrt{f_1} (-f_2 dy_2 + \nu dy_1) \\ h \sqrt{f_2}( -f_1 dy_1 - \nu d y_2) \end{array}\right) \ .$$ The plus and minus frames are then related by a Lorentz rotation $$\label{eq:eq4} \Lambda \cdot \widehat{e}_- = \widehat{e}_+ \ , \quad \Lambda = h \left(\begin{array}{cc} \nu^2 -f_1f_2 & - 2 \nu \sqrt{f_1 f_2} \\ 2\nu\sqrt{f_1 f_2} & \nu^2 - f_1 f_2 \end{array}\right) \ , \quad \det \Lambda = 1 \ , \quad \Lambda \cdot \Lambda^T = \mathbb{I} \ .$$ This coset-based construction is interesting, however for calculation purposes it is enough to follow the T-duality rules for the non-centrally-extended dualisation, while replacing the structure constants entering the $\dim H \times \dim H$ matrix $F_{ab}= {\operatorname{Tr}}([ H_{a}, H_{b}] V)$ with the corresponding central extension and the centrally-extended Lagrange multipliers i.e. $V^{ext} = v_{a} H^{a} + v_{\mu }Z^{\mu}$ and $F^{ext}_{ab}= {\operatorname{Tr}}([ H_{a}, H_{b}]^{ext} V^{ext})$. Applications {#sec:examples} ============ Let us now turn to specific examples for which we construct the dual RR fluxes corresponding to various centrally-extended non-abelian T-dualities of $AdS_5 \times S^5$ using the technology outlined in section \[sec:natd\]. Here we will consider certain deformations that are well-known to correspond to TsT transformations. In appendix \[app:furtherexamples\] we consider further examples that correspond to Yang-Baxter deformations with time-like abelian and non-abelian $r$-matrices. Application 1: Non-Commutative Deformations {#ssec:app1} ------------------------------------------- The first application we consider is the string background dual to non-commutative $\mathcal{N} = 4$ super Yang-Mills [@Hashimoto:1999ut; @Maldacena:1999mh] $$\begin{aligned} \nonumber ds^2 &= \frac{du^2}{u^2} + u^2 \left( -dt^2 + dx_1^2 + \tilde h (dx^2_2 + dx^2_3) \right) + d\Omega_5^2 \ , \quad \tilde h = \frac{1}{1+ a^4 u^4} \ , \\ \label{eq:mrback} B &= a^2 \tilde h u^4 dx_2 \wedge dx_3 \ , \quad \exp 2 \Phi = g_0^2 \tilde h \ , \\ \nonumber F_3 &= -\frac{4}{g_0} a^2 u^3 dt\wedge dx_1 \wedge du \ , \quad F_5= \frac{4}{g_0} \tilde h u^3 (1+\star) \, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ .\end{aligned}$$ Starting from the undeformed background $$\label{eq:undefads5} \begin{aligned} ds^2 &= \frac{du^2}{u^2} + u^2 \left( -dt^2 + dx_1^2 + dx^2_2 + dx^2_3 \right) + d\Omega_5^2 \ , \quad \exp 2 \Phi = g_0^2 \ , \\ F_5& = \frac{4}{g_0} u^3 (1+\star) \, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ , \end{aligned}$$ we now consider the non-abelian T-dual with respect to the central extension of $U(1)^2$, where the $U(1)^2$ is generated by shifts in $x_2$ and $x_3$. Using eqs. – with $y_1 = \frac{x_3}{a^2}$, $y_2 = \frac{x_2}{a^2}$, $f_1 = f_2 = u^2$ and setting the deformation parameter $\nu=a^{-2}$ we find that the plus and minus frames are given by $$\widehat e_+ = \left(\begin{array}{c} \frac{hu}{a^4} ( a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}( a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad \widehat e_- = \left(\begin{array}{c} \frac{hu}{a^4} (- a^2 u^2 dx_2 + dx_3) \\ \frac{hu}{a^4}(- a^2 u^2 dx_3 - dx_2) \end{array}\right) \ , \quad h = \frac{a^4}{1+a^4 u^4} \ .$$ The Lorentz rotation of induces a spinorial action according to given by $$\Omega = \sqrt{\frac{h}{a^4}} \left( \mathbb{I} - a^2 u^2 \Gamma^{23} \right) \ .$$ Now let us consider the duality transformation of the five-form RR flux supporting the $AdS_5 \times S^5$ geometry . The self-dual five-form flux can be written as $F_5 = (1+\star) f_5$, where $$f_5 = \frac{4}{g_0}u^3 du \wedge dt \wedge d x_1\wedge dx_2 \wedge dx_3 \equiv \frac{4}{g_0} e^u \wedge e^0 \wedge e^1 \wedge e^2 \wedge e^3 \ .$$ The corresponding poly-form of eq.  is then given by $${\cal P} = 4 \Gamma^{u 0 1 2 3} - 4 \Gamma^{56789} \ .$$ The transformation of the poly-form under T-duality is given by $$\widehat{{\cal P}}= {\cal P} \cdot \Omega^{-1} = 4\sqrt{\frac{h}{a^4}} \Gamma^{u 0 1 2 3} - 4\sqrt{\frac{h}{a^4}}a^2 u^2 \Gamma^{u 0 1 } + \text{duals} \ .$$ Extracting the dual background from the above data we find $$\label{eq:mrback2} \begin{aligned} \widehat{ds}^2 &= \frac{du}{u^2} + u^2 \big( -dt^2 + dx_1^2 + \frac{h}{a^4} ( dx_2^2 + dx_3^2 ) \big) + d\Omega_5^2 \ , \\ \widehat{B}&= -a^{-2}\frac{h}{a^4} dx_2 \wedge dx_3 \ , \quad \exp(2\widehat{\Phi}) = (g_0a^2)^2 \frac{h}{a^4} \ , \\ \widehat{F}_3&= -\frac{4}{g_0 a^2} a^2 u^3 du \wedge dt \wedge dx_1 \ , \quad \widehat{F}_5 = \frac{4}{g_0 a^2} \frac{h}{a^4} u^3 (1+\star)\, du \wedge dt \wedge dx_1 \wedge dx_2 \wedge dx_3 \ . \\ \end{aligned}$$ Noting that $\tilde h = a^{-4}h$, we then immediately see that this is precisely the background up to the constant shift of the dilaton $g_0 \to g_0 a^{-2}$. A small subtlety is that while there is precise agreement between $H = dB$ in and , the $B$-field itself differs by a gauge: $$\widehat{B} =- a^{-2} \frac{1}{1+a^4 u^4} dx_2 \wedge dx_3 = -a^{-2}dx_2 \wedge dx_3 + \frac{ a^2 u^4}{1+a^4 u^4}dx_2 \wedge dx_3 \ .$$ This is always the case in these comparisons [@Hoare:2016wsk; @Borsato:2016pas] and from now on by agreement we always mean up to a gauge term in the $B$-field. Application 2: Marginal Deformations {#ssec:app2} ------------------------------------ ${\cal N}=4$ super Yang-Mills with gauge group $SU(N)$ admits a class of marginal deformations that preserve ${\cal N}= 1$ supersymmetry [@Leigh:1995ep]. The corresponding superpotential for these theories is $$W = \kappa {\operatorname{Tr}}\Big( \Phi_1 [\Phi_2, \Phi_3]_q + \frac{h}{4}\big( \sum_{i=1}^3 \Phi_i^2 \big) \Big) \ ,$$ in which the commutator is $q$-deformed i.e. $[\Phi_i, \Phi_j]_{q} = \Phi_i \Phi_j - q \Phi_j \Phi_i$. For the case where $h=0$ and $q= e^{i \beta}$ with $\beta$ real, known as the $\beta$-deformation, the seminal work of Lunin and Maldacena [@Lunin:2005jy] provides the gravitational dual background constructed via a TsT solution generating technique consisting of a sequence of T-duality, coordinate shift and T-duality. In this case integrability has been shown on both the string [@Frolov:2005ty; @Frolov:2005dj; @Alday:2005ww] and gauge side [@Roiban:2003dw; @Berenstein:2004ys; @Frolov:2005ty; @Beisert:2005if] of the AdS/CFT correspondence. The cubic deformation ($q=1$ and $h \neq 0$) is far less understood, with integrability not expected and, as of now, no known complete gravitational dual constructed. A more general class of non-supersymmetric deformations[^4] of this gauge theory are defined by a scalar potential $$V= {\operatorname{Tr}}\Big( |[\Phi_1, \Phi_2]_{q_3}|^2 + |[\Phi_2, \Phi_3]_{q_1}|^2 + |[\Phi_3, \Phi_1]_{q_2}|^2 \Big) + {\operatorname{Tr}}\Big( \sum_{i=1}^3 [\Phi_i , \bar{\Phi}_i] \Big)^2 \ ,$$ where $q_i = e^{-2\pi i \gamma_i}$. This three parameter deformation, known as the $\gamma$-deformation, enjoys integrability both in the gauge theory [@Frolov:2005iq] and in the worldsheet $\sigma$-model with the target space given by the postulated gravitational dual background constructed in [@Frolov:2005dj]. Upon setting all three deformation parameters equal this reduces to the $\beta$-deformation with enhanced ${\cal N}=1$ supersymmetry and hence we will proceed with the general case. Rather remarkably the string $\sigma$-model in the $\gamma$-deformed target space can be obtained as Yang-Baxter $\sigma$-model [@Kyono:2016jqy; @Osten:2016dvf]. Let us consider the bosonic sector, restricting our attention to the five-sphere of $AdS_5\times S^5$; the $AdS$ factor plays no role in what follows. It is convenient to follow [@Frolov:2005dj] and parametrise the $S^5$ in coordinates adapted to the $U(1)^3$ isometry $$\label{eq:metgam} ds_{S^5}^2 = d\alpha^2 + \S_\alpha^2 d\xi^2 + \C_\alpha^2 d\phi_1^2 + \S_\alpha^2 \C_\xi^2 d\phi_2^2 + \S_\alpha^2 \S_\xi^2 d\phi_3^2 = \sum_{i= 1\dots 3} dr_i^2 + r_i^2 d\phi_i^2 \ ,$$ where $r_1 = \C_\alpha$, $r_2= \S_\alpha \C_\xi$, $r_3= \S_\alpha \S_\xi$ with $\C_x$ and $\S_x$ denoting $\cos x$ and $\sin x$ respectively. The sphere can be realised as the coset $SU(4)/SO(5)$ for which a particular coset representative is given by $$\label{eq:paragam} g = e^{\frac{1}{2} \sum_{m=1}^3 \phi^m h_m } e^{-\frac{\xi}{2} \gamma^{13}} e^{\frac{i}{2} \alpha \gamma^1} \ ,$$ where $ \gamma^{13}$ and $\gamma^1$ are certain $SU(4)$ generators (see appendix \[app:algconv\] for conventions) and $h_i$ are the three Cartan generators. Letting $P$ be the projector onto the coset and $J_\pm = g^{-1} \partial_\pm g$ pull backs of the left-invariant one-form, the $S^5$ $\sigma$-model Lagrangian is $${\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ ,$$ with the parametrisation giving the $\sigma$-model with target space metric . Starting with the $r$-matrix $$r= \frac{\nu_1}{4} h_2 \wedge h_3 + \frac{\nu_3}{4} h_1 \wedge h_2 + \frac{\nu_2}{4} h_3 \wedge h_1 \ ,$$ it was shown in [@Matsumoto:2014nra; @vanTongeren:2015soa] that the NS sector of the Yang-Baxter $\sigma$-model matches the $\gamma$-deformed target space explicitly given by $$\label{eq:gammadef} \begin{aligned} ds^2 & = ds^2_{AdS}+ \sum_{i= 1\dots 3} ( dr_i^2 + G r_i^2 d\phi_i^2) + G r_1^2 r_2^2 r_3^2 \Big( \sum_{i= 1\dots 3} \nu_i dr_i \Big)^2 \ ,\\ B & = G ( r_1^2 r_2^2 \nu_3 d\phi_1 \wedge d\phi_2 + r_1^2 r_3^2 \nu_2 d\phi_3 \wedge d\phi_1 + r_2^2 r_3^2 \nu_1 d\phi_2 \wedge d\phi_3 ) \ , \\ \end{aligned}$$ with $$G^{-1}\equiv \lambda= 1+ r_1^2 r_2^2 \nu_3^2 + r_3^2 r_1^2 \nu_2^2 + r_2^2 r_3^2 \nu_1^2 \ ,$$ where the parameters $\nu_i$ are related to the $\gamma_i$ of the field theory by a factor of the $AdS$ radius [@Frolov:2005dj], which we suppress throughout. We would like to interpret this in terms of the centrally-extended (non-)abelian T-duality introduced in section \[sec:centralext\]. To do so we find it expedient to make a basis transformation of the Cartan generators; let us assume $\nu_3 \neq 0$ and define $$\tilde{h}_1 = h_1 - \frac{\nu_1}{\nu_3} h_3 \ , \quad \tilde{h}_2 = h_2 - \frac{\nu_2}{\nu_3} h_3 \ , \quad \tilde{h}_3 = h_3+ \frac{\nu_1}{\nu_3} h_3 + \frac{\nu_2}{\nu_3} h_3 \ .$$ In this basis the $r$-matrix simply reads $$r= \frac{\nu_3}{4} \tilde{h}_1 \wedge \tilde{h}_2 \ .$$ We also introduce a new set of angles such that $\tilde{h}_i \tilde{\phi}_i = h_i \phi_i$ (where the sum over $i$ is implicit). Written in this way it is clear that we should consider a centrally-extended (non-)abelian T-duality along the $\tilde{h}_1$ and $\tilde{h}_2$ directions. To proceed we defined a slightly exotic set of frame fields for the $S^5$, adapted to the dualisation as described $$\begin{aligned} e^\alpha &= d \alpha \ , \quad e^\xi = \sin \alpha d\xi \ , \quad e^1 = \frac{1}{\varphi \sqrt{\lambda-1} } \left( r_1^2 \varphi^2 d\phi_1 - r_2^2 r_3^2 \nu_1 \nu_2 d\phi_2 - r_2^2 r_3^2 \nu_1 \nu_3 d\phi_3 \right) \ , \\ e^2 & = \frac{1}{\varphi } \left( r_2^2 \nu_3 d\phi_2 - r_3^2 \nu_2 d\phi_3 \right) \ , \quad e^3 = \frac{r_1 r_2 r_3}{\sqrt{\lambda -1} }\sum_{i} \nu_i d\phi_i \ , \end{aligned}$$ where $\varphi=(r_2^2 \nu_3^2 + r_3^3 \nu_2^2)^{\frac{1}{2}}$. Though these frames depend on $\nu_i$ the overall metric remains the round $S^5$ independent of $\nu_i$. The advantage of this basis is that the T-dualisation acts only on the $e_1$ and $e_2$ directions. We non-abelian T-dualise with respect to the central extension of $\tilde{h}_1$ and $\tilde{h}_2$ making the gauge fixing choice $$\hat{g} = e^{\frac{1}{2} \tilde{\phi}_3 \tilde{h}_3 } e^{-\frac{\xi}{2} \gamma^{13}} e^{\frac{i}{2} \alpha \gamma^1}$$ and by parametrising the Lagrange multiplier parameters as $$v_1 = - \frac{2 }{\nu_{3}} \tilde{\phi}_2 \ , \quad v_2 =\frac{2 }{\nu_{3}} \tilde{\phi}_1 \ , \quad v_3 = \frac{4}{\nu_3} \ , \quad dv_3= 0 \ .$$ After some work one finds the dual metric is exactly that of eq.  with a $B$-field matching up to a gauge transformation.[^5] The dual dilaton is given by $$e^{\widehat{\Phi} - \phi_0} = \frac{\nu_3}{4 \sqrt{\lambda}} \ .$$ The frame fields produced by dualisation, using eq. , are $$\begin{aligned} \widehat{e}^{\,\alpha} &= e^\alpha \ , \quad \widehat{e}^{\,\xi} = e^\xi \ , \quad \widehat{e}^{\,3} = e^3 \ , \\ \widehat{e}^{\,1} &\equiv \widehat{e}^{\,1}_{+} = \frac{1}{ \lambda \varphi \sqrt{\lambda-1} } \left( r_1^2 \varphi^2 d\phi_1 - r_2^2 (r_3^2 \nu_1 \nu_2 + (\lambda-1)\nu_3 ) d\phi_2 - r_3^2 (r_2^2 \nu_1 \nu_3 - (\lambda-1)\nu_2 d\phi_3 \right) \ , \\ \widehat{e}^{\,2} &\equiv \widehat{e}^{\,2}_{+} = \frac{1}{ \lambda \varphi } \left( r_1^2 \varphi^2 d\phi_1 + r_2^2 ( \nu_3 - \nu_1 \nu_2 r_3^2 ) d\phi_2 - r_3^2( \nu_2 + \nu_1 \nu_3 r_2^2 ) d\phi_3 \right) \ . \end{aligned}$$ Following the dualisation procedure the Lorentz transformation in eq.  is given by $$\Lambda = \frac{1}{\lambda} \left(\begin{array}{cc} 2-\lambda & - 2 \sqrt{\lambda - 1} \\ 2 \sqrt{\lambda -1} & 2- \lambda \end{array} \right) \ ,$$ for which the corresponding action on spinors is simply $$\Omega =\frac{1}{\sqrt{\lambda}} \mathbb{I} - \frac{\sqrt{\lambda -1 }}{\sqrt{\lambda} }\Gamma^{12} \ .$$ Then acting on the poly-form we ascertain the T-dual fluxes $$\begin{aligned} \widehat{F}_3&= -4 e^{-\phi_0} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge \left( \nu_1 d\phi_1 + \nu_2 d\phi_2 + \nu_3 d\phi_3 \right) \ , \\ \widehat{F}_5 &= (1 +\star) \frac{4 e^{-\phi_0}}{\lambda} r_1 r_2 r_3 \, e^\alpha \wedge e^\xi \wedge d\phi_1 \wedge d\phi_2 \wedge d\phi_3 \ , \end{aligned}$$ in complete agreement with the results of [@Frolov:2005dj]. To close this section let us make a small observation. For the $\beta$-deformation $\nu_1 = \nu_2 = \nu_3 \equiv \gamma$ there a special simplification that happens when $\gamma = \frac{1}{n}$, $n\in \mathbb{Z}$. In this case the deformed gauge theory is equivalent to that of D3 branes on the discrete torsion orbifold $\mathbb{C}^3/\Gamma$ with $\Gamma = \mathbb{Z}_n \times \mathbb{Z}_n$. These cases are also special in the dualisation procedure above. Notice that the Lagrange multiplier $v$ corresponding to the central extension is inversely proportional to $\gamma$ and hence the orbifold points correspond to cases where $v$ is integer quantised. Moreover, recalling that non-abelian T-duality with respect to a centrally-extended $U(1)^2$ is equivalent to first adding a total derivative $B$-field, i.e. making a large gauge transformation, and then T-dualising with respect to $U(1)^2$, where the required total derivative is again given by the expression in footnote \[foot:bdiff\], we find that at the orbifold points ($\nu_1 = \nu_2 = \nu_3 \equiv \gamma = \frac1n$) the integral of this total derivative $$\frac1{4\pi^2} \int B_2 = \frac{n}{12\pi^2} \int (d\phi_2 \wedge d\phi_3 + d \phi_1 \wedge d \phi_2 + d\phi_3 \wedge d\phi_1) = n \ ,$$ is also integer quantised. Application 3: Dipole Deformations {#ssec:app3} ---------------------------------- Dipole theories [@Bergman:2000cw; @Bergman:2001rw] are a class of non-local field theories obtained from regular (or even non-commutative) field theories by associating to each non-gauge field $\Phi_a$ a vector $L^\mu_a$ and replacing the product of fields with a non-commutative product $$(\Phi_1 \tilde\star \Phi_2 )(x) \equiv \Phi_1(x- \frac{1}{2} L_2) \Phi_2 (x+ \frac{1}{2} L_1) \ .$$ Whilst intrinsically non-local, these theories can be mapped to local field theories with a tower of higher-order corrections. For small $L$ the leading correction is the coupling to a dimension 5 operator, which for $\mathcal{N} = 4$ SYM was identified in [@Bergman:2000cw] as $$\Delta {\cal L} = L^\mu \cdot {\cal O}_\mu \ , \quad {\cal O}_\mu^{IJ} = \frac{i}{g^2_{YM}} \textrm{tr}\left(F_\mu{}^\nu \Phi^{[I} D_{\mu} \Phi^{J]}+ (D_\mu \Phi^K)\Phi^{[K}\Phi^I\Phi^{J]} \right) \ .$$ In [@Bergman:2001rw] the supergravity dual to this dipole deformation was constructed. When aligned in the $x^3$ direction the dipole vector $L$ specifies a constant element in $ \mathfrak{su}(4)$ which defines in the $\bf{4}$ a $4\times 4$ traceless hermitian matrix $U$ and in the $\bf{6}$ a $6\times 6$ real antisymmetric matrix $M$. In terms of these matrices the supergravity metric is given by [@Bergman:2001rw] $$ds^2 = \frac{R^2}{z^2} \left( -dt^2 + dx_1^2 + dx_2^2 + f_{1}^{-1}z^2 d x_3^2 \right) + R^2 \left( d\textrm{n}^Td\textrm{n} + \lambda^2 f_{1}^{-1} (\textrm{n}^T M d \textrm{n})^2 \right) \ ,$$ where $\textrm{n}$ is a unit vector in $\mathbb{R}^6$, $\lambda = R^4 (\alpha^\prime)^{-2}= 4 \pi g^2_{YM}N$ and $$f_1 = \frac{z^2}{R^2}+ \lambda^2 \textrm{n}^T M^T M \textrm{n} \ .$$ The deformation acts in both $S^5$ and $AdS^5$. The eigenvalues of a $6 \times 6$ real antisymmetric matrix are three imaginary numbers and their complex conjugates. If we take three of the independent eigenvalues of $M$ to be equal, $M^T M$ is a positive constant, $l^2/\lambda^2$, times the identity matrix, and hence $$f_1 = z^2 + l^2 \ ,$$ where we have set $R = 1$. Though this case preserves no supersymmetry, it does yield a simple metric on the five-sphere; viewed as a $U(1)$ fibration over $\mathbb{C}\mathbf{P}^2$ (given in appendix \[app:algconv\] in eq. ) the deformation acts to change the radius of this fibration such that it depends on the function $f_1$ [@Bergman:2001rw], which now only depends on the $AdS$ radial coordinate. To arrive at this dipole deformation via centrally-extended non-abelian T-duality we gauge the central extension of the $U(1)^{2}$ subgroup generated by $\{ \mathfrak{P}_{3} , (S_{12}+ S_{34}+ S_{56}) \}$. We gauge fix the coset representative $$\hat{g} = g_{AdS_{5}}\oplus g_{S^{5}} \ ,\quad x_{3}\rightarrow 0 \ , \quad \phi \rightarrow 0 \ ,$$ where $g_{AdS_{5}}$ is the parametrisation relevant for the Poincaré patch and $g_{S^{5}}$ is given in eq. . The Lagrange multipliers are then parametrised as $$v_{1} = \frac{\phi}{l} \ , \quad v_{2}= \frac{x_{3}}{l} \ , \quad v_{3} = \frac{1}{l} \ .$$ Following the general formulae one arrives at the T-dual frame fields $$\widehat{e}^{\,1}_{\pm} = \frac{z}{z^{2} + l^{2}} \big( dx_{3} \pm l \Psi \big) \ , \quad \widehat{e}^{\,2}_{\pm} = \frac{z}{z^{2} + l^{2}} \big( -z \Psi \pm \frac{l}{z} dx_{3} \big) \ ,$$ in which $\Psi$ is the global one-form corresponding to the $U(1)$ fibration defined in eq. . It is a simple matter to extract the Lorentz rotation in the spinor representation $$\Omega = \frac{1}{\sqrt{z^{2}+ l^{2}} }\left(z \mathbb{I} - l \Gamma^{12} \right) \ .$$ Here $\Gamma_{12}$ refers to the directions in tangent space given by frames $\widehat{e}^{\,1}$ and $\widehat{e}^{\,2}$. This is a product of two gamma matrices, one with legs in $S^{5}$ and the other in $AdS_{5}$. Therefore, the action of $\Omega$ only produces a five-form in the dualised target space. In fact since, for example, $z\widehat{e}^{\,2}_+ - l \widehat{e}^{\,1}_+ = - z \Psi$ one finds that $F_5$ is only altered by an overall constant scaling that could be re-absorbed into a shift of the dilaton. The final result is the target space geometry $$\begin{aligned} \widehat{ds}^{2} &= \frac{1}{z^{2}} \left( -dx_{0}^{2}+ dx_{1}^{2}+dx_{2}^{2 } + dz^{2}\right) + ds^{2}_{\mathbb{C}\mathbf{P}^2} + \frac{1}{z^{2}+l^{2}} dx_{3}^{2 } + \frac{z^{2}}{z^{2}+l^{2}}\Psi^{2} \ , \\ \widehat{B} &= \frac{l}{z^{2}+ l^{2}}\Psi \wedge dx_{3} + \frac{1}{l} dx_{3} \wedge d\phi \ , \quad e^{ 2(\widehat{\Phi}-\phi_{0})} = \frac{z^{2}l^{2}}{z^{2}+l^{2}} \ , \quad \widehat{F}_5 = \frac{1}{l} F_5 \ . \end{aligned}$$ Modulo a gauge transformation in $\widehat{B}$ this agrees with the geometry of [@Bergman:2001rw]. Concluding Comment {#sec:conclusions} ================== In this article we have demonstrated that the holographic dual of many known deformations of gauge theories can be understood in terms of non-abelian T-duality, extending the construction in the NS sector of [@Hoare:2016wsk; @Borsato:2016pas] to the RR sector. In section \[sec:examples\] we tested the construction on a number of examples: a non-commutative deformation, the $\gamma$-deformation, a dipole deformation and, in appendix \[app:furtherexamples\], a unimodular non-abelian deformation and a jordanian deformation. There are a number of interesting open directions. Our construction involved only bosonic generators of the $\mathfrak{psu}(2,2|4)$ algebra of the superstring; it would be interesting to extend this to more general $r$-matrices, including those containing fermionic generators. Furthermore, to formalise the relation between the Yang-Baxter deformations and non-abelian dualities it would be useful to understand how the spinor rotation defining the deformed RR fluxes in the former [@Borsato:2016ose] is related to that in the latter, which was the subject of the present article. Additionally, one would like to understand whether solutions of the modified cYBE (i.e. $\eta$-deformations and their Poisson-Lie dual $\lambda$-deformations) can be understood in this framework. Finally, and perhaps optimistically, one might hope that generalised notions of T-duality can be employed to find gravitational duals of other non-integrable marginal deformations of gauge theories. Acknowledgements ================ It is a pleasure to thank Saskia Demulder, Carlos Núñez, Arkady Tseytlin, Linus Wulff and Konstantinos Zoubos for discussions concerning aspects of this work. The work of BH is partially supported by grant no. 615203 from the European Research Council under the FP7. The work of DT is supported in part by the Belgian Federal Science Policy Office through the Interuniversity Attraction Pole P7/37, and in part by the “FWO-Vlaanderen” through the project G020714N and a postdoctoral fellowship, and by the Vrije Universiteit Brussel through the Strategic Research Program “High-Energy Physics”. (Modified) Supergravity Conventions {#app:sugra} =================================== In this appendix we summarise our conventions for the (modified) Type IIB supergravity equations. Similar equations exist for Type IIA. Let us define the following beta-functions $$\label{eq:betafunctions} \begin{aligned} \beta_{mn}^{G} &= R_{mn} + 2\nabla_{m n }\Phi -\frac{1}{4}H_{mpq}H_{n}{}^{pq} \\ & - e^{2\Phi}\Big( \frac{1}{2}({F_1}^2)_{mn}+\frac{1}{4}({F_3}^2)_{mn} +\frac{1}{96}({F_5}^2)_{mn} - \frac{1}{4}g_{mn} \big( F_1^2 +\frac{1}{6}F_{3}^2 \big)\Big) \ , \\ \beta_{mn}^B &= d[e^{-2\Phi} \star H] - F_1\ww \star F_3 - F_3 \ww F_5 \ , \\ \beta^\Phi &= R+ 4 \nabla^2 \Phi - 4 (\partial \Phi)^2 - \frac{1}{12} H^2 \ . \end{aligned}$$ For a globally scale invariant $\sigma$-model the beta-functions for the metric and $B$-field vanish up to diffeomorphisms and gauge transformations. There should then be similar second-order equations for the RR fluxes. The Type IIB supergravity equations, i.e. the critical string equations, are given by $$\label{eq:sugraeq} \begin{aligned} \beta^G_{mn} & = 0 \ , \quad \beta^B_{mn} = 0 \ , \quad \beta^\Phi = 0 \ , \\ d\cF_1 & = d\Phi \ww \cF_1 \ , \quad d\star\cF_1 + H\ww \star \cF_3 = d\Phi \ww\star \cF_1 \ , \\ d\cF_3 - H\ww \cF_1 &= d\Phi \ww \cF_3 \ , \quad d\star \cF_3 + H\ww \star \cF_5 = d\Phi \ww \star \cF_3 \ ,\\ d\cF_5 - H\ww \cF_3&= d\Phi \ww \cF_5 \ , \quad \cF_5 = \star \cF_5 \ , \\ \end{aligned}$$ where we have defined $\cF = e^\Phi F$. There exists a modification to the supergravity equations that still imply the global scale invariance conditions, but now depend on an additional Killing vector of the background $I$. These modified supergravity equations can be understood as follows. We start from a solution of the Type II supergravity equations for which the metric, $B$-field and weighted RR fluxes $\cF$ have a $U(1)$ isometry corresponding to shifts in the coordinate $y$, but where the dilaton breaks this isometry via a piece linear in $y$, i.e. $\Phi = cy + \ldots$. The supergravity equations only depend on $d\Phi$ and hence we can ask what happens if we formally T-dualise in $y$. The dual background then solves the modified equations with the Killing vector corresponding to shifts in the dual coordinate to $y$ [@Arutyunov:2015mqj]. Alternatively they follow from the requirement that the Type II Green-Schwarz string action is $\kappa$-symmetric [@Wulff:2016tju]. Recently they have also been formulated in an $O(d,d)$ invariant manner, as a modification of Type II double field theory [@Sakatani:2016fvh]. The modified Type IIB supergravity equations are $$\label{eq:modsugra1} \begin{aligned} \beta_{mn}^{G} &= - \nabla_m W_n - \nabla_n W_m \ , \quad e^{2\Phi} \beta_{mn}^{B} = 2 \star d W +2 W \wedge \star H \ , \\ \beta^{\Phi} &= 4 \star d \star W -4 \star( (W+ 2 d\Phi) \ww \star W) \ , \\ d\cF_1 & = Z\ww \cF_1 + \star (I \ww \star \cF_3) \ , \quad d\star\cF_1 + H\ww \star \cF_3 = Z\ww\star \cF_1 \ , \quad \star (I \ww \star \mathcal{F}_1) = 0 \ , \\ d\cF_3 - H\ww \cF_1 &= Z\ww \cF_3 + \star (I \ww \star \cF_5) \ , \quad d\star \cF_3 + H\ww \star \cF_5 = Z\ww \star \cF_3- \star( I \wedge \cF_1 ) \ , \\ d\cF_5 - H\ww \cF_3&= Z\ww \cF_5- \star (I \ww \cF_3) \ , \quad \cF_5 = \star \cF_5 \ , \end{aligned}$$ where $I$ is a one-form corresponding to a certain Killing vector of the background, i.e. $$\label{eq:modsugra2} \mathcal{L}_I G= \mathcal{L}_I B = \mathcal{L}_I \Phi = \mathcal{L}_I \cF_{1,3,5} = 0 \ ,$$ and the one-forms $Z$, $X$ and $W$ are constructed from $I$ and $\Phi$ $$\label{eq:modsugra3} Z = d\Phi - \iota_I B \ , \quad X= I + Z \ , \quad W = X - d\Phi = I - \iota_I B \ .$$ It is important to note that for the modified system of equations to be invariant under the gauge freedom $B \to B + d \Lambda$ (where for simplicity we assume that $\mathcal{L}_I \Lambda = 0$) the “dilaton” field $\Phi$ must now transform as $\Phi \to \Phi - \iota_I \Lambda$, and hence is not unique. This can be understood by starting from a Weyl-invariant background with a dilaton linear in an isometric direction $y$, $\Phi = c y + \ldots$. If we shift $y$ by an arbitrary function of the transverse coordinates this ansatz is preserved, however the explicit form of the dilaton is changed. After “T-dualising” in $y$ this coordinate redefinition then maps to a gauge transformation under which the dual “dilaton” field now transforms. Conventions for Algebras {#app:algconv} ======================== In this appendix we outline our conventions for the algebras $\mathfrak{so}(4,2)$ and $\mathfrak{so}(6)$ for which we largely adopt those of [@Arutyunov:2009ga]. For $SO(4,2)$ we start by defining the $\gamma$ matrices $$\gamma_0 = i \sigma_3 \otimes \sigma_0 \ , \quad \gamma_1= \sigma_2 \otimes \sigma_2 \ , \quad \gamma_2= -\sigma_2 \otimes \sigma_1\ , \quad \gamma_3= -\sigma_1 \otimes \sigma_0 \ , \quad \gamma_4= \sigma_2 \otimes \sigma_3 \ ,$$ in terms of which the generators of $SO(4,2)$ are given by $$T_{ij} = \frac{1}{4} [\gamma_{i}, \gamma_j] \ , \quad T_{i5} = \frac{1}{2} \gamma_i \ , \quad i,j = 0, \ldots, 4 \ .$$ The $SO(4,1)$ subgroup is generated by $T_{ij}$ for $i,j = 0, \ldots, 4$. The projector onto the orthogonal complement is given by $$P(X)= - {\operatorname{Tr}}(X T_{0,5}) T_{0,5} + \sum_{i=1}^4 {\operatorname{Tr}}(X T_{i,5}) T_{i,5} \ .$$ A useful adapted basis when considering Poincaré patch is $$\mathfrak{D}= T_{45} \ , \quad \mathfrak{P}_\mu = T_{\mu 5} - T_{\mu 4} \ , \quad \mathfrak{K}_\mu = T_{\mu 5} + T_{\mu 4} \ , \quad \mathfrak{M}_{\mu \nu} = T_{\mu \nu} \ , \quad \mu = 0,\ldots, 3 \ .$$ We also use $\mathfrak{M}_{+i} = \mathfrak{M}_{0i} +\mathfrak{M}_{1i} $ for $i=2,3$. The bosonic $AdS_5$ $\sigma$-model is given by $${\cal L } = {\operatorname{Tr}}(J_+ P(J_-) ) \ ,$$ for $J= g^{-1}dg$ and when the gauged fixed group element is parametrised as $$\label{eq:gAdS} g= \exp\left[ \eta^{\mu \nu} x_\mu \mathfrak{P}_\nu \right] z^{\mathfrak{D} } \ ,$$ the target space metric is given on the Poincaré patch by $$ds^2=\frac{1}{z^2} \left( dz^2 + \eta^{\mu \nu} dx_\mu dx_\nu \right) \ .$$ As usual the coordinate $u$ used in section \[ssec:app1\] is related to $z$ by $u = z^{-1}$. In the examples that we consider we dualise with respect to a subalgebra $\mathfrak{h}\subset \mathfrak{so}(4,2)$ which does not necessarily need to be a subgroup of the $\mathfrak{so}(4,1)$ subalgebra specified above. For $\mathfrak{so}(6)\cong \mathfrak{su}(4)$ we supplement $\gamma_i$ $i=1,\ldots, 4$, defined above with $\gamma_5 = -i \gamma_0$ and construct the (anti-hermitian) generators $$S_{ij} = \frac{1}{4} [\gamma_{i}, \gamma_j] \ , \quad S_{i6} = \frac{i}{2} \gamma_i \ , \quad i,j = 1, \ldots, 5 \ .$$ The Cartan subalgebra is generated by $$h_1= i {\operatorname{diag}}(1,1,-1,-1) \ , \quad h_2= i {\operatorname{diag}}(1,-1,1,-1) \ , \quad h_3= i {\operatorname{diag}}(1,-1,-1,1)$$ We take the $\mathfrak{so}(5)$ subalgebra to be generated by $S_{ij}$ for $i=1,\ldots,5$, such that the projector onto the orthogonal complement of this subgroup is $$P(X) = \sum_{i=1}^5{\operatorname{Tr}}( X\cdot S_{i6}) S_{i6} \ ,$$ where here ${\operatorname{Tr}}$ stands for the negative of the matrix trace. A coset representative for $SO(6)/SO(5)$ can be chosen as $$g = \exp[\frac{1}{2} \phi^m h_m ] \exp[-\frac{\xi}{2} \gamma^{13}] \exp[\frac{i\alpha}{2} \gamma^1] \ ,$$ leading to the $\sigma$-model parametrisation of $S^5$ employed in section \[ssec:app2\]. An alternative parametrisation is given by $$\label{eq:CPparam} g= \exp[\frac{i\phi}{2} \gamma_5] \cdot \big[s \, \mathbb{I} + \frac{it}{2} \big( e^{i\phi} \alpha (\gamma_1 - i \gamma_2) + e^{-i\phi} \bar\alpha (\gamma_1 + i \gamma_2) + e^{i\phi} \beta (\gamma_3- i \gamma_4) + e^{-i\phi}\bar\beta (\gamma_3+ i \gamma_4) \big) \big] \ ,$$ where $$r = 1+ |\alpha|^2 + |\beta|^2 \ , \quad s^2 = \frac{1}{2\sqrt{r} }(1+ \sqrt{r}) \ , \quad t^2 = \frac{1}{2\sqrt{r} (1+ \sqrt{r})} \ .$$ These coordinates give a metric on $S^5$ that makes manifest the structure of $S^5$ as a $U(1)$ fibration over $\mathbb{C}\mathbf{P}^2$ $$\label{eq:CPform} \begin{aligned} &ds^2_{S^5} = ds^{2}_{\mathbb{C}\mathbf{P}^2} + \Psi^{2}\ , \quad ds^{2}_{\mathbb{C}\mathbf{P}^2} = \frac{1}{r}( |d\alpha|^2 + |d\beta|^2 ) - \frac{1}{r^2}| \omega |^2 \ , \\ & \Psi= d\phi + \frac{1}{r} \Im( \omega) \ , \quad \omega = \bar\alpha d \alpha + \bar\beta d \beta \ . \end{aligned}$$ The global one-form $\Psi = \sum_{i=1\dots 3} x_i dy_i - y_i d x_i$ where $z_i = x_i + i y_i$ are coordinates on $\mathbb{C}^3$ given by $z_1 = \frac{1}{\sqrt{r}} e^{i \phi}$, $z_2 = \frac{\alpha}{\sqrt{r}} e^{i \phi}$, $z_3 = \frac{\beta}{\sqrt{r}} e^{i \phi}$. One can think of $\Psi$ as a contact form whose corresponding Reeb vector has orbits which are the $S^1$ fibres. For computational purposes we note that frame fields for $\mathbb{C}\mathbf{P}^2$ can be found in e.g. [@Eguchi:1980jx]. When dealing with the dipole deformation in section \[ssec:app3\] we will need the full ten-dimensional space-time. This is readily achieved by taking a block diagonal decomposition, i.e. $g = g_{AdS_{5}} \oplus g_{S^5}$, with the generators of $\mathfrak{su}(2,2) \oplus \mathfrak{su}(4)$ given by $8 \times 8$ matrices, with the $\mathfrak{su}(2,2)$ and $\mathfrak{su}(4)$ generators in the upper left and lower right $4 \times 4$ blocks respectively. Traces are then replaced with “supertrace” (the bosonic restriction of the supertrace on $\mathfrak{psu}(2,2|4)$) given by the matrix trace of the upper $\mathfrak{su}(2,2)$ block minus the matrix trace of the lower $\mathfrak{su}(4)$ block. Further Examples of Deformations in AdS5 {#app:furtherexamples} ======================================== In section \[sec:examples\] we considered non-abelian T-dualities with respect to a centrally-extended two-dimensional abelian algebra, demonstrating that this is equivalent to a TsT transformation of the full supergravity background. There are additional classes of deformations that can be constructed as non-abelian T-duals. These come from considering particular non-semisimple subalgebras of $\mathfrak{su}(2,2) \oplus \mathfrak{su}(4)$, whose existence relies on the non-compactness of $\mathfrak{su}(2,2)$. There are a number of such algebras that are non-abelian and admit central extensions [@Borsato:2016ose], such that when we T-dualise the metric with respect to this centrally-extended subalgebra we find a deformation of the original metric [@Hoare:2016wsk; @Borsato:2016pas] that coincides with a certain Yang-Baxter deformation. To illustrate this richer story we present a summary of two examples showing how the techniques described in this paper also apply, i.e. the R-R fluxes following from non-abelian T-duality agree with those of the Yang-Baxter $\sigma$-model. An $r$-matrix $$r = r^{ab} T_a \wedge T_b \ ,$$ is said to be non-abelian if $[T_a , T_b]\neq 0$ for at least some of the generators. An $r$-matrix is said to be unimodular if $$r^{ab} [T_a, T_b] = 0 \ .$$ For a solution of the classical Yang-Baxter equation the unimodularity of the $r$-matrix is equivalent to the unimodularity ($f_{ab}{}^b = 0$) of the corresponding subalgebra. In [@Borsato:2016ose] it was shown that the background defined by a Yang-Baxter $\sigma$-model based on a non-unimodular non-abelian $r$-matrix is not a supergravity background, but rather solves the modified supergravity described above. The first example we discuss corresponds to a non-abelian but unimodular $r$-matrix, while the second is a non-unimodular $r$-matrix. Unimodular r-matrix {#sapp:340} ------------------- The first example corresponds to an $r$-matrix considered in [@Borsato:2016ose] $$r = \eta~ \mathfrak{M}_{23}\wedge \mathfrak{P}_1 + \zeta~ \mathfrak{P}_2\wedge \mathfrak{P}_3 \ .$$ This is non-abelian e.g. $[\mathfrak{M}_{23}, \mathfrak{P}_2]= - \mathfrak{P}_3$, but since $[\mathfrak{M}_{23}, \mathfrak{P}_1]= [\mathfrak{P}_{2}, \mathfrak{P}_3]=0$ it is unimodular. In [@Borsato:2016ose] it was shown that the corresponding deformation is nevertheless equivalent to two non-commuting TsT transformations, with a non-linear coordinate redefinition in between. On the other hand it was discussed from the perspective of non-abelian T-duality in [@Hoare:2016wsk] where the relevant subalgebra was $\mathfrak{h}= \{ \mathfrak{M}_{23}, \mathfrak{P}_1 , \mathfrak{P}_2, \mathfrak{P}_3 \}$. The gauge freedom can be used to fix the coset representative in eq.  to $\hat{g} = e^{-x_{0 }\mathfrak{P}_{0}} z^{\mathfrak{D}}$, but there remains one residual gauge symmetry which is used to fix a Lagrange multiplier to zero. The Lagrange multipliers are parametrised by $$v_1= -\frac{x_1}{\eta} + \frac{r^2}{2 \zeta} \ , \quad v_2= \frac{\theta}{\eta}\ , \quad v_3=\frac{r}{\zeta}\ ,\quad v_4= 0\ , \quad v_5= \frac{1}{\eta}\ , \quad v_6= \frac{1}{\zeta} \ ,$$ where $v_5$ and $v_6$ correspond to the two central generators and $r$ and $\theta$ are polar coordinates on the $x_2 ,x_3$ plane. Applying the non-abelian T-duality technology one finds the dual geometry is $$\begin{aligned} \widehat{ds}^2 &=\frac{1}{z^2} \left( dz^2 - dx_0^2\right) + \widehat{e}_\pm \cdot \widehat{e}_\pm + ds^2_{S^5} \ , \\ \widehat{e}^{\,1}_+ &= \frac{dx_1 \left(\zeta ^2+z^4\right)+\eta r \left(-\zeta dr +r z^2 d\theta \right)}{z f } \ , \\ \widehat{e}^{\,2}_+& = \frac{z \left(\zeta dr+\eta r dx_1-r z^2 d \theta \right)}{f} \ , \\ \widehat{e}^{\,3}_+&= \frac{-dr \left(\eta ^2 r^2+z^4\right)+\zeta \eta r dx_1-\zeta r z^2 d\theta }{z f}\ , \end{aligned}$$ where $f= \zeta ^2+\eta ^2 r^2+z^4$, while the remaining NS fields are $$\widehat{B} = \frac{-\zeta \eta r dr \wedge d\theta + \left(\zeta ^2+z^4\right) dx_1\wedge d \theta }{\eta f } \ , \quad e^{-2(\widehat{\Phi} - \phi_0)} = \frac{f}{\zeta ^2 \eta ^2 z^4} \ .$$ The Lorentz rotation $\Lambda e_- = e_+$ is given by $$\Lambda = \frac{1}{f} \left( \begin{array}{ccc} z^4+\zeta ^2-r^2 \eta ^2 & -2 r z^2 \eta & 2 r \zeta \eta \\ 2 r z^2 \eta & z^4-\zeta ^2-r^2 \eta ^2 & - 2 z^2 \zeta \\ 2 r \zeta \eta & 2 z^2 \zeta & z^4-\zeta ^2+r^2 \eta ^2 \\ \end{array} \right) \ ,$$ with the corresponding spinor representation $$\Omega = \frac{1}{\sqrt{f} } \left( z^2 \mathbb{I} - r \eta \Gamma^{12} - \zeta \Gamma^{23} \right) \ .$$ This completes the IIB supergravity solution with the three-form and five-form flux $$\begin{aligned} F_3 &=\frac{4 e^{-\phi_0} }{z^5 \zeta \eta }\left( \zeta dx_0\wedge dx_1 \wedge dz - r \eta dx_0 \wedge dr \wedge dz \right) \ , \\ F_5&= (1+\star) \frac{-4 e^{-\phi_0} r }{ z \zeta \eta f} dx_0\wedge dx_1 \wedge dr \wedge dz\wedge d\theta \ , \end{aligned}$$ in agreement with the expressions following from the Yang-Baxter $\sigma$-model [@Borsato:2016ose]. Non-unimodular r-matrix {#sapp:366a} ----------------------- The final example we consider is an $r$-matrix that can be found by infinitely boosting the Drinfel’d-Jimbo solution to the modified classical Yang-Baxter equation for $\mathfrak{su}(2,2)$ [@Hoare:2016hwh] $$\label{eq:rmatnm} r= \eta \left(\mathfrak{D} \wedge \mathfrak{P}_0 + \mathfrak{M}_{01}\wedge \mathfrak{P}_1 + \mathfrak{M}_{+2}\wedge \mathfrak{P}_2 + \mathfrak{M}_{+3}\wedge \mathfrak{P}_3 \right) \ .$$ This $r$-matrix of jordanian type and the corresponding deformations of the $AdS_5 \times S^5$ superstring were first studied in [@Kawaguchi:2014qwa; @Kawaguchi:2014fca]. Furthermore, the $r$-matrix is non-unimodular and the corresponding dualisation of $AdS_5$ with respect to the non-abelian subalgebra $$\mathfrak{h}= \{\mathfrak{D} , \mathfrak{P}_0 , \mathfrak{M}_{01}, \mathfrak{P}_1, \mathfrak{M}_{+2} , \mathfrak{P}_2 , \mathfrak{M}_{+3} , \mathfrak{P}_3\} \ ,$$ is afflicted with a mixed gravity/gauge anomaly (i.e. $n_a= f_{ab}{}^b \neq 0$) [@Elitzur:1994ri]. The algebra $\mathfrak{h}$ admits a single central extension with the commutator of each pair of generators in being extended by the same generator. Since all directions are dualised the coset representative is fully fixed to $\hat{g}=1$ leaving three further gauge fixings to be made on the dynamical Lagrange multipliers. We parametrise these as $$v_1= \frac{x_0}{\eta} \ , \quad v_2= \frac{-1+z}{\eta} \ , \quad v_3 = \frac{x_1}{\eta} \ , \quad v_5 + i v_7 = \frac{r e^{i\theta}}{\eta} \ , \quad v_4=v_6=v_8=0 \ , \quad v_9 = \frac{1}{\eta} \ ,$$ where $v_9$ corresponds to the central direction. The dual metric is given by $$\begin{aligned} &\widehat{ds}^2 =e_\pm^i \eta_{ij} e_\pm^i + \widehat{ds}^2_{S^5} \ , \quad \eta_{ij} = \textrm{diag} (1,-1,1, 1,1) \ , \\ &\widehat{e}^{\,1}_+= \frac{1}{p}(-\eta dx_0 + z dz) \ , \quad \widehat{e}^{\,2}_+= \frac{1}{p}(-z dx_0 + \eta dz) \ , \quad \widehat{e}^{\,3}_+= -\frac{z}{q}(z^2 dx_1 + r \eta dr) \ ,\\ & \widehat{e}^{\,4}_+ + i \widehat{e}^{\,5}_+ = \frac{e^{i\theta}}{q}\left( r z \eta dx_1 -z^3 dr - \frac{i q r}{z} d\theta \right) \ , \end{aligned}$$ where $p= z^2-\eta^2$ and $q= z^4+ r^2 \eta^2$. The remaining NS fields are $$\widehat{B} = \frac{z}{p \eta}dz\wedge dx_0 + \frac{r\eta}{q} dr \wedge dx_1 \ , \quad e^{-2(\widehat{\Phi}-\phi_0)} = \frac{p q z^2}{\eta^8} \ .$$ The $SO(1,4)$ Lorentz rotation has a block diagonal decomposition $\Lambda = \Lambda_1 \oplus \Lambda_2$ with $$\Lambda_1 = \frac{1}{p}\left(\begin{array}{cc} z^2+\eta^2 & 2 z \eta \\ 2 z \eta & z^2 +\eta^2 \end{array} \right) \ , \ \ \ \Lambda_2= \frac{1}{q}\left(\begin{array}{ccc} z^4 -r^2 \eta^2 & 2 r z^2 \eta {\cal C}_\theta & 2 r z^2 \eta {\cal S}_\theta \\ -2 r z^2 \eta {\cal C}_\theta & z^4 - r^2 \eta^2 {\cal C}_{2\theta} & -r^2 \eta^2 {\cal S}_{2\theta} \\ -2 r z^2 \eta {\cal S}_\theta &- r^2 \eta^2 {\cal S}_{2\theta} & z^4 + r^2 \eta^2 {\cal C}_{2\theta} \end{array}\right) \ .$$ The corresponding spinor rotation $\Omega = \Omega_1\cdot\Omega_2$ is given by (recalling the signature is such that $(\Gamma^{2})^{2}= - \mathbb{I} $ whilst the remaining $(\Gamma^{i})^{2}= \mathbb{I}$) $$\begin{aligned} \Omega_1 = \frac{1}{\sqrt{p}} \left( z \mathbb{I} + \eta \Gamma^{12} \right) \ , \quad \Omega_2 = \frac{1}{\sqrt{q}} \left(z^2 \mathbb{I} + r\eta \cos\theta \Gamma^{34} + r\eta \sin\theta \Gamma^{35} \right) \ . \end{aligned}$$ This gives the fluxes $$\begin{aligned} F_1&= \frac{ 4e^{-\phi_0}}{\eta^2} r^2 d\theta \ , \quad F_3 = \frac{4e^{-\phi_0} r z^4}{\eta^3 q } dx_1\wedge dr\wedge d\theta - \frac{ 4e^{-\phi_0} r^2 z}{\eta^3 p} dx_0\wedge dz \wedge d\theta \ , \\ F_5&= (1+\star) \frac{ - 4e^{-\phi_0} r z^5}{\eta^4 q p} dx_0\wedge dx_1\wedge dr \wedge dz \wedge d\theta \ . \end{aligned}$$ These fluxes do not solve their Bianchi identities, nor their equations of motions. 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M. Spradlin, T. Takayanagi and A. Volovich, “String theory in beta deformed spacetimes,” JHEP [**0511**]{} (2005) 039 \[[[arXiv:hep-th/0509036](http://arxiv.org/abs/hep-th/0509036)]{}\]. S. A. Frolov, R. Roiban and A. A. Tseytlin, ‘Gauge-string duality for (non)supersymmetric deformations of $\mathcal{N} = 4$ super Yang-Mills theory,’’ Nucl. Phys. B [**731**]{} (2005) 1 \[[[arXiv:hep-th/0507021](http://arxiv.org/abs/hep-th/0507021)]{}\]. A. Bergman and O. J. Ganor, “Dipoles, twists and noncommutative gauge theory,” JHEP [**0010**]{} (2000) 018 \[[[arXiv:hep-th/0008030](http://arxiv.org/abs/hep-th/0008030)]{}\]. A. Bergman, K. Dasgupta, O. J. Ganor, J. L. Karczmarek and G. Rajesh, “Nonlocal field theories and their gravity duals,” Phys. Rev. D [**65**]{} (2002) 066005 \[[[arXiv:hep-th/0103090](http://arxiv.org/abs/hep-th/0103090)]{}\]. R. Borsato and L. Wulff, “Target space supergeometry of $\eta$ and $\lambda$-deformed strings,” JHEP [**1610**]{} (2016) 045 \[[[arXiv:1608.03570](http://arxiv.org/abs/1608.03570)]{}\]. G. Arutyunov and S. Frolov, “Foundations of the $AdS_5 \times S^5$ Superstring. Part I,” J. Phys. A [**42**]{} (2009) 254003 \[[[arXiv:0901.4937](http://arxiv.org/abs/0901.4937)]{}\]. T. Eguchi, P. B. Gilkey and A. J. Hanson, “Gravitation, Gauge Theories and Differential Geometry,” Phys. Rept. [**66**]{} (1980) 213. B. Hoare and S. J. van Tongeren, “On jordanian deformations of $AdS_5$ and supergravity,” J. Phys. A [**49**]{} (2016) no.43, 434006 \[[[arXiv:1605.03554](http://arxiv.org/abs/1605.03554)]{}\]. I. Kawaguchi, T. Matsumoto and K. Yoshida, “A Jordanian deformation of AdS space in type IIB supergravity,” JHEP [**1406**]{} (2014) 146 \[[[arXiv:1402.6147](http://arxiv.org/abs/1402.6147)]{}\]. D. Orlando, S. Reffert, J. i. Sakamoto and K. Yoshida, “Generalized type IIB supergravity equations and non-abelian classical $r$-matrices,” J. Phys. A [**49**]{} (2016) no.44, 445403 \[[[arXiv:1607.00795](http://arxiv.org/abs/1607.00795)]{}\]. [^1]: A more precise field theoretic explanation of what this limit means has been proposed in [@Lozano:2016kum]. [^2]: In some cases it can be that this doesn’t fully fix the gauge and additional fixing should be imposed on the Lagrange multipliers $ V= v_a \tilde{H}^a$, details of this are discussed in [@Lozano:2011kb]. [^3]: An explicit demonstration of the RR transformation law in the context of supersymmetry in $SU(2)$ non-Abelian T-duality can be found in[@Kelekci:2014ima]. [^4]: Care needs be taken in the interpretation of this deformation. Away from the supersymmetric point the $\gamma_i$ deformation is not conformal due a running coupling of a double-trace operator [@Fokken:2013aea] and indeed the gravitational dual has a tachyon [@Spradlin:2005sv]. [^5]: As with the previous example the $B$-field obtained by the central extension dualisation procedure differs by a closed piece $\Delta B = \frac{1}{\nu_1^2+ \nu_2^2 + \nu_3^2} \left( \nu_1 d\phi_2\wedge d\phi_3 + \nu_3 d\phi_1\wedge d\phi_2+ \nu_2 d\phi_3\wedge d\phi_1 \right)$.\[foot:bdiff\]

ArXiv

--- abstract: 'Within the Glauber formalism and a BUU transport model we analyze the $\eta$-photoproduction data from nuclei and evaluate the in-medium $\eta N$ cross section. Our results indicate that the $\eta N$ cross section is almost independent of the $\eta$ energy up to 200 MeV.' author: - | M. Effenberger and A. Sibirtsev\ Institut für Theoretische Physik, Universität Giessen\ D-35392 Giessen, Germany date: title: 'The energy dependence of the in-medium $\eta N$ cross section evaluated from $\eta$-photoproduction [^1]' --- =-10mm Introduction ============ For a long time $\eta$-meson production in nuclei has been of interest as a source of information about the $\eta$-nucleus final state interaction. The present knowledge about the $\eta N$ interaction even in the vacuum comes from either simple analysis of the inverse $\pi N \to \eta N$ reaction or as a free parameter fitted to experimental data by theoretical calculations [@Liu; @Wilkin; @Chiavassa; @Sibirtsev1]. Note that the value of the $\eta N$ scattering length is still an open problem and there is not actual agreement between a bulk of theoretical investigations. The analysis of $\eta$-production from $pA$ collisions indicated strong sensitivity of the calculations to the prescription of the $\eta$-meson final state interaction [@Chiavassa; @Golubeva; @Sibirtsev2]. It was found that the $\eta$-energy spectrum [@Chiavassa2] is mostly influenced by the variation of the $\eta N$ cross section [@Golubeva; @Sibirtsev2]. However the experimental data [@Chiavassa2] had large uncertainties and there was no continuation of the systematical studies. Recent measurements on $\eta$-meson photoproduction in nuclei [@Krusche] are more detailed and accurate. Among the theoretical investigations [@Lee; @Carrasco; @Effenberger] only the calculations within the Distorted Wave Impulse Approximation (DWIA) from Lee et al.[@Lee] are able to reproduce the experimental data by incorporating the $\eta$-nucleus potential proposed by Bennhold and Tanabe [@Ben]. The present paper is organized as follows. In section \[glauber\] we use the Glauber formalism to extract the in-medium $\eta N$ cross section from the experimental data. In section \[buu\] these results are compared to Boltzmann-Uehling-Uhlenbeck transport model calculations. The sensitivity of the theoretical results to the prescription of $\eta N$ scattering is investigated. Analysis within the Glauber Model {#glauber} ================================= In an incoherent approximation the cross section of $\eta$-meson photoproduction off nuclei is given by $$\label{app1} {\sigma}_{\gamma \eta}^A = {\sigma}_{\gamma \eta }^p \times \left[ Z + \zeta (A-Z) \right]$$ with $A$, $Z$ being the mass and charge of the target,respectively, while the factor $\zeta=2/3$ [@Krusche2] stands for the ratio of the elementary $\eta$-photoproduction cross sections from $\gamma n$ and $\gamma p$ reactions. In nuclei the cross section differs from the approximation (\[app1\]) due to nuclear effects. ([*i*]{}) The Fermi motion of nucleons as well as ([*ii*]{}) Pauli blocking are important at energies below and close to the reaction threshold in free space [@Cassing1; @Salcedo]. We should also take into account [*(iii)*]{} the modification of the $N^*$-resonance by the nuclear medium [@Carrasco; @Effenberger]. However the most important effect is [*(iv)*]{} the strong final state interaction of $\eta $-mesons in nuclear matter. The deviation of the $A$-dependence of the ${\sigma}_{\gamma A \rightarrow \eta X}$ from $A^1$ mostly reflects the strength of the final state interaction. Here we present an analysis of the experimental data on $\gamma A \rightarrow \eta X$ reactions in order to extract the in-medium cross section ${\sigma}_{\eta N}$. Our approach is based on the Glauber model [@Glauber] and first was developed by Margolis [@Margolis1] for evaluation of the $\rho N$ cross section from both incoherent and coherent $\rho$-meson photoproduction off nuclei. The most detailed description and application of the Glauber model to photoproduction reactions may be found from review of Bauer, Spital and Yennie [@Bauer]. A similar formalism is adopted for studying color transparency [@Bertch; @Kopeliovich1], where the in-medium cross sections is treated as a function of a transverse separation of the hadronic wave function. In the Glauber model the cross section of the incoherent $\eta$-meson photoproduction reads $$\label{fact} {\sigma}_{\gamma \eta }^A= {\sigma}_{\gamma \eta }^p \frac {Z + \zeta (A-Z)} { A} \times A_{eff}$$ where $$\label{aef} A_{eff} = \frac {1} {2 \pi} \int_0^{+\infty} d{\bf b} \int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z) \int_0^{2\pi} d\phi \ exp \left[-{\sigma}_{\eta N} \oint d\xi \ {\rho}({\bf r}_{\xi}) \right]$$ Here ${\rho}(r)$ is the single particle density function, which was taken of Fermi type with parameters for each nucleus from [@Jager]. The last integration in Eq. (\[aef\]) being over the path of the produced $\eta$-meson $$r_{\xi}^2 = (b+ \xi cos\phi sin\theta)^2+(\xi sin\phi sin\theta )^2 +(z + \xi cos\theta )^2$$ Here $\theta$ is the emission angle of the $\eta$-meson relative to $\gamma$-momentum. Eq. (\[aef\]) is similar to those from [@Vercellin; @Hufner] and in the low energy limit, i.e. by integration over the $\eta$-emission angle $\theta$ becomes as [@Benhar] $$\label{ave} A_{eff} = \int_0^{+\infty} d{\bf b} \int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z) exp \left( -{\sigma}_{\eta N} \int_z^{\infty} d\xi \ {\rho}({\bf b}, \xi) \right)$$ In the high energy limit, i.e. with the small angle scattering approximation $\theta =0$, Eq. (\[aef\]) reduces to simple formula from [@Margolis2] $$\label{eq2} A_{eff}= \frac {1} {{\sigma}_{\eta N}} \int_0^{\infty} d{\bf b} \ \left( 1- exp \left[ -{\sigma}_{\eta N} \ \int_{-\infty}^{+\infty} dz \ {\rho}({\bf b},z) \right] \right)$$ The nuclear transparency is defined now as $$T^A = \frac {{\sigma}_{\gamma \eta }^A } {{\sigma}_{\gamma \eta }^p \times \left[ Z + \zeta (A-Z) \right]}$$ and in the Glauber model it is simply given by $$T^A = A_{eff} / A$$ being the function of the target mass $A$, emission angle $\theta$ and in-medium $\eta N$ cross section ${\sigma}_{\eta N}$. Note that this model neglect the in-medium effects  [*(i-iii)*]{}, and takes only the final state interactions into account. We analyze now the recent MAMI data [@Krusche] on $\eta$-photoproduction from $^{12}C$, $^{40}Ca$, $^{93}Nb$ and $^{207}Pb$ at $E_{\gamma}<$800 MeV in order to resolve the dependence (\[aef\]) with respect to the target mass. The $\eta$-production threshold on a free nucleon lies at $\simeq$706 MeV, thus our analysis is expected to be valid for $E_{\gamma} \geq 750$ MeV, in order to minimize effects [*(i-ii)*]{}. Moreover, to minimize the uncertainties related to [*(iii)*]{}, which also are valid at high $E_{\gamma}$, we analyze the ratios of the differential cross sections integrated over the $\eta$-meson emission angle as $$\label{rat} R(A/^{12}C) = \frac{d{\sigma}_{\gamma \eta }^A} {dT} \left( \frac {d{\sigma}_{\gamma \eta }^{^C}}{dT} \right)^{-1}$$ We thus assume that the medium modifications of the $N^{\star}$-resonance are almost the same for all nuclear targets. The ratios (\[rat\]) are shown in Fig. \[fi1\] for several kinetic energies of $\eta$-mesons and as a function of the target mass. The lines indicate our calculations performed for different $\eta N$ cross sections. The model results are integrated over the $\theta$. Note that for ${\sigma}_{\eta N}$=0 the ratio (\[rat\]) saturates at $R=A/C$ as was expected neglecting the final state interaction. We now fit the experimental ratios for each $T_{\eta}$ by minimizing the ${\chi}^2$ in order to evaluate ${\sigma}_{\eta N}$. A similar analysis was perfomed recently by Kharzeev et al. [@Kharzeev] for the evaluation of the $J/\Psi$-nucleon cross section. Fig.\[fi2\] illustrates the minimization procedure and shows sensitivity of the data to the variation of $\eta N$ cross section. We fixed the confidence level that gives the value of the reduced ${\chi}^2/n >2$ can be expected no more than 10% of the time. With respect to the statistical errors of the experimental data the minimization produces three types of results. Namely, 1) with extraction of ${\sigma}_{\eta N}$ and indication its uncertainty, 2) with evaluation only the lower limit for ${\sigma}_{\eta N}$ or 3) with obtaining the minima behind the confidence level. Fig. \[fi3\] shows our final results in comparison with the experimental data and illustrate excellent agreement for wide range of the $\eta$-energies. Nevertheless we keep in mind the uncertainties in evaluating of $\eta N$ cross section and collect the ${\sigma}_{\eta N}$ in Fig. \[fi4\] as function of the $\eta$ energy and indication of confidence level. Note that within present analysis we evaluate the inelastic (or absorption) $\eta$-nucleon cross section, because the elastic scattering does not remove the $\eta$-meson from the total flux, which was detected experimentally. Our results indicate almost constant in-medium $\eta N$ cross section as function of the $\eta$-energy in strong contradiction with the ${\sigma}_{\eta N}$ from the scattering in vacuum. To make a more definite conclusion about the suppression of the $\eta N$ cross section in nuclear matter we need an accurate data on the coherent $\eta$-photoproduction off nuclei. The coherent reactions are more sensitive to the nuclear transparency ($\propto A_{eff}^2$ [@Bauer; @Margolis2; @Bochman]) and might solve the uncertanties of the present analysis performed with the Glauber model. Results from BUU calculations {#buu} ============================= In order to verify the results from the previous section we use a BUU transport model [@Cassing1; @Bertsch1; @Teis1] to calculate energy differential $\eta$-photoproduction cross sections in nuclei. This allows to drop several assumptions needed for the Glauber calculations. Fermi motion and Pauli blocking are taken into account as well for the primary $\eta$-production process as for the final state interaction of the produced particles. In Ref. [@Effenberger] the BUU model was used to calculate $\eta$-photoproduction in nuclei with the resonance model for the $\eta$ final state interaction from Ref. [@Teis1]. Here the $\eta$-rescattering was described by intermediate excitations of N(1535) resonances. The elastic and inelastic $\eta N$ cross sections calculated with this model are shown in Fig. \[abs\] with the solid lines. It turned out that this model was able to describe the total $\eta$-photoproduction cross sections reasonably well but failed in the description of angular and energy differential cross sections. Compared to the experimental data the calculated cross sections were shifted to smaller angles and larger $\eta$-energies for all considered target nuclei and all photon energies up to 800 MeV. In Fig. \[new\] the solid line shows the calculation of an energy differential $\eta$-photoproduction cross section on $^{40}$Ca with this model. It was already reported in Ref. [@Effenberger] that the discrepancy to the experimental data is cured by using an energy independent $\eta N$ cross section. The corresponding result is shown in Fig. \[new\] by the line labelled ’constant cross sections (1)’ where an inelastic cross section $\sigma^\eta_{in}=30\,$mb and an elastic cross section $\sigma^\eta_{el}=20\,$mb was used. Now we want to study the influence of different prescriptions for $\eta N$ scattering. The dashed line in Fig. \[new\] indicates the results calculated with a modified $\eta$-rescattering model: $$\begin{aligned} \label{modi} {\sigma}_{\eta N \to \pi N} &=& \frac{q_{\pi}}{q_{\eta}s} \ \frac{c^2 }{c^2+q_{\eta}^2} \ 40 \, {\rm mb \,GeV^2} \, ,\, c=0.3\,{\rm GeV} \\ {\sigma}_{\eta N \to \eta N} &=& \frac{45 \, {\rm mb\,GeV^2}}{s} \nonumber\end{aligned}$$ where $q_{\pi}$, $q_{\eta}$ are the cms momenta of the $\pi$- and $\eta$-meson, respectively, and $s$ stands for the squared invariant energy. The inelastic cross section was obtained by fitting the experimental data for the reaction $\pi N \to \eta N$, while the elastic cross section was assumed to be equal to the one in the resonance model at an $\eta$-energy of 125 MeV. The resulting elastic and inelastic cross sections are shown in Fig. \[abs\] with the dashed line. As can be seen from Fig. \[new\] (dashed line) this model improves the description of the energy differential cross section for small $\eta$-energies but still overestimates the cross section for higher energies. We have also used the $\eta N$ cross sections from Green and Wycech [@Green] and Lee et al. [@Lee]. The calculation of Green and Wycech is based on the K-matrix method and includes the $S_{11}(1535)$ and $S_{11}(1650)$ resonances. Following the authors this model is valid up to an invariant energy of about 100 MeV from $\eta N$ threshold which corresponds to an $\eta$ kinetic energy in the nucleon rest frame of 160 MeV. Lee et al. use a parameterization of the $\eta N$ scattering amplitude that is based on the calculation of Bennhold and Tanabe [@Ben]. This model contains the $P_{11}(1440)$, $D_{13}(1520)$ and $S_{11}(1535)$ resonances and might therefore be limited to an $\eta$ kinetic energy of about 100 MeV. The total cross sections within these models are shown in Fig. \[abs\]. Both models give about the same inelastic cross section which is basically due to the fact that in both models the dominating inelastic channel is given by the process $\eta N \to N \pi$. The experimental data [@Landolt] for this reaction, obtained by detailed balance from $\pi^- p \to \eta n$, are also shown. But one should note that the theoretical curves contain additional, even though small, contributions from $\eta N \to N \pi \pi$ and therefore can not directly be compared to these data points. The elastic $\eta N$ cross section in both models is very different which is an indication for the large theoretical uncertainties in the models for $\eta N$ scattering even in the vacuum. The corresponding results of the BUU calculations for photoproduction are given in Fig. \[new\]. As in the calculations within the resonance model and the model from Eq. (\[modi\]) we again fail to describe the shape of the energy differential cross section. The same holds for all target nuclei and photon energies as well as for the angular differential cross sections. Apart from the resonance model [@Teis1] all models give a satisfactory description of the cross section for $\eta$-energies below 50 MeV. The failure of the resonance model is due to the fact that this model was fitted to a larger class of elementary processes and a wider kinematical range and overestimated the cross section for $\eta N \to N \pi$ in the considered energy range. In Fig. \[new\] we also show the result of a model calculation with a constant inelastic cross section $\sigma_{in}^\eta=30\,$mb where we neglected elastic $\eta N$ scattering (curve labelled ’constant cross sections (2)’). Compared to the previous calculation with constant cross sections that included an elastic cross section the energy differential cross section is shifted to larger energies and fails to describe the data. Moreover the integrated cross section is slightly larger because the elastic cross section increases, in average, the length of the path of the produced etas through the nucleus and therefore reduces the number of etas that escape from the nucleus. One sees that in our model a constant inelastic $\eta N$ cross section alone is not sufficient to describe the data but an elastic cross section is also needed. Since the models for $\eta N$ scattering in the vacuum show a rather strong decrease of the total $\eta N$ cross section with $\eta$-energy we are not able to reproduce the data for $\eta$-photoproduction with any of these models within our transport model approach. For $\eta$-energies larger than 100 MeV we need an $\eta N$ cross section that is significantly larger than the one from the vacuum models while for lower energies our calculations are not very sensitive to the size of the cross section. A possible explanation is that the vacuum models [@Green; @Ben] are simply not applicable to the considered energy range. After all, due to the Fermi motion of the nucleons, the $\eta N$ cross section up to an invariant energy of $\sqrt{s}=1.74\,$GeV ($T_\eta=446\,$MeV in the nucleon rest frame) enters the calculations for an eta with a kinetic energy of 250 MeV in the rest frame of the nucleus. Our findings are in line with the Glauber analysis from section \[glauber\] and Ref. [@Krusche] which need a constant inelastic cross section of 30 mb in order to describe the mass dependence of the energy differential cross sections. However, Lee et al. [@Lee] were able to reproduce energy differential cross sections within the DWIA framework by using vacuum $\eta N$ cross sections. One crucial difference to our calculation is that in their calculation the outgoing nucleon in the elementary photoproduction process $\gamma N \to N \eta$ is set on-shell while in our semi-classical treatment the elementary process takes place instantaneously with following propagation of the produced particles through the nucleus. The potential energy which is needed to set the nucleon on-shell clearly shifts the $\eta$-spectrum to lower energies. A priori it is not obvious which of the two prescriptions is better suited to model the physical reality. Only a DWIA calculation along the line of Ref. [@Li] without the local approximation of Ref. [@Lee] could clarify this question. An indication for a larger $\eta N$ cross section at higher energies is the fact that we are able to describe energy and angular differential cross section for $\eta$-photoproduction simultaneously by using a constant cross section [@Effenberger] while in the calculations of Lee et al. the angular differential cross sections are shifted to smaller angles compared to the data. Summary ======= We have analyzed the $\eta$-photoproduction in nuclei within the framework of the Glauber model and a BUU transport model approach. Using the standard Glauber theory we investigate the $A$-dependence of the reaction $\gamma A \to \eta X$ in order to extract the data on $\eta$-meson final state interaction in nuclei. It was found that the in-medium $\eta N$ cross section is almost energy independent from $\eta N \to \pi N$ threshold up to $\eta$-kinetic energy of 130 MeV. Within a BUU transport model calculation we are able to reproduce energy and angular differential data for $\eta$ photoproduction only by using an energy independent $\eta N$ cross section but not with any available $\eta N$ vacuum cross section. However, the effect of the nucleon potential that can not be treated in our semi-classical calculation in a correct way might have an impact on that conclusion. Acknowledgement =============== The authors are grateful to B. Krusche and H. Stroeher for productive discussions. They especially like to thank U. Mosel for valuable suggestions and a careful reading of the manuscript. [99]{} R.S. Bhalerao and L.C. Liu, Phys. Rev. Lett. 54 (1985) 865. C. Wilkin, Phys. Rev. C 47 (1993) R938. E. Chiavassa et al. Z. Phys. A 344 (1993) 345. A.A. Sibirtsev, Phys. Scr. RS 21 (1993) 167. Ye.S. Golubeva, A.S. Ilijnov, E.Y. Paryev and I.A. Pshenichnov, Z. Phys. A 345 (1993) 223. A.A. Sibirtsev, Phys. At. Nucl. 56 (1993) 608. E. Chiavassa et al., Nuovo Cim. A 107 (1994) 1195. M. Roebig-Landau, J. Ahrens, G. Anton et al., Phys. Lett. B 373 (1996) 45. F.X. Lee, L.E. Wright, C. Bennhold and L. Tiator, Nucl. Phys. A 603 (1996) 345. R.C. Carrasco, Phys. Rev. C 48 (1993) 2333. M. Effenberger, A. Hombach, S. Teis and U. Mosel, Nucl. Phys. A 614 (1997) 501. C. Bennhold and H. Tanabe, Nucl. Phys. A 530 (1991) 625. B. Krusche et al., Phys. Lett. B 358 (1995) 40. W. Cassing, V. Metag, U. Mosel and K. Niita, Phys. Rep. 188 (1990) 363. L.L. Salcedo, E. Oset, M.J. Vicente-Vacas and C. Garcia-Recio, Nucl. Phys. A 484 (1988) 557. R.J. Glauber and J. Mathiae, Nucl. Phys. B 21 (1970) 135. B. Margolis, Phys. Lett. B 26 (1968) 254. T.H. Bauer, R.D. Spital and D.R. Yennie, Rev. Mod. Phys. 50 (1978) 261. J. Bertch, S.J. Brodsky, A.S. Golhaber and J.G. Gunion, Phys. Rev. Lett. 47 (1981) 297. B.Z. Kopeliovich and B.G. Zakharov, Phys. Lett. 264 (1991) 434. C.W. Jager, C. Vries and H. Vries, At. Data Nucl. Data Tables 14 (1974) 480. E. Vercellin, E. Chiavassa, G. Dellacasa et al., Nuovo Cim. A 106 (1993) 861. J. Hüfner, B. Kopeliovich and J. Nemchik, nucl-th/9605007 (1996) O. Benhar, B.Z. Kopeliovich, C. Mariotti, N.N. Nikolaev and B.G. Zaharov, Phys. Rev. Lett. 69 (1992) 1156. K.S. Kölbig and B. Margolis, Nucl. Phys. B 6 (1968) 85. D. Kharzeev, C. Lourenco, M. Nardi and H. Satz, Z. Phys. C 74 (1997) 307. G. Bochman, B. Margolis and C.L. Tang, Phys. Lett. B 30 (1969) 254. G.F. Bertsch, H. Kruse and S. Das Gupta, Phys. Rev. C 29 (1984) 673. S. Teis, W. Cassing, M. Effenberger, A. Hombach, U. Mosel and Gy. Wolf, Z. Phys. A 356 (1997) 421. A.M. Green and S. Wycech, Phys. Rev. C 55 (1977) 2167. Baldini et al., Landolt-Börnstein, Band 12 (Springer, Berlin, 1987). X. Li et al., Phys. Rev. C 48 (1993) 816. [^1]: Supported by Forschungszentrum Jülich, GSI, BMBF and DFG

ArXiv

--- author: - 'Yiwei Sun, , Suhang Wang$^{\mathsection}$ ,Xianfeng Tang, Tsung-Yu Hsieh, Vasant Honavar' title: Node Injection Attacks on Graphs via Reinforcement Learning --- [^1] [^1]: $^{\mathsection}$Corresponding Author

ArXiv

--- abstract: 'While Jeffreys priors usually are well-defined for the parameters of mixtures of distributions, they are not available in closed form. Furthermore, they often are improper priors. Hence, they have never been used to draw inference on the mixture parameters. We study in this paper the implementation and the properties of Jeffreys priors in several mixture settings, show that the associated posterior distributions most often are improper, and then propose a noninformative alternative for the analysis of mixtures.' author: - 'Clara Grazian[^1]' - 'Christian P. Robert[^2]' title: 'Jeffreys priors for mixture estimation: properties and alternatives' --- Introduction {#intro} ============ Bayesian inference in mixtures of distributions has been studied quite extensively in the literature. See, e.g., [@maclachlan:peel:2000] and [@fruhwirth:2006] for book-long references and [@lee:marin:mengersen:robert:2008] for one among many surveys. From a Bayesian perspective, one of the several difficulties with this type of distribution, $$\label{eq:theMix} \sum_{i=1}^k p_i\,f(x|\theta_i)\,,\quad \sum_{i=1}^k p_i=1\,,$$ is that its ill-defined nature (non-identifiability, multimodality, unbounded likelihood, etc.) leads to restrictive prior modelling since most improper priors are not acceptable. This is due in particular to the feature that a sample from may contain no subset from one of the $k$ components $f(\cdot|\theta_i)$ (see. e.g., [@titterington:smith:makov:1985]). Albeit the probability of such an event is decreasing quickly to zero as the sample size grows, it nonetheless prevents the use of independent improper priors, unless such events are prohibited [@diebolt:robert:1994]. Similarly, the exchangeable nature of the components often induces both multimodality in the posterior distribution and convergence difficulties as exemplified by the [*label switching*]{} phenomenon that is now quite well-documented [@celeux:hurn:robert:2000; @stephens:2000b; @jasra:holmes:stephens:2005; @fruhwirth:2006; @geweke:2007; @puolamaki:kaski:2009]. This feature is characterized by a lack of symmetry in the outcome of a Monte Carlo Markov chain (MCMC) algorithm, in that the posterior density is exchangeable in the components of the mixture but the MCMC sample does not exhibit this symmetry. In addition, most MCMC samplers do not concentrate around a single mode of the posterior density, partly exploring several modes, which makes the construction of Bayes estimators of the components much harder. When specifying a prior over the parameters of , it is therefore quite delicate to produce a manageable and sensible non-informative version and some have argued against using non-informative priors in this setting (for example, [@maclachlan:peel:2000] argue that it is impossible to obtain proper posterior distribution from fully noninformative priors), on the basis that mixture models were ill-defined objects that required informative priors to give a meaning to the notion of a component of . For instance, the distance between two components needs to be bounded from below to avoid repeating the same component over and over again. Alternatively, the components all need to be informed by the data, as exemplified in [@diebolt:robert:1994] who imposed a completion scheme (i.e., a joint model on both parameters and latent variables) such that all components were allocated at least two observations, thereby ensuring that the (truncated) posterior was well-defined. [@wasserman:2000] proved ten years later that this truncation led to consistent estimators and moreover that only this type of priors could produce consistency. While the constraint on the allocations is not fully compatible with the i.i.d. representation of a mixture model, it naturally expresses a modelling requirement that all components have a meaning in terms of the data, namely that all components genuinely contributed to generating a part of the data. This translates as a form of weak prior information on how much one trusts the model and how meaningful each component is on its own (by opposition with the possibility of adding meaningless artificial extra-components with almost zero weights or almost identical parameters). While we do not seek Jeffreys priors as the ultimate prior modelling for non-informative settings, being altogether convinced of the lack of unique reference priors [@robert:2001; @robert:chopin:rousseau:2009], we think it is nonetheless worthwile to study the performances of those priors in the setting of mixtures in order to determine if indeed they can provide a form of reference priors and if they are at least well-defined in such settings. We will show that only in very specific situations the Jeffreys prior provides reasonable inference. In Section \[sec:jeffreys\] we provide a formal characterisation of properness of the posterior distribution for the parameters of a mixture model, in particular with Gaussian components, when a Jeffreys prior is used for them. In Section \[sec:prosper\] we will analyze the properness of the Jeffreys prior and of the related posterior distribution: only when the weights of the components (which are defined in a compact space) are the only unknown parameters it turns out that the Jeffreys prior (and so the relative posterior) is proper; on the other hand, when the other parameters are unknown, the Jeffreys prior will be proved to be improper and in only one situation it provides a proper posterior distribution. In Section \[sec:alternative\] we propose a way to realize a noninformative analysis of mixture models and introduce improper priors for at least some parameters. Section \[sec:concl\] concludes the paper. Jeffreys priors for mixture models {#sec:jeffreys} ================================== We recall that the Jeffreys prior was introduced by [@jeffreys:1939] as a default prior based on the Fisher information matrix $$\pi^\text{J}(\theta) \propto |I(\theta)|^{{\nicefrac{1}{2}}}\,,$$ whenever the later is well-defined; $I(\cdot)$ stand for the expected Fisher information matrix and the symbol $|\cdot|$ denotes the determinant. Although the prior is endowed with some frequentist properties like matching and asymptotic minimal information [@robert:2001 Chapter 3], it does not constitute the ultimate answer to the selection of prior distributions in non-informative settings and there exist many alternative such as reference priors [@berger:bernardo:sun:2009], maximum entropy priors [@rissanen:2012], matching priors [@ghosh:carlin:srivastava:1995], and other proposals [@kass:wasserman:1996]. In most settings Jeffreys priors are improper, which may explain for their conspicuous absence in the domain of mixture estimation, since the latter prohibits the use of most improper priors by allowing any subset of components to go “empty" with positive probability. That is, the likelihood of a mixture model can always be decomposed as a sum over all possible partitions of the data into $k$ groups at most, where $k$ is the number of components of the mixture. This means that there are terms in this sum where no observation from the sample brings any amount of information about the parameters of a specific component. Approximations of the Jeffreys prior in the setting of mixtures can be found, e.g., in [@figueiredo:jain:2002], where the Authors revert to independent Jeffreys priors on the components of the mixture. This induces the same negative side-effect as with other independent priors, namely an impossibility to handle improper priors. [@rubio:steel:2014] provide a closed-form expression for the Jeffreys prior for a location-scale mixture with two components. The family of distributions they consider is $$\dfrac{2\epsilon}{\sigma_1}f\left(\frac{x-\mu}{\sigma_1}\right)\mathbb{I}_{x<\mu}+ \dfrac{2(1-\epsilon)}{\sigma_2}f\left(\frac{x-\mu}{\sigma_2}\right) \mathbb{I}_{x>\mu}$$ (which thus hardly qualifies as a mixture, due to the orthogonality in the supports of both components that allows to identify which component each observation is issued from). The factor $2$ in the fraction is due to the assumption of symmetry around zero for the density $f$. For this specific model, if we impose that the weight $\epsilon$ is a function of the variance parameters, $ \epsilon=\nicefrac{\sigma_1}{\sigma_1+\sigma_2}, $ the Jeffreys prior is given by $ \pi(\mu,\sigma_1,\sigma_2) \propto \nicefrac{1}{\sigma_1\sigma_2\{\sigma_1+\sigma_2\}}. $ However, in this setting, [@rubio:steel:2014] demonstrate that the posterior associated with the (regular) Jeffreys prior is improper, hence not relevant for conducting inference. (One may wonder at the pertinence of a Fisher information in this model, given that the likelihood is not differentiable in $\mu$.) [@rubio:steel:2014] also consider alternatives to the genuine Jeffreys prior, either by reducing the range or even the number of parameters, or by building a product of conditional priors. They further consider so-called non-objective priors that are only relevant to the specific case of the above mixture. Another obvious explanation for the absence of Jeffreys priors is computational, namely the closed-form derivation of the Fisher information matrix is almost inevitably impossible. The reason is that integrals of the form $$-\int_{\mathcal{X}} \frac{\partial^2 \log \left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]}{\partial \theta_i \partial \theta_j}\left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]^{-1} d x$$ (in the special case of component densities with a single parameter) cannot be computed analytically. We derive an approximation of the elements of the Fisher information matrix based on Riemann sums. The resulting computational expense is of order $\mathrm{O}(d^2)$ if $d$ is the total number of (independent) parameters. Since the elements of the information matrix usually are ratios between the component densities and the mixture density, there may be difficulties with non-probabilistic methods of integration. Here, we use Riemann sums (with $550$ points) when the component standard deviations are sufficiently large, as they produce stable results, and Monte Carlo integration (with sample sizes of $1500$) when they are small. In the latter case, the variability of MCMC results seems to decrease as $\sigma_i$ approaches $0$. Properness for prior and posterior distributions {#sec:prosper} ================================================ Unsurprisingly, most Jeffreys priors associated with mixture models are improper, the exception being when only the weights of the mixture are unknown, as already demonstrated in [@bernardo:giron:1988]. We will characterize properness and improperness of Jeffreys priors and derived posteriors, when some or all of the parameters of distributions from location-scale families are unknown. These results are established both analytically and via simulations, with sufficiently large Monte Carlo experiments checking the behavior of the approximated posterior distribution. Characterization of Jeffreys priors {#subsec:priors} ----------------------------------- ### Weights of mixture unknown A representation of the Jeffreys prior and the derived posterior distribution for the weights of a 3-component mixture model is given in Figure \[weights-priorpost\]: the prior distribution is much more concentrated around extreme values in the support, i.e., it is a prior distribution conservative in the number of important components. ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the weights of a Gaussian mixture model with 3-components are unknown (on the top) and of the derived posterior distribution (with known means equal to -1, 0 and 2 respectively and known standard devitations equal to 1, 5 and 0.5 respectively). The red cross represents the true values.[]{data-label="weights-priorpost"}](weights-priorpost){width="6.5cm" height="7.5cm"} \[lem:weights\] When the weights $p_i$ are the only unknown parameters in , the corresponding Jeffreys prior is proper. Figure \[weights-boxplots\] shows the boxplots for the means of the approximated posterior distribution for the weights of a three-component Gaussian mixture model. ![Boxplots of the estimated means of the three-component mixture model $0.25\mathcal{N}(-10,1)+0.65\mathcal{N}(0,5) +0.10\mathcal{N}(15,0.5)$ for 50 simulated samples of size $100$, obtained via MCMC with $10^5$ simulations. The red crosses represent the true values of the weights.[]{data-label="weights-boxplots"}](weights-boxplot.pdf){width="6.5cm" height="7.5cm"} The generic element of the Fisher information matrix is (for $i,j=\{1,\ldots,k-1\}$) $$\int_\mathcal{X} \frac{(f_i(x)-f_k(x))(f_j(x)-f_k(x))}{\sum_{l=1}^k p_l f_l(x)} d x \label{eq:ww-prior}$$ when we consider the parametrization in $(p_1,\ldots,p_{k-1})$, with $$p_k=1-p_1-\cdots-p_{k-1}\,.$$ We remind that, since the Fisher information matrix is a positive semi-definite, its determinant is bounded by the product of the terms in the diagonal, thanks to the Hadamard’s inequality. Therefore, we may consider the diagonal term, $$\begin{aligned} \int_\mathcal{X} \frac{(f_i(x)-f_k(x))^2}{\sum\limits_{l=1}^k p_l f_l(x)} d x &= \int_{f_i(x)\ge f_k(x)} \frac{(f_i(x)-f_k(x))^2}{\sum\limits_{l=1}^k p_l f_l(x)} d x\\ &\quad + \int_{f_i(x)\le f_k(x)} \frac{|(f_i(x)-f_k(x))^2|}{\sum\limits_{l=1}^k p_l f_l(x)} d x\\ &= \int_{f_i(x)\ge f_k(x)} \frac{f_i(x)-f_k(x)}{\sum\limits_{l=1}^k p_l f_l(x)} \{f_i(x)-f_k(x)\}d x\\ &\quad + \int_{f_i(x)\le f_k(x)} \Big| \frac{f_i(x)-f_k(x)}{\sum\limits_{l=1}^k p_l f_l(x)} \Big| |f_i(x)-f_k(x)| d x\\ &= \frac{1}{p_i}\,\int_{f_i\ge f_k} \frac{p_i\{f_i(x)-f_k(x)\}}{p_i\{f_i(x)-f_k(x)\}+\sum\limits_{l\ne i,k} p_l \{f_l(x)-f_k(x)\}+f_k(x)}\\ &\qquad\qquad \{f_i(x)-f_k(x)\}d x\\ &\quad + \frac{1}{p_i}\,\int_{f_i\le f_k} \Big|\frac{p_i\{f_i(x)-f_k(x)\}}{p_i\{f_i(x)-f_k(x)\}+\sum\limits_{l\ne i,k} p_l \{f_l(x)-f_k(x)\}+f_k(x)} \Big| \\ &\qquad\qquad |f_i(x)-f_k(x)| d x\\ &\le \frac{1}{p_i}\int_{f_i(x)\ge f_k(x)} \{f_i(x)-f_k(x)\}d x + \frac{1}{p_i}\int_{f_i(x)\le f_k(x)} | f_i(x)-f_k(x) |d x\\ &= \frac{2}{p_i}\int_{f_i(x)\ge f_k(x)} \{f_i(x)-f_k(x)\}d x\end{aligned}$$ since both integrals are equal. Therefore, the Jeffreys prior will be bounded by the square root of the product of the terms in the diagonal of the Fisher information matrix $$\pi^J(\mathbf{p}) \propto \prod_{i=1}^k p_i^{-\frac{1}{2}}$$ which is a generalization to $k$ components of the prior provided in [@bernardo:giron:1988] for $k=2$ (however, [@bernardo:giron:1988] find the reference prior for the limiting case when all the components have pairwise disjoint supports, while for the opposite limiting case where all the components converge to the same distribution, the Jeffrey’s prior is the uniform distribution on the $k$-dimensional simplex). This reasoning leads [@bernardo:giron:1988] to conclude that the usual $\mathcal{D}(\lambda_1,\ldots,\lambda_k)$ Dirichlet prior with $\lambda_i \in [\nicefrac{1}{2},1]$ for $\forall i=1,\cdots,k$ seems to be a reasonable approximation. They also prove that the Jeffreys prior for the weights $p_i$ is convex, with a argument based on the sign of the second derivative. As a remark, the configuration shown in proof of Lemma \[lem:weights\] is compatible with the Dirichlet configuration of the prior proposed by [@rousseau:mengersen:2011]. The shape of the Jeffreys prior for the weights of a mixture model depends on the type of the components. Figure \[weights-GMM\], \[weights-GtMM\] and \[weights-GtMM-df\] show the form of the Jeffreys prior for a 2-component mixture model for different choices of components. It is always concentrated around the extreme values of the support, however the amount of concentration around $0$ or $1$ depends on the information brought by each component. In particular, Figure \[weights-GMM\] shows that the prior is much more symmetric as there is symmetry between the variances of the distribution components, while Figure \[weights-GtMM\] shows that the prior is much more concentrated around 1 for the weight relative to the normal component if the second component is a Student t distribution. Finally Figure \[weights-GtMM-df\] shows the behavior of the Jeffreys prior when the first component is Gaussian and the second is a Student t and the number of degrees of freedom is increasing. As expected, as the Student t is approaching a normal distribution, the Jeffreys prior becomes more and more symmetric. ![Approximations of the marginal prior distributions for the first weight of a 2-component Gaussian mixture model, $p\,\mathcal{N}(-10,1)+(1-p)\,\mathcal{N}(10,1)$ (black), $p\,\mathcal{N}(-1,1)+(1-p)\,\mathcal{N}(1,1)$ (red) and $p\,\mathcal{N}(-10,1)+(1-p)\,\mathcal{N}(10,10)$ (blue).[]{data-label="weights-GMM"}](weights-prior-comparison-GMM){width="6.5cm" height="7.5cm"} ![Approximations of the marginal prior distributions for the first weight of a 2-component mixture model where the first component is Gaussian and the second is Student t, $p\,\mathcal{N}(-10,1)+(1-p)\,\mathrm{t}(df=1,10,1)$ (black), $p\,\mathcal{N}(-1,1)+(1-p)\,\mathrm{t}(df=1,1,1)$ (red) and $p\,\mathcal{N}(-10,1)+(1-p)\,\mathrm{t}(df=1,10,10)$ (blue).[]{data-label="weights-GtMM"}](weights-prior-comparison-GtMM){width="6.5cm" height="7.5cm"} ![Approximations of the marginal prior distributions for the first weight of a 2-component mixture model where the first component is Gaussian and the second is Student t with an increasing number of degrees of freedom.[]{data-label="weights-GtMM-df"}](weights-prior-comparison-GtMM-df){width="6.5cm" height="7.5cm"} ### Location and scale parameters of a mixture model unknown If the components of the mixture model are distributions from a location-scale family and the location or scale parameters of the mixture components are unknown, this turns the mixture itself into a location-scale model. As a result, model may be reparametrized by following [@mengersen:robert:1996], in the case of Gaussian components $$\label{reparMix} p\mathcal{N}(\mu,\tau^2)+(1-p)\mathcal{N}(\mu+\tau\delta,\tau^2\sigma^2)$$ namely using a reference location $\mu$ and a reference scale $\tau$ (which may be, for instance, the location and scale of a specific component). Equation may be generalized to the case of $k$ components as $$\begin{aligned} \label{eq:k_reparMix} p\mathcal{N}(\mu,\tau^2)&+\sum_{i=1}^{k-2} (1-p) (1-q_1) \cdots (1-q_{i-1})q_i \mathcal{N}(\mu+\tau\theta_1+\cdots+\tau\cdots\sigma_{i-1}\theta_i,\tau^2\sigma_1^2\cdots\sigma_i^2) \nonumber \\ &\qquad {} + (1-p)(1-q_1)\cdots (1-q_{k-2})\mathcal{N}(\mu+\tau\theta_1+\cdots+\tau\cdots\sigma_{k-2}\theta_{k-1},\tau^2\sigma_1^2\cdots\sigma_{k-1}^2)\end{aligned}$$ In this way, the mixture model is more cleary a location-scale model, which implies that the Jeffreys prior is flat in the location and powered as $\tau^{-d/2}$ if $d$ is the total number of parameters of the components, respectively [@robert:2001 Chapter 3], as we will see in the following. \[lem:meansd-prior\] If the parameters of the components of a mixture model are either location or scale parameters, the corresponding Jeffreys prior is improper. In the proof of Lemma \[lem:meansd-prior\], we will consider a Gaussian mixture model and then extend the results to the general situation of components from a location-scale family. ### Unknown location parameters {#unknown-location-parameters .unnumbered} We first consider the case where the means are the only unknown parameters of a Gaussian mixture model $$g_X(x)=\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)$$ The generic elements of the expected Fisher information matrix are, in the case of diagonal and off-diagonal terms respectively: $$\mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_i^2}\right]=\frac{p_i^2}{\sigma_i^4} \bigintsss_{-\infty}^\infty \frac{\left[ (x-\mu_i) \mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_i \partial \mu_j}\right]=\frac{p_i p_j}{\sigma_i^2 \sigma_j^2} \bigintsss_{-\infty}^\infty \frac{(x-\mu_i) \mathfrak{n}(x|\mu_i,\sigma_i^2)(x-\mu_j) \mathfrak{n}(x|\mu_j,\sigma_j^2) }{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ Now, consider the change of variable $t=x-\mu_i$ in the above integrals, where $\mu_i$ is thus the mean of the $i$-th Gaussian component ($i\in\{1,\cdots,k\}$). The above integrals are then equal to $$\begin{aligned} \mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_j^2}\right] &= \frac{p_j^2}{\sigma_j^4} \bigintsss_{-\infty}^\infty \frac{\left[ (t-\mu_j+\mu_i) \mathfrak{n}(t|\mu_j-\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(t|\mu_l-\mu_i,\sigma_l^2)} d x\\ \mathbb{E}\left[- \frac{\partial^2 \log g_X(X)}{\partial \mu_j \partial \mu_m}\right] &= \frac{p_j p_m}{\sigma_j^2 \sigma_m^2} \bigintsss_{-\infty}^\infty \frac{(t-\mu_j+\mu_i) \mathfrak{n}(x|\mu_j,\sigma_j^2)(t-\mu_m+\mu_i) \mathfrak{n}(t|\mu_m-\mu_i,\sigma_m^2) }{\sum_{l=1}^k p_l \mathfrak{n}(t|\mu_l-\mu_i,\sigma_l^2)} d x\\ \label{eq:means-prior}\end{aligned}$$ Therefore, the terms in the Fisher information only depend on the differences $\delta_j=\mu_i-\mu_j$ for $j \in \{1,\cdots,k \}$. This implies that the Jeffreys prior is improper since a reparametrization in ($\mu_i,\mathbf{\delta}$) shows the prior does not depend on $\mu_i$. This feature will reappear whenever the location parameters are unknown. When considering the general case of components from a location-scale family, this feature of improperness of the Jeffreys prior distribution is still valid, because, once reference location-scale parameters are chosen, the mixture model may be rewritten as $$\label{eq:mix-locscale} p_1 f_1(x|\mu,\tau)+\sum_{i=2}^k p_i f_i(\frac{a_i+ x}{b_i} |\mu,\tau,a_i,b_i).$$ Then the second derivatives of the logarithm of model behave as the ones we have derived for the Gaussian case, i.e. they will depend on the differences between each location parameter and the reference one, but not on the reference location itself. Then the Jeffreys prior will be constant with respect to the global location parameter. When considering the reparametrization , the Jeffreys prior for $\delta$ for a fix $\mu$ has the form: $$\pi^J(\delta|\mu)\propto \left[ \int_\mathfrak{X}\frac{\left[{(1-p)x\exp\{-\frac{x^2}{2}\}}\right]^2}{{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}}+{(1-p)\exp\{-\frac{x^2}{2}\}}} d x \right]^{\frac{1}{2}}$$ and the following result may be demonstrated. The Jeffreys prior of $\delta$ conditional on $\mu$ when only the location parameters are unknown is improper. The improperness of the conditional Jeffreys prior on $\delta$ depends (up to a constant) on the double integral $$\begin{aligned} \int_\Delta \int_\mathfrak{X} c \frac{\left[(1-p)x\exp\{-\frac{x^2}{2}\}\right]^2}{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+(1-p)\exp\{-\frac{x^2}{2}\}} d x d\delta.\end{aligned}$$ The order of the integrals is allowed to be changed, then $$\begin{aligned} \int_\mathfrak{X} x^2 \int_\Delta \frac{\left[(1-p)\exp\{-\frac{x^2}{2}\}\right]^2}{p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+(1-p)\exp\{-\frac{x^2}{2}\}} d\delta d x \end{aligned}$$ Define $f(x)=(1-p)e^{-\frac{x^2}{2}}=\frac{1}{d}$. Then $$\begin{aligned} \int_\mathcal{X} x^2 \int_\Delta \frac{1}{d^2 p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+d} d\delta d x \end{aligned}$$ Since the behavior of $\left[d^2 p\sigma\exp\{-\frac{\sigma^2(x+\frac{\delta}{\sigma\tau})^2}{2}\}+d\right]$ depends on $\exp\{-\delta^2\}$ as $\delta$ goes to $\infty$, we have that $$\int_{-\infty} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta > \int_{A} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta$$ because the integrand function is positive. Then $$\int_{A} ^{+\infty} \frac{1}{\exp\{-\delta^2\}+d} d\delta > \int_{A} ^{+\infty} \frac{1}{\varepsilon+d} d\delta = +\infty$$ Therefore the conditional Jeffreys prior on $\delta$ is improper. Figure \[fig:priorpost-diff\] compares the behavior of the prior and the resulting posterior distribution for the difference between the means of a two-component Gaussian mixture model: the prior distribution is symmetric and it has different behaviors depending on the value of the other parameters, but it always stabilizes for large enough values; the posterior distribution appears to always concentrate around the true value. ![Approximations (on a grid of values) of the Jeffreys prior (on the natural scale) of the difference between the means of a Gaussian mixture model with only the means unknown (left) and of the derived posterior distribution (on the right, the red line represents the true value), with known weights equal to $(0.5,0.5)$ (black lines), $(0.25,0.75)$ (green and blue lines) and known standard deviations equal to $(5,5)$ (black lines), $(1,1)$ (green lines) and $(7,1)$ (blue lines).[]{data-label="fig:priorpost-diff"}](priorpost-diff.pdf){width="6.5cm" height="7.5cm"} ### Unknown scale parameters {#unknown-scale-parameters .unnumbered} Consider now the second case of the scale parameters being the only unknown parameters. First, consider a Gaussian mixture model and suppose the mixture model is composed by only two components; the Jeffreys prior for the scale parameters is defined as $$\begin{aligned} \pi^J(\sigma_1,\sigma_2)&\propto \left\{ \frac{p_1^2}{\sigma_1^2} \bigintsss_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right) \mathfrak{n}(x|\mu_1,\sigma_1^2)\right]^2}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right. \nonumber \\ &\cdot \left.{} \frac{p_2^2}{\sigma_2^2} \bigintsss_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right) \mathfrak{n}(x|\mu_2,\sigma_2^2)\right]^2}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right. \nonumber \\ &- \left.{} \left[\frac{p_1 p_2}{\sigma_1 \sigma_2} \bigintsss_{-\infty}^\infty \frac{ \left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right) \left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right) \mathfrak{n}(x|\mu_1,\sigma_1^2)\mathfrak{n}(x|\mu_2,\sigma_2^2)}{\sum_{l=1}^2 p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x \right]^2\right\}^\frac{1}{2} \end{aligned}$$ Since the Fisher information matrix is positive definite, it is bounded by the product on the diagonal, then we can write: $$\begin{aligned} \pi^J(\sigma_1,\sigma_2)&\leq c \frac{p_1 p_2}{\sigma_1\sigma_2}\left\{ \bigintsss_{-\infty}^\infty \frac{\left(\frac{(x-\mu_1)^2}{\sigma_1^2}-1\right)^2 \frac{1}{\sigma_1^2} \exp\left\{ -\frac{(x-\mu_1)^2}{\sigma_1^2}\right\}}{\frac{p_1}{\sigma_1}\exp\left\{-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right\}+\frac{p_2}{\sigma_2}\exp\left\{-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right\}} d x \right. \nonumber \\ &\cdot \left.{} \bigintsss_{-\infty}^\infty \frac{\left(\frac{(x-\mu_2)^2}{\sigma_2^2}-1\right)^2 \frac{1}{\sigma_2^2} \exp\left\{ -\frac{(x-\mu_2)^2}{\sigma_2^2}\right\}}{\frac{p_1}{\sigma_1}\exp\left\{-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right\}+\frac{p_2}{\sigma_2}\exp\left\{-\frac{(x-\mu_2)^2}{2\sigma_2^2}\right\}} d x \right\}^\frac{1}{2} \end{aligned}$$ In particular, if we reparametrize the model by introducing $\sigma_1=\tau$ and $\sigma_2=\tau \sigma$ and study the behavior of the following integral $$\begin{aligned} \label{eq:scaleprior} \bigintsss_0^{\infty} \bigintsss_0^{\infty} & c \frac{p_1 p_2}{\tau\sigma}\left\{ \bigintsss_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\cdot \left.{} \left\{ \bigintsss_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}}\right\} d u \right\}^\frac{1}{2} d \tau d \sigma\end{aligned}$$ where the internal integrals with respect to $z$ and $u$ converge with respect to $\sigma$ and $\tau$, then the behavior of the external integrals only depends on $\frac{1}{\tau\sigma}$. Therefore they do not converge. This proof can be easily extended to the case of $k$ components: the behavior of the prior depends on the inverse of the product of the scale parameters, which implies that the prior is improper. Moreover this proof may be easily extended to the general case of mixtures of location-scale distributions , because the second derivatives of the logarithm of the model will depend on factors $b_i^{-2}$ for $i \in {1,\cdots,k}$. When the square root is considered, it is evident that the integral will not converge. Figures \[fig:sd-priorpost-clm\] and \[fig:sd-priorpost-asym\] show the prior and the posterior distributions of the scale parameters of a two-component mixture model for some situations with different weights and different means. ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-1,1)$.[]{data-label="fig:sd-priorpost-clm"}](lsd-priorpost-clm2){width="6.5cm" height="7.5cm"} ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-2,7)$.[]{data-label="fig:sd-priorpost-asym"}](lsd-priorpost-farm2){width="6.5cm" height="7.5cm"} Summarized results of the posterior approximation obtained via a random-walk Metropolis-Hastings algorithm by exploring the posterior distribution associated with the Jeffreys prior on the standard deviations are shown in Figures \[fig:sd2-bxp\] and \[fig:sd3-bxp\], which display boxplots of the posterior means: provided a sufficiently high sample size, simulations exhibit a convergent behavior. ![Boxplots of posterior means of the standard deviations of the two-component mixture model $0.50\mathcal{N}(-1,1) + 0.50\mathcal{N}(2,0.5)$ for 50 replications of the experiment and a sample size equal to $10$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd2-bxp"}](sd2-boxplot){width="6.5cm" height="7.5cm"} ![Boxplots of posterior means of the standard deviations of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment and a sample size equal to $50$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd3-bxp"}](sd3-boxplot){width="6.5cm" height="7.5cm"} ### Location and scale parameters unknown. Consider now the case where both location and scale parameters are unknown. Once again, each element of the Fisher information matrix is an integral in which a change of variable $x-\mu_i$ can be used, for some choice of $\mu_i,\ ,i=1,\cdots,k$ so that each term only depends on the difference $\delta_j=\mu_i-\mu_j$; the elements are $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \sigma_i^2}\right]=\frac{p_i^2}{\sigma_i^2} \int_{-\infty}^\infty \frac{\left[ \left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \sigma_i \partial \sigma_j}\right]=\frac{p_i p_j}{\sigma_i \sigma_j} \int_{-\infty}^\infty \frac{ \left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \left(\frac{(x-\mu_j)^2}{\sigma_j^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\mathfrak{n}(x|\mu_j,\sigma_j^2)}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \mu_i \partial \sigma_i}\right]=\frac{p_i^2}{\sigma_i^3} \int_{-\infty}^\infty \frac{ \left(x-\mu_i\right)\left(\frac{(x-\mu_i)^2}{\sigma_i^2}-1\right) \left[\mathfrak{n}(x|\mu_i,\sigma_i^2)\right]^2}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ $$\mathbb{E}\left[- \frac{\partial^2 log f_X(X)}{\partial \mu_i \partial \sigma_j}\right]=\frac{p_i p_j}{\sigma_i \sigma_j} \int_{-\infty}^\infty \frac{ \frac{(x-\mu_i)}{\sigma_i^2 \sigma_j} \left(\frac{(x-\mu_j)^2}{\sigma_j^2}-1\right) \mathfrak{n}(x|\mu_i,\sigma_i^2)\mathfrak{n}(x|\mu_j,\sigma_j^2)}{\sum_{l=1}^k p_l \mathfrak{n}(x|\mu_l,\sigma_l^2)} d x$$ ### Location and scale parameters unknown {#location-and-scale-parameters-unknown .unnumbered} When considering all the parameters unknown, the form of the Jeffreys prior may be partly defined by considering the mixture model as a location-scale model, for which a general solution exists; see [@robert:2001]. When all the parameters of a Gaussian mixture model are unknown, the Jeffreys prior is constant in $\mu$ and powered as $\tau^{-d/2}$, where $d$ is the total number of components parameters. We have already proved the Jeffreys prior is constant on the global mean (first proof of Lemma \[lem:meansd-prior\]). Consider a two-component mixture model and the reparametrization . With some computations, it is straightforward to derive the Fisher information matrix for this model, partly shown in Table \[tab:FishInfo\_repar\], where each term is multiplied for a term which does not depend on $\tau$. . \[tab:FishInfo\_repar\] **$\sigma$** **$\delta$** **p** **$\mu$** **$\tau$** -------------- -------------- -------------- ------- ------------- ------------- **$\sigma$** 1 1 $\tau^{-1}$ $\tau^{-1}$ **$\delta$** 1 1 $\tau^{-1}$ $\tau^{-1}$ **p** 1 1 $\tau^{-1}$ $\tau^{-1}$ **$\mu$** $\tau^{-1}$ $\tau^{-1}$ $\tau^{-2}$ $\tau^{-2}$ **$\tau$** $\tau^{-1}$ $\tau^{-1}$ $\tau^{-2}$ $\tau^{-2}$ : Factors depending on $\tau$ of the Fisher information matrix for the reparametrized model Therefore, the Fisher information matrix considered as a function of $\tau$ is a block matrix. From well-known results in linear algebra, if we consider a block matrix $$M= \begin{bmatrix} A & B \\ C & D \end{bmatrix}$$ then its determinant is given by $\det(M)=\det(A-BD^{-1}C)\det(D)$. In the case of a two-component mixture model, $\det(D)\propto\tau^{-4}$, while $\det(A-BD^{-1}C)\propto 1$ (always seen as functions of $\tau$ only). Then the Jeffreys prior for a two-component location-scale mixture model is proportional to $\tau^{-2}$. This result may be easily generalized to the case of $k$ components. Posterior distributions of Jeffreys priors ------------------------------------------ We now derive analytical and computational characterizations of the posterior distributions associated with Jeffreys priors for mixture models. Simulated examples are used to support the analytical results. For this purpose, we have repeated simulations from the models $$0.50\mathcal{N}(\mu_1,1) + 0.50\mathcal{N}(\mu_2,0.5) \label{eq:2mix}$$ and $$0.25\mathcal{N}(\mu_1,1) + 0.65\mathcal{N}(\mu_0,0.5) + 0.10\mathcal{N}(\mu_2,5) \label{eq:3mix}$$ where $\mu_1$ and $\mu_2$ are chosen to be either close ($\mu_1=-1$, $\mu_2=2$) or well separated ($\mu_1=-10$, $\mu_2=15$) and $\mu_0=0$. The Tables shown in the following will analyze the behavior of simulated Markov chains with the goal to approximate the posterior distribution. Even if the output of a MCMC method is not conclusive to assess the properness of the target distribution, it may give a hint on improperness: if the target is improper, an MCMC chain cannot be positive recurrent but instead either null-recurrent or transient [@robert:casella:2004], then it should show convergence problems, as trends or difficulties to move from a particular region. Therefore, simulation studies will be used to support analytical results on properness or improperness of the posterior distribution. In the following, we will say that the results are stable if they show a convergent behavior, i.e. they move around the true values which have generated the data. In particular, an approximation is stable if the proportion of experiments for which the chains show no trend and acceptance rates around the expected values (20%-40%, which means that there are not regions where the chain have difficulties to move from) is 0. The following results are based on Gaussian mixture models, anyway, the Jeffreys prior has a behavior common to all the location-scale families, as shown in Section \[subsec:priors\], as well as the likelihood function; therefore the results may be generalized to any location-scale family. ### Location parameters unknown A first numerical study where the Jeffreys prior and its posterior are computed on a grid of parameter values confirms that, provided the means only are unknown, the prior is constant on the difference between the means and takes higher and higher values as the difference between them increases. However, the posterior distribution is correctly concentrated around the true values for a sufficiently high sample size and it exhibits the classical bimodal nature of such posteriors [@celeux:hurn:robert:2000]. In Figure \[fig:mean-priorpost\], the posterior distribution appears to be perfectly symmetric because the other parameters (weights and standard deviations) have been fixed as identical. ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the means of a Gaussian mixture model with two components are unknown (on the top) and of the derived posterior distribution (with known weights both equal to 0.5 and known standard deviations both equal to 5).[]{data-label="fig:mean-priorpost"}](means-prior){width="6.5cm" height="7.5cm"} Tables \[tab:post2means\] and \[tab:post3means\] show that, when considering a two-component Gaussian mixture model, the results are stabilizing for a sample size equal to $10$ if the components are close and they are always stable if the means are far enough; on the other hand, huge sample sizes (around $100$ observations) are needed to have always converging chains for a three-component mixture model (even if, when the components are well-separated a sample size equal to $10$ seems to be enough to have stable results). When $k=2$, the posterior distribution derived from the Jeffreys prior when only the means are unknown is proper. \[lem:mean-post\] The conditional Jeffreys prior for the means of a Gaussian mixture model is $$\begin{aligned} \pi^J(\mu|p,\sigma) &\propto \frac{p_1 p_2}{\sigma_1^2 \sigma_2^2}\left\{ \int_{-\infty}^{+\infty}\frac{\left[ t\mathfrak{n}(0,\sigma_1)\right]^2}{p_1\mathfrak{n}(0,\sigma_1)+p_2\mathfrak{n}(\delta,\sigma_2)} d t \right. \nonumber \\ &{} \left. \times \int_{-\infty}^{+\infty} \frac{\left[ u\mathfrak{n}(0,\sigma_2)\right]^2}{p_1\mathfrak{n}(-\delta,\sigma_1)+p_2\mathfrak{n}(0,\sigma_2)} d u \right. \nonumber \\ &{} \left. -\left(\int_{-\infty}^{+\infty} \frac{ t\mathfrak{n}(0,\sigma_1) (t-\delta)\mathfrak{n}(\delta,\sigma_2)}{p_1\mathfrak{n}(0,\sigma_1)+p_2\mathfrak{n}(\delta,\sigma_2)} d t\right)^2 \right\}^\frac{1}{2}\end{aligned}$$ where $\delta=\mu_2-\mu_1$. The posterior distribution is then defined as $$\prod_{j=1}^n \left[p_1\mathfrak{n}(\mu_1,\sigma_1)+p_2\mathfrak{n}(\mu_2,\sigma_2)\right]\pi^J(\mu_1,\mu_2|p,\sigma)$$ The likelihood may be rewritten (without loss of generality, by considering $\sigma_1=\sigma_2=1$, since they are known) as $$\begin{aligned} L(\theta)&=\prod_{j=1}^n \left[p_1\mathfrak{n}(\mu_1,1)+p_2\mathfrak{n}(\mu_2,1)\right] \nonumber \\ &= \frac{1}{(2\pi)^\frac{n}{2}}\left[p_1^n e^{-\frac{1}{2}\sum_{i=1}^n (x_i-\mu_1)^2}+\sum_{j=1}^n p_1^{n-1}p_2e^{-\frac{1}{2}\sum_{i\neq j} (x_i-\mu_1)^2-\frac{1}{2}(x_j-\mu_2)^2}\right. \nonumber \\ &{} \left. +\sum_{j=1}^n \sum_{k\neq j} p_1^{n-2}p_2^2 e^{-\frac{1}{2}\sum_{i\neq j,k} (x_i-\mu_1)^2-\frac{1}{2}\left[(x_j-\mu_2)^2+(x_k-\mu_2)^2 \right]} \right. \nonumber \\ &{} \left. +\cdots+p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2}\right] \label{eq:mixlik}\end{aligned}$$ Then, for $|\mu_1|\rightarrow\infty$, $L(\theta)$ tends to the term $p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2}$ that is constant for $\mu_1$. Therefore we can study the behavior of the posterior distribution for this part of the likelihood to assess its properness. This explains why we want the following integral to converge: $$\int_{\mathbb{R}\times\mathbb{R}} p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_2)^2} \pi^J(\mu_1,\mu_2) d\mu_1 d\mu_2$$ which is equal to (by the change of variable $\mu_2-\mu_1=\delta$) $$\int_{\mathbb{R}\times\mathbb{R}} p_2^n e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_1-\delta)^2} \pi^J(\mu_1,\delta) d\mu_1 d\delta$$ We have seen that the prior distribution only depends on the difference between the means $\delta$: $$\begin{aligned} &\int_\mathbb{R} p_2^n \int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\mu_1-\delta)^2}d\mu_1 \pi^J(\delta)d\delta \nonumber \\ &\propto \int_\mathbb{R} \int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2 +\mu_1\sum_{j=1}^n(x_j-\delta)-\frac{1}{2}n\mu_1^2} d\mu_1 \pi^J(\delta)d\delta \nonumber \\ &=\int_\mathbb{R} \left[\int_\mathbb{R} e^{\mu_1\sum_{j=1}^n(x_j-\delta)-\frac{1}{2}n\mu_1^2} d\mu_1\right] e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2} \pi^J(\delta)d\delta \nonumber \\ &=\int_\mathbb{R} e^{-\frac{1}{2}\sum_{j=1}^n (x_j-\delta)^2+\sum_{j=1}^n\frac{(x_j-\delta)}{2n}} \pi^J(\delta)d\delta \nonumber \\ &\approx \int_\mathbb{R} e^{-\frac{n-1}{2}\delta^2} \pi^J(\delta)d\delta \label{eq:postmean}\end{aligned}$$ The prior on $\delta$ depends on the determinant of the corresponding Fisher information matrix that is positive definite, then it is bounded by the product of the Fisher information matrix diagonal entries: $$\footnotesize{ \pi(\delta)\leq \frac{p_1 p_2}{\sigma_1 \sigma_2}\left\{ \bigintsss_{-\infty}^{+\infty} \frac{\left[t\mathfrak{n}(0,\sigma_1^2)\right]^2}{p_1 \mathfrak{n}(0,\sigma_1^2)+p_2\mathfrak{n}(\delta,\sigma_2^2)} d t \times \bigintsss_{-\infty}^{+\infty} \frac{\left[u\mathfrak{n}(0,\sigma_2^2)\right]^2}{p_1 \mathfrak{n}(-\delta,\sigma_1^2)+p_2\mathfrak{n}(0,\sigma_2^2)} d u \right\}^\frac{1}{2} } \label{eq:deltaprior}$$ where we have used the proof of lemma \[lem:meansd-prior\] and a change of variable $(t-\delta)=u$ in the second integral. As $\delta \rightarrow \pm \infty$ this quantity is constant with respect to $\delta$. Therefore the integral is convergent for $n \geq 2$. Unfortunately this result can not be extended to the general case of $k$ components. When $k>2$, the posterior distribution derived from the Jeffreys prior when only the means are unknown is improper. \[lem:meankcomp-post\] In the case of $k\neq 2$ components, the Jeffreys prior for the location parameters is still constant with respect to a reference mean (for example, $\mu_1$). Therefore it depends on the difference parameters $(\delta_2=\mu_2-\mu_1,\delta_3=\mu_3-\mu_1,\cdots,\delta_k=\mu_k-\mu_1)$. The Jeffreys prior will be bounded by the product on the diagonal, which is an extension of Equation : $$\begin{aligned} \pi^J(\delta_2,\cdots,\delta_k) &\leq c \left\{ \bigintsss_{-\infty}^\infty \frac{[t\mathfrak{n}(0,\sigma_1^2)]^2}{p_1\mathfrak{n}(0,\sigma_1^2)+\cdots+p_k \mathfrak{n}(\delta_k,\sigma_k^2)}d t \right. \nonumber \\ & \left. {} \cdots \bigintsss_{-\infty}^\infty \frac{[u \mathfrak{n}(0,\sigma_k^2)]^2}{p_1\mathfrak{n}(-\delta_k,\sigma_1^2)+\cdots+p_k \mathfrak{n}(0,\sigma_k^2)} d u \right\}^\frac{1}{2}.\end{aligned}$$ If we consider the case as in Lemma \[lem:mean-post\], where only the part of the likelihood depending on e.g. $\mu_2$ may be considered, the convergence of the following integral has to be studied: $$\int_\mathbb{R} \cdots \int_\mathbb{R} e^{-\frac{n-1}{2}\delta_2^2} \pi^J(\delta_2,\cdots,\delta_k) d \delta_2 \cdots d \delta_k$$ In this case, however, the integral with respect to $\delta_2$ may converge, nevertheless the integrals with respect to $\delta_j$ with $j\neq 2$ will diverge, since the prior tends to be constant for each $\delta_j$ as $|\delta_j| \rightarrow \infty$. This results confirms the idea that each part of the likelihood gives information about at most the difference between the location of the respective components and the reference locations, but not on the locations of the other components. [|cccc|ccc|]{} & & & & & &\ & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=2: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior (on the left) and a prior constant on the means (on the right).[]{data-label="tab:post2means"} \ & 0.2505 & 0.88 & 1.8182 & 0.2709 & 0.72 & 1.9968\ & 0.2656 & 0.94 & 1.6804 & 0.2782 & 0.58 & 1.9613\ & 0.2986 & 0.56 & 1.3097 & 0.2812 & 0.18 & 1.9824\ & 0.2879 & 0.48 & 1.2918 & 0.2830 & 0.14 & 1.8358\ & 0.3066 & 0.16 & 1.1251 & 0.3090 & 0.00 & 1.9363\ & 0.3052 & 0.24 & 1.1205 & 0.3103 & 0.02 & 1.7994\ & 0.3181 & 0.02 & 1.0149 & 0.3521 & 0.00 & 1.3923\ & 0.3101 & 0.02 & 1.0244 & 0.3369 & 0.00 & 1.5219\ & 0.3460 & 0.00 & 0.9914 & 0.3627 & 0.00 & 1.2933\ & 0.3418 & 0.00 & 1.0097 & 0.3913 & 0.00 & 1.1970\ & 0.3881 & 0.00 & 0.9948 & 0.4097 & 0.00 & 1.1032\ & 0.4556 & 0.00 & 1.0005 & 0.4515 & 0.00 & 1.0303\ & 0.5090 & 0.00 & 1.0008 & 0.5090 & 0.00 & 1.0007\ & 0.5603 & 0.00 & 1.0006 & 0.5305 & 0.00 & 1.0002\ & 0.4915 & 0.00 & 1.0006 & 0.2327 & 0.00 & 1.0042\ & & & & & &\ & 0.2752 & 0.00 & 1.0838 & 0.2736 & 0.00 & 1.0474\ & 0.2692 & 0.00 & 1.0313 & 0.2546 & 0.00 & 1.0313\ & 0.2969 & 0.00 & 1.1385 & 0.3152 & 0.00 & 1.0167\ & 0.2938 & 0.00 & 1.0138 & 0.2920 & 0.00 & 0.9968\ & 0.3066 & 0.00 & 1.2207 & 0.3470 & 0.00 & 0.9975\ & 0.3350 & 0.00 & 1.1055 & 0.3473 & 0.00 & 0.9920\ & 0.3154 & 0.00 & 1.1374 & 0.3583 & 0.00 & 1.0092\ & 0.3309 & 0.00 & 1.1566 & 0.3512 & 0.00 & 0.9893\ & 0.3338 & 0.00 & 1.1820 & 0.3601 & 0.00 & 1.0112\ & 0.3579 & 0.00 & 1.1796 & 0.3840 & 0.00 & 1.0136\ & 0.3950 & 0.00 & 1.1615 & 0.4190 & 0.00 & 1.0096\ & 0.4879 & 0.00 & 1.1682 & 0.4659 & 0.00 & 1.0059\ & 0.5083 & 0.00 & 1.2123 & 0.4957 & 0.00 & 1.0017\ & 0.5570 & 0.00 & 1.1996 & 0.4777 & 0.00 & 0.9976\ & 0.3463 & 0.00 & 1.2161 & 0.1792 & 0.00 & 1.0010\ [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} & ** ---------------- Chains towards high values ---------------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\mu$ unknown, k=3: as in Table \[tab:post2means\] for two three-component Gaussian mixture models, with close and far means, only for the Jeffreys prior.[]{data-label="tab:post3means"} \ & 0.2366 & 1.00 & 2.5175\ & 0.2608 & 1.00 & 2.8447\ & 0.2455 & 0.98 & 1.3749\ & 0.2446 & 1.00 & 1.3807\ & 0.2330 & 1.00 & 1.4062\ & 0.2480 & 0.98 & 1.2411\ & 0.2684 & 0.94 & 1.2535\ & 0.2784 & 0.98 & 1.2744\ & 0.2904 & 0.68 & 1.1168\ & 0.3214 & 0.74 & 1.1217\ & 0.3819 & 0.32 & 1.0616\ & 0.3774 & 0.10 & 1.0383\ & 0.4407 & 0.04 & 1.0108\ & 0.4935 & 0.00 & 1.0018\ & 0.5577 & 0.00 & 1.0068\ & 0.5511 & 0.00 & 1.0006\ & & &\ & 0.2641 & 1.00 & 2.1786\ & 0.2804 & 1.00 & 2.1039\ & 0.2813 & 0.82 & 1.1173\ & 0.2840 & 0.84 & 1.0412\ & 0.2887 & 0.84 & 1.1050\ & 0.2865 & 0.82 & 1.0840\ & 0.3248 & 0.66 & 1.0982\ & 0.3277 & 0.76 & 1.1177\ & 0.2998 & 0.00 & 1.2604\ & 0.3038 & 0.00 & 1.3149\ & 0.2869 & 0.00 & 1.3533\ & 0.3762 & 0.00 & 1.2479\ & 0.4283 & 0.00 & 1.3791\ & 0.5251 & 0.00 & 1.2585\ & 0.5762 & 0.00 & 1.4779\ & 0.4751 & 0.00 & 1.2161\ ### Scale parameters unknown The posterior distribution derived from the Jeffreys prior when only the standard deviations are unknown is improper. \[lem:sd-post\] Consider equation generalized to the case of $\sigma_1$ and $\sigma_2$ unknown: then when we integrate the posterior distribution with respect to $\sigma_1$ and $\sigma_2$, the complete integral may be split into several integrals then summed up. In particular, if we consider the first part of the likelihood (which only depends on the first component of the mixture) and use the change of variable used in , we have: $$\begin{aligned} \bigintsss_0^{\infty} \bigintsss_0^{\infty} & c \frac{p_1^n}{\tau^n}\frac{p_1 p_2}{\tau\sigma} \exp \left\{-\frac{1}{2\tau^2}\sum_{i=1}^n(x_i-\mu_1)^2\right\} \nonumber \\ & \times{} \left\{ \bigintsss_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\times \left.{} \bigintsss_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}} d u \right\}^\frac{1}{2} d \tau d \sigma\end{aligned}$$ The integral with respect to $\tau$ in the previous equation converges, nevertheless the likelihood does not provide information for $\sigma$, then the integral with respect to $\sigma$ diverges and the posterior will be improper. This results may be easily extented to the case of $k$ components: there is a part of the likelihood which only depends on the global scale parameter and is not informative for the ay other components; the form of the integral will remain the same, with integrations with respect to $\sigma_1,\sigma_2,\cdots,\sigma_k$ which do not converge. When only the standard deviations are unknown, the Jeffreys prior is concentrated around $0$. Nevertheless, the posterior distribution shown in Figures \[fig:sd-priorpost-farm\] turns out to be concentrated around the true values of the parameters for a sufficient high sample size (in the figures, $n$ is always equal to $100$). ![Approximations (on a grid of values) of the Jeffreys prior (on the log-scale) when only the standard deviations of a Gaussian mixture model with 2 components are unknown (on the top) and of the derived posterior distribution (with known weights both equal to $0.5$ and known means equal to $(-5,5)$). The blue cross represents the maximum likelihood estimates.[]{data-label="fig:sd-priorpost-farm"}](lsd-priorpost2){width="6.5cm" height="7.5cm"} Figures \[fig:sd-priorpost-clm\] and \[fig:sd-priorpost-asym\] show the prior and the posterior distributions of the scale parameters of a two-component mixture model for some situations with different weights and different means. ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-1,1)$.[]{data-label="fig:sd-priorpost-clm"}](lsd-priorpost-clm2){width="6.5cm" height="7.5cm"} ![Same as Figure \[fig:sd-priorpost-farm\] but with known weights equal to $(0.25,0.75)$ and known means equal to $(-2,7)$.[]{data-label="fig:sd-priorpost-asym"}](lsd-priorpost-farm2){width="6.5cm" height="7.5cm"} Summarized results of the posterior approximation obtained via a random-walk Metropolis-Hastings algorithm by exploring the posterior distribution associated with the Jeffreys prior on the standard deviations are shown in Figures \[fig:sd2-bxp\] and \[fig:sd3-bxp\], which display boxplots of the posterior means: provided a sufficiently high sample size, simulations exhibit a convergent behavior. ![Boxplots of posterior means of the standard deviations of the two-component mixture model $0.50\mathcal{N}(-1,1) + 0.50\mathcal{N}(2,0.5)$ for 50 replications of the experiment and a sample size equal to $10$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd2-bxp"}](sd2-boxplot){width="6.5cm" height="7.5cm"} ![Boxplots of posterior means of the standard deviations of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment and a sample size equal to $50$, obtained via MCMC with $10^5$ simulations. The red cross represents the true values.[]{data-label="fig:sd3-bxp"}](sd3-boxplot){width="6.5cm" height="7.5cm"} Repeated simulations show that, for a Gaussian mixture model with two components, a sample size equal to $10$ is necessary to have convergent results, while for a three-component Gaussian mixture model with a sample size equal to $50$ is still possible to have chains stuck to values of standard deviations close to $0$. Table \[tab:post2sd\] and \[tab:post3sd\] show results for repeated simulations in the cases of two-component and three-component Gaussian mixture models with unknown standard deviations, respectively, where the means that generate the data may be close or far from one another. In Table \[tab:post2sd\] it seems that the chains tend to be convergent for sample sizes smaller than $10$, but in Table \[tab:post3sd\] one may see that even with a high sample size (equal to $50$) it may happens, for $k=3$, that the chains are stuck to very small values of standard deviations and this fact confirms what we have proved with Lemma \[lem:sd-post\]. [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\sigma$ unknown, $k=2$: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:post2sd"} \ & 0.2414 & 0.02 & 1.2245\ & 0.1875 & 0.02 & 1.1976\ & 0.2403 & 0.00 & 1.0720\ & 0.2233 & 0.02 & 1.1269\ & 0.2475 & 0.00 & 1.0553\ & 0.2494 & 0.02 & 1.0324\ & 0.2465 & 0.00 & 1.0093\ & 0.2449 & 0.00 & 1.0026\ & 0.2476 & 0.00 & 0.9960\ & 0.2541 & 0.00 & 0.9959\ & 0.2480 & 0.00 & 0.9946\ & 0.2364 & 0.00 & 1.0052\ & 0.2510 & 0.00 & 0.9981\ & 0.3033 & 0.00 & 0.9994\ & 0.4314 & 0.00 & 0.9999\ & 0.4353 & 0.00 & 1.0001\ & & &\ & 0.2262 & 0.14 & 1.09202\ & 0.2384 & 0.10 & 1.0536\ & 0.2542 & 0.02 & 1.0281\ & 0.2502 & 0.04 & 0.9932\ & 0.2550 & 0.00 & 0.9981\ & 0.2554 & 0.00 & 0.9569\ & 0.2473 & 0.00 & 0.9929\ & 0.2481 & 0.00 & 0.9888\ & 0.2402 & 0.00 & 0.9969\ & 0.2431 & 0.00 & 0.9988\ & 0.2416 & 0.00 & 0.9998\ & 0.2453 & 0.04 & 1.0016\ & 0.2550 & 0.00 & 0.9992\ & 0.2359 & 0.00 & 0.9999\ & 0.3000 & 0.00 & 1.0001\ & 0.3345 & 0.00 & 1.0000\ [|cccc|]{} & & &\ & ** --------- Ave. Accept. Rate --------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : $\sigma$ unknown, $k=3$: as in table \[tab:post2sd\] for two three-components Gaussian mixture models, with close and far means.[]{data-label="tab:post3sd"} \ & 0.0441 & 0.88 & 0.1206\ & 0.0659 & 0.72 & 1.0638\ & 0.0621 & 0.70 & 1.1061\ & 0.1013 & 0.54 & 1.0655\ & 0.0781 & 0.52 & 1.0880\ & 0.0729 & 0.60 & 1.1003\ & 0.1506 & 0.26 & 1.0516\ & 0.1689 & 0.18 & 1.0493\ & 0.2322 & 0.10 & 1.0478\ & 0.2366 & 0.00 & 1.0125\ & 0.4407 & 0.02 & 1.0061\ & 0.2666 & 0.00 & 1.0021\ & 0.3871 & 0.00 & 1.0003\ & 0.4353 & 0.00 & 1.0001\ & & &\ & 0.0222 & 0.78 & 1.0045\ & 0.0610 & 0.44 & 1.0427\ & 0.0567 & 0.52 & 1.0317\ & 0.0779 & 0.46 & 1.0147\ & 0.0862 & 0.32 & 1.0244\ & 0.1312 & 0.26 & 1.0027\ & 0.1472 & 0.18 & 1.0350\ & 0.15884 & 0.14 & 1.0170\ & 0.2331 & 0.06 & 1.0092\ & 0.2464 & 0.04 & 1.0062\ & 0.2498 & 0.00 & 1.0017\ & 0.2567 & 0.00 & 1.0008\ & 0.2594 & 0.00 & 0.9999\ & 0.3073 & 0.00 & 1.2161\ ### Location and weight parameters unknown. Figure \[fig:MW-bxp\] shows the boxplots of repeated simulations when both the weights and the means are unknown. It is evident that the posterior chains are concentrated around the true values, neverthless some chains (the 14% of the replications) show a drift to very high values (in absolute value) and this behavior suggests improperness of the posterior distribution. ![Boxplots of posterior means of the weigths and the means of the three-component mixture model $0.25\mathcal{N}(-1,1) + 0.65\mathcal{N}(0,0.5) + 0.10\mathcal{N}(2,5)$ for 50 replications of the experiment, obtained via MCMC with $10^5$ simulations. The red cross represents the true value.[]{data-label="fig:MW-bxp"}](MW-boxplot){width="6.5cm" height="7.5cm"} ### All the parameters unknown {#sub:post} Improperness of the prior does not imply improperness of the posterior, obviously, but requires a careful checking of whether or not the posterior is proper, however the proof of Lemma \[lem:sd-post\] gives an hint about the actual properness of the posterior distribution when all the parameters are unknown. The posterior distribution derived from the Jeffreys prior when all the parameters are unknown is improper. \[lem:all-post\] Consider the elements on the diagonal of the Fisher information matrix; again, since the Fisher information matrix is positive definite, the determinant is bounded by the product of the terms in the diagonal. Consider a reparametrization into $\tau=\sigma_1$ and $\tau\sigma=\sigma_2$. Then it is straightforward to see that the integral of this part of the prior distribution will depend on a term $(\tau)^{-(d+1)}(\sigma)^{-d}$. Again, as in the proof of Lemma \[lem:sd-post\], when composing the prior with the part of the likelihood which only depends on the first component, this part does not provide information about the parameters $\sigma$ and the integral will diverge. In particular, the integral of the first part of the posterior distribution relative to the part of the likelihood dependent on the first component only and on the product of the diagonal terms of the Fisher information matrix for the prior when considering a two-component mixture model is $$\begin{aligned} \int_0^1 & \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \int_0^{\infty} \int_0^{\infty} c \frac{p_1^n}{\tau^n}\frac{p_1^2 p_2^2}{\tau^3\sigma^2} \exp \left\{-\frac{1}{2\tau^2}\sum_{i=1}^n(x_i-\mu_1)^2\right\} \nonumber \\ & \times{} \left\{ \int_{-\infty}^{\infty} \frac{\left[ \sigma\exp\left\{-\frac{(\tau\sigma y + \delta)^2}{2\tau^2} \right\} - \exp\left\{-\frac{y^2}{2} \right\}\right]^2}{p_1 \sigma \exp\left\{-\frac{(\tau \sigma y + \delta)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{y^2}{2}\right\}} d y \right. \nonumber \\ & \times \left.{} \int_{-\infty}^{\infty} \frac{z^2 \exp(-z^2)}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau-\delta)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ & \times \left.{} \int_{-\infty}^\infty \frac{w^2 \exp\left\{ -w^2\right\}}{p_1 \sigma \exp\left\{-\frac{(\tau \sigma w+\delta)^2}{2\tau^2\sigma^2}\right\}+p_2\exp\left\{-\frac{w^2}{2}\right\}} d w \right. \nonumber \\ & \times \left.{} \int_{-\infty}^\infty \frac{\left(z^2-1\right)^2 \exp\left\{ -z^2\right\}}{p_1 \exp\left\{-\frac{z^2}{2}\right\}+\frac{p_2}{\sigma}\exp\left\{-\frac{(z\tau+\mu_1-\mu_2)^2}{2\tau^2\sigma^2}\right\}} d z \right. \nonumber \\ &\times \left.{} \int_{-\infty}^\infty \frac{\left(u^2-1\right)^2 \exp\left\{-u^2\right\}}{p_1\sigma \exp\left\{-\frac{(u\tau\sigma+\mu_2-\mu_1)^2}{2\tau^2}\right\}+p_2\exp\left\{-\frac{u^2}{2}\right\}} d u \right\}^\frac{1}{2} d \tau d \sigma d \mu_1 d \mu_2 d p_1 \nonumber\end{aligned}$$ When considering the integrals relative to the Jeffreys prior, they do not represent an issue for convergence with respect to the scale parameters, because exponential terms going to $0$ as the scale parameters tend to $0$ are present. However, when considering the part out of the previous integrals, a factor $\sigma^-2$ whose behavior is not convergent is present. Then this particular part of the posterior distribution is not integrating. When considering the case of $k$ components, the integral will always inversily depends on $\sigma_1, \sigma_2,\cdots, \sigma_{k-1}$ and then the posterior will always be improper. As a note aside, it is worth noting that the usual separation between parameters proposed by Jeffreys himself in the multidimensional problems does not change the behavior of the posterior, because even if the Fisher information matrix is decomposed as $$I(\theta)=\left( \begin{matrix} I_1(\theta_1) & 0 \\ 0 & I_2(\theta_2) \end{matrix} \right)$$ for any possible combination of the parameters $\theta=(p,\mu_1,\mu_2,\sigma_1,\sigma_2)$ (note that $\theta_1$ and $\theta_2$ are vectors and $I(\theta_1)$ and $I(\theta_2)$ are diagonal or non-diagonal matrices), the product of the elements in the diagonal (considered in the proof) will be the same. A comparison with maximum likelihood estimation obtained via EM has shown that the Bayesian estimates obtained via MCMC and by using a Jeffreys prior seems to better identify the true values which have generated the data for a sufficient high sample size. Table \[tab:EMvsBayes\] shows the comparison between the ML and the Bayesian estimates (for repeated simulations, the initial values for the MCMC algorithm have been randomly chosen to have a sufficiently high likelihood level). The log-likelihood value of the ML estimates is always lower that the log-likelihood value of the Bayesian estimates. The better performance of the Bayesian algorithm is only shown for practical reasons, since we have already proved the posterior distribution is improper. Figure \[priorlik\] shows that the MCMC algorithm accepts moves with an increasing likelihood value, until this value stabilizes around $-210$. The same happens for the prior level. **Parameters** **ML** **Bayes** **True** ---------------- ------------------------ ------------------------ ------------------ -- -- $\mu$ (-7.245,13.308,14.999) (-10.003,0.307,14.955) (-10,0,15) $\sigma$ (0.547,5.028,0.154) (1.243,3.642,0.607) (1.0,5.0,0.5) w (0.350,0.016,0.634) (0.258,0.106,0.636) (0.25,0.10,0.65) : Comparison between ML estimates and Bayesian estimates obtained by using a Jeffreys prior for a 3-components Gaussian mixture model.[]{data-label="tab:EMvsBayes"} ![Values of (Jeffreys) prior (above) and likelihood function (below) for the accepted moves of the MCMC algorithm which estimate the posterior distribution of the parameters of a 3-components Gaussian mixture model and a sample size equal to 1000.[]{data-label="priorlik"}](wmeansd1-priorlik){width="6.5cm" height="7.5cm"} For small sample sizes, the chains tend to get stuck when very small values of standard deviations are accepted. Table \[tab:allunkn2\] and \[tab:allunkn3\] show the results for different sample sizes and different scenarios (in particular, the situations when the means are close or far from each other are considered) for a mixture model with two and three components respectively. The second and the third columns show the reason why the chain goes into trouble: sometimes the chains do not converge and tend towards very high values of means, sometimes the chains get stuck to very small values of standard deviations. [|ccccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=2, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: results of 50 replications of the experiment for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior.[]{data-label="tab:allunkn2"} \ & 0.1119 & 0.54 & 0.74 & 3.5280\ & 0.1241 & 0.56 & 0.74 & 3.6402\ & 0.0927 & 0.56 & 0.70 & 3.2180\ & 0.0693 & 0.54 & 0.70 & 3.1380\ & 0.1236 & 0.42 & 0.72 & 3.3281\ & 0.1081 & 0.44 & 0.84 & 2.8173\ & 0.1172 & 0.40 & 0.78 & 2.1455\ & 0.1107 & 0.40 & 0.70 & 1.8998\ & 0.1273 & 0.44 & 0.74 & 1.8269\ & 0.1253 & 0.42 & 0.76 & 1.2876\ & 0.1218 & 0.36 & 0.82 & 1.2949\ & 0.1278 & 0.38 & 0.66 & 1.2587\ & & & &\ & 0.1650 & 0.18 & 0.30 & 3.7712\ & 0.2218 & 0.12 & 0.20 & 3.1400\ & 0.1836 & 0.12 & 0.36 & 3.1461\ & 0.2313 & 0.08 & 0.08 & 3.5102\ & 0.1942 & 0.14 & 0.12 & 3.5585\ & 0.2290 & 0.04 & 0.02 & 3.0718\ & 0.2320 & 0.04 & 0.02 & 2.9825\ & 0.2305 & 0.08 & 0.02 & 2.9122\ & 0.2264 & 0.06 & 0.00 & 2.9571\ & 0.2292 & 0.08 & 0.04 & 1.0612\ & 0.2005 & 0.12 & 0.04 & 1.0804\ & 0.2343 & 0.00 & 0.02 & 1.0146\ [|ccccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=3, ($\mathbf{p}$, $\mu$, $\sigma$) unknown: as in table \[tab:allunkn2\] for two three-component Gaussian mixture models with close and far means.[]{data-label="tab:allunkn3"} \ & 0.0302 & 0.76 & 0.44 & 2.9095\ & 0.0368 & 0.76 & 0.48 & 3.2507\ & 0.0290 & 0.80 & 0.30 & 3.1318\ & 0.0578 & 0.62 & 0.54 & 3.0043\ & 0.0488 & 0.74 & 0.52 & 2.5798\ & 0.0426 & 0.70 & 0.44 & 2.3023\ & 0.0572 & 0.66 & 0.38 & 1.7497\ & 0.0464 & 0.66 & 0.48 & 1.4032\ & 0.0706 & 0.52 & 0.44 & 1.9303\ & 0.0556 & 0.66 & 0.36 & 1.3588\ & 0.0610 & 0.74 & 0.44 & 1.3588\ & 0.0654 & 0.48 & 0.46 & 1.2161\ & & & &\ & 0.0644 & 0.60 & 0.10 & 5.9707\ & 0.0631 & 0.64 & 0.18 & 2.0557\ & 0.0726 & 0.54 & 0.08 & 2.9351\ & 0.1745 & 0.22 & 0.12 & 2.9193\ & 0.1809 & 0.32 & 0.04 & 95.793\ & 0.1724 & 0.28 & 0.14 & 2.5938\ & 0.1948 & 0.24 & 0.14 & 3.1566\ & 0.1718 & 0.26 & 0.08 & 2.8595\ & 0.2110 & 0.16 & 0.06 & 1.8595\ & 0.1880 & 0.24 & 0.10 & 1.2165\ & 0.1895 & 0.20 & 0.12 & 1.2133\ & 0.2468 & 0.08 & 0.02 & 1.0146\ Since the improperness of the posterior distribution is due to the scale parameters, we may use a reparametrization of the problem as in Equation and use a proper prior on the parameter $\sigma$, for example, by following [@robert:mengersen:1999] $$p(\sigma)=\frac{1}{2}\mathcal{U}_{[0,1]}(\sigma)+\frac{1}{2}\frac{1}{\mathcal{U}_{[0,1]}(\sigma)}.$$ and the Jeffreys prior for all the other parameters $(\mathbf{p},\mu,\delta,\tau)$ conditionally on $\sigma$. Since the improperness of the posterior distribution is due to the scale parameters, we may use a reparametrization of the problem as in Equation and use a proper prior on the parameter $\sigma$, for example, by following [@robert:mengersen:1999] $$p(\sigma)=\frac{1}{2}\mathcal{U}_{[0,1]}(\sigma)+\frac{1}{2}\frac{1}{\mathcal{U}_{[0,1]}(\sigma)}.$$ and the Jeffreys prior for all the other parameters $(\mathbf{p},\mu,\delta,\tau)$ conditionally on $\sigma$. Actually, using a proper prior on $\sigma$ does not avoid convergence trouble, as demonstrated by Table \[tab:sigmaprop2\], which shows that, even if the chains with respect to the standard deviations are not stuck around $0$ when using a proper prior for $\sigma$ in the reparametrization proposed by [@robert:mengersen:1999], the chains with respect to the locations parameters demonstrate a divergent behavior. [|c|cccc|]{} & & & &\ & ** --------- Ave. Accept. Rate --------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} & ** ---------------------- Ave. lik($\theta^{fin}$) / lik($\theta^{true}$) ---------------------- : k=2, ($\mathbf{p}$, $\mu$, $\delta$, $\tau$,$\sigma$) unknown, proper prior on $\sigma$: results of 50 replications of the experiment by using a proper prior on $\sigma$ and the Jeffreys prior for the other parameters conditionally on it for both close and far means with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0 and the average ratio between the log-likelihood of the last accepted values and the true values in the 50 replications when using the Jeffreys prior. []{data-label="tab:sigmaprop2"} \ 5 & 0.2094 & 0.02 & 0.92 & 1.4440\ 6 & 0.2152 & 0.00 & 0.98 & 1.3486\ 7 & 0.2253 & 0.00 & 0.92 & 1.3290\ 8 & 0.2021 & 0.00 & 0.94 & 1.2258\ 9 & 0.1828 & 0.00 & 0.84 & 1.2666\ 10 & 0.2087 & 0.00 & 0.88 & 1.1770\ 11 & 0.1854 & 0.00 & 0.94 & 1.2088\ 12 & 0.1829 & 0.00 & 0.86 & 1.2153\ 13 & 0.1658 & 0.00 & 0.92 & 1.1682\ 14 & 0.2017 & 0.00 & 0.86 & 1.2043\ 15 & 0.1991 & 0.00 & 0.88 & 1.2002\ 20 & 0.1851 & 0.00 & 0.76 & 1.1688\ & & & &\ 5 & 0.2071 & 0.00 & 0.70 & 1.5741\ 6 & 0.2021 & 0.00 & 0.68 & 1.4384\ 7 & 0.1947 & 0.00 & 0.60 & 1.3597\ 8 & 0.2054 & 0.00 & 0.44 & 1.2869\ 9 & 0.2093 & 0.00 & 0.46 & 1.3064\ 10 & 0.2271 & 0.00 & 0.20 & 1.1618\ 11 & 0.2030 & 0.00 & 0.32 & 1.1996\ 12 & 0.2178 & 0.00 & 0.24 & 1.1494\ 13 & 0.2812 & 0.00 & 0.18 & 1.1215\ 14 & 0.1880 & 0.00 & 0.08 & 1.0717\ 15 & 0.2511 & 0.00 & 0.06 & 1.0594\ 20 & 0.2359 & 0.00 & 0.00 & 1.0166\ A noninformative alternative to Jeffreys prior {#sec:alternative} ============================================== The information brought by the Jeffreys prior does not seem to be enough to conduct inference in the case of mixture models. The computation of the determinant creates a dependence between the elements of the Fisher information matrix in the definition of the prior distribution which makes it difficult to find slight modifications of this prior that would lead to a proper posterior distribution. For example, using a proper prior for part of the scale parameters and the Jeffreys prior conditionally on them does not avoid impropriety, as we have demonstrated in Section \[sub:post\]. The literature covers attempts to define priors which add a small amount of information that is sufficient to conduct the statistical analysis without overwhelming the information contained in the data. Some of these are related to the computational issues in estimating the parameters of mixture models, as in the approach of [@casella:mengersen:robert:titterington:2002], who find a way to use perfect slice sampler by focusing on components in the exponential family and conjugate priors. A characteristic example is given by [@richardson:green:1997], who propose weakly informative priors, which are data-dependent (or empirical Bayes) and are represented by flat normal priors over an interval corresponding to the range of the data. Nevertheless, since mixture models belong to the class of ill-posed problems, the influence of a proper prior over the resulting inference is difficult to assess. Another solution found in [@mengersen:robert:1996] proceeds through the reparametrization and introduces a reference component that allows for improper priors. This approach then envisions the other parameters as departures from the reference and ties them together by considering each parameter $\theta_i$ as a perturbation of the parameter of the previous component $\theta_{i-1}$. This perspective is justified by the fact that the $(i-1)$-th component is not informative enough to absorb all the variability in the data. For instance, a three-component mixture model gets rewritten as $$\begin{aligned} p\mathcal{N}(\mu,\tau^2)&+(1-p)q\mathcal{N}(\mu+\tau\theta,\tau^2\sigma_1^2) \\ &\quad {} + (1-p)(1-q)\mathcal{N}(\mu+\tau\theta+\tau\sigma\epsilon,\tau^2\sigma_1^2\sigma_2^2)\end{aligned}$$ where one can impose the constraint $1 \geq \sigma_1 \geq \sigma_2$ for identifiability reasons. Under this representation, it is possible to use an improper prior on the global location-scale parameter $(\mu,\tau)$, while proper priors must be applied to the remaining parameters. This reparametrization has been used also for exponential components by [@gruet:philippe:robert:1999] and Poisson components by [@robert:titterington:1998]. Moreover, [@roeder:wasserman:1997] propose a Markov prior which follows the same resoning of dependence between the parameters for Gaussian components, where each parameter is again a perturbation of the parameter of the previous component $\theta_{i-1}$. This representation suggests to define a global location-scale parameter in a more implicit way, via a hierarchical model that considers more levels in the analysis and choose noninformative priors at the last level in the hierarchy. More precisely, consider the Gaussian mixture model $$\label{eq:hierarc1} g(x|\boldsymbol{\theta})=\sum_{i=1}^K p_i \mathfrak{n}(x|\mu_i,\sigma_i).$$ The parameters of each component may be considered as related in some way; for example, the observations have a reasonable range, which makes it highly improbable to face very different means in the above Gaussian mixture model. A similar argument may be used for the standard deviations. Therefore, at the second level of the hierarchical model, we may write $$\begin{aligned} \label{eq:hierarc2} \mu_i & \stackrel{iid}{\sim} \mathcal{N}(\mu_0, \zeta_0) \nonumber \\ \sigma_i & \stackrel{iid}{\sim} \frac{1}{2} \mathcal{U}(0,\zeta_0) + \frac{1}{2}\frac{1}{\mathcal{U}(0,\zeta_0)} \nonumber \\ \mathbf{p} & \sim Dir\left(\frac{1}{2},\cdots,\frac{1}{2}\right) \end{aligned}$$ which indicates that the location parameters vary between components, but are likely to be close, and that the scale parameters may be lower or bigger than $\zeta_0$, but not exactly equal to $\zeta_0$. The weights are given a Dirichlet prior (or in the case of just two components, a Beta prior) independently from the components’ parameters. At the third level of the hierarchical model, the prior may be noninformative: $$\begin{aligned} \label{eq:hierarc3} \pi(\mu_0,\zeta_0) \propto \frac{1}{\zeta_0}\end{aligned}$$ As in @mengersen:robert:1996 the parameters in the mixture model are considered tied together; on the other hand, this feature is not obtained via a representation of the mixture model itself, but via a hierarchy in the definition of the model and the parameters. The posterior distribution derived from the hierarchical representation of the Gaussian mixture model associated with , and is proper. Consider the composition of the three levels of the hierarchical model described in equations , and : $$\begin{aligned} \label{eq:hierarch_post} \pi(\boldsymbol{\mu},\boldsymbol{\sigma},\mu_0,\zeta_0;\mathbf{x}) & \propto L(\mu_1,\mu_2,\sigma_1,\sigma_2;\mathbf{x}) p^{-1/2} (1-p)^{-1/2} \nonumber \\ & {} \times \frac{1}{\zeta_0} \frac{1}{2\pi\zeta_0^2} \exp\left\{- \frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2}\right\} \nonumber \\ & {} \times \left[ \frac{1}{2}\frac{1}{\zeta_0} \mathbb{I}_{[\sigma_1\in(0,\zeta_0)]}(\sigma_1) + \frac{1}{2}\frac{\zeta_0}{\sigma_1^2} \mathbb{I}_{[\sigma_1\in(\zeta_0,+\infty)]}(\sigma_1) \right] \nonumber \\ & {} \times \left[ \frac{1}{2}\frac{1}{\zeta_0} \mathbb{I}_{[\sigma_2\in(0,\zeta_0)]}(\sigma_2) + \frac{1}{2}\frac{\zeta_0}{\sigma_2^2} \mathbb{I}_{[\sigma_2\in(\zeta_0,+\infty)]}(\sigma_2) \right]\end{aligned}$$ where $L(\cdot;\mathbf{x})$ is given by Equation . Once again, we can initialize the proof by considering only the first term in the sum composing the likelihood function for the mixture model. Then the product in may be split into four terms corresponding to the different terms in the scale parameters’ prior. For instance, the first term is $$\begin{aligned} \int_0^\infty & \int_{-\infty}^\infty \int_\mathbb{R}\int_\mathbb{R} \int_\mathbb{R^+} \int_\mathbb{R^+} \int_0^1 \frac{1}{\sigma_1^n} p_1^n \exp \left\{- \frac{\sum_{i=1}^n (x_i-\mu_1)^2}{2\sigma_1^2} \right\} \nonumber \\ & {} \times \frac{1}{\zeta_0^3} \exp\left\{-\frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2} \right\} \nonumber \\ & {} \times \frac{1}{4}\frac{1}{\zeta_0} \frac{1}{\zeta_0} \mathbb{I}_{[\sigma_1 \in (0,\zeta_0)]}(\sigma_1) \mathbb{I}_{[\sigma_2 \in (0,\zeta_0)]}(\sigma_2) d p d\sigma_1 d\sigma_2 d\mu_1 d\mu_2 d \mu_0 d\zeta_0 \end{aligned}$$ and the second one $$\begin{aligned} \int_0^\infty & \int_{-\infty}^\infty \int_\mathbb{R}\int_\mathbb{R} \int_\mathbb{R^+} \int_\mathbb{R^+} \int_0^1 \frac{1}{\sigma_1^n} p_1^n \exp \left\{- \frac{\sum_{i=1}^n (x_i-\mu_1)^2}{2\sigma_1^2} \right\} \nonumber \\ & {} \times \frac{1}{\zeta_0^3} \exp\left\{-\frac{(\mu_1-\mu_0)^2 (\mu_2-\mu_0)^2}{2\zeta_0^2} \right\} \nonumber \\ & {} \times \frac{1}{4}\frac{1}{\zeta_0} \frac{\zeta_0}{\sigma_2^2} \mathbb{I}_{[\sigma_1 \in (0,\zeta_0)]}(\sigma_1) \mathbb{I}_{[\sigma_2 \in (\zeta_0,\infty)]}(\sigma_2) d p d\sigma_1 d\sigma_2 d\mu_1 d\mu_2 d \mu_0 d\zeta_0 .\end{aligned}$$ The integrals with respect to $\mu_1$, $\mu_2$ and $\mu_0$ converge, since the data are carrying information about $\mu_0$ through $\mu_1$. The integral with respect to $\sigma_1$ converges as well, because, as $\sigma_1 \rightarrow 0$, the exponential function goes to $0$ faster than $\frac{1}{\sigma_1^n}$ goes to $\infty$ (integrals where $\sigma_1>\zeta_0$ are not considered here because this reasoning may easily extend to those cases). The integrals with respect to $\sigma_2$ converge, because they provide a factor proportional to $\zeta_0$ and $1/\zeta_0$ respectively which simplifies with the normalizing constant of the reference distribution (the uniform in the first case and the Pareto in second one). Finally, the term $1/\zeta_0^4$ resulting from the previous operations has its counterpart in the integrals relative to the location priors. Therefore, the integral with respect to $\zeta_0$ converges. The part of the posterior distribution relative to the weights is not an issue, since the weights belong to the corresponding simplex. Table \[tab:hierMM\] shows the results given by simulation from the posterior distribution of the hierarchical mixture model and confirms that the chains always converge. [|c|ccccccc|]{} & & & & & & &\ & ** --------- Ave. Accept. Rate --------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** ----------------- Chains stuck at small values of $\sigma$ ----------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** ---------------- Chains towards high values of $\mu$ ---------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Mean l($\theta^{fin}$)/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Median l($\theta^{fin}$)/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Mean max(l($\theta$))/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} & ** -------------------- Median max(l($\theta$))/ l($\theta^{true}$) -------------------- : Hierarchical Mixture model: results of 50 replications of the experiment for a two and a three-component Gaussian mixture model with a Monte Carlo approximation of the posterior distribution based on $10^5$ simulations and a burn-in of $10^4$ simulations. The table shows the average acceptance rate, the proportion of chains diverging towards higher values, the proportion of chains stuck at values of standard deviations close to 0, the mean and the median log-likelihood of the last accepted values and the mean and the median maximum log-likelihood of the accepted values.[]{data-label="tab:hierMM"} \ 3 & 0.1947 & 0.00 & 0.00 & 1.1034 & 0.9825 & 0.0838 & 0.5778\ 4 & 0.2295 & 0.00 & 0.00 & 1.0318 & 1.0300 & 0.4678 & 0.5685\ 5 & 0.2230 & 0.00 & 0.00 & 0.9572 & 0.9924 & 0.8464 & 0.7456\ 6 & 0.2275 & 0.00 & 0.00 & 0.9870 & 0.9641 & 0.6614 & 0.6696\ 7 & 0.2112 & 0.00 & 0.00 & 1.0658 & 1.0043 & 0.8406 & 0.7848\ 8 & 0.2833 & 0.00 & 0.00 & 1.0077 & 1.0284 & 0.8268 & 0.8495\ 9 & 0.2696 & 0.00 & 0.00 & 1.0741 & 1.0179 & 0.8854 & 0.8613\ 10 & 0.2266 & 0.00 & 0.00 & 1.1446 & 0.9968 & 0.9589 & 0.8508\ 15 & 0.1982 & 0.00 & 0.00 & 1.0201 & 0.9959 & 0.9409 & 0.9280\ 20 & 0.2258 & 0.00 & 0.00 & 1.2023 & 1.0145 & 0.9172 & 0.9400\ 30 & 0.2073 & 0.00 & 0.00 & 0.9888 & 1.0022 & 1.0424 & 0.9656\ 50 & 0.2724 & 0.00 & 0.00 & 1.0493 & 1.0043 & 1.0281 & 0.9859\ 100 & 0.2739 & 0.00 & 0.00 & 1.0932 & 1.0025 & 1.0805 & 0.9932\ 200 & 0.3031 & 0.00 & 0.00 & 1.1610 & 1.0036 & 1.1519 & 0.9964\ 500 & 0.2753 & 0.00 & 0.00 & 1.1729 & 1.0023 & 1.1694 & 0.9989\ 1000 & 0.2317 & 0.00 & 0.00 & 1.1800 & 1.0021 & 1.1772 & 0.9994\ & & & & & & &\ 3 & 0.2840 & 0.00 & 0.00 & 1.1316 & 1.0503 & 0.3432 & 0.2950\ 4 & 0.2217 & 0.00 & 0.00 & 1.0326 & 0.9452 & 0.6699 & 0.6624\ 5 & 0.2144 & 0.00 & 0.00 & 1.0610 & 1.0421 & 0.6858 & 0.6838\ 6 & 0.2258 & 0.00 & 0.00 & 1.0908 & 0.9683 & 0.6472 & 0.6355\ 7 & 0.1843 & 0.00 & 0.00 & 1.0436 & 0.9915 & 0.7878 & 0.8008\ 8 & 0.2760 & 0.00 & 0.00 & 1.0276 & 1.0077 & 0.7996 & 0.7958\ 9 & 0.2028 & 0.00 & 0.00 & 1.0025 & 1.0145 & 0.7830 & 0.8016\ 10 & 0.2116 & 0.00 & 0.00 & 1.0426 & 1.0015 & 0.8752 & 0.8591\ 15 & 0.2023 & 0.00 & 0.00 & 1.0247 & 1.0063 & 0.8810 & 0.8871\ 20 & 0.2211 & 0.00 & 0.00 & 1.0281 & 1.0104 & 0.9290 & 0.9268\ 30 & 0.2242 & 0.00 & 0.00 & 1.1978 & 1.0123 & 1.0841 & 0.9508\ 50 & 0.2513 & 0.00 & 0.00 & 1.0543 & 1.0142 & 1.0148 & 0.9775\ 100 & 0.2768 & 0.00 & 0.00 & 1.0563 & 1.0206 & 1.0324 & 0.9955\ 200 & 0.2910 & 0.00 & 0.00 & 1.0325 & 1.0118 & 1.0200 & 0.9993\ 500 & 0.2329 & 0.00 & 0.00 & 1.0943 & 1.0079 & 1.0882 & 1.0002\ 1000 & 0.2189 & 0.00 & 0.00 & 1.1068 & 1.0105 & 1.1212 & 1.0110\ Figures \[fig:hierc\_densmean2\_3\_8\]–\[fig:hierc\_densmean3\_30\_1000\] show the results how a simulations study to approximate the posterior distribution of the means of a two or three-component mixture model, compared to the true values (red vertical lines) and for different sample sizes, from $n=3$ to $n=1000$. ![Distribution of the posterior means for the hierarchical mixture model with two components, global mean $\mu_0=0$ and global variance $\zeta_0=5$, based on $50$ replications of the experiment with different sample sizes, black and blue lines for the marginal posterior distribution of $\mu_1$ and $\mu_2$ respectively.[]{data-label="fig:hierc_densmean2_3_8"}](densmeans2_3_8) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_9_14"}](densmeans2_9_14) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_15_20"}](densmeans2_15_20) ![Same caption as in Figure \[fig:hierc\_densmean2\_3\_8\].[]{data-label="fig:hierc_densmean2_30_1000"}](densmeans2_30_1000) ![Distribution of the posterior means for the hierarchical mixture model with three components, global mean $\mu_0=0$ and global variance $\zeta_0=5$, based on $50$ replications of the experiment with different sample sizes (the red lines stands for the true values, black, green and blue lines for the marginal posterior distributions of $\mu_1$, $\mu_2$ and $\mu_3$ respectively).[]{data-label="fig:hierc_densmean3_3_8"}](densmeans3_3_8) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_9_14"}](densmeans3_9_14) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_15_20"}](densmeans3_15_20) ![Same caption as in Figure \[fig:hierc\_densmean3\_3\_8\].[]{data-label="fig:hierc_densmean3_30_1000"}](densmeans3_30_1000) Implementation features {#sec:implant} ======================= The computing expense due to derive the Jeffreys prior for a set of parameter values is in $\mathrm{O}(d^2)$ if $d$ is the total number of (independent) parameters. Each element of the Fisher information matrix is an integral of the form $$-\int_{\mathcal{X}} \frac{\partial^2 \log \left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]}{\partial \theta_i \partial \theta_j}\left[\sum_{h=1}^k p_h\,f(x|\theta_h)\right]^{-1} d x$$ which has to be approximated. We have applied both numerical integration and Monte Carlo integration and simulations show that, in general, numerical integration obtained via Gauss-Kronrod quadrature (see [@piessens:1983] for details), has more stable results. Neverthless, when one or more proposed values for the standard deviations or the weights is too small, the approximations tend to be very dependent on the bounds used for numerical integration (usually chosen to omit a negligible part of the density) or the numerical approximation may not be even applicable. In this case, Monte Carlo integration seems to have more stable, where the stability of the results depends on the Monte Carlo sample size. Figure \[fig:MCvsNUM\_incrN\] shows the value of the Jeffreys prior obtained via Monte Carlo integration of the elements of the Fisher information matrix for an increasing number of Monte Carlo simulations both in the case where the Jeffreys prior is concentrated (where the standard deviations are small) and where it assumes low values. The value obtained via Monte Carlo integration is then compared with the value obtained via numerical integration. The sample size relative to the point where the graph stabilizes may be chosen to perform the approximation. A similar analysis is shown in Figures \[fig:MCvsNUM\_bpl1\] and \[fig:MCvsNUM\_bpl2\] which provide the boxplots of $100$ replications of the Monte Carlo approximations for different numbers of simulations (on the *x*-axis); one can choose to use the number of simulations which lead to a reasonable or acceptable variability of the results. ![Jeffreys prior obtained via Monte Carlo integration (and numerical integration, in *red*) for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ (above) and for the model $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$ (below).[]{data-label="fig:MCvsNUM_incrN"}](MCvsNUM-increaN) ![Boxplots of 100 replications of the procedure which approximates the Fisher information matrix via Monte Carlo integration to obtain the Jeffreys prior for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ for sample sizes from $500$ to $3000$. The value obtained via numerical integration is represented by the red line.[]{data-label="fig:MCvsNUM_bpl1"}](MCvsNUM-boxpl1) ![Same caption as in Figure \[fig:MCvsNUM\_bpl1\] for the model $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$.[]{data-label="fig:MCvsNUM_bpl2"}](MCvsNUM-boxpl2) Since the approximation problem is one-dimensional, another numerical solution could be based on the sums of Riemann; Figure \[fig:MCvsNUMSRvsINTR\] shows the comparison between the results of the Gauss-Kronrod quadrature procedure and a procedure based on sums of Riemann for an increasing number of points considered in a region which contain the $99.999\%$ of the data density. Moreover, Figure \[fig:MCvsRIEMbxp\] shows the comparison between the approximation to the Jeffreys prior obtained via Monte Carlo integration and via the sums of Riemann: it is clear that the sums of Riemann lead to more stable results in comparison with Monte Carlo integration. On the other hand, they can be applied in more situations than the Gauss-Kromrod quadrature, in particular, in cases where the standard deviations are very small (of order $10^{-2}$). Nevertheless, when the standard deviations are smaller than this, one has to pay attention on the features of the function to integrate. In fact, the mixture density tends to concentrate around the modes, with regions of density close to 0 between them. The elements of the Fisher informtation matrix are, in general, ratios between the components’ densities and the mixture density, then in those regions an indeterminate form of type $\frac{0}{0}$ is obtained; Figure \[fig:FishInfoelem\] represents the behavior of one of these elements when $\sigma_i \rightarrow 0$ for $i=1,\cdots,k$. ![Comparison between the Jeffreys prior density obtained via integration in the Fisher information matrix via Gauss-Kronrod quadrature and sums of Riemann for the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ (above) and $\frac{1}{3}\mathcal{N}(-1,0.2)+\frac{1}{3}\mathcal{N}(0,0.2)+\frac{1}{3}\mathcal{N}(1,0.2)$ (below).[]{data-label="fig:MCvsNUMSRvsINTR"}](SRvsINTR-increaN) ![Boxplots of 100 replications of the procedure based on Monte Carlo integration (above) and sums of Riemann (below) which approximates the Fisher information matrix of the model $0.25\mathcal{N}(-10,1)+0.10\mathcal{N}(0,5)+0.65\mathcal{N}(15,7)$ for sample sizes from $500$ to $1700$. The value obtained via numerical integration is represented by the red line (in the graph below, all the approximations obtained with more than $550$ knots give the same result, exactly equal to the one obtained via Gauss-Kronrod quadrature).[]{data-label="fig:MCvsRIEMbxp"}](MCvsRIEM-boxpl1) ![The first element on the diagonal of the Fisher information matrix relative to the first weight of the two-component Gaussian mixture model $0.5 \mathcal{N}(-1,0.01)+0.5 \mathcal{N}(2,0.01)$.[]{data-label="fig:FishInfoelem"}](elem) Thus, we have decided to use the sums of Riemann (with a number of points equal to $550$) to approximate the Jeffreys prior when the standard deviations are sufficiently big and Monte Carlo integration (with sample sizes of $1500$) when they are too small. In this case, the variability of the results seems to decrease as $\sigma_i$ approaches $0$, as shown in Figure \[fig:MCsmallsd\]. ![Approximation of the Jeffreys prior (in log-scale) for the two-component Gaussian mixture model $0.5 \mathcal{N}(-1,\sigma)+0.5\mathcal{N}(2,\sigma)$, where $\sigma$ is taken equal for both components and decreasing.[]{data-label="fig:MCsmallsd"}](MCsmallsigma) We have chosen to consider Monte Carlo samples of size equal to $1500$ because both the value of the approximation and its standard deviations are stabilizing. An adaptive MCMC algorithm has been used to define the variability of the kernel density functions used to propose the moves. During the burnin, the variability of the kernel distributions has been reduced or increased depending on the acceptance rate, in a way such that the acceptance rate stay between $20\%$ and $40\%$. The transitional kernel used have been truncated normals for the weights, normals for the means and log-normals for the standard deviations (all centered on the values accepted in the previous iteration). Conclusion {#sec:concl} ========== This thorough analysis of the Jeffreys priors in the setting of Gaussian mixtures shows that mixture distributions can also be considered as an ill-posed problem with regards to the production of non-informative priors. Indeed, we have shown that most configurations for Bayesian inference in this framework do not allow for the standard Jeffreys prior to be taken as a reference. While this is not the first occurrence where Jeffreys priors cannot be used as reference priors, the wide range of applications of mixture distributions weights upon this discovery and calls for a new paradigm in the construction of non-informative Bayesian procedures for mixture inference. 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[^1]: Corresponging Author: Memotef, Sapienza Università di Roma. Via del Castro Laurenziano, 9, 00161, Roma, Italy. CEREMADE Université Paris-Dauphine, Paris, France. e-mail: [email protected] [^2]: CEREMADE Université Paris-Dauphine, University of Warwick and CREST, Paris. e-mail: [email protected].

ArXiv

[**[Old Galaxies in the Young Universe]{}**]{} [**[A. Cimatti$^1$, E. Daddi$^2$, A. Renzini$^2$, P. Cassata$^3$, E. Vanzella$^{3}$, L. Pozzetti$^4$, S. Cristiani$^5$, A. Fontana$^6$, G. Rodighiero$^3$, M. Mignoli$^4$, G. Zamorani$^4$ ]{}**]{}\ $^1$ INAF - Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125, Firenze, Italy\ $^2$ European Southern Observatory, Karl-Schwarzschild-Str. 2, D-85748, Garching, Germany\ $^3$ Dipartimento di Astronomia, Università di Padova, Vicolo dell’Osservatorio, 2, I-35122 Padova, Italy\ $^4$ INAF - Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127, Bologna, Italy\ $^5$ INAF - Osservatorio Astronomico di Trieste, Via Tiepolo 11, I-34131 Trieste, Italy\ $^6$ INAF - Osservatorio Astronomico di Roma, via dell’Osservatorio 2, Monteporzio, Italy [ **More than half of all stars in the local Universe are found in massive spheroidal galaxies$^{1}$, which are characterized by old stellar populations$^{2,3}$ with little or no current star formation. In present models, such galaxies appear rather late as the culmination of a hierarchical merging process, in which larger galaxies are assembled through mergers of smaller precursor galaxies. But observations have not yet established how, or even when, the massive spheroidals formed$^{2,3}$, nor if their seemingly sudden appearance when the Universe was about half its present age (at redshift $z \approx 1$) results from a real evolutionary effect (such as a peak of mergers) or from the observational difficulty of identifying them at earlier epochs. Here we report the spectroscopic and morphological identification of four old, fully assembled, massive ($>10^{11}$ solar masses) spheroidal galaxies at $1.6<z<1.9$, the most distant such objects currently known. The existence of such systems when the Universe was only one-quarter of its present age, shows that the build-up of massive early-type galaxies was much faster in the early Universe than has been expected from theoretical simulations$^{4}$.** ]{} In the $\Lambda$CDM scenario$^5$, galaxies are thought to build-up their present-day mass through a continuous assembly driven by the hierarchical merging of dark matter halos, with the most massive galaxies being the last to form. However, the formation and evolution of massive spheroidal early-type galaxies is still an open question. Recent results indicate that early-type galaxies are found up to $z\sim1$ with a number density comparable to that of local luminous E/S0 galaxies$^{6,7}$, suggesting a slow evolution of their stellar mass density from $z\sim1$ to the present epoch. The critical question is whether these galaxies do exist in substantial number$^{8,9}$ at earlier epochs, or if they were assembled later$^{10,11}$ as favored by most renditions of the hierarchical galaxy formation scenario$^{4}$. The problem is complicated also by the difficulty of identifying such galaxies due to their faintness and, for $z>1.3$, the lack of strong spectral features in optical spectra, placing them among the most difficult targets even for the largest optical telescopes. For example, while star-forming galaxies are now routinely found up to $z\sim6.6$$^{12}$, the most distant spectroscopically confirmed old spheroid is still a radio–selected object at $z=1.552$ discovered almost a decade ago$^{13,14}$. One way of addressing the critical question of massive galaxy formation is to search for the farthest and oldest galaxies with masses comparable to the most massive galaxies in the present-day universe ($10^{11-12}$ M$_{\odot}$), and to use them as the “fossil” tracers of the most remote events of galaxy formation. As the rest-frame optical – near-infrared luminosity traces the galaxy mass$^{15}$, the $K_s$-band ($\lambda \sim 2.2\,\mu$m in the observer frame) allows a fair selection of galaxies according to their mass up to $z\sim 2$. Following this approach, we recently conducted the K20 survey$^{16}$ with the Very Large Telescope (VLT) of the European Southern Observatory (ESO). Deep optical spectroscopy was obtained for a sample of 546 objects with $K_s<20$ (Vega photometric scale) and extracted from an area of 52 arcmin$^2$, including 32 arcmin$^2$ within the GOODS–South field $^{17}$ (hereafter the GOODS/K20 field). The spectroscopic redshift ($z_{spec}$) completeness of the K20 survey is 92%, while the available multi-band photometry ($BVRIzJHK_s$) allowed us to derive the spectral energy distribution (SED) and photometric redshift ($z_{phot}$) of each galaxy. The K20 survey spectroscopy was complemented with the ESO/GOODS public spectroscopy (Supplementary Table 1). The available spectra within the GOODS/K20 field were then used to search for old, massive galaxies at $z>1.5$. We spectroscopically identified four galaxies with $18 \lesssim K_s \lesssim 19$ and $1.6 \lesssim z_{spec} \lesssim 1.9$ which have rest-frame mid-UV spectra with shapes and continuum breaks compatible with being dominated by old stars and $R-K_s \gtrsim 6$ (the colour expected at $z>1.5$ for old passively evolving galaxies due to the combination of old stellar populations and k-correction effects$^{9}$). The Supplementary Table 1 lists the main galaxy information. The spectrum of each individual object allows a fairly precise determination of the redshift based on absorption features and on the overall spectral shape (Fig. 1). The co-added average spectrum of the four galaxies (Fig. 2–3) shows a near-UV continuum shape, breaks and absorption lines that are intermediate between those of a F2 V and a F5 V star$^{18}$, and typical of about 1-2 Gyr old synthetic stellar populations$^{19,20}$. It is also very similar to the average spectrum of $z\sim1$ old Extremely Red Objects$^7$ (EROs), and slightly bluer than that of the $z\sim0.5$ SDSS red luminous galaxies$^{21}$ and of the $z=1.55$ old galaxy LBDS 53w091$^{13}$. However, it is different in shape and slope from the average spectrum of $z\sim1$ dusty star-forming EROs$^7$. The multi-band photometric SED of each galaxy was successfully fitted without the need for dust extinction, and using a library of simple stellar population (SSP) models$^{19}$ with a wide range of ages, $Z=Z_{\odot}$ and Salpeter IMF. This procedure yielded best-fitting ages of 1.0-1.7 Gyr, the mass-to-light ratios and hence the stellar mass of each galaxy, which results in the range of 1–3$\times 10^{11}$ $h_{70}^{-2}$ M$_{\odot}$. $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ (with $h_{70} \equiv H_0/70$), $\Omega_{\rm m}=0.3$ and $\Omega_{\Lambda}=0.7$ are adopted. In addition to spectroscopy, the nature of these galaxies was investigated with the fundamental complement of [*Hubble Space Telescope*]{}+ ACS ([*Advanced Camera for Surveys*]{}) imaging from the GOODS public [*Treasury Program*]{}$^{17}$. The analysis of the ACS high-resolution images reveals that the surface brightness distribution of these galaxies is typical of elliptical/early-type galaxies (Fig. 4). Besides pushing to $z\sim1.9$ the identification of the highest redshift elliptical galaxy, these objects are very relevant to understand the evolution of galaxies in general for three main reasons: their old age, their high mass, and their substantial number density. Indeed, an average age of about 1-2 Gyr ($Z=Z_{\odot}$) at $<\! z\!>\sim 1.7$ implies that the onset of the star formation occurred not later than at $z\sim 2.5-3.4$ ($z\sim 2-2.5$ for $Z=2.5Z_{\odot}$). These are strict lower limits because they follow from assuming instantaneous bursts, whereas a more realistic, prolonged star formation activity would push the bulk of their star formation to an earlier cosmic epoch. As an illustrative example, the photometric SED of ID 646 ($z=1.903$) can be reproduced (without dust) with either a $\sim$1 Gyr old instantaneous burst occurred at $z \sim 2.7$, or with a $\sim$2 Gyr old stellar population with a star formation rate declining with $exp(-t/ \tau)$ ($\tau=0.3$ Gyr). In the latter case, the star formation onset would be pushed to $z \sim 4$ and half of the stars would be formed at $z \sim 3.6$. In addition, with stellar masses $M_*>10^{11} h_{70}^{-2} M_\odot$, these systems would rank among the most massive galaxies in the present-day universe, suggesting that they were fully assembled already at this early epoch. Finally, their number density is considerably high. Within the comoving volume relative to 32 arcmin$^2$ and $1.5<z<1.9$ (40,000 $h_{70}^{-3}$ Mpc$^3$), the comoving density of such galaxies is about $10^{-4}$ $h_{70}^{3}$ Mpc$^{-3}$, corresponding to a stellar mass density of about $2 \times 10^{7}$ $h_{70}$ M$_{\odot}$Mpc$^{-3}$, i.e. about 10% of the local ($z=0$) value$^{22}$ for masses greater than $10^{11}$ M$_{\odot}$. This mass density is comparable to that of star-forming $M_*>10^{11}M_\odot$ galaxies at $z\sim 2$ $^{23}$, suggesting that while the most massive galaxies in the local universe are now old objects with no or weak star formation, by $z\sim 2$ passive and active star-forming massive galaxies coexist in nearly equal number. Although more successful than previous models, the most recent realizations of semi-analytic hierarchical merging simulations still severely underpredict the density of such old galaxies: just one old galaxy with $K_s<20$, $R-K_s>6$, and $z>1.5$ is present in the mock catalog$^{4}$ for the whole five times wider GOODS/CDFS area. As expected for early-type galaxies$^{9,24}$, the three galaxies at $z\sim 1.61$ may trace the underlying large scale structure. In this case, our estimated number density may be somewhat biased toward a high value. On the other hand, the number of such galaxies in our sample is likely to be a lower limit due to the spectroscopic redshift incompleteness. There are indeed up to three more candidate old galaxies in the GOODS/K20 sample with $18.5 \lesssim K_s \lesssim 19.5$, $1.5 \lesssim z_{phot} \lesssim 2.0$, $5.6 \lesssim R-K_s \lesssim 6.8$ and compact HST morphology. Thus, in the GOODS/K20 sample the fraction of old galaxies among the whole $z>1.5$ galaxy population is 15$\pm$8% (spectroscopic redshifts only), or up to 25$\pm$11% if also all the 3 additional candidates are counted. It is generally thought that the so-called “redshift desert” (i.e. around $1.4<z<2.5$) represents the cosmic epoch when most star formation activity and galaxy mass assembly took place$^{25}$. Our results show that, in addition to actively star forming galaxies$^{26}$, also a substantial number of “fossil” systems already populate this redshift range, and hence remain undetected in surveys biased towards star-forming systems. The luminous star-forming galaxies found at $z>2$ in sub-mm$^{27}$ and near-infrared$^{23,28}$ surveys may represent the progenitors of these old and massive systems. 1\. Fukugita, M., Hogan, C.J., Peebles, P.J.E. The Cosmic Baryon Budget. Astrophys. J. 503, 518-530 (1998).\ 2. Renzini, A. Origin of Bulges. In “The formation of galactic bulges”, ed. C.M. Carollo, H.C. Ferguson, R.F.G. Wyse, Cambridge University Press, p.9-26 (1999).\ 3. Peebles, P.J.E. When did the Large Elliptical Galaxies Form? In “A New Era in Cosmology”, ASP Conference Proceedings, Vol. 283. ed. N. Metcalfe and T. Shanks, Astronomical Society of the Pacific, 2002., p.351-361 (2002).\ 4. Somerville, R.S. et al. The Redshift Distribution of Near-Infrared-Selected Galaxies in the Great Observatories Origins Deep Survey as a Test of Galaxy Formation Scenarios. Astrophys. J., 600, L135-139 (2004).\ 5. Freedman, W.L. & Turner, M.S. Colloquium: Measuring and understanding the universe. Reviews of Modern Physics, 75, 1433-1447 (2003).\ 6. Im, M. et al. The DEEP Groth Strip Survey. X. Number Density and Luminosity Function of Field E/S0 Galaxies at $z<1$. Astrophys. J. 571, 136-171 (2002).\ 7. Cimatti, A. et al. The K20 survey. I. Disentangling old and dusty star-forming galaxies in the ERO population. Astron. Astrophys. 381, L68-73 (2002).\ 8. Benitez, N. et al. Detection of Evolved High-Redshift Galaxies in Deep NICMOS/VLT Images. Astrophys. J. 515, L65-69 (1999).\ 9. Daddi, E. et al. Detection of strong clustering of extremely red objects: implications for the density of $z>1$ ellipticals. Astron. Astrophys. 361, 535-549 (2000).\ 10. Zepf, S.E. Formation of elliptical galaxies at moderate redshifts. Nature 390, 377-380 (1997).\ 11. Rodighiero G., Franceschini A., Fasano G. Deep Hubble Space Telescope imaging surveys and the formation of spheroidal galaxies. Mon. Not. R. Astron. Soc. 324, 491-497 (2001).\ 12. Taniguchi, Y. et al. Lyman$\alpha$ Emitters beyond Redshift 5: The Dawn of Galaxy Formation. Journal of the Korean Astronomical Society 36, no.3, 123-144 (2003).\ 13. Dunlop, J.S. et al. A 3.5-Gyr-old galaxy at redshift 1.55. Nature, 381, 581-584 (1996).\ 14. Spinrad, H., Dey, A., Stern, D., Dunlop, J., Peacock, J., Jimenez, R., Windhorst, R. LBDS 53W091: an Old, Red Galaxy at z=1.552. Astrophys. J., 484, 581-601 (1997).\ 15. Gavazzi, G., Pierini, D., Boselli, A., The phenomenology of disk galaxies. Astron. Astrophys. 312, 397-408 (1996).\ 16. Cimatti, A. et al. The K20 survey. III. Photometric and spectroscopic properties of the sample. Astron. Astrophys. 392, 395-406 (2002).\ 17. Giavalisco, M. et al., The Great Observatories Origins Deep Survey: Initial Results from Optical and Near-Infrared Imaging. Astrophys. J. 600, L93-98 (2004).\ 18. Pickles, A.J. A Stellar Spectral Flux Library: 1150-25000 Å. PASP 110, 863-878 (1998).\ 19. Bruzual, G. & Charlot, S. Stellar population synthesis at the resolution of 2003. Mon. Not. R. Astron. Soc. 344, 1000-1028 (2003).\ 20. Jimenez, R. et al. Synthetic stellar populations: single stellar populations, stellar interior models and primordial proto-galaxies, Mon. Not. R. Astron. Soc. 349, 240-254 (2004).\ 21. Eisenstein, D.J. et al. Average Spectra of Massive Galaxies in the Sloan Digital Sky Survey. Astrophys. J. 585, 694-713 (2003).\ 22. Cole, S. et al. The 2dF galaxy redshift survey: near-infrared galaxy luminosity functions. Mon. Not. R. Astron. Soc., 326, 255-273 (2001).\ 23. Daddi, E. et al., Near-Infrared Bright Galaxies at $z\sim2$. Entering the Spheroid Formation Epoch ? Astrophys. J., 600, L127-131 (2004).\ 24. Davis, M., Geller, M.J. Galaxy Correlations as a Function of Morphological Type. Astrophys. J., 208, 13-19 (1976).\ 25. Dickinson, M., Papovich, C., Ferguson, H.C., Budavari, T. The Evolution of the Global Stellar Mass Density at $0<z<3$. Astrophys. J., 587, 25-40 (2003).\ 26. Steidel, C.C. et al. A Survey of Star-Forming Galaxies in the z=1.4-2.5 ‘Redshift Desert’: Overview. Astrophys. J. 604, 534-550 (2004).\ 27. Genzel, R., Baker, A.J., Tacconi, L.J., Lutz, D., Cox, P.; Guilloteau, S., Omont, A. Spatially Resolved Millimeter Interferometry of SMM J02399-0136: A Very Massive Galaxy at $z=2.8$. Astrophys. J. 584, 633-642 (2003).\ 28. Franx, M. et al. A Significant Population of Red, Near-Infrared-selected High-Redshift Galaxies. Astrophys. J., 587, L79-L83 (2003).\ 29. Pignatelli, E. & Fasano, G. GASPHOT: A Tool for Automated Surface Photometry of Galaxies. Astrophys. Sp. Sci. 269, 657-658 (1999).\ 30. Peng, C.Y., Ho, L.C., Impey, C.D., Rix, H.-W., Detailed Structural Decomposition of Galaxy Images. Astron. J., 124, 266-293 (2002).\ Correspondence and requests for material should be sent to Andrea Cimatti ([email protected]). This work is based on observations made at the European Southern Observatory, Paranal, Chile, and with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy (AURA). We thank Rachel Somerville for information on the GOODS/CDFS mock catalog. We are grateful to the GOODS Team for obtaining and releasing the HST and FORS2 data. [**Figure 1**]{} [*The individual and average spectra of the detected galaxies.*]{} From bottom to top: the individual spectra smoothed to a 16 Å  boxcar (26 Å  for ID 237) and the average spectrum of the four old galaxies ($z_{average}=1.68$). The red line is the spectrum of the old galaxy LBDS 53w091 ($z=1.55$) used to search for spectra with a similar continuum shape. Weak features in individual spectra (e.g. MgII$\lambda$2800 and the 2640 Å  continuum break, B2640) become clearly visible in the average spectrum. The object ID 235 has also a weak \[OII\]$\lambda$3727 emission (not shown here). The spectra were obtained with ESO VLT+FORS2, grisms 200I (R($1^{\prime\prime})\sim$400) (ID 237) and 300I (R($1^{\prime\prime})\sim$600) (IDs 235,270,646), 1.0$^{\prime\prime}$ wide slit and $\lesssim 1^{\prime\prime}$ seeing conditions. The integrations times were 3 hours for ID 237, 7.8 hours for IDs 235 and 270, and 15.8 hours for ID 646. For ID 646, the ESO/GOODS public spectrum was co-added to our K20 spectrum (see Supplementary Tab. 1). “Dithering” of the targets along the slits was applied to remove efficiently the CCD fringing pattern and the strong OH sky lines in the red. The data reduction was done with the IRAF software package (see$^{16}$). The spectrophotometric calibration of all spectra was achieved and verified by observing several standard stars. The average spectrum, corresponding to 34.4 hours integration time, was obtained by co-adding the individual spectra convolved to the same resolution, scaled to the same arbitrary flux (i.e. with each spectrum having the same weight in the co-addition), and assigning wavelength–dependent weights which take into account the noise in the individual spectra due to the OH emission sky lines. ![[]{data-label="fig1"}](cimatti_fig1.ps){width="15cm"} [**Figure 2**]{} [*The detailed average spectrum of the detected galaxies.*]{} A zoom on the average spectrum (blue) compared with the synthetic spectrum$^{19}$ of a 1.1 Gyr old simple stellar population (SSP) with solar metallicity ($Z=Z_{\odot}$) and Salpeter IMF (red). The observed average spectrum was compared to a library of synthetic SSP template spectra$^{19,20}$ with a range of ages of 0.1-3.0 Gyr with a step of 0.1 Gyr, and with assumed metallicities $Z$=0.4$\times$, 1.0$\times$, and 2.5$\times Z_{\odot}$. The best fit age for each set of synthetic templates was derived through a $\chi^2$ minimization over the rest-frame wavelength range 2300–3400 Å. The rms as a function of wavelength used in the $\chi^2$ procedure was estimated from the average spectrum computing a running mean rms with a step of 1 Å  and a box size of 20 Å, corresponding to about three times the resolution of the observed average spectrum. The median signal-to-noise ratio is $\sim$20 per resolution element in the 2300–3400 Å  range. The wavelength ranges including the strongest real features (i.e. absorptions and continuum breaks) were not used in the estimate of the rms. The resulting reduced $\chi^2$ is of the order of unity for the best fit models. In the case of solar metallicity, the ranges of ages acceptable at 95% confidence level are $1.0^{+0.5}_{-0.1}$ Gyr and $1.4^{+0.5}_{-0.4}$ Gyr for SSP models of$^{19}$ and$^{20}$ respectively (see also Fig. 3, top panel). Ages $\sim 50\%$ younger or older are also acceptable for $Z=2.5Z_{\odot}$ or $Z=0.4Z_{\odot}$ respectively. The 2640 Å  and 2900 Å  continuum break$^{13}$ amplitudes measured on the average spectrum are B2640=1.8$\pm$0.1 and B2900=1.2$\pm$0.1. These values are consistent with the ones expected in SSP models$^{19-20}$ for ages around 1–1.5 Gyr and solar metallicity. For instance, the SSP model spectrum shown here has B2640=1.84 and B2900=1.27. ![[]{data-label="fig2"}](cimatti_fig2.ps){width="15cm"} [**Figure 3**]{} [*The comparison between the average spectrum and a set of spectral templates.*]{} The average spectrum (blue) compared to a set of template spectra. From bottom: F2 V (green) and F5 V (red) stellar spectra$^{18}$ with $Z=Z_{\odot}$, the composite spectrum (red) of 726 luminous red galaxies at $0.47<z<0.55$ selected from the SDSS$^{21}$ (available only for $\lambda>2600$ Å), the average spectra of $z\sim1$ old (red) and dusty star-forming (green) EROs$^7$, SSP synthetic spectra$^{19}$ ($Z=Z_{\odot}$, Salpeter IMF) with ages of 0.5 Gyr (magenta), 1.1 Gyr (green) and 3.0 Gyr (red). ![[]{data-label="fig3"}](cimatti_fig3.ps){width="15cm"} [**Figure 4**]{} [*The morphological properties of the detected galaxies.*]{} Images of the four galaxies taken with the [*Hubble Space Telescope*]{} +ACS through the F850LP filter (from GOODS data$^{17}$) which samples the rest-frame $\sim$3000-3500 Å  for $1.6<z<2$. The images are in logarithmic grey–scale and their size is $2^{\prime\prime} \times 2^{\prime\prime}$, corresponding to $\sim 17 \times 17$ kpc for the average redshift $z=1.7$ and the adopted cosmology. At a visual inspection, the galaxies show rather compact morphologies with most of the flux coming from the central regions. A fit of their surface brightness profiles was performed with a “Sersic law” ($\propto r^{1/n}$) convolved with the average point spread function extracted from the stars in the ACS field and using the GASPHOT$^{29}$ and GALFIT$^{30}$ software packages. Objects ID 237 and ID 646 have profiles with acceptable values of $n$ in the range of $4<n<6$, i.e., typical of elliptical galaxies, object ID 270 is better reproduced by a flatter profile ($1<n<2$), whereas a more ambiguos result is found for the object showing some evidence of irregularities in the morphology (ID 235, $1<n<3$). These latter objects may be bulge-dominated spirals but no bulge/disk decomposition was attempted. Ground-based near-infrared images taken under 0.5$^{\prime\prime}$ seeing conditions with the ESO VLT+ISAAC through the $K_s$ filter (rest-frame $\sim$6000-8000 Å) show very compact morphologies, but no surface brightness fitting was done. ![[]{data-label="fig4"}](cimatti_fig4.ps){width="15cm"} [**SUPPLEMENTARY TABLE 1**]{}\ \ [cccccccl]{}\ IAU & K20 & R.A. (J2000) & Dec (J2000) & $K_s$ & $R-K_s$ & $z$ & Spectrum\ ID & ID & h m s & $\circ$ $\prime$ $\prime\prime$& & & &\ \ J033210.79-274627.8 & 235 & 03 32 10.776 & -27 46 27.73 & 17.98$\pm$0.04 & 6.47$\pm$0.10&1.610&K20\ J033210.52-274628.9 & 237 & 03 32 10.507 & -27 46 28.84 & 19.05$\pm$0.05 & 6.83$\pm$0.28&1.615&K20\ J033212.53-274629.2 & 270 & 03 32 12.525 & -27 46 29.16 & 18.74$\pm$0.05 & 5.99$\pm$0.10&1.605&K20\ J033233.85-274600.2 & 646 & 03 32 33.847 & -27 46 00.24 & 19.07$\pm$0.07 & 5.99$\pm$0.10&1.903&K20+GOODS\ \ [**Supplementary Table 1**]{}\ \ IAU ID: official identification number in the GOODS–South catalog (z-band)\ (http://www.stsci.edu/science/goods/catalogs).\ \ K20 ID: identification number in the K20 survey catalog (http://www.arcetri.astro.it/$\sim$k20/).\ \ R.A., Dec: Right Ascension and Declination at equinox J2000 based on the public ESO/GOODS $K_s$-band VLT+ISAAC image.\ \ $K_s$: K20 survey total magnitude in the $K_s$-band (Vega scale).\ \ $R-K_s$ color (Vega scale) in 2$^{\prime\prime}$ diameter aperture.\ \ $z$: spectroscopic redshift.\ \ Spectrum: K20: K20 survey, GOODS: public ESO/GOODS VLT+FORS2 spectroscopy (Vanzella et al., in preparation; http://www.eso.org/science/goods).

ArXiv

--- author: - 'A. Gusdorf' - 'S. Anderl' - 'R. Güsten' - 'J. Stutzki' - 'H.-W. Hübers' - 'P. Hartogh' - 'S. Heyminck' - 'Y. Okada' bibliography: - 'biblio.bib' date: 'Received September 15, 1996; accepted March 16, 1997' title: 'Probing MHD Shocks with high-$J$ CO observations: W28F' --- [Observing supernova remnants (SNRs) and modelling the shocks they are associated with is the best way to quantify the energy SNRs re-distribute back into the Interstellar Medium (ISM).]{} [We present comparisons of shock models with CO observations in the F knot of the W28 supernova remnant. These comparisons constitute a valuable tool to constrain both the shock characteristics and pre-shock conditions.]{} [New CO observations from the shocked regions with the APEX and SOFIA telescopes are presented and combined. The integrated intensities are compared to the outputs of a grid of models, which were combined from an MHD shock code that calculates the dynamical and chemical structure of these regions, and a radiative transfer module based on the large velocity gradient’ (LVG) approximation.]{} [We base our modelling method on the higher *J* CO transitions, which unambiguously trace the passage of a shock wave. We provide fits for the blue- and red-lobe components of the observed shocks. We find that only stationary, C-type shock models can reproduce the observed levels of CO emission. Our best models are found for a pre-shock density of 10$^4$ cm$^{-3}$, with the magnetic field strength varying between 45 and 100 $\mu$G, and a higher shock velocity for the so-called blue shock ($\sim$25 km s$^{-1}$) than for the red one ($\sim$20 km s$^{-1}$). Our models also satisfactorily account for the pure rotational H$_2$ emission that is observed with *Spitzer*.]{} Introduction ============ The interstellar medium (ISM) is in constant evolution, ruled by the energetic feedback from the cosmic cycle of star formation and stellar death. At the younger stages of star formation (bipolar outflows), and after the death of massive stars (SNRs), shock waves originating from the star interact with the ambient medium. They constitute an important mechanical energy input, and lead to the dispersion of molecular clouds and to the compression of cores, possibly triggering further star formation. Studying the signature of these interactions in the far-infrared and sub-mm range is paramount for understanding the physical and chemical conditions of the shocked regions and the large-scale roles of these feedback mechanisms. Supernovae send shock waves through the ISM, where they successively carve out large hot and ionised cavities. They subsequently emit strong line radiations (optical/UV), and eventually interact with molecular clouds, driving lower-velocity shocks. Similar to their bipolar outflow equivalents, these shocks heat, compress, and accelerate the ambient medium before cooling down through molecular emission (@Vandishoeck93, @Yuan11, hereafter Y11). Valuable information has been provided by ISO [@Cesarsky99; @Snell05] and *Spitzer* (@Neufeld07, hereafter N07), but neither of those instruments provided sufficient spectral resolution to allow for a detailed study of the shock mechanisms. High-$J$ CO emission is one of the most interesting diagnostics of SNRs. CO is indeed a stable and abundant molecule, and an important contributor to the cooling of these regions, whose high-frequency emission is expected to be a pure’ shock tracer. Observations of the latter must be carried out from above the Earth’s atmosphere. As part of a multi-wavelength study of MHD shocks that also includes *Herschel* data, we present here the first velocity-resolved CO (11–10) observations towards a prominent SNR-driven shock with the GREAT spectrometer onboard SOFIA, and combine them with new lower-$J$ ones in a shock-model analysis. The supernova remnant W28 ========================= W28 is an old ($>$10$^{4.5}$ yr, @Claussen99) SNR in its radiative phase of evolution, with a non-thermal radio shell centrally filled with thermal X-ray emission. Lying in a complex region of the Galactic disk at a distance of 1.9$\pm$0.3 kpc [@Velazquez02], its structure in the 327 MHz radio continuum represents a bubble-like shape of about 40$\times$30 pc [@Frail93]. Early on, molecular line emission peaks, not associated with star formation activity, but revealing broad lines, were suggested as evidence for interaction of the remnant with surrounding molecular clouds [@Wootten81]. Later studies spatially resolved the shocked CO gas layers from the ambient gas [@Frail98; @Arikawa99]. OH maser spots line up with the post-shock gas layers [@Frail94; @Claussen97; @Hoffman05], for which the strongest masers VLBA polarisation studies yield line-of-sight magnetic field strengths of up to 2 mG [@Claussen99]. Pure rotational transitions of H$_2$ have been detected with ISO [@Reach00] and were more recently observed with *Spitzer*, better resolved spatially and spectrally, by N07 and Y11. Recently, very high energy (TeV) $\gamma$-ray emission has been detected by HESS [@Aharonian08], Fermi [@Abdo10], and AGILE [@Giuliani10], spatially slightly extended and coincident with the bright interaction zones, W28-E and -F. If interpreted as the result of hadronic cosmic ray interactions in the dense gas ($\pi^0$ decay), a cosmic ray density enhancement by an order of magnitude is required (which is supplied/accelerated by the SNR). Sub-mm CO observations of W28F {#sub:opcooow} ------------------------------ APEX[^1] [@Guesten06] observations towards W28F were conducted in 2009 and will be the subject of a forthcoming publication (Gusdorf et al., in prep.). For the present study, we used 100$'' \times$100$''$ maps in the $^{13}$CO (3–2), CO (3–2), (4–3), (6–5), and (7–6) transitions, described in Appendix \[sec:tao\]. ![Overlay of the velocity-integrated CO (6–5) (colour background) with the CO (3–2) (white contours) emission observed with the APEX telescope. For both lines, the intensity was integrated between -30 and 40 km s$^{-1}$. The wedge unit is K km s$^{-1}$ in antenna temperature. The CO (3–2) contours are from 30 to 160 $\sigma$, in steps of 10$\sigma$ = 16 K km s$^{-1}$. The half-maximum contours of the CO (3–2) and (6–5) maps are indicated in red and black, respectively. The dark blue circle indicates the position and beam size of the SOFIA/GREAT observations. The APEX beam sizes of our CO (3–2), (4–3), (6–5), and (7–6) observations are also provided (upper right corner light green circles, see also Table \[tablea1\]). The maps are centred at (R.A.$_{[\rm{J}2000]}$=$18^h01^m52\fs3$, Dec$_{[\rm{J}2000]}$=$-23^\circ19'$25$''$). The black and light blue hexagons mark the position of the OH masers observed by @Claussen97 and @Hoffman05.[]{data-label="figure1"}](figure1.eps){width="9cm"} In Fig. \[figure1\] the velocity-integrated CO (6–5) broad-line emission of W28F is shown overlaid with the CO (3–2) emission (white contours): a north-south elongated structure of about 100$''$ height and 30$''$ width traces the same warm accelerated post-shocked gas. In our high-resolution CO (6–5) data the structure is resolved, though probably still sub-structured similar to what is seen in H$_2$, e.g., Y11. Comparison with the distributions of excited H$_2$ and OH masers (whose locations also mark the leading edge of the non-thermal radio shell) suggests a textbook morphology of an SNR-molecular cloud interaction: the shock propagates E-NE into the ambient cloud that extends east for several arcmins. Hot H$_2$ and OH masers mark the first signposts of the shock-compressed gas. Farther downstream, the gas cooling is seen prominently in warm CO. The shock impact appears edge-on, but the fact that high - projected - streaming velocities are indeed observed (-30 km s$^{-1}$ with respect to the ambient cloud) requires a significant inclination angle. ![CO transitions observed in the position (+7$''$,-26$''$) indicated in Fig. \[figure1\]: APEX (3–2), black (corresponding $^{13}$CO, green); (4–3), pink; (6–5), dark blue; (7–6), light blue; and SOFIA (11–10), red. The $^{13}$CO (3–2) and CO (11–10) profiles were multiplied by three for comparison purposes. Respective CO spectral resolutions are 0.212, 0.318, 0.159, 0.272, and 0.693 km s$^{-1}$, and 0.664 km s$^{-1}$ for $^{13}$CO (3–2). []{data-label="figure2"}](figure2.eps){width="9cm"} We have selected the most prominent position in the southern extension of W28F, marked with the blue circle in Fig.1, for our shock analysis. This position was also covered by *Spitzer*, offering a set of complementary H$_2$ data. Fig. \[figure2\] shows the APEX spectra obtained towards this position, in main beam temperature units, all convolved to the 23.7$''$ beam of the SOFIA observations (Sect. \[sub:oadr\]). Absorption notwithstanding, the spectra show Gaussian profiles, with all line wings extending in blue (-30 km s$^{-1}$) and red (+15 km s$^{-1}$) velocities with respect to the ambient cloud ($\sim$10 km  s$^{-1}$). The higher the line frequency, the narrower the line profile, with a typical linewidth of about 20 km s$^{-1}$ for CO (7–6). We detect red-shifted line-of-sight absorption features in all lines up to CO (6–5), the deepest arising at the ambient cloud velocity. Off-position contamination results in minor absorption features at 20 and 25 km s$^{-1}$ in the (3–2) and (4–3) profiles. Comparison with the $^{13}$CO(3-2) profile is also shown in Fig. \[figure2\]. An analysis of the line temperature ratio of $^{12}$CO/$^{13}$CO yields an optical thickness value of 3–7 in the wings of the $^{12}$CO (3–2), assuming a typical interstellar abundance ratio of 50–60 (e.g., @Langer93). Far-infrared CO spectroscopy with GREAT/SOFIA {#sub:oadr} --------------------------------------------- The observations towards W28F were conducted with the GREAT[^2] spectrometer (@Heyminck12) during SOFIA’s flight from Stuttgart to Washington on September 21 2011. Only one position could be observed, towards the southern tip of the shocked cloud at offset (+7$''$,-26$''$) (Fig. \[figure1\]). The CO (11–10) line was tuned to the frequency 1267.015 GHz LSB. The receiver was connected to a digital FFT spectrometer (@Klein12) providing a bandwidth of 1.5 GHz with a spectral resolution of 0.05 km s$^{-1}$. The observations were performed in double beam-switching mode, with an amplitude of 80$''$ (or a throw of 160$''$) at the position angle of 135$^\circ$ (NE–SW), and a phase time of 0.5 sec. The nominal focus position was updated regularly against temperature drifts of the telescope structure. The pointing was established with the optical guide cameras to an accuracy of $\sim$5$''$. The beam width $\Theta_{\rm{mb}}$ is 23.7$''$; the main beam and forward efficiencies are $\eta_{\rm{mb}}$ = 0.54 and $\eta_{\rm f}$ = 0.95. The integration time was 13 min ON source, for a final r.m.s of 0.66 K. The data were calibrated with the KOSMA/GREAT calibrator (@Guan12), removing residual telluric lines, and subsequently processed with the CLASS software[^3]. The CO (11–10) spectrum, overlaid on the sub-mm lines in Fig. \[figure2\], reveals a markedly different profile: weak emission only is seen from the high-velocity gas, while the line is more prominent at low velocities. The profile basically follows the shape of the optically thin $^{13}$CO(3-2). Discussion {#sec:dis} ========== The observations {#sub:theobs} ---------------- Although most likely part of a single original shock clump, we separated the profiles into a blue lobe (-30 to 7.5-12.5 km s$^{-1}$) and a red lobe (7.5-12.5 to 40 km s$^{-1}$), and fited the data independently. The uncertainty of the upper (lower) limit of those ranges reflects our lack of knowledge of the ambient velocity component. In Fig. \[figure3\] we plot the velocity-integrated intensities of the CO lines, all convolved to the same angular resolution of 23.7$''$, against the rotational quantum number of their upper level in a so-called spectral line energy distribution’ (or SLED, filled black squares with errorbars). The underlying assumption is that the filling factor is the same for all CO lines, which is validated by the similarity between their emitting regions (see for instance Fig. \[figure1\], where the half-maximum contours of CO (3–2) and (6–5) coincide at the available resolution). For Fig. \[figure3\], the assumption was made that all observed transitions present a circular emission region of radius 25$''$, corresponding to a filling factor of 0.53 (compatible with our maps, see Fig. \[figure1\]). The upper (lower) panel shows the diagram associated with the blue (red) shock component. The lower limit to the integrated intensity corresponds to the integration of the observed profile in the smallest velocity range, whereas upper limits were obtained by integrating Gaussian fits to the observation on the largest velocity range. The fits were adjusted to recover the shock flux lost through absorption and based on the un-absorbed parts of the profiles. Although yielding high errorbars on our measurements, specially for low $J_{\rm{up}}$ values, this method also provides the most conservative approach to our blue-red decomposition of the CO emission. The models {#sub:themod} ---------- We then compared the resulting integrated intensity diagrams to modelled ones. To build those, we used a radiative transfer module based on the large velocity gradient’ (LVG) approximation to characterise the emission from the CO molecule over outputs generated by a state-of-the-art model that calculates the structure of one-dimension, stationary shock layers (or approximations of non-stationary layers). This method has already been used and extensively introduced in @Gusdorf082. Since then, the LVG module has been modified to incorporate the latest collisional rate coefficients of CO with H$_2$ computed by @Yang10, but the shock model is the same. Based on a set of input parameters (pre-shock density, shock velocity, type, and age, and magnetic field parameter value $b$ such as B\[$\mu$G\] = $b~\times~\sqrt{n_{\rm H}~[{\rm cm}^{-3}]}$), it calculates the structure of a shock layer, providing dynamical (velocity, density, temperature), and chemical (fractional abundances of more than 125 species linked by over 1000 reactions) variables values at each point of the layer. The relevant outputs are then used by our LVG module, which calculates the level population and line emissivities of the considered molecule. In the present case, the line temperatures were computed for transitions of CO up to (40–39) at each point of the shock layer, over which they were then integrated to form our model’s SLED. Because of the importance of H$_2$ (key gas coolant, abundant molecule), its radiative transfer was treated within the shock model. Its abundance was calculated at each point of the shock layer, based on the processes listed in @Lebourlot02, and the populations of the 150 first levels were also calculated inside the shock code, their contribution to the cooling being included in the dynamical calculations. It is hence also possible to associate an H$_2$ excitation diagram to each model of the grid, see Sect. \[sub:excdia\]. Our model grid consists of a large sample of integrated intensity diagrams, obtained for stationary C- and J-type, and non-stationary CJ-type shocks. It covers four pre-shock densities (from 10$^3$ to 10$^6$ cm$^{-3}$), $0.45 \leq b \leq 2$, over a range of velocities spanning from 5 km s$^{-1}$ up to the maximum, critical’ value above which a C-shock can no longer be maintained, which depends on the other parameters. We then independently compared the observed blue- and the red-shock component to the whole grid diagrams. We used a $\chi^2$ routine that was set up to provide the best fits to our purest shock-tracing lines: CO (6–5), (7–6), and (11–10), to reduce any effect induced by ambient emission contamination on the lower-lying transitions. The results {#sub:theres} ----------- ![Best-model comparisons between CO observations and models for each observed shock: the blue- (upper panel) and the red- (lower panel) components. Observations are marked by the black squares, our individual best-shock models are in blue line and circles ($\zeta_1$) or diamonds ($\zeta_2$), the warm layer that we used to compensate the ambient emission affecting the (3–2) and (4–3) transitions in green line and triangles, and the sum of each of these components is represented by a red line and circles or diamonds.[]{data-label="figure3"}](figure3.eps){width="30.00000%"} The upper (lower) panel of Fig. \[figure3\] shows our best-fitting models for the blue (red) component. We found that only stationary, C-type shock models can reproduce the levels of observed CO (6–5), (7–6) and (11-10) integrated intensities. In both cases, the pre-shock density is 10$^4$ cm$^{-3}$, for which a stationary state is typically reached for a shock age of 10$^4$ years, in agreement with the age of the remnant of 3.5$\times 10^4$ years quoted by @Giuliani10. The modelled shock velocities, 25 and 20 km s$^{-1}$, respectively, are rough upper limits to the observed ones, an expected conclusion given the one-dimensional nature of our models. Additionally, our models provide respective constraints to the magnetic field component perpendicular to the shock layers, with parameter $b$ of 1 and 0.45. The difference between those two values can be explained by a projection effect. In both cases, the direct post-shock density is of the order of 2-3$\times 10^5$cm$^{-3}$, yielding expected post-shock values for this component of 0.05-0.55 mG, compatible with those inferred by @Claussen99 (2 mG for the total post-shock magnetic field), and by @Hoffman05 (0.3–1.1 mG based on the OH maser measurements in the positions indicated in Fig. \[figure1\] located within the blue beam). Finally, our models produce typical OH column densities of a few 10$^{16}$cm$^{-2}$. This complies with the list of requirements set out by @Lockett99 for exciting the observed OH 1720 MHz masers, together with the C- nature of the considered shocks, as well as the temperature and density conditions they create. The respective CO column density found in each of the red and blue shock models is 11 and 5$\times 10^{17}$cm$^{-3}$. However, our shock models failed to account for the important observed levels of CO (3–2) and (4–3) emission. We found it possible to model this lower-$J$ surplus emission by adding a thin layer of gas to our shock model results. Its corresponding emission was calculated with our LVG module used in a homogeneous slab’ mode, and indeed generated significant emission only from the (3–2) and (4–3) lines, as can also be seen in Fig. \[figure3\]. We used a linewidth of 10 km s$^{-1}$ and a density of $n_{\rm H} = 10^4$ cm$^{-3}$ with a fractional abundance of $X(\rm{CO}) = 10^{-4}$ for the blue and red cases. We found that this CO layer’ is warm (35 K and 15 K for the blue and red components, respectively), and adds to the pure-shock CO column density in both components, from 2$\times$10$^{17}$ cm$^{-2}$ for the blue component, and 10$^{17}$ cm$^{-2}$ for the red one. Our final result, the sum of the C-shock and CO layer’, is also shown in Fig. \[figure3\]. To understand the physical origin of this emitting layer, we varied the cosmic ray ionisation rate in our best-fitting models from its solar value $\zeta_1 = 5 \times 10^{-17}$ s$^{-1}$ to the $\zeta_2 = 3.4 \times 10^{-16}$ s$^{-1}$ indicated by @Hewitt09 in the specific case of W28. In our modelling, this modification impacts on the chemistry (higher ionisation fraction) and the physics (warmer gas) of our outputs. The resulting shock profile is more narrow (hence generating slightly less CO emission, see Fig. \[figure3\]), with higher post-shock temperatures close to the aforementioned CO layer. Because one of the limitations of our shock model is the absence of significant velocity gradient in the post-shock region, the warm CO layer component might correspond to a post-shock layer of gas mildly heated by the energetic radiation that seems to exist in the region (UV or more energetic, as characterised by @Hewitt09). The rigorous inclusion of these radiative effects in our shock model is still in progress. In Sect. \[sec:tso\] we present a consistency check based on the use of the *Spitzer* H$_2$ observations. We thank the SOFIA engineering and operations teams, whose support has been essential during basic science flights, and the DSI telescope engineering team. Based \[in part\] on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy. SOFIA Science Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. S. Anderl acknowledges support by the Deutsche Forschungsgemeinschaft within the SFB 956 “Conditions and impact of star formation”, the International Max Planck Research School (IMPRS) for Radio and Infrared Astronomy at the Universities of Bonn and Cologne, and the Bonn-Cologne Graduate School of Physics and Astronomy. A. Gusdorf acknowledges support by the grant ANR-09-BLAN-0231-01 from the French [*Agence Nationale de la Recherche*]{} as part of the SCHISM project. The APEX observations {#sec:tao} ===================== APEX observations towards the supernova remnant W28F were conducted in several runs in the year 2009 (in May, June, August and October). We used of a great part of the suite of heterodyne receivers available for this facility: APEX–2 @Risacher06, FLASH460 @Heyminck06, and CHAMP$^+$ @Kasemann06 [@Guesten08], in combination with the MPIfR fast Fourier transform spectrometer backend (FFTS, @Klein06). The central position of all observations was set to be $\alpha_{[\rm{J}2000]}$=$18^h01^m51\fs78$, $\beta_{[\rm{J}2000]}$=$-23^\circ18'58$$50$. Focus was checked at the beginning of each observing session, after sunrise and/or sunset on Mars, or on Jupiter. Line and continuum pointing was locally checked on RAFGL1922, G10.47B1, NGC6334-I or SgrB2(N). The pointing accuracy was found to be of the order of 5$''$ rms, regardless of the receiver that was used. Table \[tablea1\] contains the main characteristics of the observed lines and corresponding observing set-ups: frequency, beam size, sampling, used receiver, observing days, forward and beam efficiency, system temperature, spectral resolution, and finally the velocity interval that was used to generate the integrated intensity maps. The observations were performed in position-switching/raster mode using the APECS software @Muders06. The data were reduced with the CLASS software (see http://www.iram.fr/IRAMFR/GILDAS). For all observations, the maximum number of channels available in the backend was used (8192), except for CO (4–3), for which only 2048 channels were used, leading to the spectral resolutions indicated in Table \[tablea1\]. Maps were obtained for all considered transitions, covering the field introduced in Fig. \[figure1\] and put in the perspective of the whole SNR in Fig. \[figuresup\]. ![Location of the field covered by our CO observations on the larger-scale radio continuum image at 327 MHz taken from @Claussen97: entire SNR in the left panel, zoom in the right panel. The shown CO observations are the (6–5) map, also displayed in Fig. \[figure1\]. []{data-label="figuresup"}](figuresup.eps){width="9cm"} line CO (3–2) CO (4–3) CO (6–5) CO (7–6) $^{13}$CO (3–2) ------ ---------- ---------- ---------- ---------- ----------------- The H$_2$ observations {#sec:tso} ====================== The dataset ----------- ![Overlay of the map of CO (6–5) emission observed by the APEX telescope (colour background) with the H$_2$ 0–0 S(5) emission (white contours), observed with the *Spitzer* telescope. The wedge unit is K km s$^{-1}$ (antenna temperature) and refers to the CO observations. The H$_2$ 0–0 S(5) contours are from 50 to 210 $\sigma$, in steps of $20 \sigma \simeq 1.6 \times 10^{-4}$ erg cm$^{-2}$ s$^{-1}$ sr$^{-1}$. The green contour defines the half-maximum contour of this transition. Like in Fig. \[figure1\], the blue circle indicates the beam size of the SOFIA/GREAT observations, on the position of the centre of the circle, marked by a blue dot. The beam and pixel sizes of the CO (6–5), and H$_2$ 0–0 S(5) observations are also provided in yellow (big and small, respectively) circles in the lower right corner. The black contour delineates the half-maximum contour of the H$_2$ 0–0 S(2) transition. The field is smaller than in Fig. \[figure1\], and the (0,0) position is that of the *Spitzer* observations (see Appendix \[sec:tso\]). []{data-label="figurea1"}](figurea1.eps){width="6cm"} As a consistency check for our models, we used of the *Spitzer*/IRS observations of the H$_2$ pure rotational transitions (0–0 S(0) up to S(7)), reported and analysed in N07 and Y11. Although the original dataset also includes other ionised species, we chose to use only the H$_2$ data. The raw product communicated to us by David Neufeld contains rotational transitions maps, with 1.2$''$ per pixel, centred on $\alpha_{[\rm{J}2000]}$=$18^h01^m52\fs32$, $\beta_{[\rm{J}2000]}$=$-23^\circ19'24$$92$. Fig. \[figurea1\] shows an overlay of our APEX CO (6–5) map with the H$_2$ 0–0 S(5) region observed by *Spitzer*. The figure shows coinciding maxima between the two datasets in the selected position, and a slightly different emission distribution. This might be the effect of the better spatial resolution of the H$_2$ data, which reveal more peaks than in CO. This overlay also shows the slightly different morphology of the emission of the S(2) (half maximum contour in black) and S(5) (half maximum contour in green) transitions, at the available resolution. Excitation diagram {#sub:excdia} ------------------ We performed a consistency check on our modelling (see Sect. \[sec:dis\]) using the excitation diagram derived for the selected emission region. The H$_2$ excitation diagram displays ln($N_{\varv j}/g_j$) as a function of $E_{\varv j}/k_{\rm B}$, where $N_{\varv j}$ (cm$^{-2}$) is the column density of the rovibrational level ($v, J$), $E_{\varv j}/k_{\rm B}$ is its excitation energy (in K), and $g_j = (2j+1)(2I+1)$ its statistical weight (with $I=1$ and $I=0$ in the respective cases of ortho- and para-H$_2$). If the gas is thermalised at a single temperature, all points in the diagram fall on a straight line. The selected position for the present study was introduced in Sect. \[sub:opcooow\], and can be seen in Figs. \[figure1\] and \[figurea1\]. The column density of the higher level of each considered transition was extracted by averaging the line intensities in a 11.85$''$ radius circular region, consistent with our handling of the CO data. In the process, we corrected the line intensities for extinction, adopting the visual extinction values from N07, $A_{\rm \varv} = 3-4$, and using the interstellar extinction law of @Rieke85. The resulting excitation diagram can be seen in Fig. \[figurea2\]. We note that the intensity of the H$_2$ 0–0 S(6) transition ($J = 8$) cannot be determined reliably, because the line is blended with a strong 6.2 $\mu$m PAH feature. Comparisons with our models --------------------------- Unlike our CO observations, the *Spitzer*/IRS observations are not spectrally resolved (owing to the relatively low resolving power of the short low module in the range 60–130). On the other hand, given the minimum energy of the levels excited in the H$_2$ transitions (for 0–0 S(0), $E_{\rm u} \sim 509.9$ K), we can make the double assumption that the lines are not contaminated by ambient emission or self-absorption, and that the measured line intensities are the result from the blue- and red-shock present in the line of sight. Therefore, one must compare the observed H$_2$ level populations to what is generated by the sum of our best-fitting blue- and red- shock models. Fig. \[figurea2\] shows the comparison between the excitation diagram derived from the sum of our two CO best-fitting models and the observed one. The H$_2$ excitation diagram comprising the sum of our CO best-fitting models is shown in both panels in red circles, and provides a satisfying fit to the observations. With the aim of improving the quality of this fit, we also studied the influence of the initial ortho-to-para ratio (OPR) value, upper panel, and of the cosmic ray ionisation rate value, lower panel. In the upper panel, for a cosmic ray ionisation rate value adopted as the solar one, $\zeta_1 = 5 \times 10^{-17}$ s$^{-1}$, the influence of the initial OPR value is studied. In their models, N07 inferred a value of 0.93 for the warm’ component ($\sim$322 K). On the other hand, the value associated to their hot’ ($\sim$1040 K) component could not be estimated, owing to the large uncertainty associated to the determination of the H$_2$ 0–0 S(6) flux. In our shock models, the OPR value is consistently calculated at each point of the shocked layer, and is mostly the result of conversion reactions between H$_2$ and H, H$^+$ or H$_3^+$. The heating associated to the passage of the wave increases the OPR value towards the equilibrium value of 3.0. Nevertheless, this value is reached only when it is also the initial one. The N07 situation is then probably adequately approximated in our models where the OPR initial value is less than 3. However, the influence of this parameter is minimal on the excitation diagram, where it only seems to create a slight saw-tooth pattern between the odd and even values of $J$. We finally investigated the effects of the high-energy radiation field on the excitation diagram in the lower panel of Fig. \[figurea2\]. The limits of our modelling are reached because the only handle that we have to study this effect on our outputs is the variation of the cosmic ray ionisation rate value, which affects the corresponding chemistry (higher ionisation fraction) and physics (warmer gas). A proper treatment of the energetic radiation components listed in @Hewitt09 should indeed incorporate UV pumping or self-shielding, as would be the case in PDR regions (e.g., @Habart11), as well as chemical and H$_2$ excitation effects by X-ray (see @Dalgarno99) and cosmic rays (e.g., @Ferland08). Keeping these limits in mind, we found that even strong modifications of the cosmic ray ionisation rate from the solar value of 5$\times$10$^{-17}$ s$^{-1}$ to the more extreme value of 10$^{-14}$ s$^{-1}$ have only very limited effects on the excitation diagram. The rather convincing fits to the H$_2$ data seem to indicate that these excitation effects by energetic photons could well be minimal in our case, but we repeat that their proper inclusion to our models is work in progress. ![H$_2$ excitation diagram comparisons. *Upper panel:* evolution of the modelled excitation diagram obtained for $\zeta_1 = 5 \times 10^{-17}$ s$^{-1}$, varying the initial value of the OPR from the unrealistic, extreme-case 0 value to its equilibrium one, 3. *Lower panel:* influence of the cosmic ray ionisation rate variation on our modelled excitation diagram, with the initial OPR set to 3.[]{data-label="figurea2"}](figurea2.eps){width="9cm"} [^1]: This publication is partly based on data acquired with the Atacama Pathfinder EXperiment (APEX). APEX is a collaboration between the Max-Planck-Institut für Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory. [^2]: GREAT is a development by the MPI für Radioastronomie and the KOSMA$/$Universität zu Köln, in cooperation with the MPI für Sonnensystemforschung and the DLR Institut für Planetenforschung. [^3]: http://www.iram.fr/IRAMFR/GILDAS

ArXiv

--- abstract: 'We investigate scalar perturbations from inflation in braneworld cosmologies with extra dimensions. For this we calculate scalar metric fluctuations around five dimensional warped geometry with four dimensional de Sitter slices. The background metric is determined self-consistently by the (arbitrary) bulk scalar field potential, supplemented by the boundary conditions at both orbifold branes. Assuming that the inflating branes are stabilized (by the brane scalar field potentials), we estimate the lowest eigenvalue of the scalar fluctuations – the radion mass. In the limit of flat branes, we reproduce well known estimates of the positive radion mass for stabilized branes. Surprisingly, however, we found that for de Sitter (inflating) branes the square of the radion mass is typically negative, which leads to a strong tachyonic instability. Thus, parameters of stabilized inflating braneworlds must be constrained to avoid this tachyonic instability. Instability of “stabilized” de Sitter branes is confirmed by the [ BraneCode]{} numerical calculations in the accompanying paper [@branecode]. If the model’s parameters are such that the radion mass is smaller than the Hubble parameter, we encounter a new mechanism of generation of primordial scalar fluctuations, which have a scale free spectrum and acceptable amplitude.' author: - 'Andrei V. Frolov' - Lev Kofman title: 'Can Inflating Braneworlds be Stabilized?' --- Introduction ============ One of the most interesting recent developments in high energy physics has been the picture of braneworlds. Higher dimensional formulations of braneworld models in superstring/M theory, supergravity and phenomenological models of the mass hierarchy have the most obvious relevance to cosmology. In application to the very early universe this leads to braneworld cosmology, where our 3+1 dimensional universe is a 3d curved brane embedded in a higher-dimensional bulk [@review]. Early universe inflation in this picture corresponds to 3+1 (quasi) de Sitter brane geometry, so that the background geometry is simply described by the five dimensional warped metric with four dimensional de Sitter slices $$\label{warp} ds^2 = a^2(w)\left[dw^2 - dt^2 + e^{2Ht}d\vec{x}^2\right].$$ For simplicity we use spatially flat slicing of the de Sitter metric $ds^2_4$. The conformal warp factor $a(w)$ is determined self-consistently by the five-dimensional Einstein equations, supplemented by the boundary conditions at two orbifold branes. We assume the presence of a single bulk scalar field $\varphi$ with the potential $V(\varphi)$ and self-interaction potentials $U_\pm(\varphi)$ at the branes. The potentials can be pretty much arbitrary as long as the phenomenology of the braneworld is acceptable. The class of metrics (\[warp\]) with bulk scalars and two orbifold branes covers many interesting braneworld scenarios including the Hořava-Witten theory [@HW; @Lukas], the Randall-Sundrum model [@RS1; @RS2] with phenomenological stabilization of branes [@GW; @Dewolfe], supergravity with domain walls, and others [@FTW; @FFK]. We will consider models where by the choice of the bulk/brane potentials the inter-brane separation (the so-called radion) can be fixed, i.e. models in which branes could in principle be stabilized. The theory of scalar fluctuations around flat stabilized branes, involving bulk scalar field fluctuations $\delta\varphi$, scalar 5d metric fluctuations and brane displacements, is well understood [@Tanaka:2000er]. Similar to Kaluza-Klein (KK) theories, the extra-dimensional dependence can be separated out, and the problem is reduced to finding the eigenvalues of a second-order differential equation for the extra-dimensional ($w$-dependent) part of the fluctuation eigenfunctions subject to the boundary conditions at the branes. The lowest eigenvalue corresponds to the radion mass, which is positive $m^2>0$ and exceeds the TeV scale or so [@Csaki:1999mp]. Tensor fluctuations around flat stabilized branes are also stable. Brane inflation, like all inflationary models, generates long wavelength cosmological perturbations from the vacuum fluctuations of all light (i.e. with mass less than the Hubble parameter $H$) degrees of freedom. The theory of metric fluctuations around the background geometry (\[warp\]) with inflating (de Sitter) branes is more complicated than that for the flat branes. For tensor fluctuations (gravitational waves), the lowest eigenvalue of the extra dimensional part of the tensor eigenfunction is zero, $m=0$, which corresponds to the usual 4d graviton. As it was shown in [@LMW; @gw], massive KK gravitons have a gap in the spectrum; the universal lower bound on the mass is $m \ge \sqrt{3 \over 2}\, H$. This means that massive KK tensor modes are not generated from brane inflation. Massless scalar and vector projections of the bulk gravitons are absent, so only the massless 4d tensor mode is generated. Scalar cosmological fluctuations from inflation in the braneworld setting (\[warp\]) have been considered in many important works [@Mukohyama:2000ui; @Kodama:2000fa; @Langlois:2000ia; @vandeBruck:2000ju; @Koyama:2000cc; @Deruelle:2000yj; @Gen:2000nu; @Mukohyama:2001ks]. The theory of scalar perturbations in braneworld inflation with bulk scalars is even more complicated than for tensor perturbations. This is because one has to consider 5d scalar metric fluctuations and brane displacements induced not only by the bulk scalar field fluctuations $\delta\varphi$, but also by the fluctuations $\delta \chi$ of the inflaton scalar field $\chi$ living at the brane. In fact, most papers on scalar perturbations from brane inflation concentrated mainly on the inflaton fluctuations $\delta \chi$, while the bulk scalar fluctuations were not included. This was partly because in the earlier papers on brane inflation people considered a single brane embedded in an AdS background without a bulk scalar field, and partly because for braneworlds with two stabilized branes there was an expectation that the fluctuations of the bulk scalar would be massive and thus would not be excited during inflation. In this letter we focus on the bulk scalar field fluctuations, assuming for the sake of simplicity that the inflaton fluctuations $\delta \chi$ are subdominant. We consider a relatively simple problem of scalar fluctuations around curved (de Sitter) branes, involving only bulk scalar field fluctuations $\delta\varphi$. We find the extra-dimensional eigenvalues of the scalar fluctuations subject to boundary conditions at the branes, focusing especially on the radion mass $m^2$ for the inflating branes. In particular, we investigate the presence or absence of a gap in the KK spectrum of scalar fluctuations in view of the tensor mode result. Our results are a generalization of the known results for flat stabilized branes [@Tanaka:2000er], which we reproduce in the limit where the branes are flattening $H \to 0$. Bulk Equations ============== The five-dimensional braneworld models with a scalar field in the bulk are described by the action $$\begin{aligned} \label{eq:action} S &=& M_5^3 \int \sqrt{-g}\, d^5 x\, \left\{R - (\nabla\varphi)^2 - 2V(\varphi)\right\} \nonumber\\ && -2 M_5^3 \sum \int \sqrt{-q}\, d^4 x\, \left\{ [{{\cal K}}] + U(\varphi)\right\},\end{aligned}$$ where the first term corresponds to the bulk and the sum contains contributions from each brane. The jump of the extrinsic curvature $[{\cal K}]$ provides the junction conditions across the branes (see equation (\[eq:jc\]) below). Variation of this action gives the bulk Einstein $G_{AB}=T_{AB}(\varphi)$ and scalar field $\Box\varphi=V_{,\varphi}$ equations. For the (stationary) warped geometry (\[warp\]) they are \[eq:bg\] $$\begin{aligned} &\displaystyle \varphi'' + 3\frac{a'}{a} \varphi' - a^2 V' = 0,&\label{eq:bg:phi}\\ &\displaystyle \frac{a''}{a} = 2\, \frac{a'^2}{a^2} - H^2 - \frac{\varphi'^2}{3},&\label{eq:bg:a}\\ &\displaystyle 6\left(\frac{a'^2}{a^2} - H^2\right) = \frac{\varphi'^2}{2} - a^2 V,&\label{eq:bg:c}\end{aligned}$$ where the prime denotes the derivative with respect to the extra dimension coordinate $w$. The first two equations are dynamical, and the last is a constraint. The solutions of equations (\[eq:bg\]) were investigated in detail in [@FFK]. Now we consider scalar fluctuations around the background (\[warp\]). The perturbed metric can be written in the longitudinal gauge as $$\label{eq:metric:pert} ds^2 = a(w)^2 \left[(1+2\Phi) dw^2 + (1+2\Psi)ds_4^2\right].$$ The linearized bulk Einstein equations and scalar field equation relate two gravitational potentials $\Phi(x^A)$, $\Psi(x^A)$ and bulk scalar field fluctuations $\delta\varphi(x^A)$. The off-diagonal Einstein equations require that $$\Psi = - \frac{\Phi}{2},$$ similar to four-dimensional cosmology, although the coefficient is different. The symmetry of the background guarantees separation of variables, so that perturbations can be decomposed with respect to four-dimensional scalar harmonics, e.g. $$\label{eq:sep} \Phi(x^A) = \sum\limits_m \Phi_m(w) Q_m(t, \vec x),$$ where the eigenvalues $m$ (constant of separation) appear as the four-dimensional masses ${^4}\Box Q_m = m^2 Q_m$, where ${^4}\Box$ is the D’Alembert operator on the 4d de Sitter slice. The four-dimensional massive scalar harmonics $Q_m$ can be further decomposed as $Q_m(t,\vec x) = \int f_k^{(m)}(t)\, e^{i \vec k \vec x}\, d^3k$. The temporal mode functions $f_k^{(m)}(t)$ obey the equation $$\label{eq:4} \ddot{f} + 3H\dot{f} + \left( e^{-2Ht}k^2 + m^2 \right) f = 0,$$ where dot denotes time derivative, and we dropped the labels $k$ and $m$ for brevity. Out of the remaining linearized Einstein equation we get the following equations for the extra-dimensional mode functions $\Phi_m(w)$ and $\delta\varphi_m(w)$ \[eq:pert\] $$\begin{aligned} (a^2 \Phi)' &=& \frac{2}{3} a^2 \varphi'\, \delta\varphi,\\ \left(\frac{a}{\varphi'}\, \delta\varphi\right)' &=& \left(1 - \frac{3}{2} \frac{m^2+4H^2}{\varphi'^2}\right) a \Phi,\end{aligned}$$ where we again omitted the label $m$ for transparency. These are very similar to the scalar perturbation equations in four-dimensional cosmology with a scalar field [@mukhanov], except for some numerical coefficients and powers of $a(w)$ (because the spacetime dimensionality is higher), and up to time to extra spatial dimension exchange. Indeed, we can introduce the higher-dimensional analog of the Mukhanov’s variable. However, in the presence of the curvature term $H^2$, the eigenvalue $m^2$ enters the second order equation for it in a complicated way, similar to that in the 4d problem with non-zero spatial curvature, see e.g. [@Garriga:1999vw]. We can introduce another convenient variable $u_m = \sqrt{\frac{3}{2}}\frac{a^{3/2}}{\varphi'}\, \Phi_m$. Then the two first order differential equations (\[eq:pert\]) can be combined into a single Schrödinger-type equation $$\label{v} u_m'' + \Big( m^2+4H^2 - V_{\text{eff}}(w) \Big) u_m = 0$$ with the effective potential $V_{\text{eff}} = \frac{z''}{z} + \frac{2}{3}\varphi'^2$, where we defined $z = \left(\frac{2}{3} a \varphi'^2\right)^{-\frac{1}{2}}$. There are two main differences relative to the four dimensional cosmology. First, in the latter case, FRW geometry with [*flat*]{} 3d spatial slices is usually considered, while the five dimensional brane inflation metric has [*curved*]{} 4d slices, which results in extra terms like $4H^2$ in equation (\[v\]). Second, here we are dealing not with an *initial* but a *boundary* value problem, with associated boundary conditions for perturbations at the branes on the edges. After we derive the boundary conditions, we will calculate the KK spectrum of the eigenvalues $m$. Brane Embedding and Boundary Conditions ======================================= The embedding of each brane is described by $w=w_{\pm}+\xi_\pm(x^a)$, where $\xi_\pm$ is the transverse displacement of the perturbed brane and $w_\pm$ is the position of the unperturbed brane. Holonomic basis vectors along the brane surface are $e_{(a)}^A \equiv \frac{\partial x^A}{\partial x^a} = \Big(\xi_{,a}, \delta_a^A\Big)$, while the unit normal to the brane is $n_{A} = a \Big(1+\Phi, -\xi_{,a} \delta_A^a\Big)$. The induced four-metric on the brane $d\sigma^2 = q_{ab} dx^a dx^b$ does not feel the brane displacement (to linear order) and is conformally flat $$\label{eq:induced} d\sigma^2 = a^2(1-\Phi)\left[-dt^2 + e^{2Ht}d\vec{x}^2\right].$$ The junction conditions for the metric and the scalar field at the brane are $$\label{eq:jc} [{{\cal K}}_{ab} - {{\cal K}}q_{ab}] = U(\varphi) q_{ab}, \hspace{1em} [n\cdot\nabla\varphi] = \frac{\partial U}{\partial \varphi},$$ where the extrinsic curvature is defined by ${{\cal K}}_{ab} = e_{(a)}^A e_{(b)}^B n_{A;B}$. We will only need its trace, which up to linear order in perturbations is $$\label{eq:k} {{\cal K}}= 4\frac{a'}{a^2} - 2 \frac{(a^2\Phi)'}{a^3} - \frac{{^4}\Box\xi}{a}.$$ For the background geometry (under the assumption of reflection symmetry across the branes), equations (\[eq:jc\]) reduce to $$\label{eq:jc:bg} \frac{a'}{a^2} = \mp \frac{U}{6}, \hspace{1em} \frac{\varphi'}{a} = \pm \frac{U'}{2}.$$ For the perturbed geometry, the traceless part of the extrinsic curvature must vanish in the absence of matter perturbations on the brane. Since it contains second cross-derivatives of $\xi$, the brane displacement $\xi$ is severely restricted. Basically, this means that the oscillatory modes of brane displacement are not excited without matter support at the brane. While there could possibly be global displacements of the brane, they do not interest us, so in the following we set $\xi=0$. Of course, for the more complete problem which includes fluctuations $\delta \chi$ of the “inflaton” field on the brane, the displacement $\xi$ does not vanish. Using expression (\[eq:k\]) for the trace of the extrinsic curvature, the first of equations (\[eq:jc\]) gives us the junction condition for linearized perturbations at the two branes $(a^2 \Phi)'\big|_{w_{\pm}} = \pm \frac{1}{3}\, U' a^3\, \delta\varphi \big|_{w_{\pm}}$. However, this junction condition does not really place any further restrictions on the bulk field perturbations, as it identically follows from the bulk perturbation equations (\[eq:pert\]) and the background junction condition (\[eq:jc:bg\]). Rather, this junction condition would relate the brane displacement $\xi$ to the matter perturbations on the brane if they were not absent. The second of equations (\[eq:jc\]) gives us a physically relevant boundary condition for the bulk field perturbations $$(\delta\varphi' - \varphi' \Phi)\big|_{w_{\pm}} = \pm \frac{1}{2}\, U'' a\, \delta\varphi \big|_{w_{\pm}}.$$ Using the bulk equations (\[eq:pert\]), this can be rewritten in a more suggestive form $$\label{eq:bc} \left(\frac{a}{\varphi'}\, \delta\varphi\right)\Bigg|_{w_{\pm}} = \frac{3}{2} \frac{m^2+4H^2}{a\varphi'^2} \frac{a^2 \Phi}{\frac{a^2 V'}{\varphi'} - 4 \frac{a'}{a} \mp a U_\pm''} \Bigg|_{w_{\pm}}.$$ The eigenvalues $m^2$ of bulk perturbation equations subject to the boundary condition (\[eq:bc\]) form a KK spectrum, which we find numerically. We considered several examples of the potentials $V$ and $U_{\pm}$, and found no universal positive mass gap. Moreover, for the most interesting models we found negative $m^2$. To understand the KK spectrum of $m^2$, we make a simplification of the boundary condition (\[eq:bc\]) which will allow us to treat the eigenvalue problem analytically, and which well corresponds to a spirit of brane stabilization [@GW]. Indeed, *rigid stabilization* of branes is thought to be achieved by taking $U''$ (i.e. the brane mass of the field) very large, so that the scalar field gets pinned down at the positions of the branes. In this case, the right hand side of (\[eq:bc\]) becomes very small, which leads to the boundary condition $$\label{eq:stab} \delta\varphi\big|_{w_\pm} = 0.$$ This by itself *does not guarantee stability*, or vanishing of the metric perturbations on the brane for that matter, as perturbations live in the bulk and only need to satisfy (\[eq:stab\]) on the branes. This poses an eigenvalue problem for the mass spectrum of the perturbation modes, which we study next. KK Mass Spectrum ================ Unlike the situation with gravitational waves [@gw], for the scalar perturbations there is no zero mode with $m=0$, nor is there a “supersymmetric” factorized form of the “Schrödinger”-like equation (\[v\]). To find the lowest mass eigenvalue, we have to use other ideas. Powerful methods for analyzing eigenvalue problems exist for normal self-adjoint systems [@kamke]. To use them, we transform our eigenvalue problem (\[eq:pert\]) and (\[eq:stab\]) into the self-adjoint form. While the second order differential equation (\[v\]) is self-adjoint, the boundary conditions for $u$ are not. Therefore, we introduce a new variable $Y=u/z=a^2\Phi$ and impose the boundary conditions (\[eq:stab\]) to obtain the boundary value problem \[eq:evp\] $$\begin{aligned} \label{eq:evp:de} &{{\cal D}}Y \equiv -(gY')' + fY = \lambda gY,&\\ \label{eq:evp:bc} &Y'(w_{\pm}) = 0,&\end{aligned}$$ where we have introduced the short-hand notation $f = 1/a$, $g = z^2 = \left(\frac{2}{3} a \varphi'^2\right)^{-1}$, and $\lambda = m^2+4H^2$. Since the boundary value problem (\[eq:evp\]) is self-adjoint, it is guaranteed that the eigenvalues $\lambda$ are real and non-negative, $\lambda \ge 0$. To estimate the lowest eigenvalue $\lambda_1$ of the eigenvalue problem (\[eq:evp\]), we apply the Rayleigh’s formula [@kamke], which places a rigorous upper bound on $\lambda_1$ $$\lambda_1 \le \frac{\int F {{\cal D}}F\, dw}{\int g F^2\, dw},$$ where $F$ can be *any* function satisfying the boundary conditions (\[eq:evp:bc\]), and does not have to be a solution of (\[eq:evp:de\]). Taking a trial function $F=1$, we have $$\lambda_1 \le \frac{\int f\, dw}{\int g\, dw}.$$ This bound on the lowest mass eigenvalue is our main result: $$\label{eq:bound} m^2 \le -4H^2 + \frac{2}{3} \frac{\int \frac{dw}{a}}{\int \frac{dw}{a\varphi'^2}}.$$ In practice, $F=1$ is a pretty good guess for the lowest eigenfunction, so the bound (\[eq:bound\]) is usually close to saturation (up to a few percent accuracy in some cases), as we have observed in direct computations using a numerical eigenvalue finder. The right hand side of equation (\[eq:bound\]) has the structure $-4H^2 + m_0^2(H)$, where the second term is a functional of $H$ (including the implicit $H$-dependence of the warp factor $a$). In the limit of flat branes $H \to 0$ we have only the second, positive term. In this limit our expression agrees with the estimation of the radion mass $m_0^2$ for flat branes, obtained in various approximations [@Csaki:1999mp; @Tanaka:2000er; @Mukohyama:2001ks]. A non-vanishing $H$ alters $m^2$ through both terms. The most drastic alteration of $m^2$ due to $H$ comes from the big negative term $ -4H^2$. For the particular case of two de Sitter branes embedded in 5d AdS without a bulk scalar this negative term was noticed in [@Gen:2000nu]. Tachyonic Instability of the Radion for Inflating Branes ======================================================== The most striking feature of the mass bound (\[eq:bound\]) is that $m^2$ for de Sitter branes is typically negative. Trying, for instance, to do Goldberger-Wise stabilization of braneworlds with inflating branes while taking bulk gradients $\varphi'^2$ small enough to ignore their backreaction (as it is commonly done for flat branes) is a sure way to get a tachyonic radion mass: an estimate of the integrals gives $m^2 \le -4H^2 + O(\varphi'^2)$, which will go negative if the bulk scalar field is negligible $\varphi'^2 \ll H^2$. In what follows we consider two situations. In this section, we consider braneworld models where $m^2$ is negative and mostly defined by the first term $-4H^2$ in equation (\[eq:bound\]). In the next section, we consider the case where both terms in equation (\[eq:bound\]) are tuned to be comparable and the net radion mass is smaller than the Hubble parameter $|m^2| \leq H^2$. In the last section we will discuss how these two cases may be dynamically connected. Suppose we start with a braneworld with curved de Sitter branes, and we find the mass squared of the radion to be negative. The extra-dimensional eigenfunction $\Phi_m(w)$ is regular in the interval $w_{-} \leq w \leq w_{+}$. Let us turn, however, to the four-dimensional eigenfunction $Q_m(t, \vec x)$. Bearing in mind the evolution of the quantum fluctuations of the bulk field, we choose the positive frequency vacuum-like initial conditions in the far past $t \to -\infty $, $f_k(t) \simeq \frac{1}{\sqrt{2k}} e^{ik\eta}$, $\eta=\int dt\, e^{-Ht}$. For the tachyonic mode $m^2 <0$ the solution to equation (\[eq:4\]) with this initial condition is given in terms of Hankel functions $f_k^{(m)}(\eta)=\frac{\sqrt{\pi}}{2} H |\eta|^{3/2} {\cal H}^{(1)}_{\mu} (k\eta)$, with the index $\mu=\sqrt{\frac{9}{4}+\frac{|m^2|}{H^2}}$. The late-time asymptotic of this solution diverges exponentially as $t \to \infty$ ($\eta \to 0$) $$\label{asym} f_k^{(m)}(t) \propto \exp \left[\left( {\sqrt{\frac{9}{4}+\frac{|m^2|}{H^2}} - \frac{3}{2}} \right) H t \right].$$ Thus the negative tachyon mass of the radion $|m^2| \sim 4H^2$ leads to a strong exponential instability of scalar fluctuations $\Phi \propto e^{Ht}$. This instability is observed using a completely different method in the accompanying [BraneCode]{} paper [@branecode], where we give a fully non-linear numerical treatment of inflating branes which were initially set to be stationary by the potentials $U_{\pm}(\varphi)$, and without any simplifications like approximating boundary condition (\[eq:bc\]) with (\[eq:stab\]). Tachyonic instability of the radion for inflating branes means that, in general, [*braneworlds with inflation are hard to stabilize.*]{} From the point of view of 4d effective theory one would expect brane stabilization at energies lower than the mass of the flat brane radion $m_0^2$, which is roughly equal to the second term in (\[eq:bound\]). If the energy scale of inflation $H$ is larger than $m_0$, $H^2 \gg m_0^2$, this expectation is incorrect. Successful inflation (lasting more than $65 H^{-1}$) requires the radion mass $m^2$ to be not too negative $$\label{criter} m^2 \gtrsim - \frac{H^2}{20}.$$ This is possible if both terms in (\[eq:bound\]) are of the same order. In the popular braneworld models the radion mass in the low energy limit, $m_0$, is of order of a TeV. For these models the scale of “stable” inflation would be the same order of magnitude, $H \sim \text{TeV}$. Although there is no evidence that this scale of inflation is too low, it is not a comfortable scale from the point of view of the theory of primordial perturbations from inflation. It is interesting to note that the system of curved branes may dynamically re-configure itself to reach a state where the condition (\[criter\]) is satisfied. In the case of the bulk scalar field $\varphi$ acting alone, for quadratic potentials $U_{\pm}$ suitable for brane stabilization, there may be two stationary warped geometry solutions (\[warp\]) with two different values of $H$. The solution with the larger Hubble parameter $H$ might be dynamically unstable due to the tachyonic instability of the radion, which we described above. The second solution with the lower $H$ which satisfies (\[criter\]) might be stable. A fully non-linear study of this model was performed numerically with the [BraneCode]{} and is reported in the accompanying paper [@branecode]. It shows that, indeed, the tachyonic instability violently re-configures the starting brane state with the larger $H$ into the stable brane state with the lower $H$. This re-configuration of the brane system has a spirit of the Higgs mechanism. If we add an “inflaton” scalar field $\chi$ located at the brane, its slow roll contributes to the decrease of $H$. Thus, for the “stable” brane we have a radion mass (\[criter\]). This condition includes the case when the radion is lighter than $H$, $|m^2| < H^2$. Even if the radion tachyonic instability is avoided, the light radion leads us to the other side of the story, a new mechanism of generation of scalar fluctuations from inflation associated with the radion. Induced Scalar Metric Perturbations at the Observable Brane =========================================================== Suppose that the radion mass is smaller than $H$, $|m^2| \ll H^2$, so that from (\[asym\]) we get the amplitude of the temporal mode function $f_k^{(m)}(t)$ in the late time asymptotic frozen at the level $f_k^{(m)}(t) \simeq \frac{H}{\sqrt{2}k^{3/2}}$. This is nothing but the familiar generation of inhomogeneities of a light scalar field from its quantum fluctuations during inflation. Therefore an observer at the observable brane will encounter long wavelength scalar metric fluctuations generated from braneworld inflation. The four dimensional metric describing scalar fluctuations around an inflating background is usually written as $$\label{eq:induced1} d\sigma^2 = - (1+2\widetilde{\Phi}) d\tilde{t}^2 + (1-2\widetilde{\Psi})e^{2\widetilde{H}\tilde{t}}d\tilde{x}^2,$$ where $\widetilde{\Phi}$ and $\widetilde{\Psi}$ are scalar metric fluctuations. The induced four-metric on the brane (\[eq:induced\]) in our problem can be rewritten in this standard form (\[eq:induced1\]) if we absorb the (constant) warp factor $a(w_{+})$ in the redefined time $\tilde{t} = at$ and spatial coordinates $\tilde{x} = a\vec{x}$ and rescale the Hubble parameter $\widetilde{H} = H/a$. Then we see that the induced scalar perturbations on the brane are $$\label{prop} \widetilde{\Psi} = -\widetilde{\Phi} = \frac{1}{2}\, \Phi.$$ The sign of the first equality here is opposite to what we usually have for $3+1$ dimensional inflation with a scalar field. It implies that the 4d Weyl tensor of the induced metric vanishes, as the induced fluctuations are conformally flat. The conformal structure of fluctuations (\[prop\]) is typical [@km87] for a $R^2$ inflation in the Starobinsky model [@star]. It is not a surprise, because for the scale of inflation comparable to the mass $m_0$ of the flat brane radion we expect higher derivative corrections to the 4d effective gravity on the brane. Indeed, the massive radion corresponds to a higher derivative 4d gravity [@Mukohyama:2001ks]. The amplitude and spectrum of induced fluctuations is defined by $\Phi$. From the mode decomposition (\[eq:sep\]) we get $$\label{ampl} k^{3/2}\, \widetilde{\Phi}_k \simeq \Phi_m (w_+) \, \frac{H}{M_4},$$ where $\Phi_m(w_+)$ is the amplitude of the extra-dimensional eigenmode at the observable brane, normalized in such a way that the fluctuations $\Phi(w, t, {\vec x})$ are canonically quantized on the 4d slice, namely $M_5^3 \int \frac{3}{2} \frac{a^3}{\varphi'^2}\, |\Phi_m(w)|^2 \, dw = 1$. The normalization $M_4$ of the 4d mode functions follows from canonical quantization of the perturbed action (\[eq:action\]); the usual 4d Planck mass $M_p$ is expected to be recovered in the effective field theory on the observable brane [@Tanaka:2000er]. The scalar metric fluctuations induced by the bulk scalar field fluctuations are scale-free and have the amplitude $k^{3/2} \widetilde{\Phi}_k \propto \frac{H}{M_p}$, with the numerical coefficient depending on the details of the warped geometry. The nature of these fluctuations is very different from those in $(3+1)$-dimensional inflation, where the inflaton scalar field is time dependent. Induced scalar fluctuations do not require “slow-roll” properties of the potentials $V$ and $U_{\pm}$. The underlying background bulk scalar field has no time-dependence, but only $y$ dependence. Thus, generation of induced scalar metric fluctuations from braneworld inflation is a new mechanism for producing cosmological inhomogeneities. If we add another, inflaton field $\chi$ localized at the brane, we should expect that fluctuations of both fields, the bulk scalar $\delta \varphi$ and the inflaton $\delta \chi$, contribute to the metric perturbations. We can conjecture that the net fluctuations will be similar to those derived in the combined model with $R^2$ gravity and a scalar field [@kls]. Discussion ========== Let us discuss the physical interpretation and the meaning of our result. Stabilization of flat branes is based on the balance between the gradient $\phi'$ of the bulk scalar field and the brane potentials $U(\phi)$ which keeps $\phi$ pinned down to its values $\phi_i$ at the branes. The interplay between different forces becomes more delicate if the branes are curved. The warped configuration of curved branes has the lowest eigenvalue for scalar fluctuations around it $$\label{crit} m^2=-4H^2+m_0^2(H) \ .$$ The term $m_0^2(H)$ is a functional of $H$, and depends on the parameters of the model. If parameters are such that $m^2$ becomes negative due to excessive curvature $\sim H^2$, the brane configuration becomes unstable. This is analogous to an instability of a simple elastic mechanical system supported by the balance of opposite forces, which arises for a certain range of the underlying parameters. Tachyonic instability of curved branes has serious implications for the theory of inflation in braneworlds. It may be not so easy to have a realization of inflation in the braneworld picture without taking care of parameters of the model. Inflation where $m^2$ in (\[crit\]) is negative and $|m^2|$ is larger than $H^2$ is a short-lived stage because of this instability. After inflation, the late time evolution should bring the brane configuration to (almost) flat stabilized branes in the low energy limit. This by itself requires fine tuning of the potentials $V$ and $U_{\pm}$ to provide stabilization. Stabilization at the inflation energy scale requires extra fine tuning to get rid of the tachyonic effect. Working with a single bulk scalar field, it is probably not easy to simultaneously achieve stabilization not only at low energy, but also at the high energy scale of inflation, to insure that $|m^2| \ll H^2$, and to provide a graceful exit from inflation. One may expect that introduction of another scalar field $\chi$ on the brane can help to have stabilization both at the scale of inflation and in the low energy limit. If we can achieve brane stabilization during inflation by suppression of the tachyonic instability, we encounter a byproduct effect. Light modes of radion fluctuations inevitably contribute to the induced scalar metric perturbations. Therefore the theory of braneworld inflation has an additional mechanism of generation of primordial cosmological perturbations. This new mechanism is different from that of the usual 4d slow roll inflation. It appears that one of the most interesting potential applications of our effect is a mechanism for reducing the 4d effective cosmological constant at the brane. Indeed, in terms of brane geometry, the 4d cosmological constant is related to the 4d curvature of the brane. Suppose we have two solutions of the background equations (\[eq:bg\]) with higher and lower values of the curvature of de Sitter brane, which is proportional to $H^2$. (The existence of two solutions for certain choices of parameters of the Goldberger-Wise type potentials used for brane stabilization can be demonstrated, see [@FFK; @branecode].) Suppose that the solution with the larger value of brane curvature is unstable. Then the brane configuration will violently restructure into the other static configuration, which is characterized by the lower value of brane curvature where the tachyonic instability is absent. The branes are flattening, which for a 4d observer means the lowering of the cosmological constant. It will be interesting to investigate how this mechanism works for brane configurations with several scalar fields or potentials which can admit more than two static solutions. The problem of the cosmological constant from a braneworld perspective (as a flat brane) was discussed in the literature. There was a suggestion that the flat brane is a special solution of the bulk gravity/dilaton system with a single brane [@Arkani-Hamed:2000eg; @Kachru:2000hf], the claim which was later dismissed [@Forste:2000ft]. In our setup, we consider two branes in order to screen the naked bulk singularity, which was one of the factors spoiling the models [@Arkani-Hamed:2000eg; @Kachru:2000hf]. The new element which emerges from our study is the instability of the curved branes. Acknowledgments {#acknowledgments .unnumbered} =============== We are grateful to R. Brandenberger, J. Cline, C. Deffayet, J. Garriga, A. Linde, S. Mukohyama, D. Pogosyan and V. Rubakov for valuable discussions. We are especially indebted to our collaborators on the [BraneCode]{} project, G. Felder, J. Martin and M. Peloso. 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ArXiv

--- abstract: 'In this paper we continue the development of quantum holonomy theory, which is a candidate for a fundamental theory based on gauge fields and non-commutative geometry. The theory is build around the $\mathbf{QHD}(M)$ algebra, which is generated by parallel transports along flows of vector fields and translation operators on an underlying configuration space of connections, and involves a semi-final spectral triple with an infinite-dimensional Bott-Dirac operator. Previously we have proven that the square of the Bott-Dirac operator gives the free Hamilton operator of a Yang-Mills theory coupled to a fermionic sector in a flat and local limit. In this paper we show that the Hilbert space representation, that forms the backbone in this construction, can be extended to include many-particle states.' --- **** On the Fermionic Sector of Quantum Holonomy Theory 6ex Johannes <span style="font-variant:small-caps;">Aastrup</span>$^{a}$[^1] & Jesper Møller <span style="font-variant:small-caps;">Grimstrup</span>$^{b}$[^2]\ 3ex $^{a}\,$*Mathematisches Institut, Universität Hannover,\ Welfengarten 1, D-30167 Hannover, Germany.*\ $^{b}\,$*QHT Gruppen, Copenhagen, Denmark.*\ [*This work is financially supported by Ilyas Khan,\ St. EdmundÕs College, Cambridge, United Kingdom and by\ Tegnestuen Haukohl & Køppen, Copenhagen, Denmark.*]{} 3ex Introduction ============ In this paper we continue the development of [Quantum Holonomy Theory]{}, which is a candidate for a fundamental theory based on gauge fields and formulated within the framework of non-commutative geometry and spectral triples. The basic idea in Quantum Holonomy Theory is to start with an algebra that encodes the canonical commutation relations of a gauge theory in an integrated and non-local fashion. The algebra in question is called the quantum holonomy-diffeomorphisms algebra, denoted $\mathbf{QHD}(M)$, which was first presented in [@Aastrup:2014ppa] and which is generated by parallel transports along flows of vector fields and by translation operators on an underlying configuration space of gauge connections. In [@Aastrup:2015gba] it was demonstrated that this algebra encodes the canonical commutation relations of a gauge theory. Once the $\mathbf{QHD}(M)$ has been identified the question arises whether it has non-trivial Hilbert space representations. This question was answered in the affirmative in [@Aastrup:2017vrm] where we proved that separable and strongly continuous Hilbert space representations of the $\mathbf{QHD}(M)$ exist in any dimensions. A key feature of these Hilbert space representations is that they are non-local. They are labelled by a scale $\tau$, which we tentatively interpret as the Planck scale and which essentially serves as a UV-regulator by suppressing modes in the ultra-violet. This UV-suppression does not break any spatial symmetries, i.e. these representations are isometric. In [@Aastrup:2017atr] we constructed an infinite-dimensional Bott-Dirac operator that interacts with an algebra generated by holonomy-diffeomorphisms alone, denoted by $\mathbf{HD}(M)$, and proved that this Bott-Dirac operator together with the aforementioned Hilbert space representation forms a semi-finite spectral triple over a configuration space of connections. In that paper we also demonstrated that the square of the Bott-Dirac operator coincides in a local and flat limit with the free Hamilton operator of a gauge field coupled to a fermionic sector, a result which opens the door to an interpretation of quantum holonomy theory in terms of a quantum field theory on a curved background. In this paper we continue the analysis of these Hilbert space representations. One feature of the Bott-Dirac operator is that it naturally introduces the CAR algebra into the construction via an infinite-dimensional Clifford algebra. This CAR algebra has a natural interpretation in terms of a fermionic sector due to the aforementioned result that the square of the Bott-Dirac operator includes the Hamilton of a free fermion. One drawback of the Hilbert space representation constructed in [@Aastrup:2017vrm] is that it only involves what amounts to one-particle states. In other words, the Hilbert space representation does not act on the CAR algebra itself. In this paper we construct such a Hilbert space representation of the $\mathbf{QHD}(M)$ algebra. The result that such a representation exist solidifies the interpretation that quantum holonomy theory should be understood as a quantum theory of gauge fields coupled to fermions.\ This paper is organised as follows: We begin by introducing the $\mathbf{HD}(M)$ and $\mathbf{QHD}(M)$ algebras in section 2 and the infinite-dimensional Bott-Dirac operator in section 3. We then review the Hilbert space representation constructed in [@Aastrup:2017vrm] in section 4. Finally we construct in section 5 a new Hilbert space representation where the $\mathbf{QHD}(M)$ algebra acts on the Fock space. We end with a discussion in section 6. The $\mathbf{HD}(M)$ and $\mathbf{QHD}(M)$ algebras {#sektion2} =================================================== In this section we introduce the algebras $\mathbf{HD}(M)$ and $\mathbf{QHD}(M)$, which are generated by parallel transports along flows of vector-fields and for the latter part also by translation operators on an underlying configuration space of connections. The $\mathbf{HD}(M)$ algebra was first defined in [@Aastrup:2012vq; @AGnew] and the $\mathbf{QHD}(M)$ algebra in [@Aastrup:2014ppa]. In the following we shall define these algebras in a local and a global version.\ Let $M$ be a compact manifold and let $\ca$ be a configuration space of gauge connections that takes values in the Lie-algebra of a compact gauge group $G$. A holonomy-diffeomorphism $e^X\in \mathbf{HD}(M)$ is a parallel transport along the flow $t\to \exp_t(X)$ of a vector field $X$. To see how this works we first let $\gamma$ be the path $$\gamma (t)=\exp_{t} (X) (m)$$ running from $m$ to $m'=\exp_1 (X)(m)$. Given a connection $\nabla$ that takes values in a $n$-dimensional representation of the Lie-algebra $\mathfrak{g}$ of $G$ we then define a map $$e^X_\nabla :L^2 (M )\otimes \mathbb{C}^n \to L^2 (M )\otimes \mathbb{C}^n$$ via the holonomy along the flow of $X$ $$(e^X_\nabla \xi )(m')= \hbox{Hol}(\gamma, \nabla) \xi (m) , \label{chopin1}$$ where $\xi\in L^2(M,\mathbb{C}^n)$ and where $\hbox{Hol}(\gamma, \nabla)$ denotes the holonomy of $\nabla$ along $\gamma$. This map gives rise to an operator valued function on the configuration space $\ca$ of $G$-connections via $$\ca \ni \nabla \to e^X_\nabla , {\nonumber}$$ which we denote by $e^X$ and which we call a holonomy-diffeomorphism[^3]. For a function $f\in C^\infty (M)$ we get another operator valued function $fe^X$ on $\ca$. We call the algebra generated by all holonomy-diffeomorphisms $e^X$ for the [*global*]{} holonomy-diffeomorphism algebra, denoted by $\mathbf{HD}_{\mbox{\tiny g}}(M)$, and we call the algebra generated by all holonomy-diffeomorphisms $f e^X$ for the [*local*]{} holonomy-diffeomorphism algebra, denoted simply by $\mathbf{HD}(M)$.\ Furthermore, a $\mathfrak{g}$ valued one-form $\oo$ induces a transformation on $\ca$ and therefore an operator $U_\omega $ on functions on $\ca$ via $$U_\omega (\xi )(\nabla) = \xi (\nabla - \omega) ,$$ which gives us the quantum holonomy-diffeomorphism algebras, denoted either by $\mathbf{QHD}_{\mbox{\tiny g}}(M)$, which is the algebra generated by $\mathbf{HD}_{\mbox{\tiny g}}(M)$ and all the $U_\oo$ operators, or by $\mathbf{QHD}(M)$, which is the algebra generated by $\mathbf{HD}(M)$ and all the $U_\oo$ operators (see also [@Aastrup:2014ppa]). An infinite-dimensional Bott-Dirac operator {#Bott} =========================================== In this section we introduce an infinite-dimensional Bott-Dirac operator that acts in a Hilbert space that shall later play a key role in defining a representation of the $\mathbf{QHD}(M)$ algebras. The following formulation of an infinite-dimensional Bott-Dirac operator is due to Higson and Kasparov [@Higson] (see also [@Aastrup:2017atr]).\ Let $\ch_n= L^2(\mathbb{R}^n)$, where the measure is given by the flat metric, and consider the embedding $$\varphi_n : \ch_n\rightarrow\ch_{n+1}$$ given by $$\varphi_n(\eta)(x_1,x_2,\ldots x_{n+1}) = \eta(x_1,\ldots, x_n) \left(\frac{s_{n+1}}{\tau_2\pi}\right)^{1/4}e^{- \frac{s_{n+1} x_{n+1}^2}{2\tau_2}}, \label{ref}$$ where $\{s_n\}_{n\in\mathbb{N}}$ is a monotonously increasing sequence of parameters, which we for now leave unspecified[^4]. This gives us an inductive system of Hilbert spaces $$\ch_1\stackrel{\varphi_1}{\longrightarrow} \ch_2 \stackrel{\varphi_2}{\longrightarrow} \ldots \stackrel{\varphi_n}{\longrightarrow} \ch_{n+1} \stackrel{\varphi_{n+1}}{\longrightarrow}\ldots$$ and we define[^5] $L^2(\mathbb{R}^\infty) $ as the Hilbert space direct limit $$L^2(\mathbb{R}^\infty) = \lim_{\rightarrow} L^2(\mathbb{R}^n)$$ taken over the embeddings $\{\varphi_n\}_{n\in\mathbb{N}}$ given in (\[ref\]). We are now going to define the Bott-Dirac operator on $ L^2(\mathbb{R}^n)\otimes \Lambda^*\mathbb{R}^n$. Denote by $\mbox{ext}(v)$ the operator of external multiplication with $v$ on $\Lambda^*\mathbb{R}^n$, where $v$ is a vector in $\mathbb{R}^n$, and denote by $\mbox{int}(v)$ its adjoint, i.e. the interior multiplication by $v$. Denote by $\{v_i\}$ a set of orthonormal basis vectors on $\mathbb{R}^n$ and let $\bar{c}_i$ and $c_i$ be the Clifford multiplication operators given by $$\begin{aligned} {c}_i &=& \mbox{ext}(v_i) + \mbox{int}(v_i) {\nonumber}\\ \bar{c}_i &=& \mbox{ext}(v_i) - \mbox{int}(v_i) \end{aligned}$$ that satisfy the relations $$\begin{aligned} \{c_i, \bar{c}_j\} = 0, \quad \{c_i, {c_j}\} = \d_{ij}, \quad \{\bar{c}_i, \bar{c}_j\} =- \d_{ij}.\end{aligned}$$ The Bott-Dirac operator on $ L^2(\mathbb{R}^n)\otimes \Lambda^*\mathbb{R}^n$ is given by $$B_n = \sum_{i=1}^n\left( \tau_2 \bar{c}_i \frac{{\partial}}{{\partial}x_i} + s_i c_i x_i\right).$$ With $B_n$ we can then construct the Bott-Dirac operator $B$ on $L^2(\mathbb{R}^\infty)\otimes \Lambda^*\mathbb{R}^\infty$ that coincides with $B_n$ on any finite subspace $L^2(\mathbb{R}^n)$. Here we mean by $\Lambda^*\mathbb{R}^\infty$ the inductive limit $$\Lambda^*\mathbb{R}^\infty= \lim_{\rightarrow} \Lambda^*\mathbb{R}^n.$$ For details on the construction of $B$ we refer the reader to [@Higson] and to [@Aastrup:2017atr], where we also showed that the square of $B$ coincides with the free Hamilton operator of a fermion Yang-Mills theory in a flat and local limit. A representation of the $\mathbf{QHD}(M)$ algebra ================================================= In this section we write down the representation of the $\mathbf{QHD}(M)$ algebra, which was first constructed in [@Aastrup:2017vrm]. A key feature of this representation is that it involves a spatial non-locality characterised by a physical parameter $\tau_1$, which effectively acts as an ultra-violet regulator and which we in [@Aastrup:2017vrm] tentatively interpreted in terms of the Planck length.\ To obtain a representation of the $\mathbf{QHD}(M)$ algebra we let $\langle \cdot\vert\cdot\rangle_{\mbox{\tiny s}} $ denote the Sobolev norm on $\OO^1(M\otimes\mathfrak{g})$, which has the form $$\langle \omega_1\vert\omega_2\rangle_{\mbox{\tiny s}} := \int_M \big( (1+ \tau_1\Delta^{\sigma})\omega_1 , (1+ \tau_1\Delta^{\sigma})\omega_2 \big)_{T_x^*M\otimes \mathbb{C}^n} (m) dm \label{sob}$$ where the Hodge-Laplace operator $\Delta$ and the inner product $(,)_{T_x^*M\otimes \mathbb{C}^n}$ on $T_x^*M\otimes \mathbb{C}^n$ depend on a metric g and where $\tau_1$ and $\sigma$ are positive constants. Also, we choose an $n$-dimensional representation of $\mathfrak{g}$. Next, denote by $\{\xi_i\}_{i\in\mathbb{N}}$ an orthonormal basis of $\OO^1(M\otimes\mathfrak{g})$ with respect to the scalar product (\[sob\]). With this we can construct a space $L^2(\ca)$ as an inductive limit over intermediate spaces $L^2(\ca_n)$ with an inner product given by $$\begin{aligned} \langle \eta \vert \zeta \rangle_{\ca_n} &=& \int_{\mathbb{R}^n} \overline{\eta(x_1\xi_1 + \ldots + x_n \xi_n)} \zeta (x_1\xi_1 + \ldots + x_n \xi_n) dx_1\ldots dx_n \label{rn} $$ where $\eta$ and $\zeta$ are elements in $L^2(\ca_n)$, as explained in section 3, and also using the same tail behaviour as in section 3. Finally, we define the Hilbert space $$\ch_{\mbox{\bf\tiny YM}}= L^2(\ca)\otimes L^2(M, \mathbb{C}^n) \label{ymm}$$ in which we then construct the following representation of the $\mathbf{QHD}_{\mbox{\tiny l}}(M)$ algebra. First, given a smooth one-form $\oo\in\OO^1(M,\mathfrak{g})$ we write $\oo =\sum \oo_i \xi_i$. The operator $U_\chi$ acts by translation in $L^2(\ca)$, i.e. $$\begin{aligned} U_{\oo}(\eta) (\omega)&=&U_{\oo}(\eta) (x_1 \xi_1+x_2 \xi_2+ \ldots) {\nonumber}\\ &=& \eta ( (x_1+\oo_1)\xi_1+(x_2+\oo_2)\xi_2+ \ldots) \label{rep1}\end{aligned}$$ with $\eta\in L^2(\ca)$. Next, we let $f e^X\in \mathbf{HD}(M)$ be a holonomy-diffeomorphism and $\Psi(\omega,m)=\eta(\omega)\otimes \psi(m)\in\ch_{\mbox{\tiny\bf YM}}$ where $\psi(m)\in L^2(M)\otimes \mathbb{C}^n$. We write $$f e^X \Psi(\omega,m') = f(m) \eta(\omega) Hol(\gamma, \omega) \psi(m) \label{rep2}$$ where $\gamma$ is again the path generated by the vector field $X$ with $m'=\exp_1(X)(m)$. In [@Aastrup:2017vrm] and [@Aastrup:2017atr] we prove that equations (\[rep1\]) and (\[rep2\]) gives a strongly continuous Hilbert space representation of the $\mathbf{QHD}(M)$ algebra in $\ch_{\mbox{\bf\tiny YM}}$. Note that this representation is isometric with respect to the background metric $g$, see [@Aastrup:2017vrm] for details. Representing $\mathbf{QHD}_{\mbox{\tiny g}}(M)$ on the Fock Space ================================================================= The Bott-Dirac operator acts on $L^2 (\mathbb{R}^\infty )\otimes \Lambda^*\mathbb{R}^\infty$, and not on $L^2(\ca)\otimes L^2(M,\mathbb{C}^n)$ as the $\mathbf{QHD}(M)$-algebra does. The Hilbert space $L^2(\mathbb{R}^\infty)$ is, however, easily identified with $L^2 (\ca )$ via $$\mathbb{R}^n \ni (x_1,\ldots , x_n) \mapsto x_1\xi_1 +\ldots + x_n \xi_n \in \ca_n .$$ We will therefore denote $\Lambda^*\mathbb{R}^\infty$ by $\Lambda^*\ca$. We thus get an action of the Bott-Dirac operator and the $\mathbf{QHD}(M)$-algebra on $L^2(\ca)\otimes\Lambda^*\ca \otimes L^2(M,\mathbb{C}^n)$. This is somewhat unsatisfactory due to two reasons: 1. The Fermions on which the $\mathbf{QHD}(M)$-algebra acts, is a one-particle space. We could of course try to take the Fock space of $L^2(M,\mathbb{C}^n)$ instead of just $L^2(M,\mathbb{C}^n)$. 2. We have a fermionic doubling in the sense that we have the fermionic Fock space $\Lambda^*\ca$, where the bosons, i.e. the $\mathbf{QHD}(M)$-algebra, do not act at all, and then the fermions in $L^2(M,\mathbb{C}^n)$, where the bosons do act. It is therefore desirable to get an action the $\mathbf{QHD}(M)$ algebra on $L^2 (\ca )\otimes \Lambda^* \ca $. In this section we show how this can be accomplished for the $\mathbf{QHD}_{\mbox{\tiny g}}(M)$ algebra but at the present moment not for the local $\mathbf{QHD}(M)$ algebra.\ We begin with the basespace $$\mathcal{H}^\sigma =\Omega^1 (M,\mathfrak{g}) , \label{trmpp}$$ where the Hilbert space structure is again with respect to a suitable Sobolev norm (\[sob\]) in the sense that the righthand side of (\[trmpp\]) has been completed in this norm (we remind the reader that the superscript ’$\sigma$’ is the power of the Laplace operator in (\[sob\])). The main purpose is to get a unitary, connection dependent action of the group of holonomy-diffeomorhphisms in $\mathbf{HD}_{\mbox{\tiny g}}(M)$ on the Hilbert space $\mathcal{H}^\sigma$. Once we have a unitary action it extends uniquely to an action on the associated Fock space $\Lambda^* \mathcal{H}^\sigma$ via $$F_\nabla(v_1\wedge \ldots \wedge v_n)=F_\nabla (v_1)\wedge \ldots \wedge F_\nabla (v_n) ,$$ where $F$ denotes a holonomy-diffeomorphism and $\nabla$ denotes a connection. Once we have this we get a unitary action of the $\mathbf{HD}_{\mbox{\tiny g}}(M)$ algebra on $\Lambda^* \mathcal{H}^\sigma \otimes L^2(\mathcal{A})$ via $$F (\xi \otimes \eta )(\nabla ) =F_\nabla (\xi)\eta ( \nabla ) .$$ The question is of course how we get an action of $\mathbf{HD}_{\mbox{\tiny g}}(M)$ on $\mathcal{H}^\sigma$. To answer this question we let $F$ be a holonomy-diffeomorphism and let $\nabla$ be a $\mathfrak{g}$-connection. We start with the case $\sigma=0$. Let $F$ be the flow of the vector field $X$ and let $\omega \in \Omega^1 (M,\mathfrak{g})$. Let $m_1\in M$ and $m_2=\exp (X)(m_1)$, and $\gamma$ the path $t\to e^{tX}(m_1)$. Furthermore we denote by $(e^{-X})^* (\omega )$ the pullback of the one-form part of $\omega$ by the diffeomorphism $e^{ -X}$, i.e. $(e^{-X})^*$ leaves the Lie algebra $\mathfrak{g}$ unchanged. We define $$e_\nabla^{X}(\omega ) (m_2)= \hbox{Hol} (\gamma ,\nabla)\Big( (e^{-X})^*(\omega)(m_2) \Big) (\hbox{Hol} (\gamma ,\nabla))^{-1} .$$ This does not define a unitary operator, unless $\exp (X)$ is an isometric flow. Unlike in section \[sektion2\] we cannot adjust the lack of unitarity by multiplying by a suitable determinant. The problem lies in the one form part. One possible way to deal with this is to consider only holonomy-diffeomorphisms, which are isometries with respect to a chosen metric. Alternatively – and this is the option that we shall adopt – we can allow the operators to be non-unitary. In this latter case we will still get bounded operators on $\mathcal{H}^\sigma$, even when we consider the supremum over all connections. The problem is, that when we extend the action to the Fock space the operators will no longer be bounded. The unboundedness is however not so severe since the operators are bounded when we consider them only acting on a subspace of the Fock space which contains particle states with particle number bounded by a given value. For general $\sigma$’s there is a natural way to proceed: The map $1+\tau_1\Delta^\sigma: \mathcal{H}^0 \to \mathcal{H}^\sigma$ is a unitary operator, and to get the action on $\mathcal{H}^\sigma$ we simply conjugate the action we have on $\mathcal{H}^0 $ with $1+\tau_1\Delta^\sigma$. If we choose holonomy-diffeomorphisms, which are isometries, this gives a unitary action. For general $\sigma$’s, we could also just proceed directly like above, without conjugating with $1+\tau_1\Delta^\sigma$. However without this conjugation the action would not be unitary on $\mathcal{H}^\sigma$, $\sigma\not= 0$, for the isometric flows.\ Finally, this representation of the $\mathbf{HD}_{\mbox{\tiny g}}(M)$ algebra is straight forwardly extended to the full $\mathbf{QHD}_{\mbox{\tiny g}}(M)$ algebra via $$U_\oo (\xi \otimes \eta )(\nabla ) = (\xi \otimes \eta )(\nabla +\oo) , \label{NOO2}$$ for $\xi\otimes\eta\in \Lambda^* \mathcal{H}^\sigma \otimes L^2(\mathcal{A})$. This implies that we have a non-unitary action of the $\mathbf{QHD}_{\mbox{\tiny g}}(M)$ algebra on $\Lambda^*\mathcal{A}\otimes L^2 (\mathcal{A})$. The action is strongly continuous if it is restricted to finite particle states. Note that the reason that this representation does not work for the local $\mathbf{HD}(M)$ and $\mathbf{QHD}(M)$ algebras is that it is not clear what the action of a function $f\in C^\infty(M)$ should be on the $0$-forms in $\Lambda^* \mathcal{H}^\sigma$, i.e. on the vacuum. For this reason we leave out the $C^\infty(M)$ part and consider instead only global holonomy-diffeomorphisms. Discussion ========== In this paper we show that the representation of the $\mathbf{QHD}(M)$ algebra constructed in [@Aastrup:2017vrm] can be extended to include also the CAR algebra if we consider only global holonomy-diffeomorphisms. This result provides us with what we believe is a completely new interpretation of fermionic quantum field theory in terms of geometrical data of a configuration space of connections. Consider first the ordinary Dirac operator and a spin-geometry. Here the fermion can be viewed as being part of an encoding of geometrical data of the underlying manifold, i.e. a spectral triple. In our case we have instead of the 4-dimensional Dirac operator an infinite dimensional Bott-Dirac operator acting in a Hilbert space over a configuration space of connections. This means that the CAR algebra and the fermionic sector is part of an encoding of geometrical data of this configuration space.\ As we demonstrated in [@Aastrup:2017atr] quantum holonomy theory is closely related to quantum field theory, the latter being based on two basic principles: locality and Lorentz invariance. In the axiomatic approaches these principles are encoded in the Osterwalder Schrader [@Osterwalder:1973dx] axioms for the Euclidean theory and in the Garding-Wightman [@Wightman] or the Haag-Kastler [@Haag:1963dh] axioms for the Lorentzian theory. To understand the difference between the present approach and ordinary quantum field theory we need to understand the role of the ultra-violet regulator in the form of the Sobolev norm (\[sob\]), which is the central element required to secure the existence of the Hilbert space representations. There are two options: either this regulator is a traditional cut-off that should eventually be taken to zero or it is a physical feature of this particular theory. If the ultra-violet regulator is a traditional cut-off then we are firmly within the boundaries of ordinary quantum field theory albeit with a different approach and with a different toolbox. In that case the question is whether the introduction of the Bott-Dirac operator and the fact that we have a spectral triple will give us new information about the limit where the cut-off goes to zero. Similar to algebraic quantum field theory [@Haag:1992hx] this approach is not limited in its choice of background. If on the other hand the regulator is to be viewed as a physical feature then we are decidedly outside the realm of traditional quantum field theory. There are two immediate consequences: 1. The Lorenz symmetry will be broken. The Hilbert space representation based on the Sobolev norm (\[sob\]) is isometric with respect to the metric on the three-dimensional manifold but the Lorentz symmetry will not be preserved. Instead there will be a larger symmetry that involves a scale transformation. 2. The theory is non-local. Whereas ordinary quantum field theory is based on operator valued distributions the present setup does not permit sharply localised entities. This also implies that the canonical commutation relations will only be realised up to a correction at the scale of the regularisation. Clearly this breaks with all the aforementioned axiomatic systems but the question is whether it is physically feasible? We believe that it is. First of all, it is not known whether the Lorentz symmetry is an exact symmetry in Nature and indeed much experimental effort has gone into testing whether it is [@Jacobson:2004rj]. We believe that the experimental constraints are sufficiently weak to permit the type of Lorenz breaking that we propose as long as it is restricted to the Planck scale. Secondly, it is generally believed that exact locality is not realised in Nature. Simple arguments combining quantum mechanics with general relativity strongly suggest that distances shorter than the Planck length are operational meaningless [@Doplicher:1994tu]. It is generally believed that a Planck scale screening will be produced by a theory of quantum gravity but we see no reason why it cannot be generated by quantum field theory itself as a part of its representation theory. We would then of course need to address the question of which regulator to choose, since the regulator would now be a quantity, that in principle is obvervable. [**Acknowledgements**]{}\ JMG would like to express his gratitude towards Ilyas Khan, United Kingdom, and towards the engineering company Tegnestuen Haukohl & Køppen, Denmark, for their generous financial support. JMG would also like to express his gratitude towards the following sponsors: Ria Blanken, Niels Peter Dahl, Simon Kitson, Rita and Hans-Jørgen Mogensen, Tero Pulkkinen and Christopher Skak for their financial support, as well as all the backers of the 2016 Indiegogo crowdfunding campaign, that has enabled this work. Finally, JMG would like to thank the mathematical Institute at the Leibniz University in Hannover for kind hospitality during numerous visits.\ [99]{} J. Aastrup and J. M. Grimstrup, “The quantum holonomy-diffeomorphism algebra and quantum gravity,” Int. J. Mod. Phys. A [**31**]{} (2016) no.10, 1650048. J. Aastrup and J. M. Grimstrup, “Quantum Holonomy Theory,” Fortsch. Phys.  [**64**]{} (2016) no.10, 783. 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[^4]: In [@Higson] these parameters were not included, i.e. $s_n=1\forall n$. [^5]: The notation $L^2 (\mathbb{R}^\infty )$, which we are using here, is somewhat ambiguous. We are here only considering functions on $\mathbb{R}^\infty $ with a specific tail behaviour, namely the one generated by (3). We have not included this tail behaviour in the notation. See [@Aastrup:2017vrm] for further details.

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